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AMC_12__2012_AMC_12A_Problems__p21 | Let $a$, $b$, and $c$ be positive integers with $a\ge$ $b\ge$ $c$ such that
$a^2-b^2-c^2+ab=2011$ and
$a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$.
What is $a$?
$\textbf{(A)}\ 249\qquad\textbf{(B)}\ 250\qquad\textbf{(C)}\ 251\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 253$
Please reason step by step, and put your final answer wit... | <think>
Okay, so I need to solve this problem where I have two equations with three variables, a, b, c, all positive integers with a ≥ b ≥ c. The equations are:
1. $ a^2 - b^2 - c^2 + ab = 2011 $
2. $ a^2 + 3b^2 + 3c^2 - 3ab - 2ac - 2bc = -1997 $
And I need to find the value of a, with options given from 249 to 253. ... |
Okay, so I need to solve this problem where I have two equations with three variables, a, b, c, all positive integers with a ≥ b ≥ c. The equations are:
1. $ a^2 - b^2 - c^2 + ab = 2011 $
2. $ a^2 + 3b^2 + 3c^2 - 3ab - 2ac - 2bc = -1997 $
And I need to find the value of a, with options given from 249 to 253. Let me ... | To solve the given system of equations:
$$
\begin{cases}
a^2 - b^2 - c^2 + ab = 2011 \quad \text{(1)} \\
a^2 + 3b^2 + 3c^2 - 3ab - 2ac - 2bc = -1997 \quad \text{(2)}
\end{cases}
$$
with the constraints that $ a, b, c $ are **positive integers** and $ a \ge b \ge c $, we aim to find the value of $ a $.
---
### Step ... | stop | 178 | 6,786 | 0 | 32,768 |
AMC_10__2014_AMC_10A_Problems__p3 | Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for $\textdollar 2.50$ each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf co... | <think>
Okay, let me try to figure out Bridget's profit for the day. Hmm, so she bakes 48 loaves of bread. Let me start by breaking down the problem step by step.
First, she sells half of them in the morning. So, half of 48 is... let me calculate that. Half of 48 is 24. So she sells 24 loaves in the morning at $2.50 ... |
Okay, let me try to figure out Bridget's profit for the day. Hmm, so she bakes 48 loaves of bread. Let me start by breaking down the problem step by step.
First, she sells half of them in the morning. So, half of 48 is... let me calculate that. Half of 48 is 24. So she sells 24 loaves in the morning at $2.50 each. L... | To determine Bridget's profit for the day, we need to calculate her **total revenue** from selling the loaves and subtract her **total cost** of making them.
---
### **Step 1: Total Loaves Baked**
Bridget bakes **48 loaves** of bread.
---
### **Step 2: Sales Breakdown**
#### **Morning Sales**
- She sells **half** ... | stop | 195 | 4,016 | 0 | 32,768 |
AMC_12__2018_AMC_12A_Problems__p6 | For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$. What is $m+n$?
$\textbf{(A)}20\qquad\textbf{(B)}21\qquad\textbf{(C)}22\qquad\textbf{(D)}23\qquad\textbf{(E)}24$
Please reason step by step, and put your final answer within... | <think>
Okay, so I need to solve this problem where we have two positive integers m and n. The conditions given are that m + 10 < n + 1, which simplifies to m < n - 9. Also, both the mean and the median of the set {m, m+4, m+10, n+1, n+2, 2n} are equal to n. We need to find m + n. The answer choices are from 20 to 24, ... |
Okay, so I need to solve this problem where we have two positive integers m and n. The conditions given are that m + 10 < n + 1, which simplifies to m < n - 9. Also, both the mean and the median of the set {m, m+4, m+10, n+1, n+2, 2n} are equal to n. We need to find m + n. The answer choices are from 20 to 24, so mayb... | To solve the problem, we are given a set of six numbers:
$$
\{m, m+4, m+10, n+1, n+2, 2n\}
$$
with the conditions:
- $ m + 10 < n + 1 $, which simplifies to $ m < n - 9 $
- The **mean** of the set is equal to $ n $
- The **median** of the set is also equal to $ n $
We are to find the value of $ m + n $, and the ans... | stop | 147 | 6,442 | 0 | 32,768 |
USAMO__2019_USAMO_Problems__p4 | Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$, for integers $i,j$ with $0\leq i,j\leq n$, such that:
$\bullet$ for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and
$\bullet$ $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq... | <think>
Okay, so I need to figure out the number of ways to choose $(n+1)^2$ sets $S_{i,j}$ where each $S_{i,j}$ is a subset of $\{1,2,\ldots,2n\}$, with the conditions that each $S_{i,j}$ has exactly $i + j$ elements, and they are nested in the sense that if $i \leq k$ and $j \leq l$, then $S_{i,j} \subseteq S_{k,l}$.... |
Okay, so I need to figure out the number of ways to choose $(n+1)^2$ sets $S_{i,j}$ where each $S_{i,j}$ is a subset of $\{1,2,\ldots,2n\}$, with the conditions that each $S_{i,j}$ has exactly $i + j$ elements, and they are nested in the sense that if $i \leq k$ and $j \leq l$, then $S_{i,j} \subseteq S_{k,l}$.
Firs... | To solve the problem, we are given a grid of $(n+1)^2$ sets $S_{i,j}$, indexed by $0 \leq i, j \leq n$, with the following constraints:
1. Each set $S_{i,j}$ has exactly $i + j$ elements.
2. The sets are nested: if $0 \leq i \leq k \leq n$ and $0 \leq j \leq l \leq n$, then $S_{i,j} \subseteq S_{k,l}$.
---
### Under... | stop | 177 | 17,363 | 0 | 32,768 |
AMC_8__1996_AJHSME_Problems__p25 | A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?
$\text{(A)}\frac{1}{4} \qquad \text{(B)}\frac{1}{3} \qquad \text{(C)}\frac{1}{2} \qquad \text{(D)}\frac{2}{3} \qquad \text{(E)}\frac{3}{4}$
... | <think>
Okay, so I need to find the probability that a randomly chosen point inside a circular region is closer to the center than to the boundary. The answer choices are given, so maybe I can figure it out step by step.
First, let me visualize the problem. There's a circle, and I pick a point somewhere inside it. I n... |
Okay, so I need to find the probability that a randomly chosen point inside a circular region is closer to the center than to the boundary. The answer choices are given, so maybe I can figure it out step by step.
First, let me visualize the problem. There's a circle, and I pick a point somewhere inside it. I need to ... | To solve the problem, we consider a **circular region** of radius $ R $, and we are to find the **probability** that a randomly chosen point within this circle is **closer to the center** than to the **boundary** of the circle.
---
### Step 1: Understanding the Distance Conditions
Let’s denote the distance from a po... | stop | 136 | 5,113 | 0 | 32,768 |
AIME__2004_AIME_II_Problems__p3 | A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $231$ of the 1-cm cubes cannot be seen. Find the smallest possible value of $N.$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the smallest possible value of N, where N is the number of 1-cm cubes in a solid rectangular block. When viewed so that three of its faces are visible, exactly 231 cubes cannot be seen. Hmm, let me think about how to approach this.
First, let me visualize the problem. A rectangular bloc... |
Okay, so I need to find the smallest possible value of N, where N is the number of 1-cm cubes in a solid rectangular block. When viewed so that three of its faces are visible, exactly 231 cubes cannot be seen. Hmm, let me think about how to approach this.
First, let me visualize the problem. A rectangular block has l... | To solve the problem, we are given a solid rectangular block made up of $ N $ unit cubes, and when viewed such that three of its faces are visible, exactly 231 of the cubes are **not visible**. Our goal is to find the **smallest possible value of $ N $**.
---
### Step 1: Understanding the Hidden Cubes
When a rectang... | stop | 90 | 8,678 | 0 | 32,768 |
AHSME__1979_AHSME_Problems__p8 | Find the area of the smallest region bounded by the graphs of $y=|x|$ and $x^2+y^2=4$.
$\textbf{(A) }\frac{\pi}{4}\qquad \textbf{(B) }\frac{3\pi}{4}\qquad \textbf{(C) }\pi\qquad \textbf{(D) }\frac{3\pi}{2}\qquad \textbf{(E) }2\pi$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the area of the smallest region bounded by the graphs of y = |x| and x² + y² = 4. Let me start by visualizing these graphs.
First, the equation x² + y² = 4 is a circle centered at the origin with radius 2. That's straightforward. Then, y = |x| is a V-shaped graph that consists of two l... |
Okay, so I need to find the area of the smallest region bounded by the graphs of y = |x| and x² + y² = 4. Let me start by visualizing these graphs.
First, the equation x² + y² = 4 is a circle centered at the origin with radius 2. That's straightforward. Then, y = |x| is a V-shaped graph that consists of two lines: y... | To find the **smallest region bounded by the graphs** of $ y = |x| $ and $ x^2 + y^2 = 4 $, we begin by analyzing the geometry of the two curves.
---
### **Step 1: Understand the Graphs**
- The equation $ x^2 + y^2 = 4 $ represents a **circle** centered at the origin with radius 2.
- The equation $ y = |x| $ represe... | stop | 126 | 10,601 | 0 | 32,768 |
AMC_10__2002_AMC_10A_Problems__p2 | Given that a, b, and c are non-zero real numbers, define $(a, b, c) = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}$, find $(2, 12, 9)$.
$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the value of (2, 12, 9) where this operation is defined as (a, b, c) = a/b + b/c + c/a. Let me make sure I understand the problem correctly. The question defines a new operation for three non-zero real numbers a, b, c, and then asks me to compute this operation with the specific numbers ... |
Okay, so I need to find the value of (2, 12, 9) where this operation is defined as (a, b, c) = a/b + b/c + c/a. Let me make sure I understand the problem correctly. The question defines a new operation for three non-zero real numbers a, b, c, and then asks me to compute this operation with the specific numbers 2, 12, ... | To solve the problem, we are given a custom operation defined as:
$$
(a, b, c) = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}
$$
We are asked to compute the value of $(2, 12, 9)$ using this definition.
---
### Step 1: Substitute the values
Let:
- $ a = 2 $
- $ b = 12 $
- $ c = 9 $
Now substitute into the expression:
... | stop | 142 | 2,774 | 0 | 32,768 |
AMC_10__2005_AMC_10A_Problems__p11 | A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$?
$\textbf{(A) } 3\qquad \textbf{(B) } 4\qquad \textbf{(C) } 5\qquad \textbf{(D) } 6\qquad \textbf{(E) } 7$
Please reason step by s... | <think>
Okay, so I need to solve this problem about a wooden cube that's painted red on all six faces, then cut into unit cubes. The problem says that exactly one-fourth of the total number of faces of the unit cubes are red. I need to find the value of n, which is the side length of the original cube. The answer choic... |
Okay, so I need to solve this problem about a wooden cube that's painted red on all six faces, then cut into unit cubes. The problem says that exactly one-fourth of the total number of faces of the unit cubes are red. I need to find the value of n, which is the side length of the original cube. The answer choices are ... | To solve the problem, we begin by analyzing the structure and properties of the cube and its unit cubes.
---
### Step 1: Total Number of Unit Cubes and Their Faces
A cube with side length $ n $ is divided into $ n^3 $ unit cubes. Each unit cube has **6 faces**, so the **total number of faces** across all unit cubes ... | stop | 135 | 2,982 | 0 | 32,768 |
AHSME__1960_AHSME_Problems__p15 | Triangle $I$ is equilateral with side $A$, perimeter $P$, area $K$, and circumradius $R$ (radius of the circumscribed circle).
Triangle $II$ is equilateral with side $a$, perimeter $p$, area $k$, and circumradius $r$. If $A$ is different from $a$, then:
$\textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad \... | <think>
Okay, so I need to figure out this problem about two equilateral triangles, Triangle I and Triangle II. The question is about the relationships between their perimeters, areas, and circumradii. The answer choices are about whether certain ratios are always equal or only sometimes. Let me start by recalling some... |
Okay, so I need to figure out this problem about two equilateral triangles, Triangle I and Triangle II. The question is about the relationships between their perimeters, areas, and circumradii. The answer choices are about whether certain ratios are always equal or only sometimes. Let me start by recalling some proper... | To solve this problem, we analyze the relationships between the perimeters, areas, and circumradii of two equilateral triangles, Triangle I and Triangle II, with side lengths $ A $ and $ a $, respectively. The key is to compare the ratios of these quantities and determine under what conditions they are equal.
---
###... | stop | 223 | 4,507 | 0 | 32,768 |
AMC_12__2023_AMC_12B_Problems__p4 | Jackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip $25$ meters long. How many square centimeters of paper could Jackson cover with paint?
$\textbf{(A) } 162,500 \qquad\textbf{(B) } 162.5 \qquad\textbf{(C) }1,625 \qquad\textbf{(D) }1,625,000 \qquad\text... | <think>
Okay, so Jackson is using his paintbrush to make a strip of paint, and we need to figure out how many square centimeters of paper he can cover with the paint he has. Let me try to break this down step by step.
First, the problem says that the paintbrush makes a narrow strip with a width of 6.5 millimeters. Hmm... |
Okay, so Jackson is using his paintbrush to make a strip of paint, and we need to figure out how many square centimeters of paper he can cover with the paint he has. Let me try to break this down step by step.
First, the problem says that the paintbrush makes a narrow strip with a width of 6.5 millimeters. Hmm, so th... | To determine how many square centimeters of paper Jackson can cover with his paint, we need to calculate the **area** of the strip he can paint. The strip has a **width** of 6.5 millimeters and a **length** of 25 meters. Since the final answer is expected in **square centimeters**, we must convert all measurements to c... | stop | 157 | 2,928 | 0 | 32,768 |
AIME__2015_AIME_I_Problems__p1 | The expressions $A$ = $1 \times 2 + 3 \times 4 + 5 \times 6 + \cdots + 37 \times 38 + 39$ and $B$ = $1 + 2 \times 3 + 4 \times 5 + \cdots + 36 \times 37 + 38 \times 39$ are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference betw... | <think>
Okay, so I need to find the positive difference between integers A and B. Let me first make sure I understand what A and B are.
Starting with A: It's given as 1 × 2 + 3 × 4 + 5 × 6 + ... + 37 × 38 + 39. So, it seems like there's a pattern where we multiply pairs of consecutive odd and even numbers, starting fr... |
Okay, so I need to find the positive difference between integers A and B. Let me first make sure I understand what A and B are.
Starting with A: It's given as 1 × 2 + 3 × 4 + 5 × 6 + ... + 37 × 38 + 39. So, it seems like there's a pattern where we multiply pairs of consecutive odd and even numbers, starting from 1×2,... | To find the positive difference between the two expressions:
- $ A = 1 \times 2 + 3 \times 4 + 5 \times 6 + \cdots + 37 \times 38 + 39 $
- $ B = 1 + 2 \times 3 + 4 \times 5 + \cdots + 36 \times 37 + 38 \times 39 $
we begin by analyzing the structure of both expressions.
---
### Step 1: Understand the Structure
- *... | stop | 144 | 6,716 | 0 | 32,768 |
AMC_8__2008_AMC_8_Problems__p24 | Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
$\textbf{(A)}\ \frac{1}{10}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{11}{60}\qquad\textbf{(D)}\... | <think>
Okay, so I need to find the probability that the product of the numbers on a tile (which is numbered 1 through 10) and a die roll (which is a standard die, so numbers 1 through 6) is a perfect square. The answer choices are given, so maybe I can figure it out step by step.
First, let me recall that a perfect s... |
Okay, so I need to find the probability that the product of the numbers on a tile (which is numbered 1 through 10) and a die roll (which is a standard die, so numbers 1 through 6) is a perfect square. The answer choices are given, so maybe I can figure it out step by step.
First, let me recall that a perfect square i... | To solve the problem, we are asked to find the probability that the product of a randomly selected tile number (from 1 to 10) and a randomly rolled die number (from 1 to 6) is a **perfect square**.
---
### Step 1: Total Number of Outcomes
There are 10 possible tile numbers and 6 possible die numbers, so the total nu... | stop | 159 | 7,511 | 0 | 32,768 |
AMC_12__2020_AMC_12B_Problems__p14 | Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real n... | <think>
Okay, so Bela and Jenn are playing this game on the interval [0, n], where n is an integer greater than 4. They take turns choosing real numbers, with Bela going first. The rule is that each number chosen has to be more than one unit away from all previously chosen numbers. The person who can't choose such a nu... |
Okay, so Bela and Jenn are playing this game on the interval [0, n], where n is an integer greater than 4. They take turns choosing real numbers, with Bela going first. The rule is that each number chosen has to be more than one unit away from all previously chosen numbers. The person who can't choose such a number lo... | To determine the winner of the game played by Bela and Jenn on the interval $[0, n]$, we analyze the rules and the structure of the game.
---
### **Game Rules Recap**
- Players take turns choosing real numbers from the interval $[0, n]$.
- Each chosen number must be **more than one unit away** from all previously ch... | stop | 258 | 24,902 | 0 | 32,768 |
AIME__2008_AIME_I_Problems__p10 | Let $ABCD$ be an isosceles trapezoid with $\overline{AD}||\overline{BC}$ whose angle at the longer base $\overline{AD}$ is $\dfrac{\pi}{3}$. The diagonals have length $10\sqrt {21}$, and point $E$ is at distances $10\sqrt {7}$ and $30\sqrt {7}$ from vertices $A$ and $D$, respectively. Let $F$ be the foot of the altitud... | <think>
Okay, so I need to solve this geometry problem about an isosceles trapezoid. Let me try to visualize it first. The trapezoid is ABCD with AD parallel to BC, and AD is the longer base. The angle at the longer base AD is π/3, which is 60 degrees. The diagonals are each 10√21 in length. There's a point E that's 10... |
Okay, so I need to solve this geometry problem about an isosceles trapezoid. Let me try to visualize it first. The trapezoid is ABCD with AD parallel to BC, and AD is the longer base. The angle at the longer base AD is π/3, which is 60 degrees. The diagonals are each 10√21 in length. There's a point E that's 10√7 away... | To solve the problem, we analyze the geometry of the isosceles trapezoid $ABCD$ with $AD \parallel BC$, and the given angle at the longer base $AD$ is $\frac{\pi}{3}$ (i.e., $60^\circ$). The diagonals $AC$ and $BD$ are both of length $10\sqrt{21}$, and point $E$ lies at distances $10\sqrt{7}$ from $A$ and $30\sqrt{7}$ ... | stop | 201 | 19,934 | 0 | 32,768 |
AIME__2012_AIME_II_Problems__p11 | Let $f_1(x) = \frac23 - \frac3{3x+1}$, and for $n \ge 2$, define $f_n(x) = f_1(f_{n-1}(x))$. The value of $x$ that satisfies $f_{1001}(x) = x-3$ can be expressed in the form $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Please reason step by step, and put your final answer within o... | <think>
Okay, so I need to solve this problem where we have a function f₁(x) defined as 2/3 - 3/(3x + 1), and then for n ≥ 2, each fₙ(x) is defined as f₁ composed with f_{n-1}(x). The question is asking for the value of x that satisfies f_{1001}(x) = x - 3, and then express that x as a reduced fraction m/n and find m +... |
Okay, so I need to solve this problem where we have a function f₁(x) defined as 2/3 - 3/(3x + 1), and then for n ≥ 2, each fₙ(x) is defined as f₁ composed with f_{n-1}(x). The question is asking for the value of x that satisfies f_{1001}(x) = x - 3, and then express that x as a reduced fraction m/n and find m + n. Alr... | To solve the problem, we are given a recursive function definition:
- $ f_1(x) = \frac{2}{3} - \frac{3}{3x + 1} $
- For $ n \geq 2 $, $ f_n(x) = f_1(f_{n-1}(x)) $
We are asked to find the value of $ x $ such that:
$$
f_{1001}(x) = x - 3
$$
---
### Step 1: Simplify $ f_1(x) $
We begin by simplifying the expression... | stop | 129 | 15,230 | 0 | 32,768 |
AMC_12__2020_AMC_12A_Problems__p1 | Carlos took $70\%$ of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?
$\textbf{(A)}\ 10\%\qquad\textbf{(B)}\ 15\%\qquad\textbf{(C)}\ 20\%\qquad\textbf{(D)}\ 30\%\qquad\textbf{(E)}\ 35\%$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so the problem says that Carlos took 70% of a whole pie. Then Maria took one third of the remainder. We need to find out what portion of the whole pie was left. The answer choices are given as percentages, so I need to calculate the remaining percentage after both Carlos and Maria took their shares.
Firs... |
Okay, so the problem says that Carlos took 70% of a whole pie. Then Maria took one third of the remainder. We need to find out what portion of the whole pie was left. The answer choices are given as percentages, so I need to calculate the remaining percentage after both Carlos and Maria took their shares.
First, let ... | To determine what portion of the whole pie was left after Carlos and Maria took their shares, we proceed step by step using percentages.
---
### Step 1: Carlos Takes 70% of the Pie
Assume the whole pie is 100%. Carlos takes 70% of it. Therefore, the remaining portion of the pie is:
$$
100\% - 70\% = 30\%
$$
So, af... | stop | 124 | 1,601 | 0 | 32,768 |
AMC_10__2002_AMC_10A_Problems__p5 | Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
[asy] unitsize(.3cm); path c=Circle((0,2),1); filldraw(Circle((0,0)... | <think>
Okay, so I need to find the area of the shaded region in this figure. Let me try to visualize it based on the Asymptote code provided. There's a large circle, and inside it, there's an innermost circle that's tangent to six surrounding circles. Each of those six circles is also tangent to the large circle and t... |
Okay, so I need to find the area of the shaded region in this figure. Let me try to visualize it based on the Asymptote code provided. There's a large circle, and inside it, there's an innermost circle that's tangent to six surrounding circles. Each of those six circles is also tangent to the large circle and to their... | To determine the area of the shaded region in the given figure, we analyze the structure of the circles and their relationships.
---
### **Understanding the Structure**
The figure consists of:
- **One central small circle** with radius 1.
- **Six surrounding small circles**, each also with radius 1, arranged symmet... | stop | 282 | 2,509 | 0 | 32,768 |
AMC_8__1987_AJHSME_Problems__p18 | Half the people in a room left. One third of those remaining started to dance. There were then $12$ people who were not dancing. The original number of people in the room was
$\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 42 \qquad \text{(E)}\ 72$
Please reason step by step, and put ... | <think>
Okay, let's try to figure out this problem. So, the question is about people leaving a room and then some starting to dance. We need to find the original number of people. Let me break it down step by step.
First, let me restate the problem in my own words to make sure I understand it. There's a room with some... |
Okay, let's try to figure out this problem. So, the question is about people leaving a room and then some starting to dance. We need to find the original number of people. Let me break it down step by step.
First, let me restate the problem in my own words to make sure I understand it. There's a room with some number... | To solve the problem, we begin by defining the original number of people in the room as $ x $.
---
### Step 1: People Who Left
Half of the people left the room.
So, the number of people who left is:
$$
\frac{1}{2}x
$$
Therefore, the number of people remaining in the room is:
$$
x - \frac{1}{2}x = \frac{1}{2}x... | stop | 126 | 1,870 | 0 | 32,768 |
AIME__1997_AIME_Problems__p4 | circles of radii $5, 5, 8,$ and $\frac mn$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the radius of a circle that's mutually externally tangent to three other circles with radii 5, 5, and 8. The problem says that all four circles are mutually externally tangent, and the radius we need to find is m/n where m and n are coprime. Then, I have to find m + n. Alright, let me th... |
Okay, so I need to find the radius of a circle that's mutually externally tangent to three other circles with radii 5, 5, and 8. The problem says that all four circles are mutually externally tangent, and the radius we need to find is m/n where m and n are coprime. Then, I have to find m + n. Alright, let me think abo... | To solve the problem, we are given four circles with radii $ 5, 5, 8, $ and $ \frac{m}{n} $, all of which are **mutually externally tangent**. Our goal is to find the value of $ m + n $, where $ \frac{m}{n} $ is the radius of the fourth circle.
---
### Step 1: Understanding the Configuration
When four circles are **... | stop | 71 | 11,886 | 0 | 32,768 |
AHSME__1965_AHSME_Problems__p9 | The vertex of the parabola $y = x^2 - 8x + c$ will be a point on the $x$-axis if the value of $c$ is:
$\textbf{(A)}\ - 16 \qquad \textbf{(B) }\ - 4 \qquad \textbf{(C) }\ 4 \qquad \textbf{(D) }\ 8 \qquad \textbf{(E) }\ 16$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the value of c that makes the vertex of the parabola y = x² - 8x + c lie on the x-axis. Let me think about how to approach this.
First, I remember that the vertex of a parabola given by y = ax² + bx + c is at the point (-b/(2a), f(-b/(2a))). In this case, the equation is y = x² - 8x + ... |
Okay, so I need to find the value of c that makes the vertex of the parabola y = x² - 8x + c lie on the x-axis. Let me think about how to approach this.
First, I remember that the vertex of a parabola given by y = ax² + bx + c is at the point (-b/(2a), f(-b/(2a))). In this case, the equation is y = x² - 8x + c. Comp... | To determine the value of $ c $ such that the vertex of the parabola $ y = x^2 - 8x + c $ lies on the **x-axis**, we need to understand the geometric and algebraic properties of a parabola.
---
### Step 1: Identify the Vertex of the Parabola
The general form of a quadratic function is:
$$
y = ax^2 + bx + c
$$
For ... | stop | 132 | 2,694 | 0 | 32,768 |
AMC_10__2015_AMC_10B_Problems__p3 | Isaac has written down one integer two times and another integer three times. The sum of the five numbers is $100$, and one of the numbers is $28.$ What is the other number?
$\textbf{(A) }8\qquad\textbf{(B) }11\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad\textbf{(E) }18$
Please reason step by step, and put your fin... | <think>
Okay, let's try to figure out this problem. So, Isaac wrote down one integer two times and another integer three times. The total sum of these five numbers is 100, and one of the numbers is 28. We need to find the other number. The options are A) 8, B) 11, C) 14, D) 15, E) 18. Alright, let's break this down ste... |
Okay, let's try to figure out this problem. So, Isaac wrote down one integer two times and another integer three times. The total sum of these five numbers is 100, and one of the numbers is 28. We need to find the other number. The options are A) 8, B) 11, C) 14, D) 15, E) 18. Alright, let's break this down step by st... | To solve the problem, we are told that Isaac has written down **one integer two times** and **another integer three times**, resulting in a total of **five numbers**. The **sum of these five numbers is 100**, and **one of the numbers is 28**. We are to find the **other number**.
---
### Step 1: Understand the Structu... | stop | 125 | 3,569 | 0 | 32,768 |
AMC_8__2010_AMC_8_Problems__p9 | Ryan got $80\%$ of the problems correct on a $25$-problem test, $90\%$ on a $40$-problem test, and $70\%$ on a $10$-problem test. What percent of all the problems did Ryan answer correctly?
$\textbf{(A)}\ 64 \qquad\textbf{(B)}\ 75\qquad\textbf{(C)}\ 80\qquad\textbf{(D)}\ 84\qquad\textbf{(E)}\ 86$
Please reason step b... | <think>
Okay, so Ryan took three different tests, and I need to figure out what percent of all the problems he answered correctly overall. Let me think about how to approach this.
First, let me recall that to find the overall percentage, I need to find the total number of problems he got right and divide that by the ... |
Okay, so Ryan took three different tests, and I need to figure out what percent of all the problems he answered correctly overall. Let me think about how to approach this.
First, let me recall that to find the overall percentage, I need to find the total number of problems he got right and divide that by the total n... | To determine the overall percentage of problems Ryan answered correctly across all three tests, we need to compute the total number of correct answers and divide that by the total number of problems across all tests. Then, we convert that fraction into a percentage.
---
### Step 1: Calculate the number of correct ans... | stop | 155 | 1,816 | 0 | 32,768 |
AIME__2023_AIME_II_Problems__p10 | Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2 \times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3.$ One way to do this is shown below. Find the number of positive integer divisors of $N.$
\[\be... | <think>
Okay, so I need to figure out the number of ways to place the integers 1 through 12 in a 2x6 grid such that any two adjacent cells (sharing a side) have numbers whose difference isn't divisible by 3. Then, once I find that number N, I have to find how many positive integer divisors N has.
First, let me unders... |
Okay, so I need to figure out the number of ways to place the integers 1 through 12 in a 2x6 grid such that any two adjacent cells (sharing a side) have numbers whose difference isn't divisible by 3. Then, once I find that number N, I have to find how many positive integer divisors N has.
First, let me understand th... | To solve the problem, we are asked to count the number of ways to place the integers $1$ through $12$ in a $2 \times 6$ grid such that **no two adjacent cells** (sharing a side) have numbers whose **difference is divisible by 3**. Then, we are to find the number of **positive integer divisors** of the total number of s... | stop | 211 | 28,387 | 0 | 32,768 |
AMC_12__2001_AMC_12_Problems__p25 | Consider sequences of positive real numbers of the form $x, 2000, y, \dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term $2001$ appear somewhere in the sequence?
$\text{(A) }1 \qquad \text{(B) }2 \qquad \text{(C) }3 \qq... | <think>
Okay, so I need to figure out how many different values of x will make the term 2001 appear somewhere in the sequence. The sequence starts with x, 2000, y, ... and every term after the first is 1 less than the product of its two immediate neighbors. Hmm, let me try to parse this.
First, let me write down the r... |
Okay, so I need to figure out how many different values of x will make the term 2001 appear somewhere in the sequence. The sequence starts with x, 2000, y, ... and every term after the first is 1 less than the product of its two immediate neighbors. Hmm, let me try to parse this.
First, let me write down the recurren... | To solve the problem, we are given a sequence of positive real numbers of the form:
$$
x, 2000, y, \dots
$$
with the rule that **every term after the first is 1 less than the product of its two immediate neighbors**. This gives us a recurrence relation. Let’s denote the sequence as $ a_1, a_2, a_3, \dots $, where:
-... | stop | 142 | 6,885 | 0 | 32,768 |
AMC_12__2011_AMC_12B_Problems__p25 | For every $m$ and $k$ integers with $k$ odd, denote by $\left[\frac{m}{k}\right]$ the integer closest to $\frac{m}{k}$. For every odd integer $k$, let $P(k)$ be the probability that
\[\left[\frac{n}{k}\right] + \left[\frac{100 - n}{k}\right] = \left[\frac{100}{k}\right]\]
for an integer $n$ randomly chosen from the int... | <think>
Okay, so I need to find the minimum possible value of P(k) for odd integers k between 1 and 99. P(k) is the probability that [n/k] + [(100 - n)/k] equals [100/k], where n is chosen randomly from 1 to 99! (which is a huge number, but maybe the exact value doesn't matter because the probability might depend on th... | null | <think>
Okay, so I need to find the minimum possible value of P(k) for odd integers k between 1 and 99. P(k) is the probability that [n/k] + [(100 - n)/k] equals [100/k], where n is chosen randomly from 1 to 99! (which is a huge number, but maybe the exact value doesn't matter because the probability might depend on th... | length | 277 | 32,768 | 0 | 32,768 |
IMO__1978_IMO_Problems__p1 | Let $m$ and $n$ be positive integers such that $1 \le m < n$. In their decimal representations, the last three digits of $1978^m$ are equal, respectively, to the last three digits of $1978^n$. Find $m$ and $n$ such that $m + n$ has its least value.
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find positive integers m and n where 1 ≤ m < n, such that the last three digits of 1978^m and 1978^n are the same. And among all such pairs, I need to find the one where m + n is as small as possible. Alright, let me think about how to approach this.
First, the problem is about modular arith... |
Okay, so I need to find positive integers m and n where 1 ≤ m < n, such that the last three digits of 1978^m and 1978^n are the same. And among all such pairs, I need to find the one where m + n is as small as possible. Alright, let me think about how to approach this.
First, the problem is about modular arithmetic, ... | To solve the problem, we are given that $ 1978^m \equiv 1978^n \pmod{1000} $ for some positive integers $ m < n $, and we are to find such $ m $ and $ n $ such that $ m + n $ is minimized.
---
### Step 1: Reduce the Base Modulo 1000
We begin by reducing the base modulo 1000:
$$
1978 \equiv 978 \pmod{1000}
$$
So th... | stop | 102 | 13,183 | 0 | 32,768 |
AMC_12__2017_AMC_12B_Problems__p14 | An ice-cream novelty item consists of a cup in the shape of a 4-inch-tall frustum of a right circular cone, with a 2-inch-diameter base at the bottom and a 4-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height 4 inches, whose base, at the bottom, is the top base... | <think>
Okay, so I need to find the total volume of this ice cream novelty item. It has two parts: a frustum of a cone and a cone on top of it. Let me try to visualize this. The frustum is 4 inches tall, with the bottom base having a diameter of 2 inches and the top base having a diameter of 4 inches. Then there's a so... |
Okay, so I need to find the total volume of this ice cream novelty item. It has two parts: a frustum of a cone and a cone on top of it. Let me try to visualize this. The frustum is 4 inches tall, with the bottom base having a diameter of 2 inches and the top base having a diameter of 4 inches. Then there's a solid con... | To determine the total volume of the ice cream novelty item, we analyze the structure, which consists of two distinct parts:
1. **A frustum of a cone** with:
- Height $ h = 4 $ inches
- Bottom base diameter = 2 inches → radius $ r = 1 $ inch
- Top base diameter = 4 inches → radius $ R = 2 $ inches
2. **A sol... | stop | 208 | 4,712 | 0 | 32,768 |
AMC_8__1996_AJHSME_Problems__p20 | Suppose there is a special key on a calculator that replaces the number $x$ currently displayed with the number given by the formula $1/(1-x)$. For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display -1 since $1/(1-2)=-1$. Now suppose that the calculator is disp... | <think>
Okay, so there's this calculator with a special key that takes the current number x and replaces it with 1/(1 - x). The question is, if we start with 5 and press this key 100 times in a row, what number will we end up with? The options are A through E, and I need to figure out which one is correct. Let me think... |
Okay, so there's this calculator with a special key that takes the current number x and replaces it with 1/(1 - x). The question is, if we start with 5 and press this key 100 times in a row, what number will we end up with? The options are A through E, and I need to figure out which one is correct. Let me think throug... | To solve the problem, we are given a special calculator function that transforms a number $ x $ into $ \frac{1}{1 - x} $. Starting with the number 5, we are to determine the result after pressing this key **100 times in a row**.
---
### Step 1: Understand the Function
Let’s define the function:
$$
f(x) = \frac{1}{1... | stop | 187 | 2,856 | 0 | 32,768 |
AIME__2022_AIME_II_Problems__p12 | Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that\[\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2}=1.\]Find the least possible value of $a+b.$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the least possible value of a + b given these two equations involving x, y, a, and b. Let me start by writing down the problem again to make sure I have it right.
We have real numbers a, b, x, y with a > 4 and b > 1. The equations given are:
$$
\frac{x^2}{a^2} + \frac{y^2}{a^2 - 16} = ... |
Okay, so I need to find the least possible value of a + b given these two equations involving x, y, a, and b. Let me start by writing down the problem again to make sure I have it right.
We have real numbers a, b, x, y with a > 4 and b > 1. The equations given are:
$$
\frac{x^2}{a^2} + \frac{y^2}{a^2 - 16} = 1
$$
an... | To solve the problem, we are given two ellipses:
$$
\frac{x^2}{a^2} + \frac{y^2}{a^2 - 16} = 1 \quad \text{(1)}
$$
$$
\frac{(x - 20)^2}{b^2 - 1} + \frac{(y - 11)^2}{b^2} = 1 \quad \text{(2)}
$$
with the constraints $ a > 4 $ and $ b > 1 $, and we are to find the **least possible value of $ a + b $** such that the two... | stop | 123 | 21,060 | 0 | 32,768 |
AMC_12__2006_AMC_12A_Problems__p3 | The ratio of Mary's age to Alice's age is $3:5$. Alice is $30$ years old. How old is Mary?
$\textbf{(A)}\ 15\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 50$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to figure out how old Mary is based on the ratio of their ages and Alice's age. Let me start by understanding the problem again. The ratio of Mary's age to Alice's age is 3:5, and Alice is 30 years old. The question is asking for Mary's age, with multiple choices provided.
First, ratios can som... |
Okay, so I need to figure out how old Mary is based on the ratio of their ages and Alice's age. Let me start by understanding the problem again. The ratio of Mary's age to Alice's age is 3:5, and Alice is 30 years old. The question is asking for Mary's age, with multiple choices provided.
First, ratios can sometimes ... | To determine Mary's age, we are given the ratio of Mary's age to Alice's age as $3:5$, and we are told that Alice is 30 years old.
---
### Step 1: Understand the Ratio
The ratio $3:5$ means that for every 3 units of Mary's age, Alice's age corresponds to 5 units. This can be expressed as:
$$
\frac{\text{Mary's age}... | stop | 119 | 1,529 | 0 | 32,768 |
AIME__2008_AIME_II_Problems__p1 | Let $N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $N$ is divided by $1000$.
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the value of this big expression N, which is 100² + 99² - 98² - 97² + 96² + ... + 4² + 3² - 2² - 1². The pattern here is that the signs alternate in pairs: two positives, two negatives, two positives, etc. And then I have to find N mod 1000. Alright, let me try to break this down step by... |
Okay, so I need to find the value of this big expression N, which is 100² + 99² - 98² - 97² + 96² + ... + 4² + 3² - 2² - 1². The pattern here is that the signs alternate in pairs: two positives, two negatives, two positives, etc. And then I have to find N mod 1000. Alright, let me try to break this down step by step.
... | To solve the problem, we are given the expression:
$$
N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + 95^2 - 94^2 - 93^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2
$$
The pattern of signs is **two positives followed by two negatives**, repeating every four terms. This allows us to group the expression into **25 groups** of four terms ... | stop | 112 | 11,826 | 0 | 32,768 |
AHSME__1957_AHSME_Problems__p38 | From a two-digit number $N$ we subtract the number with the digits reversed and find that the result is a positive perfect cube. Then:
$\textbf{(A)}\ {N}\text{ cannot end in 5}\qquad\\ \textbf{(B)}\ {N}\text{ can end in any digit other than 5}\qquad \\ \textbf{(C)}\ {N}\text{ does not exist}\qquad\\ \textbf{(D)}\ \t... | <think>
Okay, so I need to solve this problem where we have a two-digit number N. When we subtract the number with its digits reversed from N, the result is a positive perfect cube. Then we have to figure out which of the options A to E is correct. Let me try to break this down step by step.
First, let me recall that ... |
Okay, so I need to solve this problem where we have a two-digit number N. When we subtract the number with its digits reversed from N, the result is a positive perfect cube. Then we have to figure out which of the options A to E is correct. Let me try to break this down step by step.
First, let me recall that a two-d... | To solve the problem, we begin by representing the two-digit number $ N $ as $ 10a + b $, where $ a $ is the tens digit and $ b $ is the units digit. The number with its digits reversed is $ 10b + a $. The problem states that the difference between the original number and the reversed number is a **positive perfect cub... | stop | 170 | 5,647 | 0 | 32,768 |
USAJMO__2016_USAJMO_Problems__p4 | Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from the set $\{1, 2,...,N\}$, one can still find $2016$ distinct numbers among the remaining elements with sum $N$.
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the smallest integer N such that if you remove any 2016 elements from the set {1, 2, ..., N}, there will still be 2016 distinct numbers left that add up to N. Hmm, that sounds a bit tricky, but let me try to break it down.
First, let me rephrase the problem to make sure I understand it.... |
Okay, so I need to find the smallest integer N such that if you remove any 2016 elements from the set {1, 2, ..., N}, there will still be 2016 distinct numbers left that add up to N. Hmm, that sounds a bit tricky, but let me try to break it down.
First, let me rephrase the problem to make sure I understand it. We hav... | To solve the problem, we are asked to find the **least integer** $ N $ such that **no matter which 2016 elements are removed** from the set $ \{1, 2, \ldots, N\} $, **there will always exist 2016 distinct numbers among the remaining elements** that sum to $ N $.
---
### Key Observations
- The total number of element... | stop | 86 | 30,897 | 0 | 32,768 |
AHSME__1973_AHSME_Problems__p30 | Let $[t]$ denote the greatest integer $\leq t$ where $t \geq 0$ and $S = \{(x,y): (x-T)^2 + y^2 \leq T^2 \text{ where } T = t - [t]\}$. Then we have
$\textbf{(A)}\ \text{the point } (0,0) \text{ does not belong to } S \text{ for any } t \qquad$
$\textbf{(B)}\ 0 \leq \text{Area } S \leq \pi \text{ for all } t \qquad$
$... | <think>
Okay, let me try to work through this problem step by step. First, let me make sure I understand what's being asked here.
We have a set S defined as all points (x, y) such that (x - T)^2 + y^2 ≤ T^2, where T is equal to t minus the greatest integer less than or equal to t. So, T is essentially the fractional p... |
Okay, let me try to work through this problem step by step. First, let me make sure I understand what's being asked here.
We have a set S defined as all points (x, y) such that (x - T)^2 + y^2 ≤ T^2, where T is equal to t minus the greatest integer less than or equal to t. So, T is essentially the fractional part of ... | To solve the problem, we analyze the set $ S $ defined as:
$$
S = \{(x, y) : (x - T)^2 + y^2 \leq T^2 \text{ where } T = t - [t] \}
$$
---
### Understanding the Definition of $ T $
The term $ T = t - [t] $ represents the **fractional part** of $ t $, which is always in the interval $ [0, 1) $. That is, for any real... | stop | 247 | 5,254 | 0 | 32,768 |
AMC_12__2002_AMC_12A_Problems__p9 | Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?
$\textbf{(A)}\ 12 \qquad \textbf{(B)... | <think>
Okay, so Jamal needs to save 30 files onto disks, each with 1.44 MB of space. The files are of three different sizes: 3 files are 0.8 MB each, 12 files are 0.7 MB each, and the rest are 0.4 MB each. The question is asking for the smallest number of disks needed to store all these files without splitting any fil... |
Okay, so Jamal needs to save 30 files onto disks, each with 1.44 MB of space. The files are of three different sizes: 3 files are 0.8 MB each, 12 files are 0.7 MB each, and the rest are 0.4 MB each. The question is asking for the smallest number of disks needed to store all these files without splitting any file acros... | To determine the **minimum number of disks** needed to store all 30 files, we begin by analyzing the file sizes and the disk capacity.
---
### **Step 1: Understand the File Sizes and Total Space**
We are given:
- **3 files** of **0.8 MB**
- **12 files** of **0.7 MB**
- **15 files** of **0.4 MB**
Total number of fi... | stop | 177 | 13,072 | 0 | 32,768 |
AMC_8__2009_AMC_8_Problems__p19 | Two angles of an isosceles triangle measure $70^\circ$ and $x^\circ$. What is the sum of the three possible values of $x$?
$\textbf{(A)}\ 95 \qquad \textbf{(B)}\ 125 \qquad \textbf{(C)}\ 140 \qquad \textbf{(D)}\ 165 \qquad \textbf{(E)}\ 180$
Please reason step by step, and put your final answer... | <think>
Okay, so I need to solve this problem about an isosceles triangle where two angles measure 70 degrees and x degrees. The question is asking for the sum of the three possible values of x. The answer choices are given, so maybe I can figure out the possible values of x and then add them up. Let me think step by s... |
Okay, so I need to solve this problem about an isosceles triangle where two angles measure 70 degrees and x degrees. The question is asking for the sum of the three possible values of x. The answer choices are given, so maybe I can figure out the possible values of x and then add them up. Let me think step by step.
F... | To solve the problem, we are given that **two angles of an isosceles triangle measure $70^\circ$ and $x^\circ$**, and we are to find the **sum of the three possible values of $x$**.
---
### Step 1: Understand the Triangle's Properties
An **isosceles triangle** has **at least two equal angles**. The sum of the angles... | stop | 137 | 10,914 | 0 | 32,768 |
AMC_12__2009_AMC_12A_Problems__p12 | How many positive integers less than $1000$ are $6$ times the sum of their digits?
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 12$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to figure out how many positive integers less than 1000 are 6 times the sum of their digits. The answer choices are from 0 to 12, so I need to find the correct one. Let me start by understanding the problem.
First, the numbers we are considering are positive integers less than 1000. That means ... |
Okay, so I need to figure out how many positive integers less than 1000 are 6 times the sum of their digits. The answer choices are from 0 to 12, so I need to find the correct one. Let me start by understanding the problem.
First, the numbers we are considering are positive integers less than 1000. That means we're l... | To solve the problem, we are asked to find how many **positive integers less than 1000** are **6 times the sum of their digits**.
---
### Step 1: Understand the Problem
We are looking for all positive integers $ N < 1000 $ such that:
$$
N = 6 \times (\text{sum of digits of } N)
$$
Let’s denote the sum of the digit... | stop | 107 | 7,039 | 0 | 32,768 |
AMC_10__2002_AMC_10B_Problems__p12 | For which of the following values of $k$ does the equation $\frac{x-1}{x-2} = \frac{x-k}{x-6}$ have no solution for $x$?
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to figure out for which value of k the equation (x - 1)/(x - 2) = (x - k)/(x - 6) has no solution. The options are A) 1, B) 2, C) 3, D) 4, E) 5. Alright, let me start by understanding the problem.
First, the equation is a rational equation, meaning there are variables in the denominators. So, I... |
Okay, so I need to figure out for which value of k the equation (x - 1)/(x - 2) = (x - k)/(x - 6) has no solution. The options are A) 1, B) 2, C) 3, D) 4, E) 5. Alright, let me start by understanding the problem.
First, the equation is a rational equation, meaning there are variables in the denominators. So, I need t... | To determine for which value of $ k $ the equation
$$
\frac{x - 1}{x - 2} = \frac{x - k}{x - 6}
$$
has **no solution**, we begin by analyzing the structure of the equation and solving it algebraically.
---
### Step 1: Eliminate Denominators
We cross-multiply to eliminate the denominators:
$$
(x - 1)(x - 6) = (x -... | stop | 124 | 4,663 | 0 | 32,768 |
USAJMO__2020_USAJMO_Problems__p3 | An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:
- The two $1 \times 1$ faces of each beam coinci... | <think>
Okay, so I need to figure out the minimum number of beams required to fill a 2020×2020×2020 cube under certain conditions. Let me try to understand the problem first.
First, the cube is 2020 units on each side. On each face of the cube, there's a 2020×2020 grid of unit cells. A beam is a 1×1×2020 rectangular p... |
Okay, so I need to figure out the minimum number of beams required to fill a 2020×2020×2020 cube under certain conditions. Let me try to understand the problem first.
First, the cube is 2020 units on each side. On each face of the cube, there's a 2020×2020 grid of unit cells. A beam is a 1×1×2020 rectangular prism. S... | To determine the **minimum number of beams** that can be placed inside a $2020 \times 2020 \times 2020$ cube under the given constraints, we must carefully analyze the conditions and how they constrain the placement of beams.
---
### **Understanding the Beam Structure**
Each beam is a $1 \times 1 \times 2020$ rectan... | stop | 231 | 25,297 | 0 | 32,768 |
AIME__1993_AIME_Problems__p9 | Two thousand points are given on a circle. Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label this point $3$. (See figure.) Continue this process until the labels $1,2,3\dot... | <think>
Okay, so there's this problem with 2000 points on a circle. We start labeling them by starting at point 1, then moving 2 points clockwise to label 2, then from 2 moving 3 points to label 3, and so on up to 1993. The question is asking for the smallest integer that labels the same point as 1993.
First, I need ... |
Okay, so there's this problem with 2000 points on a circle. We start labeling them by starting at point 1, then moving 2 points clockwise to label 2, then from 2 moving 3 points to label 3, and so on up to 1993. The question is asking for the smallest integer that labels the same point as 1993.
First, I need to unde... | To solve the problem, we are given a circle with 2000 points, and we label them sequentially starting from point 1, moving clockwise by an increasing number of steps. Specifically, label $ k $ is placed by moving $ k $ positions from the previous label. The goal is to find the **smallest integer** that labels the **sam... | stop | 154 | 18,329 | 0 | 32,768 |
AMC_8__1986_AJHSME_Problems__p15 | Sale prices at the Ajax Outlet Store are $50\%$ below original prices. On Saturdays an additional discount of $20\%$ off the sale price is given. What is the Saturday price of a coat whose original price is $\textdollar 180$?
$\text{(A)}$ $\textdollar 54$
$\text{(B)}$ $\textdollar 72$
$\text{(C)}$ $\textdollar 90$
$\... | <think>
Okay, let me try to figure out this problem. So, the question is about finding the Saturday price of a coat that originally costs $180. The store has sale prices that are 50% below original prices, and on Saturdays, there's an additional 20% discount off the sale price. Hmm, let me break this down step by step.... |
Okay, let me try to figure out this problem. So, the question is about finding the Saturday price of a coat that originally costs $180. The store has sale prices that are 50% below original prices, and on Saturdays, there's an additional 20% discount off the sale price. Hmm, let me break this down step by step.
First... | To determine the Saturday price of a coat originally priced at $180, we need to apply two successive discounts:
---
### **Step 1: Apply the 50% discount**
The first discount is **50% off** the original price. This means the customer pays **50% of the original price**.
$$
\text{Sale Price} = 180 \times (1 - 0.50) = ... | stop | 158 | 1,873 | 0 | 32,768 |
AMC_12__2006_AMC_12B_Problems__p24 | Let $S$ be the set of all point $(x,y)$ in the coordinate plane such that $0 \le x \le \frac{\pi}{2}$ and $0 \le y \le \frac{\pi}{2}$. What is the area of the subset of $S$ for which \[\sin^2x-\sin x \sin y + \sin^2y \le \frac34?\]
$\mathrm{(A)}\ \dfrac{\pi^2}{9} \qquad \mathrm{(B)}\ \dfrac{\pi^2}{8} \qquad \mathrm{(C... | <think>
Okay, so I need to find the area of the subset of the square S, where S is all points (x, y) with 0 ≤ x ≤ π/2 and 0 ≤ y ≤ π/2, that satisfy the inequality sin²x - sinx siny + sin²y ≤ 3/4. The answer choices are given, so maybe I can figure out the region defined by this inequality and compute its area.
First, ... |
Okay, so I need to find the area of the subset of the square S, where S is all points (x, y) with 0 ≤ x ≤ π/2 and 0 ≤ y ≤ π/2, that satisfy the inequality sin²x - sinx siny + sin²y ≤ 3/4. The answer choices are given, so maybe I can figure out the region defined by this inequality and compute its area.
First, let me ... | To solve the problem, we are given a region $ S $ in the coordinate plane defined by:
$$
0 \le x \le \frac{\pi}{2}, \quad 0 \le y \le \frac{\pi}{2}
$$
and we are to find the area of the subset of $ S $ satisfying the inequality:
$$
\sin^2 x - \sin x \sin y + \sin^2 y \le \frac{3}{4}
$$
---
### Step 1: Change of Va... | stop | 213 | 27,304 | 0 | 32,768 |
AMC_10__2007_AMC_10B_Problems__p20 | A set of $25$ square blocks is arranged into a $5 \times 5$ square. How many different combinations of $3$ blocks can be selected from that set so that no two are in the same row or column?
$\textbf{(A) } 100 \qquad\textbf{(B) } 125 \qquad\textbf{(C) } 600 \qquad\textbf{(D) } 2300 \qquad\textbf{(E) } 3600$
Please rea... | <think>
Okay, so I need to figure out how many different combinations of 3 blocks can be selected from a 5x5 grid such that no two are in the same row or column. Let me think about how to approach this.
First, let me visualize the grid. It's a 5x5 square, so there are 5 rows and 5 columns. We need to choose 3 blocks w... |
Okay, so I need to figure out how many different combinations of 3 blocks can be selected from a 5x5 grid such that no two are in the same row or column. Let me think about how to approach this.
First, let me visualize the grid. It's a 5x5 square, so there are 5 rows and 5 columns. We need to choose 3 blocks with the... | To solve the problem of selecting **3 blocks** from a **5×5 grid** such that **no two blocks are in the same row or column**, we can approach the problem using combinatorics and the concept of **permutations**.
---
### Step 1: Understanding the Constraint
We are selecting **3 blocks** from a **5×5 grid**, and the co... | stop | 145 | 6,654 | 0 | 32,768 |
AMC_12__2020_AMC_12B_Problems__p20 | Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?
$\textbf{(A)}\ \frac{9}{64} \qquad\textbf{(B)}\ \frac{289... | <think>
Okay, so I need to find the probability that two different cubes, each painted with faces independently colored black or white, can be rotated to look identical. The answer choices are given, so maybe I can figure out the number of distinct colorings up to rotation and then compute the probability accordingly. ... |
Okay, so I need to find the probability that two different cubes, each painted with faces independently colored black or white, can be rotated to look identical. The answer choices are given, so maybe I can figure out the number of distinct colorings up to rotation and then compute the probability accordingly. Let me ... | To solve the problem, we are asked to find the probability that two independently and randomly painted cubes (each face colored black or white) can be rotated to look identical. This is a classic problem in combinatorics involving group theory and symmetry.
---
### Step 1: Total Number of Colorings
Each cube has 6 f... | stop | 180 | 24,763 | 0 | 32,768 |
AHSME__1956_AHSME_Problems__p23 | About the equation $ax^2 - 2x\sqrt {2} + c = 0$, with $a$ and $c$ real constants,
we are told that the discriminant is zero. The roots are necessarily:
$\textbf{(A)}\ \text{equal and integral}\qquad \textbf{(B)}\ \text{equal and rational}\qquad \textbf{(C)}\ \text{equal and real} \\ \textbf{(D)}\ \text{equal and irra... | <think>
Okay, so I need to figure out the nature of the roots of the quadratic equation $ ax^2 - 2x\sqrt{2} + c = 0 $, given that the discriminant is zero. The options are about whether the roots are equal and integral, rational, real, irrational, or imaginary. Let me start by recalling some basics about quadratic equa... |
Okay, so I need to figure out the nature of the roots of the quadratic equation $ ax^2 - 2x\sqrt{2} + c = 0 $, given that the discriminant is zero. The options are about whether the roots are equal and integral, rational, real, irrational, or imaginary. Let me start by recalling some basics about quadratic equations.
... | To determine the nature of the roots of the quadratic equation
$$
ax^2 - 2x\sqrt{2} + c = 0,
$$
we are told that the **discriminant is zero**. Let's analyze this step by step.
---
### Step 1: Recall the Discriminant Formula
For a general quadratic equation of the form
$$
Ax^2 + Bx + C = 0,
$$
the **discrimin... | stop | 154 | 3,730 | 0 | 32,768 |
AHSME__1959_AHSME_Problems__p42 | Given three positive integers $a,b,$ and $c$. Their greatest common divisor is $D$; their least common multiple is $m$.
Then, which two of the following statements are true?
$\text{(1)}\ \text{the product MD cannot be less than abc} \qquad \\ \text{(2)}\ \text{the product MD cannot be greater than abc}\qquad \\ \text{... | <think>
Okay, so I need to figure out which two of the given statements are true regarding the greatest common divisor (GCD) and least common multiple (LCM) of three positive integers a, b, and c. The problem mentions that their GCD is D and their LCM is m. Then there are four statements, and I have to determine which ... |
Okay, so I need to figure out which two of the given statements are true regarding the greatest common divisor (GCD) and least common multiple (LCM) of three positive integers a, b, and c. The problem mentions that their GCD is D and their LCM is m. Then there are four statements, and I have to determine which two are... | To determine which two of the given statements are true, we analyze the relationship between the **greatest common divisor (GCD)** and **least common multiple (LCM)** of three positive integers $ a, b, c $, where:
- $ D = \gcd(a, b, c) $
- $ m = \text{lcm}(a, b, c) $
We are given the product $ MD = D \cdot m $, and w... | stop | 249 | 6,078 | 0 | 32,768 |
AIME__1987_AIME_Problems__p8 | What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}$?
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the largest positive integer n for which there's exactly one integer k that satisfies the inequality 8/15 < n/(n + k) < 7/13. Hmm, let me try to break this down step by step.
First, let me understand what the inequality is saying. The fraction n/(n + k) is between 8/15 and 7/13. Since 8... |
Okay, so I need to find the largest positive integer n for which there's exactly one integer k that satisfies the inequality 8/15 < n/(n + k) < 7/13. Hmm, let me try to break this down step by step.
First, let me understand what the inequality is saying. The fraction n/(n + k) is between 8/15 and 7/13. Since 8/15 is ... | To solve the problem, we are given the inequality:
$$
\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}
$$
Our goal is to find the **largest positive integer** $ n $ such that **there is exactly one integer** $ k $ satisfying the inequality.
---
### Step 1: Transform the Inequality
We start by manipulating the inequal... | stop | 73 | 32,187 | 0 | 32,768 |
AMC_10__2013_AMC_10A_Problems__p10 | A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?
$\textbf{(A)}\ 15 \qquad\textbf{(B)}\ 30 \qquad\textbf{(... | <think>
Okay, let me try to figure out this flower bouquet problem. So, there are pink roses, red roses, pink carnations, and red carnations. The question gives me some ratios and asks for the percentage of flowers that are carnations. Let me break down the information step by step.
First, let me note down all the giv... |
Okay, let me try to figure out this flower bouquet problem. So, there are pink roses, red roses, pink carnations, and red carnations. The question gives me some ratios and asks for the percentage of flowers that are carnations. Let me break down the information step by step.
First, let me note down all the given info... | To determine the percentage of flowers that are **carnations**, we are given the following information:
- **One third of the pink flowers are roses**.
- **Three fourths of the red flowers are carnations**.
- **Six tenths of the flowers are pink**.
---
### Step 1: Define Total Flowers
Let the total number of flowers... | stop | 149 | 4,413 | 0 | 32,768 |
AMC_12__2002_AMC_12P_Problems__p8 | Let $AB$ be a segment of length $26$, and let points $C$ and $D$ be located on $AB$ such that $AC=1$ and $AD=8$. Let $E$ and $F$ be points on one of the semicircles with diameter $AB$ for which $EC$ and $FD$ are perpendicular to $AB$. Find $EF.$
$\text{(A) }5 \qquad \text{(B) }5 \sqrt{2} \qquad \text{(C) }7 \qquad \te... | <think>
Okay, so I need to find the length of EF in this geometry problem. Let me try to visualize the problem first. There's a segment AB with length 26. Points C and D are on AB such that AC is 1 unit and AD is 8 units. Then there are semicircles with diameter AB, and points E and F are on one of these semicircles. T... |
Okay, so I need to find the length of EF in this geometry problem. Let me try to visualize the problem first. There's a segment AB with length 26. Points C and D are on AB such that AC is 1 unit and AD is 8 units. Then there are semicircles with diameter AB, and points E and F are on one of these semicircles. The cond... | To solve the problem, we begin by assigning coordinates to the points on the line segment $ AB $, which has a length of 26. We place the segment on the coordinate plane with point $ A $ at the origin $ (0, 0) $ and point $ B $ at $ (26, 0) $. This makes $ AB $ lie along the x-axis.
---
### Step 1: Coordinates of Poin... | stop | 169 | 5,070 | 0 | 32,768 |
IMO__1968_IMO_Problems__p2 | Find all natural numbers $x$ such that the product of their digits (in decimal notation) is equal to $x^2 - 10x - 22$.
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find all natural numbers x where the product of their digits equals x² - 10x - 22. Let me start by understanding the problem.
First, natural numbers are positive integers (1, 2, 3, ...). The product of their digits means if x is, say, 23, then the product is 2*3=6. The equation given is prod... |
Okay, so I need to find all natural numbers x where the product of their digits equals x² - 10x - 22. Let me start by understanding the problem.
First, natural numbers are positive integers (1, 2, 3, ...). The product of their digits means if x is, say, 23, then the product is 2*3=6. The equation given is product_of_... | To solve the problem, we are asked to find all **natural numbers** $ x $ such that the **product of their digits** equals the expression $ x^2 - 10x - 22 $.
---
### Step 1: Understand the Equation
Let $ P(x) $ denote the **product of the digits** of $ x $. Then the equation becomes:
$$
P(x) = x^2 - 10x - 22
$$
We ... | stop | 62 | 12,122 | 0 | 32,768 |
AMC_10__2004_AMC_10A_Problems__p14 | The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $20$ cents. If she had one more quarter, the average value would be $21$ cents. How many dimes does she have in her purse?
$\text {(A)}\ 0 \qquad \text {(B)}\ 1 \qquad \text {(C)}\ 2 \qquad \text {(D)}\ 3\qquad \text {(E)}\ 4$
Pleas... | <think>
Okay, so I need to figure out how many dimes Paula has in her purse. The problem says that the average value of all the pennies, nickels, dimes, and quarters is 20 cents. Then, if she had one more quarter, the average would be 21 cents. The answer choices are from 0 to 4, so I need to find which one is correct.... |
Okay, so I need to figure out how many dimes Paula has in her purse. The problem says that the average value of all the pennies, nickels, dimes, and quarters is 20 cents. Then, if she had one more quarter, the average would be 21 cents. The answer choices are from 0 to 4, so I need to find which one is correct.
First... | To solve the problem, we are given two key pieces of information:
1. The **average value** of all the coins in Paula's purse is **20 cents**.
2. If she adds **one more quarter**, the **average value** becomes **21 cents**.
---
### Step 1: Define Variables
Let:
- $ N $ be the number of coins in Paula's purse.
- The ... | stop | 135 | 6,716 | 0 | 32,768 |
AMC_8__2015_AMC_8_Problems__p21 | In the given figure, hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$?
[asy] draw((-4,6*sqrt(2))--(4,6*sqrt(2))); draw((-4,-6*sqrt(2))--(4,-6*sqrt(2))); draw((-8,0)--(-4,6*sqrt(2))); draw... | <think>
Okay, so I need to find the area of triangle KBC in this figure. Let me start by understanding the problem step by step.
First, the hexagon ABCDEF is equiangular. That means all its internal angles are equal. For a regular hexagon, each internal angle is 120 degrees, but since it's only mentioned as equiangul... |
Okay, so I need to find the area of triangle KBC in this figure. Let me start by understanding the problem step by step.
First, the hexagon ABCDEF is equiangular. That means all its internal angles are equal. For a regular hexagon, each internal angle is 120 degrees, but since it's only mentioned as equiangular, not... | To determine the area of triangle $ \triangle KBC $, we begin by analyzing the geometric configuration and using the given information:
---
### **Step 1: Understand the Given Information**
- **Hexagon $ ABCDEF $** is **equiangular**, meaning all internal angles are $ 120^\circ $.
- **Squares $ ABJI $** and **$ FEHG ... | stop | 650 | 10,501 | 0 | 32,768 |
AMC_8__2023_AMC_8_Problems__p14 | Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of $5$-cent, $10$-cent, and $25$-cent stamps, with exactly $20$ of each type. What is the greatest number of sta... | <think>
Okay, so Nicolas wants to send a package and needs exactly $7.10 in postage. He has 5-cent, 10-cent, and 25-cent stamps, and he has exactly 20 of each type. The question is asking for the greatest number of stamps he can use to make exactly $7.10. The answer choices are from 45 to 55, so I need to figure out wh... |
Okay, so Nicolas wants to send a package and needs exactly $7.10 in postage. He has 5-cent, 10-cent, and 25-cent stamps, and he has exactly 20 of each type. The question is asking for the greatest number of stamps he can use to make exactly $7.10. The answer choices are from 45 to 55, so I need to figure out which one... | To determine the **maximum number of stamps** Nicolas can use to make **exactly $7.10** in postage, we begin by converting the total amount into **cents**:
$$
7.10 \text{ dollars} = 710 \text{ cents}
$$
Nicolas has **20 stamps** of each of the following denominations:
- 5-cent stamps
- 10-cent stamps
- 25-cent stamps... | stop | 221 | 11,719 | 0 | 32,768 |
AIME__2000_AIME_II_Problems__p5 | Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$.
Please reason step by step, ... | <think>
Okay, so I need to figure out the number of possible five-ring arrangements on four fingers, with each finger being able to have zero or more rings, and the order of rings on each finger matters. The rings are distinguishable, so each one is unique. Then, once I find that number, n, I have to find the leftmost ... |
Okay, so I need to figure out the number of possible five-ring arrangements on four fingers, with each finger being able to have zero or more rings, and the order of rings on each finger matters. The rings are distinguishable, so each one is unique. Then, once I find that number, n, I have to find the leftmost three n... | To solve the problem, we are given **eight distinguishable rings** and asked to find the number of **five-ring arrangements** on **four fingers** (not the thumb) of one hand. The key conditions are:
- **Order of rings on each finger is significant** (i.e., the sequence of rings on a finger matters).
- **Each finger ma... | stop | 90 | 17,879 | 0 | 32,768 |
IMO__1977_IMO_Problems__p5 | Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find all pairs of natural numbers (a, b) such that when you divide a² + b² by a + b, the quotient is q and the remainder is r, and then q² + r equals 1977. Hmm, let me think about how to approach this.
First, I remember that when you divide a number by another, you can express it as dividend... | null | <think>
Okay, so I need to find all pairs of natural numbers (a, b) such that when you divide a² + b² by a + b, the quotient is q and the remainder is r, and then q² + r equals 1977. Hmm, let me think about how to approach this.
First, I remember that when you divide a number by another, you can express it as dividend... | length | 88 | 32,768 | 0 | 32,768 |
AIME__2011_AIME_II_Problems__p9 | Let $x_1, x_2, ... , x_6$ be non-negative real numbers such that $x_1 +x_2 +x_3 +x_4 +x_5 +x_6 =1$, and $x_1 x_3 x_5 +x_2 x_4 x_6 \ge {\frac{1}{540}}$. Let $p$ and $q$ be positive relatively prime integers such that $\frac{p}{q}$ is the maximum possible value of
$x_1 x_2 x_3 + x_2 x_3 x_4 +x_3 x_4 x_5 +x_4 x_5 x_6 +x_5... | <think>
Okay, so I need to find the maximum value of the expression $ S = x_1 x_2 x_3 + x_2 x_3 x_4 + x_3 x_4 x_5 + x_4 x_5 x_6 + x_5 x_6 x_1 + x_6 x_1 x_2 $ given that the sum of the variables $ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1 $ and another condition $ x_1 x_3 x_5 + x_2 x_4 x_6 \geq \frac{1}{540} $. The answer s... |
Okay, so I need to find the maximum value of the expression $ S = x_1 x_2 x_3 + x_2 x_3 x_4 + x_3 x_4 x_5 + x_4 x_5 x_6 + x_5 x_6 x_1 + x_6 x_1 x_2 $ given that the sum of the variables $ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1 $ and another condition $ x_1 x_3 x_5 + x_2 x_4 x_6 \geq \frac{1}{540} $. The answer should b... | To solve the problem, we are given six non-negative real numbers $ x_1, x_2, \ldots, x_6 $ such that:
- $ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1 $
- $ x_1 x_3 x_5 + x_2 x_4 x_6 \geq \frac{1}{540} $
We are to **maximize** the expression:
$$
S = x_1 x_2 x_3 + x_2 x_3 x_4 + x_3 x_4 x_5 + x_4 x_5 x_6 + x_5 x_6 x_1 + x_6 ... | stop | 204 | 17,060 | 0 | 32,768 |
AMC_12__2021_AMC_12A_Problems__p2 | Under what conditions is $\sqrt{a^2+b^2}=a+b$ true, where $a$ and $b$ are real numbers?
$\textbf{(A) }$ It is never true.
$\textbf{(B) }$ It is true if and only if $ab=0$.
$\textbf{(C) }$ It is true if and only if $a+b\ge 0$.
$\textbf{(D) }$ It is true if and only if $ab=0$ and $a+b\ge 0$.
$\textbf{(E) }$ It is always ... | <think>
Okay, so I need to figure out under what conditions the equation √(a² + b²) = a + b holds true, where a and b are real numbers. The options are given from A to E, and I need to choose the correct one. Let me think through this step by step.
First, let me recall that the square root of a squared term is the abs... |
Okay, so I need to figure out under what conditions the equation √(a² + b²) = a + b holds true, where a and b are real numbers. The options are given from A to E, and I need to choose the correct one. Let me think through this step by step.
First, let me recall that the square root of a squared term is the absolute v... | To determine under what conditions the equation
$$
\sqrt{a^2 + b^2} = a + b
$$
is true for real numbers $ a $ and $ b $, we begin by analyzing the structure of the equation.
---
### Step 1: Square Both Sides
To eliminate the square root, we square both sides of the equation:
$$
\left( \sqrt{a^2 + b^2} \right)^2... | stop | 157 | 4,013 | 0 | 32,768 |
AIME__2000_AIME_I_Problems__p12 | Given a function $f$ for which
\[f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)\]
holds for all real $x,$ what is the largest number of different values that can appear in the list $f(0),f(1),f(2),\ldots,f(999)?$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to figure out the largest number of different values that can appear in the list f(0), f(1), f(2), ..., f(999), given that the function f satisfies the equations f(x) = f(398 - x) = f(2158 - x) = f(3214 - x) for all real x. Hmm, let me try to unpack this step by step.
First, the function f has ... |
Okay, so I need to figure out the largest number of different values that can appear in the list f(0), f(1), f(2), ..., f(999), given that the function f satisfies the equations f(x) = f(398 - x) = f(2158 - x) = f(3214 - x) for all real x. Hmm, let me try to unpack this step by step.
First, the function f has these s... | To determine the **maximum number of different values** that can appear in the list $ f(0), f(1), f(2), \ldots, f(999) $, we are given that the function $ f $ satisfies the following symmetries for all real $ x $:
$$
f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)
$$
---
### Step 1: Understanding the Symmetries
Each ... | stop | 112 | 20,987 | 0 | 32,768 |
AMC_10__2008_AMC_10B_Problems__p3 | Assume that $x$ is a positive real number . Which is equivalent to $\sqrt[3]{x\sqrt{x}}$?
$\mathrm{(A)}\ x^{1/6}\qquad\mathrm{(B)}\ x^{1/4}\qquad\mathrm{(C)}\ x^{3/8}\qquad\mathrm{(D)}\ x^{1/2}\qquad\mathrm{(E)}\ x$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to figure out which of the given options is equivalent to the cube root of x times the square root of x. Let me write that down first to make sure I have the expression right. The original expression is ∛(x√x). Alright, so that's the cube root of x multiplied by the square root of x. Hmm, maybe ... |
Okay, so I need to figure out which of the given options is equivalent to the cube root of x times the square root of x. Let me write that down first to make sure I have the expression right. The original expression is ∛(x√x). Alright, so that's the cube root of x multiplied by the square root of x. Hmm, maybe I shoul... | To simplify the expression $\sqrt[3]{x\sqrt{x}}$, we begin by converting all radicals into exponents, which makes it easier to apply exponent rules.
---
### Step 1: Convert the square root to an exponent
We know that:
$$
\sqrt{x} = x^{1/2}
$$
So the expression becomes:
$$
\sqrt[3]{x \cdot x^{1/2}}
$$
---
### Step... | stop | 117 | 2,384 | 0 | 32,768 |
AHSME__1994_AHSME_Problems__p24 | A sample consisting of five observations has an arithmetic mean of $10$ and a median of $12$. The smallest value that the range (largest observation minus smallest) can assume for such a sample is
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 10$
Please reason s... | <think>
Okay, so I need to find the smallest possible range for a sample of five numbers where the mean is 10 and the median is 12. The range is the difference between the largest and smallest numbers. The answer choices are from 2 to 10, so I need to figure out which one is the minimum possible range.
First, let me r... |
Okay, so I need to find the smallest possible range for a sample of five numbers where the mean is 10 and the median is 12. The range is the difference between the largest and smallest numbers. The answer choices are from 2 to 10, so I need to figure out which one is the minimum possible range.
First, let me recall t... | To determine the **smallest possible range** for a sample of five observations with a **mean of 10** and a **median of 12**, we proceed step by step.
---
### Step 1: Understand the Constraints
Let the five observations be denoted in ascending order as:
$$
a \leq b \leq c \leq d \leq e
$$
Given:
- The **mean** is 1... | stop | 128 | 11,287 | 0 | 32,768 |
AIME__1989_AIME_Problems__p12 | Let $ABCD$ be a tetrahedron with $AB=41$, $AC=7$, $AD=18$, $BC=36$, $BD=27$, and $CD=13$, as shown in the figure. Let $d$ be the distance between the midpoints of edges $AB$ and $CD$. Find $d^{2}$.
[asy] defaultpen(fontsize(10)+0.8); size(175); pair A,B,C,D,M,P,Q; C=origin; B=(8,0); D=IP(CR(C,6.5),CR(B,8)); A=(4,-3); P... | <think>
Okay, so I need to find the square of the distance between the midpoints of edges AB and CD in this tetrahedron ABCD. The given edge lengths are AB=41, AC=7, AD=18, BC=36, BD=27, and CD=13. Hmm, let me think about how to approach this.
First, I remember that in a tetrahedron, the distance between midpoints of ... |
Okay, so I need to find the square of the distance between the midpoints of edges AB and CD in this tetrahedron ABCD. The given edge lengths are AB=41, AC=7, AD=18, BC=36, BD=27, and CD=13. Hmm, let me think about how to approach this.
First, I remember that in a tetrahedron, the distance between midpoints of two edg... | To find the square of the distance $ d^2 $ between the midpoints of edges $ AB $ and $ CD $ in the tetrahedron $ ABCD $, we can use vector geometry and the properties of midpoints in 3D space.
---
### Step 1: Use Vector Representation
Let the position vectors of the points be:
- $ \vec{A} $, $ \vec{B} $, $ \vec{C} $... | stop | 379 | 8,092 | 0 | 32,768 |
AMC_8__1986_AJHSME_Problems__p6 | $\frac{2}{1-\frac{2}{3}}=$
$\text{(A)}\ -3 \qquad \text{(B)}\ -\frac{4}{3} \qquad \text{(C)}\ \frac{2}{3} \qquad \text{(D)}\ 2 \qquad \text{(E)}\ 6$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to solve this fraction problem: 2 divided by (1 minus 2/3). Let me think about how to approach this. First, I remember that when you have a complex fraction like this, you need to simplify the denominator first before doing the division. Let me write it down step by step.
The original problem i... |
Okay, so I need to solve this fraction problem: 2 divided by (1 minus 2/3). Let me think about how to approach this. First, I remember that when you have a complex fraction like this, you need to simplify the denominator first before doing the division. Let me write it down step by step.
The original problem is 2 div... | To evaluate the expression:
$$
\frac{2}{1 - \frac{2}{3}}
$$
we begin by simplifying the denominator:
---
### Step 1: Simplify the Denominator
The denominator is:
$$
1 - \frac{2}{3}
$$
To subtract these, we express 1 as a fraction with denominator 3:
$$
1 = \frac{3}{3}
$$
So:
$$
1 - \frac{2}{3} = \frac{3}{3} -... | stop | 104 | 1,592 | 0 | 32,768 |
AMC_12__2022_AMC_12A_Problems__p9 | On Halloween $31$ children walked into the principal's office asking for candy. They
can be classified into three types: Some always lie; some always tell the truth; and
some alternately lie and tell the truth. The alternaters arbitrarily choose their first
response, either a lie or the truth, but each subsequent state... | <think>
Okay, let's try to tackle this Halloween candy problem. So, there are 31 children, and they are either truth-tellers, liars, or alternaters. The principal asked them three questions, and based on their answers, gave out candy. We need to figure out how many pieces of candy the truth-tellers received in total. T... |
Okay, let's try to tackle this Halloween candy problem. So, there are 31 children, and they are either truth-tellers, liars, or alternaters. The principal asked them three questions, and based on their answers, gave out candy. We need to figure out how many pieces of candy the truth-tellers received in total. The answ... | To solve this problem, we analyze the behavior of the three types of children — **truth-tellers**, **liars**, and **alternaters** — based on their responses to the three questions asked by the principal. The goal is to determine how many **candies** were given to the **truth-tellers**.
---
### Step 1: Understand the ... | stop | 280 | 7,209 | 0 | 32,768 |
AHSME__1982_AHSME_Problems__p25 | The adjacent map is part of a city: the small rectangles are blocks, and the paths in between are streets.
Each morning, a student walks from intersection $A$ to intersection $B$, always walking along streets shown,
and always going east or south. For variety, at each intersection where he has a choice, he chooses wi... | <think>
Okay, so I need to find the probability that the student goes through intersection C when walking from A to B, always moving east or south, and choosing randomly between the two when there's a choice. Let me try to visualize the map first based on the Asymptote code.
The Asymptote code draws a grid of blocks. ... |
Okay, so I need to find the probability that the student goes through intersection C when walking from A to B, always moving east or south, and choosing randomly between the two when there's a choice. Let me try to visualize the map first based on the Asymptote code.
The Asymptote code draws a grid of blocks. The mai... | To solve the problem, we analyze the movement of the student from point **A** to point **B**, always moving **east** or **south**, and choosing randomly between the two at each intersection where a choice exists. The goal is to compute the **probability** that the student passes through point **C**.
---
### Step 1: U... | stop | 516 | 23,276 | 0 | 32,768 |
USAMO__2006_USAMO_Problems__p2 | For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k+1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $N/2$.
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the minimum value of N for a given positive integer k, such that there exists a set of 2k+1 distinct positive integers. This set has a total sum greater than N, but every subset of size k has a sum at most N/2. Hmm, interesting problem. Let me try to break it down.
First, let me restate... |
Okay, so I need to find the minimum value of N for a given positive integer k, such that there exists a set of 2k+1 distinct positive integers. This set has a total sum greater than N, but every subset of size k has a sum at most N/2. Hmm, interesting problem. Let me try to break it down.
First, let me restate the pr... | To solve the problem, we are given a positive integer $ k $, and we are to find the **minimum value of $ N $** such that there exists a set of $ 2k+1 $ **distinct positive integers** with the following properties:
1. The **total sum** of the set is **greater than $ N $**.
2. The **sum of any subset of size $ k $** is ... | stop | 90 | 25,139 | 0 | 32,768 |
AHSME__1984_AHSME_Problems__p22 | Let $a$ and $c$ be fixed . For each $t$ let $(x_t, y_t)$ be the of the $y=ax^2+bx+c$. If the set of the vertices $(x_t, y_t)$ for all real numbers of $t$ is graphed on the , the is
$\mathrm{(A) \ } \text{a straight line} \qquad \mathrm{(B) \ } \text{a parabola} \qquad \mathrm{(C) \ } \text{part, but not all, of a p... | <think>
Okay, so I need to figure out what the set of vertices (x_t, y_t) of the parabola y = ax² + bx + c looks like as t varies. Wait, hold on, the problem says "for each t let (x_t, y_t) be the of the y=ax² + bx + c." Hmm, maybe there's a typo or missing part here? Let me check again.
Wait, maybe it's supposed to s... |
Okay, so I need to figure out what the set of vertices (x_t, y_t) of the parabola y = ax² + bx + c looks like as t varies. Wait, hold on, the problem says "for each t let (x_t, y_t) be the of the y=ax² + bx + c." Hmm, maybe there's a typo or missing part here? Let me check again.
Wait, maybe it's supposed to say "for... | To determine the nature of the set of vertices $(x_t, y_t)$ of the parabola $ y = ax^2 + bx + c $, we begin by recalling the formula for the vertex of a parabola in standard form.
---
### Step 1: Vertex of a Parabola
For a quadratic function of the form:
$$
y = ax^2 + bx + c
$$
the **x-coordinate** of the vertex i... | stop | 193 | 3,668 | 0 | 32,768 |
AMC_12__2009_AMC_12B_Problems__p15 | Assume $0 < r < 3$. Below are five equations for $x$. Which equation has the largest solution $x$?
$\textbf{(A)}\ 3(1 + r)^x = 7\qquad \textbf{(B)}\ 3(1 + r/10)^x = 7\qquad \textbf{(C)}\ 3(1 + 2r)^x = 7$
$\textbf{(D)}\ 3(1 + \sqrt {r})^x = 7\qquad \textbf{(E)}\ 3(1 + 1/r)^x = 7$
Please reason step by step, and put y... | <think>
Okay, so I need to figure out which of these equations has the largest solution for x, given that 0 < r < 3. Let me start by understanding each equation and how they relate to x. All the equations are of the form 3*(something)^x = 7. So, if I solve each equation for x, I can compare the solutions.
First, let m... |
Okay, so I need to figure out which of these equations has the largest solution for x, given that 0 < r < 3. Let me start by understanding each equation and how they relate to x. All the equations are of the form 3*(something)^x = 7. So, if I solve each equation for x, I can compare the solutions.
First, let me recal... | To determine which equation has the **largest solution** for $ x $, we analyze the structure of each equation:
All equations are of the form:
$$
3 \cdot (1 + \text{some expression})^x = 7
$$
We can solve for $ x $ in each case by isolating the exponential term and applying logarithms:
$$
x = \frac{\ln\left(\frac{7}... | stop | 167 | 7,462 | 0 | 32,768 |
AIME__2001_AIME_II_Problems__p2 | Each of the $2001$ students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between $80$ percent and $85$ percent of the school population, and the number who study French is between $30$ percent and $40$ percent. Let $m$ be the smallest number of students who cou... | <think>
Okay, so there's this problem about 2001 students at a high school. Each student studies either Spanish or French, and some study both. The number who study Spanish is between 80% and 85% of the school population, and the number who study French is between 30% and 40%. We need to find the smallest number of stu... |
Okay, so there's this problem about 2001 students at a high school. Each student studies either Spanish or French, and some study both. The number who study Spanish is between 80% and 85% of the school population, and the number who study French is between 30% and 40%. We need to find the smallest number of students w... | To solve the problem, we are given:
- Total number of students: $ 2001 $
- Each student studies **either** Spanish, French, or both.
- The number of students studying **Spanish** is between **80%** and **85%** of the total.
- The number of students studying **French** is between **30%** and **40%** of the total.
- We ... | stop | 132 | 7,825 | 0 | 32,768 |
AMC_10__2007_AMC_10A_Problems__p24 | centered at $A$ and $B$ each have radius $2$, as shown. Point $O$ is the of $\overline{AB}$, and $OA = 2\sqrt {2}$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively, and $EF$ is a common . What is the area of the shaded region $ECODF$?
[asy] size(5cm); pair A=(-2*sqrt(2),0), B = ... | <think>
Okay, so I need to find the area of the shaded region ECODF. Let me try to visualize the diagram based on the Asymptote code and the description. There are two circles, one centered at A and the other at B, each with radius 2. Point O is the midpoint of segment AB, and OA is given as 2√2. So, AB must be twice t... |
Okay, so I need to find the area of the shaded region ECODF. Let me try to visualize the diagram based on the Asymptote code and the description. There are two circles, one centered at A and the other at B, each with radius 2. Point O is the midpoint of segment AB, and OA is given as 2√2. So, AB must be twice that, wh... | To determine the area of the shaded region $ ECODF $, we begin by analyzing the geometric configuration and the given data.
---
### **Step 1: Coordinate Setup**
We are given:
- Two circles, each with radius 2, centered at points $ A = (-2\sqrt{2}, 0) $ and $ B = (2\sqrt{2}, 0) $.
- Point $ O $ is the midpoint of $ ... | stop | 538 | 8,418 | 0 | 32,768 |
AIME__2003_AIME_II_Problems__p7 | Find the area of rhombus $ABCD$ given that the circumradii of triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively.
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the area of rhombus ABCD. The problem says that the circumradii of triangles ABD and ACD are 12.5 and 25, respectively. Hmm, let me recall some properties of rhombuses and circumradii.
First, in a rhombus, all sides are equal in length. Also, the diagonals of a rhombus bisect each other... |
Okay, so I need to find the area of rhombus ABCD. The problem says that the circumradii of triangles ABD and ACD are 12.5 and 25, respectively. Hmm, let me recall some properties of rhombuses and circumradii.
First, in a rhombus, all sides are equal in length. Also, the diagonals of a rhombus bisect each other at rig... | To find the area of rhombus $ABCD$, we are given the circumradii of triangles $ABD$ and $ACD$ as $12.5$ and $25$, respectively.
---
### Step 1: Understand the Geometry of the Rhombus
In a rhombus:
- All sides are equal in length.
- The diagonals bisect each other at right angles.
- Let the diagonals be $AC = 2p$ and... | stop | 67 | 4,871 | 0 | 32,768 |
AMC_10__2008_AMC_10B_Problems__p16 | Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is 0.)
$\mathrm{(A)}\ {{{\frac{3} {8}}}} \qquad \mathrm{(B)}\ {{{\frac{1} {2}}}} \qquad \mathrm{(C)}\ {{{\frac{43} {72}}}}... | <think>
Okay, so I need to figure out the probability that the sum of the die rolls is odd when two fair coins are tossed once. For each head that comes up, we roll a die. And if there are no heads, the sum is 0. The answer choices are given, so maybe I can work through the problem step by step.
First, let me understa... |
Okay, so I need to figure out the probability that the sum of the die rolls is odd when two fair coins are tossed once. For each head that comes up, we roll a die. And if there are no heads, the sum is 0. The answer choices are given, so maybe I can work through the problem step by step.
First, let me understand the ... | To solve the problem, we analyze the scenario step by step, considering all possible outcomes of the coin toss and the corresponding die rolls.
---
### Step 1: Understand the Coin Toss Outcomes
We are tossing **two fair coins**, so the possible outcomes are:
- **HH** (both heads)
- **HT** (first head, second tail)
... | stop | 172 | 8,842 | 0 | 32,768 |
AMC_8__2016_AMC_8_Problems__p6 | The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names?
[asy] unitsize(0.9cm); draw((-0.5,0)--(10,0), linewidth(1.5)); draw((-0.5,1)--(10,1)); draw((-0.5,2)--(10,2)); draw((-0.5,3)--(10,3)); draw((-0.5,4)--(10,4)); draw((-0.5,5)--(10,5)); draw((-0.... | <think>
Okay, so I need to find the median length of the names from this bar graph. Let me think about how to approach this. First, the median is the middle value when all the data points are ordered from smallest to largest. Since there are 19 people, which is an odd number, the median will be the 10th value when the ... |
Okay, so I need to find the median length of the names from this bar graph. Let me think about how to approach this. First, the median is the middle value when all the data points are ordered from smallest to largest. Since there are 19 people, which is an odd number, the median will be the 10th value when the lengths... | To determine the **median length** of the names, we need to understand the data represented in the bar graph and then find the middle value when all the name lengths are ordered from smallest to largest.
---
### Step 1: Interpreting the Bar Graph
From the Asymptote code, we can extract the **frequencies** of name le... | stop | 543 | 9,652 | 0 | 32,768 |
IMO__1990_IMO_Problems__p1 | Chords $AB$ and $CD$ of a circle intersect at a point $E$ inside the circle. Let $M$ be an interior point of the segment $\overline{EB}$. The tangent line at $E$ to the circle through $D, E$, and $M$ intersects the lines $\overline{BC}$ and ${AC}$ at $F$ and $G$, respectively.
If $\frac{AM}{AB} = t$, find $\frac{EG}{E... | <think>
Okay, so I need to find the ratio EG/EF in terms of t, where t is AM/AB. Let me try to visualize the problem first. There's a circle with chords AB and CD intersecting at point E inside the circle. Then there's a point M on EB, and we're considering the tangent line at E to the circle passing through D, E, and ... |
Okay, so I need to find the ratio EG/EF in terms of t, where t is AM/AB. Let me try to visualize the problem first. There's a circle with chords AB and CD intersecting at point E inside the circle. Then there's a point M on EB, and we're considering the tangent line at E to the circle passing through D, E, and M. This... | To solve the problem, we are given a circle with chords $ AB $ and $ CD $ intersecting at point $ E $ inside the circle. A point $ M $ lies on segment $ \overline{EB} $, and the tangent line at $ E $ to the circle passing through points $ D, E, M $ intersects lines $ \overline{BC} $ and $ \overline{AC} $ at points $ F ... | stop | 136 | 8,625 | 0 | 32,768 |
AIME__1999_AIME_Problems__p5 | For any positive integer $x_{}$, let $S(x)$ be the sum of the digits of $x_{}$, and let $T(x)$ be $|S(x+2)-S(x)|.$ For example, $T(199)=|S(201)-S(199)|=|3-19|=16.$ How many values of $T(x)$ do not exceed 1999?
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to figure out how many values of T(x) do not exceed 1999. Let me start by understanding the problem again.
First, S(x) is the sum of the digits of x. Then T(x) is the absolute difference between S(x+2) and S(x). For example, they gave T(199) = |S(201) - S(199)| = |3 - 19| = 16. So, I need to f... |
Okay, so I need to figure out how many values of T(x) do not exceed 1999. Let me start by understanding the problem again.
First, S(x) is the sum of the digits of x. Then T(x) is the absolute difference between S(x+2) and S(x). For example, they gave T(199) = |S(201) - S(199)| = |3 - 19| = 16. So, I need to find all... | To solve the problem, we are given a function $ T(x) = |S(x+2) - S(x)| $, where $ S(x) $ is the sum of the digits of $ x $. The goal is to determine how many **distinct values** of $ T(x) $ do **not exceed 1999**.
---
### Step 1: Understanding the Behavior of $ T(x) $
Let’s analyze how the digit sum changes when we ... | stop | 121 | 21,300 | 0 | 32,768 |
AHSME__1952_AHSME_Problems__p12 | The sum to infinity of the terms of an infinite geometric progression is $6$. The sum of the first two terms is $4\frac{1}{2}$. The first term of the progression is:
$\textbf{(A) \ }3 \text{ or } 1\frac{1}{2} \qquad \textbf{(B) \ }1 \qquad \textbf{(C) \ }2\frac{1}{2} \qquad \textbf{(D) \ }6 \qquad \textbf{(E) \ }9\tex... | <think>
Okay, so I need to find the first term of an infinite geometric progression where the sum to infinity is 6, and the sum of the first two terms is 4 and a half, which is 9/2. The answer choices are given, so maybe I can work through the problem step by step.
First, let me recall the formula for the sum to infin... |
Okay, so I need to find the first term of an infinite geometric progression where the sum to infinity is 6, and the sum of the first two terms is 4 and a half, which is 9/2. The answer choices are given, so maybe I can work through the problem step by step.
First, let me recall the formula for the sum to infinity of ... | To solve the problem, we are given two key pieces of information about an infinite geometric progression:
1. The **sum to infinity** of the terms is $ 6 $.
2. The **sum of the first two terms** is $ 4\frac{1}{2} = \frac{9}{2} $.
---
### Step 1: Use the formula for the sum to infinity
For an infinite geometric serie... | stop | 153 | 5,544 | 0 | 32,768 |
AMC_8__2008_AMC_8_Problems__p1 | Susan had 50 dollars to spend at the carnival. She spent 12 dollars on food and twice as much on rides. How many dollars did she have left to spend?
$\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 38 \qquad \textbf{(E)}\ 50$
Please reason step by step, and put your final answer w... | <think>
Okay, let's see. Susan had 50 dollars to spend at the carnival. She spent 12 dollars on food and twice as much on rides. I need to figure out how much she has left. Hmm, let me break this down step by step.
First, she started with 50 dollars. Then she spent some money on food. The problem says she spent 12 dol... |
Okay, let's see. Susan had 50 dollars to spend at the carnival. She spent 12 dollars on food and twice as much on rides. I need to figure out how much she has left. Hmm, let me break this down step by step.
First, she started with 50 dollars. Then she spent some money on food. The problem says she spent 12 dollars on... | To determine how much money Susan had left after spending at the carnival, we begin by analyzing the information provided:
- Susan started with **$50**.
- She spent **$12** on food.
- She spent **twice as much on rides** as she did on food.
---
### Step 1: Calculate the amount spent on rides
Since she spent $12 on ... | stop | 125 | 755 | 0 | 32,768 |
AMC_12__2015_AMC_12B_Problems__p3 | Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?
$\textbf{(A)}\; 8 \qquad\textbf{(B)}\; 11 \qquad\textbf{(C)}\; 14 \qquad\textbf{(D)}\; 15 \qquad\textbf{(E)}\; 18$
Please reason step by step, and pu... | <think>
Okay, let's try to figure out this problem. So, Isaac wrote down one integer two times and another integer three times. The total sum of these five numbers is 100, and one of the numbers is 28. We need to find the other number. The options are A) 8, B) 11, C) 14, D) 15, E) 18. Alright, let's break this down ste... |
Okay, let's try to figure out this problem. So, Isaac wrote down one integer two times and another integer three times. The total sum of these five numbers is 100, and one of the numbers is 28. We need to find the other number. The options are A) 8, B) 11, C) 14, D) 15, E) 18. Alright, let's break this down step by st... | To solve the problem, we are told that Isaac has written down **one integer two times** and **another integer three times**, resulting in a total of **five numbers**. The **sum of these five numbers is 100**, and **one of the numbers is 28**. We are to find the **other number**.
---
### Step 1: Define the Variables
... | stop | 135 | 3,209 | 0 | 32,768 |
AMC_8__1988_AJHSME_Problems__p22 | Tom's Hat Shoppe increased all original prices by $25\%$. Now the shoppe is having a sale where all prices are $20\%$ off these increased prices. Which statement best describes the sale price of an item?
$\text{(A)}\ \text{The sale price is }5\% \text{ higher than the original price.}$
$\text{(B)}\ \text{The sale pri... | <think>
Okay, let me try to figure out this problem. So, Tom's Hat Shoppe first increased all original prices by 25%, and then they're having a sale where all prices are 20% off these increased prices. The question is asking which statement best describes the sale price compared to the original price. The options are A... |
Okay, let me try to figure out this problem. So, Tom's Hat Shoppe first increased all original prices by 25%, and then they're having a sale where all prices are 20% off these increased prices. The question is asking which statement best describes the sale price compared to the original price. The options are A throug... | To determine the final sale price of an item after a 25% increase followed by a 20% discount, we can analyze the problem step by step using algebraic reasoning.
---
### Step 1: Define the Original Price
Let the original price of an item be denoted by $ P $.
---
### Step 2: Apply the 25% Increase
A 25% increase me... | stop | 206 | 4,068 | 0 | 32,768 |
AMC_8__1988_AJHSME_Problems__p14 | $\diamondsuit$ and $\Delta$ are whole numbers and $\diamondsuit \times \Delta =36$. The largest possible value of $\diamondsuit + \Delta$ is
$\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 20\ \qquad \text{(E)}\ 37$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the largest possible value of ♦ + Δ, given that ♦ and Δ are whole numbers and their product is 36. The answer choices are A) 12, B) 13, C) 15, D) 20, E) 37. Let me think about how to approach this.
First, I remember that when you have two numbers with a fixed product, their sum is maxim... |
Okay, so I need to find the largest possible value of ♦ + Δ, given that ♦ and Δ are whole numbers and their product is 36. The answer choices are A) 12, B) 13, C) 15, D) 20, E) 37. Let me think about how to approach this.
First, I remember that when you have two numbers with a fixed product, their sum is maximized wh... | To solve the problem, we are given that:
- $ \diamondsuit $ and $ \Delta $ are **whole numbers**.
- $ \diamondsuit \times \Delta = 36 $
- We are to find the **largest possible value** of $ \diamondsuit + \Delta $
---
### Step 1: Understanding the Constraints
The term "whole numbers" typically refers to **non-negati... | stop | 123 | 3,260 | 0 | 32,768 |
AMC_12__2004_AMC_12B_Problems__p11 | All the students in an algebra class took a $100$-point test. Five students scored $100$, each student scored at least $60$, and the mean score was $76$. What is the smallest possible number of students in the class?
$\mathrm{(A)}\ 10 \qquad \mathrm{(B)}\ 11 \qquad \mathrm{(C)}\ 12 \qquad \mathrm{(D)}\ 13 \qquad \mathr... | <think>
Okay, so I need to figure out the smallest possible number of students in the algebra class based on the given information. Let me start by listing out all the details provided:
1. All students took a 100-point test.
2. Five students scored 100.
3. Each student scored at least 60.
4. The mean score was 76.
5. ... |
Okay, so I need to figure out the smallest possible number of students in the algebra class based on the given information. Let me start by listing out all the details provided:
1. All students took a 100-point test.
2. Five students scored 100.
3. Each student scored at least 60.
4. The mean score was 76.
5. We need... | To determine the **smallest possible number of students** in the class, we are given the following:
- Five students scored **100**.
- Each student scored **at least 60**.
- The **mean score** was **76**.
- We are to find the **minimum number of students** $ n $ that satisfies all these conditions.
---
### Step 1: Ex... | stop | 139 | 3,311 | 0 | 32,768 |
AMC_8__2004_AMC_8_Problems__p11 | The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:1. The largest isn't first, but it is in one of the first three places.
2. The smallest isn't last, but it is in one of the last three places.
3. The median isn't first or last.
What is the average of the first and last numbers?
$\textbf{(A... | <think>
Okay, let's try to figure out this problem. So, we have the numbers -2, 4, 6, 9, and 12. We need to rearrange them according to three specific rules, and then find the average of the first and last numbers. The answer choices are given, so maybe we can figure it out step by step.
First, let me list out the num... |
Okay, let's try to figure out this problem. So, we have the numbers -2, 4, 6, 9, and 12. We need to rearrange them according to three specific rules, and then find the average of the first and last numbers. The answer choices are given, so maybe we can figure it out step by step.
First, let me list out the numbers ag... | To solve the problem, we are given five numbers:
$$
-2, 4, 6, 9, 12
$$
and three rules that dictate how they must be rearranged. Our goal is to determine the **average of the first and last numbers** in a valid arrangement.
---
### Step 1: Understand the Rules
Let’s restate the rules clearly:
1. **The largest n... | stop | 181 | 6,029 | 0 | 32,768 |
AMC_10__2014_AMC_10B_Problems__p2 | What is $\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}$?
$\textbf {(A) } 16 \qquad \textbf {(B) } 24 \qquad \textbf {(C) } 32 \qquad \textbf {(D) } 48 \qquad \textbf {(E) } 64$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to solve this problem: (2³ + 2³) divided by (2⁻³ + 2⁻³). The answer choices are A through E, and I need to figure out which one is correct. Let me take it step by step.
First, let me write down the expression again to make sure I have it right:
(2³ + 2³) / (2⁻³ + 2⁻³)
Hmm, okay. Let me start ... |
Okay, so I need to solve this problem: (2³ + 2³) divided by (2⁻³ + 2⁻³). The answer choices are A through E, and I need to figure out which one is correct. Let me take it step by step.
First, let me write down the expression again to make sure I have it right:
(2³ + 2³) / (2⁻³ + 2⁻³)
Hmm, okay. Let me start by simp... | To evaluate the expression:
$$
\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}
$$
we begin by simplifying both the **numerator** and the **denominator** separately.
---
### Step 1: Simplify the Numerator
$$
2^3 + 2^3 = 2 \cdot 2^3 = 2^1 \cdot 2^3 = 2^{1+3} = 2^4 = 16
$$
---
### Step 2: Simplify the Denominator
$$
2^{-3} + 2^... | stop | 114 | 2,598 | 0 | 32,768 |
AIME__1996_AIME_Problems__p5 | Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$, $b+c$, and $c+a$. Find $t$.
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the value of $ t $ in the cubic equation $ x^3 + rx^2 + sx + t = 0 $, whose roots are $ a + b $, $ b + c $, and $ c + a $. The original cubic equation is $ x^3 + 3x^2 + 4x - 11 = 0 $ with roots $ a $, $ b $, and $ c $.
First, I remember that for a cubic equation $ x^3 + px^2 + qx + r =... |
Okay, so I need to find the value of $ t $ in the cubic equation $ x^3 + rx^2 + sx + t = 0 $, whose roots are $ a + b $, $ b + c $, and $ c + a $. The original cubic equation is $ x^3 + 3x^2 + 4x - 11 = 0 $ with roots $ a $, $ b $, and $ c $.
First, I remember that for a cubic equation $ x^3 + px^2 + qx + r = 0 $, t... | To solve the problem, we are given two cubic polynomials:
- The original polynomial:
$$
x^3 + 3x^2 + 4x - 11 = 0
$$
with roots $ a, b, c $.
- A new polynomial:
$$
x^3 + rx^2 + sx + t = 0
$$
with roots $ a + b, b + c, c + a $.
We are asked to find the value of $ t $.
---
### Step 1: Use Vieta's ... | stop | 98 | 7,892 | 0 | 32,768 |
USAMO__2000_USAMO_Problems__p4 | Find the smallest positive $n$ such that if $n$ squares of a $1000 \times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the smallest positive integer n such that if n squares of a 1000×1000 chessboard are colored, there must exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board. Hmm, interesting problem. Let me try to break it down step by step.
Fir... |
Okay, so I need to find the smallest positive integer n such that if n squares of a 1000×1000 chessboard are colored, there must exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board. Hmm, interesting problem. Let me try to break it down step by step.
First, let... | To solve the problem, we are asked to find the **smallest positive integer $ n $** such that **any coloring of $ n $ squares** on a $ 1000 \times 1000 $ chessboard **must contain three colored squares whose centers form a right triangle with sides parallel to the edges of the board**.
---
### Understanding the Triang... | stop | 83 | 20,564 | 0 | 32,768 |
AMC_10__2020_AMC_10B_Problems__p20 | Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together with its interior. For real $r\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,... | <think>
Okay, so I need to find the volume of the set S(r), which is all points within distance r of some point in the box B with dimensions 1, 3, and 4. Then express this volume as ar³ + br² + cr + d and find bc/(ad). The answer choices are given, so maybe I can figure out the coefficients step by step.
First, I reme... |
Okay, so I need to find the volume of the set S(r), which is all points within distance r of some point in the box B with dimensions 1, 3, and 4. Then express this volume as ar³ + br² + cr + d and find bc/(ad). The answer choices are given, so maybe I can figure out the coefficients step by step.
First, I remember th... | To solve the problem, we are given a right rectangular prism (box) with dimensions $1$, $3$, and $4$, and we are to compute the volume of the set $S(r)$, which consists of all points in 3D space that lie within a distance $r$ of some point in the box. This set is known as the **Minkowski sum** of the box with a ball of... | stop | 214 | 5,986 | 0 | 32,768 |
AMC_8__1987_AJHSME_Problems__p17 | Abby, Bret, Carl, and Dana are seated in a row of four seats numbered #1 to #4. Joe looks at them and says:
Bret is next to Carl.
Abby is between Bret and Carl.
However each one of Joe's statements is false. Bret is actually sitting in seat #3. Who is sitting in seat #2?
$\text{(A)}\ \text{Abby} \qquad \text{(B)}... | <think>
Okay, let's try to figure out this problem. So, we have four people: Abby, Bret, Carl, and Dana. They are seated in a row of four seats numbered 1 to 4. Joe makes two statements, but both are false. We also know that Bret is actually sitting in seat #3. The question is asking who is sitting in seat #2. The opti... |
Okay, let's try to figure out this problem. So, we have four people: Abby, Bret, Carl, and Dana. They are seated in a row of four seats numbered 1 to 4. Joe makes two statements, but both are false. We also know that Bret is actually sitting in seat #3. The question is asking who is sitting in seat #2. The options are... | To solve the problem, we are given the following:
- Four people: **Abby, Bret, Carl, and Dana**.
- Four seats: **#1 to #4**.
- **Bret is in seat #3**.
- **Joe makes two false statements**:
1. *Bret is next to Carl.*
2. *Abby is between Bret and Carl.*
- We are to determine **who is sitting in seat #2**.
---
### ... | stop | 179 | 3,566 | 0 | 32,768 |
AHSME__1962_AHSME_Problems__p20 | The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be:
$\textbf{(A)}\ 108\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 72\qquad\textbf{(D)}\ 54\qquad\textbf{(E)}\ 36$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to figure out which of the given options must be one of the angles in a pentagon where the angles are in arithmetic progression. Let me start by recalling some basics about pentagons and arithmetic progressions.
First, a pentagon has five sides, and the sum of its interior angles can be calcula... |
Okay, so I need to figure out which of the given options must be one of the angles in a pentagon where the angles are in arithmetic progression. Let me start by recalling some basics about pentagons and arithmetic progressions.
First, a pentagon has five sides, and the sum of its interior angles can be calculated usi... | To determine which angle **must** be present in a pentagon whose interior angles are in **arithmetic progression**, we begin by recalling a key geometric fact:
---
### **Step 1: Sum of Interior Angles of a Pentagon**
A pentagon has 5 sides. The sum of the interior angles of any $ n $-gon is given by:
$$
\text{Sum} ... | stop | 111 | 3,828 | 0 | 32,768 |
AMC_10__2015_AMC_10B_Problems__p13 | The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
$\textbf{(A) } 20 \qquad\textbf{(B) } \dfrac{360}{17} \qquad\textbf{(C) } \dfrac{107}{5} \qquad\textbf{(D) } \dfrac{43}{2} \qquad\textbf{(E) } \dfrac{281}{13}$
Please reason step by step, a... | <think>
Okay, so I need to find the sum of the lengths of the altitudes of the triangle formed by the line 12x + 5y = 60 with the coordinate axes. Let me start by visualizing this. The line intersects the x-axis and y-axis, forming a right triangle with the axes. First, I should find the coordinates of the vertices of ... |
Okay, so I need to find the sum of the lengths of the altitudes of the triangle formed by the line 12x + 5y = 60 with the coordinate axes. Let me start by visualizing this. The line intersects the x-axis and y-axis, forming a right triangle with the axes. First, I should find the coordinates of the vertices of this tr... | To solve the problem, we begin by identifying the triangle formed by the line $12x + 5y = 60$ and the coordinate axes. This triangle is a right triangle with vertices at the intercepts of the line with the axes and the origin.
---
### Step 1: Find the Intercepts
- **x-intercept**: Set $y = 0$ in the equation:
$$
... | stop | 153 | 4,947 | 0 | 32,768 |
AMC_8__1992_AJHSME_Problems__p25 | One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth of the remainder for the third pouring, one fifth of the remainder for the fourth pouring, etc. After how many pourings does exactly one tenth of the original water remain?
$\text{... | <think>
Okay, so I need to figure out after how many pourings exactly one tenth of the original water remains. The process is: start with a full container, pour out half, then one third of the remainder, then one fourth, etc. Each time, the fraction poured out is 1 over (n+1) where n is the number of pourings? Wait, le... |
Okay, so I need to figure out after how many pourings exactly one tenth of the original water remains. The process is: start with a full container, pour out half, then one third of the remainder, then one fourth, etc. Each time, the fraction poured out is 1 over (n+1) where n is the number of pourings? Wait, let me ch... | To solve the problem, we analyze the process of pouring water out of a container step by step, tracking the remaining amount after each step.
---
### Step-by-Step Process
Let’s denote the original amount of water as 1 unit.
- **First pouring**: Pour out $ \frac{1}{2} $ of the water.
Remaining: $ 1 - \frac{1}{2}... | stop | 146 | 2,870 | 0 | 32,768 |
AIME__2005_AIME_I_Problems__p1 | Six congruent circles form a ring with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius 30. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$ (the floor function).
Please r... | <think>
Okay, so I need to find the area K, which is the area inside the big circle C with radius 30 and outside the six smaller congruent circles arranged in a ring. Then I have to take the floor of that area, which means the greatest integer less than or equal to K. Alright, let me break this down step by step.
Firs... |
Okay, so I need to find the area K, which is the area inside the big circle C with radius 30 and outside the six smaller congruent circles arranged in a ring. Then I have to take the floor of that area, which means the greatest integer less than or equal to K. Alright, let me break this down step by step.
First, let ... | To solve the problem, we are given a large circle $ C $ with radius 30, and six congruent smaller circles arranged in a ring such that:
- Each small circle is **externally tangent** to its two neighbors.
- All small circles are **internally tangent** to the large circle $ C $.
We are to find the area $ K $ of the reg... | stop | 100 | 4,606 | 0 | 32,768 |
AHSME__1973_AHSME_Problems__p27 | Cars A and B travel the same distance. Car A travels half that distance at $u$ miles per hour and half at $v$ miles per hour. Car B travels half the time at $u$ miles per hour and half at $v$ miles per hour. The average speed of Car A is $x$ miles per hour and that of Car B is $y$ miles per hour. Then we always have
$... | <think>
Okay, so I need to figure out the relationship between the average speeds of Car A and Car B. Let me start by recalling what average speed means. Average speed is total distance divided by total time. Both cars are traveling the same distance, let's call that distance D.
First, let me handle Car A. The proble... |
Okay, so I need to figure out the relationship between the average speeds of Car A and Car B. Let me start by recalling what average speed means. Average speed is total distance divided by total time. Both cars are traveling the same distance, let's call that distance D.
First, let me handle Car A. The problem says ... | To determine the relationship between the average speeds of Car A and Car B, we begin by analyzing the given conditions and applying the definition of average speed:
---
### **Car A: Equal Distance at Different Speeds**
Car A travels **half the distance** at speed $ u $ and the other half at speed $ v $. Let the tot... | stop | 175 | 2,602 | 0 | 32,768 |
AHSME__1958_AHSME_Problems__p4 | In the expression $\frac{x + 1}{x - 1}$ each $x$ is replaced by $\frac{x + 1}{x - 1}$. The resulting expression, evaluated for $x = \frac{1}{2}$, equals:
$\textbf{(A)}\ 3\qquad \textbf{(B)}\ -3\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ -1\qquad \textbf{(E)}\ \text{none of these}$
Please reason step by step, and pu... | <think>
Okay, so I need to solve this problem where I have the expression (x + 1)/(x - 1), and each x is replaced by (x + 1)/(x - 1). Then, I have to evaluate this new expression when x is 1/2. The answer choices are given, so I need to figure out which one is correct. Let me try to break this down step by step.
First... |
Okay, so I need to solve this problem where I have the expression (x + 1)/(x - 1), and each x is replaced by (x + 1)/(x - 1). Then, I have to evaluate this new expression when x is 1/2. The answer choices are given, so I need to figure out which one is correct. Let me try to break this down step by step.
First, let m... | To solve the problem, we are given the expression:
$$
\frac{x + 1}{x - 1}
$$
and asked to **replace each occurrence of $ x $** in this expression with the expression $ \frac{x + 1}{x - 1} $, and then **evaluate the resulting expression at $ x = \frac{1}{2} $**.
---
### Step 1: Understand the Substitution
We are re... | stop | 146 | 5,673 | 0 | 32,768 |
AMC_8__2010_AMC_8_Problems__p11 | The top of one tree is $16$ feet higher than the top of another tree. The heights of the two trees are in the ratio $3:4$. In feet, how tall is the taller tree?
$\textbf{(A)}\ 48 \qquad\textbf{(B)}\ 64 \qquad\textbf{(C)}\ 80 \qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 112$
Please reason step by step, and put your final... | <think>
Okay, so I need to figure out how tall the taller tree is. The problem says that the top of one tree is 16 feet higher than the top of another tree, and their heights are in the ratio 3:4. Hmm, let me think about how to approach this.
First, let me parse the information. There are two trees. Let me call their ... |
Okay, so I need to figure out how tall the taller tree is. The problem says that the top of one tree is 16 feet higher than the top of another tree, and their heights are in the ratio 3:4. Hmm, let me think about how to approach this.
First, let me parse the information. There are two trees. Let me call their heights... | To determine the height of the taller tree, we are given two key pieces of information:
1. The **difference in height** between the two trees is **16 feet**.
2. The **ratio of their heights** is **3:4**.
---
### Step 1: Represent the Heights Algebraically
Let the height of the **shorter tree** be $ 3x $, and the he... | stop | 134 | 1,637 | 0 | 32,768 |
AIME__2017_AIME_I_Problems__p14 | Let $a > 1$ and $x > 1$ satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$.
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to solve this problem where I have two equations involving logarithms with base $ a $, and I need to find the remainder when $ x $ is divided by 1000. Let me start by writing down the given equations:
First equation: $ \log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128 $
Second equation: $ \log_... |
Okay, so I need to solve this problem where I have two equations involving logarithms with base $ a $, and I need to find the remainder when $ x $ is divided by 1000. Let me start by writing down the given equations:
First equation: $ \log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128 $
Second equation: $ \log_a(\log_... | To solve the problem, we are given two logarithmic equations involving a base $ a > 1 $ and a variable $ x > 1 $:
1. $ \log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128 $
2. $ \log_a(\log_a x) = 256 $
We are to find the remainder when $ x $ is divided by 1000.
---
### Step 1: Analyze the Second Equation
From the se... | length | 103 | 32,768 | 0 | 32,768 |
AMC_10__2003_AMC_10B_Problems__p11 | A line with slope $3$ intersects a line with slope $5$ at point $(10,15)$. What is the distance between the $x$-intercepts of these two lines?
$\textbf{(A) } 2 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 7 \qquad\textbf{(D) } 12 \qquad\textbf{(E) } 20$
Please reason step by step, and put your final answer within oxed{... | <think>
Okay, so I need to find the distance between the x-intercepts of two lines. One line has a slope of 3, and the other has a slope of 5. They intersect at the point (10, 15). Hmm, let me think about how to approach this.
First, I remember that the x-intercept of a line is the point where the line crosses the x-a... |
Okay, so I need to find the distance between the x-intercepts of two lines. One line has a slope of 3, and the other has a slope of 5. They intersect at the point (10, 15). Hmm, let me think about how to approach this.
First, I remember that the x-intercept of a line is the point where the line crosses the x-axis. Th... | To solve the problem, we are given two lines:
- One with slope $ m_1 = 3 $
- Another with slope $ m_2 = 5 $
These lines intersect at the point $ (10, 15) $. We are asked to find the **distance between the x-intercepts** of these two lines.
---
### Step 1: Find the equations of the lines
We use the **point-slope fo... | stop | 128 | 3,688 | 0 | 32,768 |
AMC_8__1996_AJHSME_Problems__p1 | How many positive factors of 36 are also multiples of 4?
$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to figure out how many positive factors of 36 are also multiples of 4. Let me start by recalling what factors and multiples are. A factor of a number is an integer that divides that number without leaving a remainder. A multiple of 4 is any number that can be expressed as 4 times some integer. S... |
Okay, so I need to figure out how many positive factors of 36 are also multiples of 4. Let me start by recalling what factors and multiples are. A factor of a number is an integer that divides that number without leaving a remainder. A multiple of 4 is any number that can be expressed as 4 times some integer. So, esse... | To solve the problem of finding how many **positive factors of 36 are also multiples of 4**, we can approach it systematically using both **prime factorization** and **direct enumeration**.
---
### Step 1: Prime Factorization of 36
We begin by expressing 36 in terms of its prime factors:
$$
36 = 2^2 \times 3^2
$$
... | stop | 93 | 2,311 | 0 | 32,768 |
AMC_10__2023_AMC_10B_Problems__p4 | Jackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip $25$ meters long. How many square centimeters of paper could Jackson cover with paint?
$\textbf{(A) } 162,500 \qquad\textbf{(B) } 162.5 \qquad\textbf{(C) }1,625 \qquad\textbf{(D) }1,625,000 \qquad\text... | <think>
Okay, so Jackson is using his paintbrush to make a strip of paint, and we need to figure out how many square centimeters of paper he can cover with the paint he has. Let me try to break this down step by step.
First, the problem says that the paintbrush makes a narrow strip with a width of 6.5 millimeters. Hm... |
Okay, so Jackson is using his paintbrush to make a strip of paint, and we need to figure out how many square centimeters of paper he can cover with the paint he has. Let me try to break this down step by step.
First, the problem says that the paintbrush makes a narrow strip with a width of 6.5 millimeters. Hmm, so t... | To determine how many square centimeters of paper Jackson can cover with his paint, we need to calculate the **area** of the strip he can paint. The strip has a **length** of 25 meters and a **width** of 6.5 millimeters. Since the final answer is required in **square centimeters**, we must convert all measurements to c... | stop | 157 | 2,101 | 0 | 32,768 |
IMO__1975_IMO_Problems__p4 | When $4444^{4444}$ is written in decimal notation, the sum of its digits is $A$. Let $B$ be the sum of the digits of $A$. Find the sum of the digits of $B$. ($A$ and $B$ are written in decimal notation.)
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the sum of the digits of B, where B is the sum of the digits of A, and A is the sum of the digits of 4444^4444. That seems pretty intense. Let me break it down step by step.
First, let me understand the problem. We have this huge number, 4444 raised to the power of 4444. Then, we need t... |
Okay, so I need to find the sum of the digits of B, where B is the sum of the digits of A, and A is the sum of the digits of 4444^4444. That seems pretty intense. Let me break it down step by step.
First, let me understand the problem. We have this huge number, 4444 raised to the power of 4444. Then, we need to compu... | To solve the problem, we are given a very large number: $ 4444^{4444} $, and we are asked to compute the sum of its digits, call this $ A $. Then, we compute the sum of the digits of $ A $, call this $ B $, and finally, the sum of the digits of $ B $. The goal is to find the final result.
---
### Step 1: Understandin... | stop | 89 | 9,824 | 0 | 32,768 |
AIME__2013_AIME_II_Problems__p11 | Let $A = \{1, 2, 3, 4, 5, 6, 7\}$, and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$.
Please reason step by step, and put your final answer within oxed{}. | <think>
Okay, so I need to find the number of functions $ f: A \rightarrow A $ where $ A = \{1, 2, 3, 4, 5, 6, 7\} $ such that $ f(f(x)) $ is a constant function. Then, I have to find this number modulo 1000. Alright, let me start by understanding the problem.
First, let me recall that a function from a set to itself ... | null | <think>
Okay, so I need to find the number of functions $ f: A \rightarrow A $ where $ A = \{1, 2, 3, 4, 5, 6, 7\} $ such that $ f(f(x)) $ is a constant function. Then, I have to find this number modulo 1000. Alright, let me start by understanding the problem.
First, let me recall that a function from a set to itself ... | length | 106 | 32,768 | 0 | 32,768 |
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