prompt stringlengths 11 3.72k | messages listlengths 2 2 |
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Find the minimum $n \in \mathbb{Z}$ such that $5^n + n^5$ is divisible by 13. | [
{
"content": "Find the minimum $n \\in \\mathbb{Z}$ such that $5^n + n^5$ is divisible by 13.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find the smallest integer n, positive or negative, such that 5^n + n^5 is divisible by 13. Hmm, modular arithmetic probably. So the prob... |
For how many integers $0 \leq n < 2013$ is the expression $n^4 + 2n^3 - 20n^2 + 2n - 21$ divisible by $2013$? | [
{
"content": "For how many integers $0 \\leq n < 2013$ is the expression $n^4 + 2n^3 - 20n^2 + 2n - 21$ divisible by $2013$?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find how many integers n between 0 and 2012 (inclusive) make the expression n⁴ + 2n³ - 20n² + 2n - 21 divisible b... |
Let \( z = z(x, y) \) be an implicit function defined by \( 2 \sin(x + 2y - 3z) = x + 2y - 3z \). Find \( \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \). | [
{
"content": "Let \\( z = z(x, y) \\) be an implicit function defined by \\( 2 \\sin(x + 2y - 3z) = x + 2y - 3z \\). Find \\( \\frac{\\partial z}{\\partial x} + \\frac{\\partial z}{\\partial y} \\).",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find the sum of the partial der... |
Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. The points $D, E,$ and $F$ are the midpoints of $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$ respectively. Let $X \ne E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA + XB + XC$? | [
{
"content": "Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. The points $D, E,$ and $F$ are the midpoints of $\\overline{AB}$, $\\overline{BC}$, and $\\overline{AC}$ respectively. Let $X \\ne E$ be the intersection of the circumcircles of $\\triangle BDE$ and $\\triangle CEF$. What is $XA + XB + XC$?",
"... |
How many ordered pairs \((x, y)\) with \(x\) and \(y\) being positive integers from 1 to 23, inclusive, have the property that \(x^2 + y^2 + x + y\) is a multiple of 6? | [
{
"content": "How many ordered pairs \\((x, y)\\) with \\(x\\) and \\(y\\) being positive integers from 1 to 23, inclusive, have the property that \\(x^2 + y^2 + x + y\\) is a multiple of 6?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of ordered pairs (x, y) where b... |
Let \(a\) and \(b\) be positive integers satisfying \(\frac{ab+1}{a+b} < \frac{3}{2}\). The maximum possible value of \(\frac{a^3b^3+1}{a^3+b^3}\) is \(\frac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p+q\). | [
{
"content": "Let \\(a\\) and \\(b\\) be positive integers satisfying \\(\\frac{ab+1}{a+b} < \\frac{3}{2}\\). The maximum possible value of \\(\\frac{a^3b^3+1}{a^3+b^3}\\) is \\(\\frac{p}{q}\\), where \\(p\\) and \\(q\\) are relatively prime positive integers. Find \\(p+q\\).",
"role": "user"
},
{
"... |
We throw $n$ raisins on a circular cake randomly and independently. What is the probability that the cake can be cut into two equal halves such that all raisins lie on the same piece? | [
{
"content": "We throw $n$ raisins on a circular cake randomly and independently. What is the probability that the cake can be cut into two equal halves such that all raisins lie on the same piece?",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. The problem is about throwing n raisins ra... |
If \( x, y, \) and \( z \) are acute angles such that \( \cos{x} = \tan{y} \), \( \cos{y} = \tan{z} \), and \( \cos{z} = \tan{x} \), find the value of \( k \) in the equation \( 1234\sin{x} + 1342\sin{y} + 1423\sin{z} = k\sin{18^\circ} \). | [
{
"content": "If \\( x, y, \\) and \\( z \\) are acute angles such that \\( \\cos{x} = \\tan{y} \\), \\( \\cos{y} = \\tan{z} \\), and \\( \\cos{z} = \\tan{x} \\), find the value of \\( k \\) in the equation \\( 1234\\sin{x} + 1342\\sin{y} + 1423\\sin{z} = k\\sin{18^\\circ} \\).",
"role": "user"
},
{
... |
Determine the values of \(a\) and \(b\) for which the system of equations
\[
\begin{cases}
x - 3y = a + 3 \\
2x - 6y = b \\
3x - 9y = b + 2 \\
3x - 6y = a
\end{cases}
\]
has exactly one solution. | [
{
"content": "Determine the values of \\(a\\) and \\(b\\) for which the system of equations\n\\[\n\\begin{cases}\nx - 3y = a + 3 \\\\\n2x - 6y = b \\\\\n3x - 9y = b + 2 \\\\\n3x - 6y = a\n\\end{cases}\n\\]\nhas exactly one solution.",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this sys... |
Compute the number of positive four-digit multiples of $11$ whose sum of digits (in base $10$) is divisible by $11$. | [
{
"content": "Compute the number of positive four-digit multiples of $11$ whose sum of digits (in base $10$) is divisible by $11$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many four-digit numbers that are multiples of 11 also have the sum of their digits divisible... |
If the sides of a triangle are $2, 3, 4$, what is the radius of the circle circumscribing the triangle? | [
{
"content": "If the sides of a triangle are $2, 3, 4$, what is the radius of the circle circumscribing the triangle?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the radius of the circle that circumscribes a triangle with sides 2, 3, and 4. Hmm, circumscribed circle... oh righ... |
For any number \( x \), let \( \lfloor x \rfloor \) denote the greatest integer less than or equal to \( x \). A sequence \( a_1, a_2, \cdots \) is given by \[ a_n = \left\lfloor \sqrt{2n} + \frac{1}{2} \right\rfloor. \] How many values of \( k \) are there such that \( a_k = 2010 \)? | [
{
"content": "For any number \\( x \\), let \\( \\lfloor x \\rfloor \\) denote the greatest integer less than or equal to \\( x \\). A sequence \\( a_1, a_2, \\cdots \\) is given by \\[ a_n = \\left\\lfloor \\sqrt{2n} + \\frac{1}{2} \\right\\rfloor. \\] How many values of \\( k \\) are there such that \\( a_k =... |
Find the smallest positive integer \( x \) such that when divided by \( 10 \), it leaves a remainder of \( 9 \); when divided by \( 9 \), it leaves a remainder of \( 8 \); when divided by \( 7 \), it leaves a remainder of \( 6 \); when divided by \( 5 \), it leaves a remainder of \( 4 \); and so on, until when divided ... | [
{
"content": "Find the smallest positive integer \\( x \\) such that when divided by \\( 10 \\), it leaves a remainder of \\( 9 \\); when divided by \\( 9 \\), it leaves a remainder of \\( 8 \\); when divided by \\( 7 \\), it leaves a remainder of \\( 6 \\); when divided by \\( 5 \\), it leaves a remainder of \... |
Find the number of distinct abelian groups of order $40$ up to isomorphism. | [
{
"content": "Find the number of distinct abelian groups of order $40$ up to isomorphism.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of distinct abelian groups of order 40 up to isomorphism. Hmm, let me recall what I know about abelian groups and their classificati... |
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