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stringlengths 19
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|---|---|---|---|
In a 4x5 grid, how many different paths are there from the bottom left (0, 0) to the top right (3, 4), provided you can move right or up by any number of cells (like a chess rook) at each step, subject to the constraint that you cannot move through the following cells: (0, 3), (1, 0), (1, 1), (0, 2)? If it's impossible, answer with -1.
|
-1
|
[
{
"content": "In a 4x5 grid, how many different paths are there from the bottom left (0, 0) to the top right (3, 4), provided you can move right or up by any number of cells (like a chess rook) at each step, subject to the constraint that you cannot move through the following cells: (0, 3), (1, 0), (1, 1), (0, 2)? If it's impossible, answer with -1.",
"role": "user"
}
] |
math
|
What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {h: 2, q: 1, g: 4}, shuffle the letters in the word, what is the probability of no letter 'h' occupy any of their original positions?
If the probability can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.
|
34
|
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {h: 2, q: 1, g: 4}, shuffle the letters in the word, what is the probability of no letter 'h' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
math
|
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures $3$ by $4$ by $5$ units. Given that the volume of this set is $\frac{m + n\pi}{p},$ where $m, n,$ and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p.$
|
505
|
[
{
"content": "Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures $3$ by $4$ by $5$ units. Given that the volume of this set is $\\frac{m + n\\pi}{p},$ where $m, n,$ and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p.$",
"role": "user"
}
] |
math
|
If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $512x^3 + 125 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, then what is $a^2+b^2+c^2+d^2+e^2+f^2$?
|
6410
|
[
{
"content": "If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $512x^3 + 125 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, then what is $a^2+b^2+c^2+d^2+e^2+f^2$?",
"role": "user"
}
] |
math
|
In a 5x5 grid, how many different paths are there from the bottom left (0, 0) to the top right (4, 4), provided you can move right or up by any number of cells (like a chess rook) at each step, subject to the constraint that you cannot move through the following cells: (1, 0), (1, 1), (3, 4), (3, 2)? If it's impossible, answer with -1.
|
3
|
[
{
"content": "In a 5x5 grid, how many different paths are there from the bottom left (0, 0) to the top right (4, 4), provided you can move right or up by any number of cells (like a chess rook) at each step, subject to the constraint that you cannot move through the following cells: (1, 0), (1, 1), (3, 4), (3, 2)? If it's impossible, answer with -1.",
"role": "user"
}
] |
math
|
Wendy eats sushi for lunch. She wants to eat six pieces of sushi arranged in a $2 \times 3$ rectangular grid, but sushi is sticky, and Wendy can only eat a piece if it is adjacent to (not counting diagonally) at most two other pieces. In how many orders can Wendy eat the six pieces of sushi, assuming that the pieces of sushi are distinguishable?
|
360
|
[
{
"content": "Wendy eats sushi for lunch. She wants to eat six pieces of sushi arranged in a $2 \\times 3$ rectangular grid, but sushi is sticky, and Wendy can only eat a piece if it is adjacent to (not counting diagonally) at most two other pieces. In how many orders can Wendy eat the six pieces of sushi, assuming that the pieces of sushi are distinguishable?",
"role": "user"
}
] |
math
|
Find the smallest positive integer $n$, such that there exist $n$ integers $x_1, x_2, \dots , x_n$ (not necessarily different), with $1\le x_k\le n$, $1\le k\le n$, and such that
\[x_1 + x_2 + \cdots + x_n =\frac{n(n + 1)}{2},\quad\text{ and }x_1x_2 \cdots x_n = n!,\]
but $\{x_1, x_2, \dots , x_n\} \ne \{1, 2, \dots , n\}$.
|
9
|
[
{
"content": "Find the smallest positive integer $n$, such that there exist $n$ integers $x_1, x_2, \\dots , x_n$ (not necessarily different), with $1\\le x_k\\le n$, $1\\le k\\le n$, and such that\n\\[x_1 + x_2 + \\cdots + x_n =\\frac{n(n + 1)}{2},\\quad\\text{ and }x_1x_2 \\cdots x_n = n!,\\]\nbut $\\{x_1, x_2, \\dots , x_n\\} \\ne \\{1, 2, \\dots , n\\}$.",
"role": "user"
}
] |
math
|
Let $T$ be a right triangle with sides having lengths $3$, $4$, and $5$. A point $P$ is called [i]awesome[/i] if P is the center of a parallelogram whose vertices all lie on the boundary of $T$. What is the area of the set of awesome points?
|
\frac{3}{2}
|
[
{
"content": "Let $T$ be a right triangle with sides having lengths $3$, $4$, and $5$. A point $P$ is called [i]awesome[/i] if P is the center of a parallelogram whose vertices all lie on the boundary of $T$. What is the area of the set of awesome points?",
"role": "user"
}
] |
math
|
Find the largest natural number $ n$ for which there exist different sets $ S_1,S_2,\ldots,S_n$ such that:
$ 1^\circ$ $ |S_i\cup S_j|\leq 2004$ for each two $ 1\leq i,j\le n$ and
$ 2^\circ$ $ S_i\cup S_j\cup S_k\equal{}\{1,2,\ldots,2008\}$ for each three integers $ 1\le i<j<k\le n$ .
|
32
|
[
{
"content": "Find the largest natural number $ n$ for which there exist different sets $ S_1,S_2,\\ldots,S_n$ such that:\r\n\r $ 1^\\circ$ $ |S_i\\cup S_j|\\leq 2004$ for each two $ 1\\leq i,j\\le n$ and\r $ 2^\\circ$ $ S_i\\cup S_j\\cup S_k\\equal{}\\{1,2,\\ldots,2008\\}$ for each three integers $ 1\\le i<j<k\\le n$ .",
"role": "user"
}
] |
math
|
Through the point \( P(1,1) \), draw a line \( l \) such that the midpoint of the chord intercepted by the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) on this line is precisely \( P \). Find the equation of the line \( l \).
|
4x + 9y = 13
|
[
{
"content": "Through the point \\( P(1,1) \\), draw a line \\( l \\) such that the midpoint of the chord intercepted by the ellipse \\(\\frac{x^2}{9} + \\frac{y^2}{4} = 1\\) on this line is precisely \\( P \\). Find the equation of the line \\( l \\).",
"role": "user"
}
] |
math
|
Does there exist a positive integer $m$ such that the equation $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{a b c} = \frac{m}{a + b + c}$ has infinitely many solutions in positive integers?
|
12
|
[
{
"content": "Does there exist a positive integer $m$ such that the equation $\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} + \\frac{1}{a b c} = \\frac{m}{a + b + c}$ has infinitely many solutions in positive integers?",
"role": "user"
}
] |
math
|
Find the product of the divisors of $72$.
|
139,\!314,\!069,\!504
|
[
{
"content": "Find the product of the divisors of $72$.",
"role": "user"
}
] |
math
|
It is known that, for all positive integers $k,$
\[1^{2}+2^{2}+3^{2}+\cdots+k^{2}=\frac{k(k+1)(2k+1)}{6}. \]Find the smallest positive integer $k$ such that $1^{2}+2^{2}+3^{2}+\cdots+k^{2}$ is a multiple of $200.$
|
112
|
[
{
"content": "It is known that, for all positive integers $k,$ \r\n\\[1^{2}+2^{2}+3^{2}+\\cdots+k^{2}=\\frac{k(k+1)(2k+1)}{6}. \\]Find the smallest positive integer $k$ such that $1^{2}+2^{2}+3^{2}+\\cdots+k^{2}$ is a multiple of $200.$ ",
"role": "user"
}
] |
math
|
The number of students in Michael's graduating class is more than 100 and fewer than 200. The number is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. How many students are in Michael's graduating class?
|
182
|
[
{
"content": "The number of students in Michael's graduating class is more than 100 and fewer than 200. The number is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. How many students are in Michael's graduating class?",
"role": "user"
}
] |
math
|
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