problem stringlengths 14 3.7k | expected_answer stringlengths 0 500 | original_expected_answer stringlengths 0 4.52k | changed_answer_to_majority bool 2
classes | data_source stringclasses 2
values | messages listlengths 2 78 | metadata dict | license stringclasses 2
values | used_in listlengths 1 1 | uuid stringlengths 36 36 | url stringlengths 0 129 | user_url stringlengths 0 80 | user_name stringlengths 0 36 | tools listlengths 1 1 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Let \(a, b, c, d\) be four positive integers. The integer \(n\) is the smallest positive integer such that the sum of every rational number of the form \(\frac{1}{a^2 + b^3 + c^8 + d^n}\) does not exceed an integer \(M\). What is \(n\)? | 25 | true | aops | [
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"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\(a, b, c, d\\) be four positive integers. The integer \\(n\\) is the smallest positive integer such that the sum of every rational number of the form \\(\\frac{1}{a^2 + b^3 + c^8 + d^n}\\) do... | {
"reason_high_no_tool": {
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"count": 8,
"pass": 8
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"count": 8,
"pass": 8
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"accuracy": 0.75,
"count": 8,
"pass": 6
},
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"accuracy": 0.125,
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... | cc-by-4.0 | [
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] | ||||
Find all functions \( f \colon \mathbb{R} \to \mathbb{R} \) such that
\[
(f(x) + f(y))(f^2(y) - f(y)f(z) + f^2(z)) = xf(x^2 + z^2) - (y - z)f(yz) + y^3 - f(x)f(y)f(z)
\]
for all \( x, y, z \in \mathbb{R} \). | \text{No such function exists.} | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all functions \\( f \\colon \\mathbb{R} \\to \\mathbb{R} \\) such that \n\\[\n(f(x) + f(y))(f^2(y) - f(y)f(z) + f^2(z)) = xf(x^2 + z^2) - (y - z)f(yz) + y^3 - f(x)f(y)f(z)\n\\]\nfor all \\( x,... | {
"reason_high_no_tool": {
"accuracy": 0.875,
"count": 8,
"pass": 7
},
"reason_high_with_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_low_no_tool": {
"accuracy": 0.8571,
"count": 7,
"pass": 6
},
"reason_low_with_tool": {
"accuracy": 0.625,
"count": 8... | cc-by-4.0 | [
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Determine all prime numbers $p$ and all positive integers $x$ and $y$ such that \[x^3 + y^3 = p(xy + p)\] | (p,x,y)\in\{(7,4,5),(7,5,4),(13,2,7),(13,7,2),(19,1,8),(19,8,1)\} | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine all prime numbers $p$ and all positive integers $x$ and $y$ such that \\[x^3 + y^3 = p(xy + p)\\]",
"name": "",
"reasoning_content": "",
"role": "user",
"tool_call_id": ""... | {
"reason_high_no_tool": {
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"count": 8,
"pass": 8
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"accuracy": 1,
"count": 8,
"pass": 8
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"accuracy": 0.125,
"count": 8,
"pass": 1
},
"reason_low_with_tool": {
"accuracy": 0.875,
"count": 8,
... | cc-by-4.0 | [
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] | ||||
Let \( m \) and \( n \) be positive integers with \( m \leq n \). Two friends, \( A \) and \( B \), play a cooperative game where \( A \) starts with \( a = 0 \). Player \( B \) then reduces \( a + m \) (mod \( n \)) to get a new number \( b \) between \( 0 \) and \( n - 1 \). Player \( A \) then reduces \( b + n \) (m... | \text{They win iff } n=m\ \text{or}\ n\ge 2m. | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( m \\) and \\( n \\) be positive integers with \\( m \\leq n \\). Two friends, \\( A \\) and \\( B \\), play a cooperative game where \\( A \\) starts with \\( a = 0 \\). Player \\( B \\) th... | {
"reason_high_no_tool": {
"accuracy": 0.875,
"count": 8,
"pass": 7
},
"reason_high_with_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_low_no_tool": {
"accuracy": 0.1429,
"count": 7,
"pass": 1
},
"reason_low_with_tool": {
"accuracy": 0.75,
"count": 8,... | cc-by-4.0 | [
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},
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] | ||||
Find a recurrence relation for the number of ways to distribute $n$ identical coins to $k$ different people so that each person receives between two and four coins. | A(n,k)=A(n-2,k-1)+A(n-3,k-1)+A(n-4,k-1) | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind a recurrence relation for the number of ways to distribute $n$ identical coins to $k$ different people so that each person receives between two and four coins.",
"name": "",
"reasoning... | {
"reason_high_no_tool": {
"accuracy": 0.75,
"count": 8,
"pass": 6
},
"reason_high_with_tool": {
"accuracy": 0.875,
"count": 8,
"pass": 7
},
"reason_low_no_tool": {
"accuracy": 0.875,
"count": 8,
"pass": 7
},
"reason_low_with_tool": {
"accuracy": 0.625,
"count":... | cc-by-4.0 | [
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] | ||||
Given that \( p \) is a prime number and \( x \) and \( y \) are natural numbers such that \(\frac{p^x - 1}{p-1} = 2^y\), find the number of divisors of \( xy \). | 4 | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven that \\( p \\) is a prime number and \\( x \\) and \\( y \\) are natural numbers such that \\(\\frac{p^x - 1}{p-1} = 2^y\\), find the number of divisors of \\( xy \\).",
"name": "",
"... | {
"reason_high_no_tool": {
"accuracy": 0.25,
"count": 8,
"pass": 2
},
"reason_high_with_tool": {
"accuracy": 0.75,
"count": 8,
"pass": 6
},
"reason_low_no_tool": {
"accuracy": 0.625,
"count": 8,
"pass": 5
},
"reason_low_with_tool": {
"accuracy": 0.625,
"count": ... | cc-by-4.0 | [
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] | ||||
Given the sequence of natural numbers \(a_n\) such that \(a_{n+3} = a_{n+2}(a_{n+1} + a_{n})\), find \(a_7\) if \(a_6 = 8820\). | 2469600 | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the sequence of natural numbers \\(a_n\\) such that \\(a_{n+3} = a_{n+2}(a_{n+1} + a_{n})\\), find \\(a_7\\) if \\(a_6 = 8820\\).",
"name": "",
"reasoning_content": "",
"role": "u... | {
"reason_high_no_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_high_with_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_low_no_tool": {
"accuracy": 0.625,
"count": 8,
"pass": 5
},
"reason_low_with_tool": {
"accuracy": 0.875,
"count": 8,
... | cc-by-4.0 | [
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] | ||||
Is there a triangle ABC where the three sides are consecutive integers and one angle is three times another angle? Can a more general conclusion be made? | \text{No – a triangle with three consecutive integers as sides cannot have an angle three times another.} | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIs there a triangle ABC where the three sides are consecutive integers and one angle is three times another angle? Can a more general conclusion be made?",
"name": "",
"reasoning_content": ... | {
"reason_high_no_tool": {
"accuracy": 0.75,
"count": 8,
"pass": 6
},
"reason_high_with_tool": {
"accuracy": 0.75,
"count": 8,
"pass": 6
},
"reason_low_no_tool": {
"accuracy": 0.7143,
"count": 7,
"pass": 5
},
"reason_low_with_tool": {
"accuracy": 0.625,
"count":... | cc-by-4.0 | [
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},
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}
] | ||||
Find all real solutions to the system:
\[ x^3 - 3x = y \]
\[ y^3 - 3y = z \]
\[ z^3 - 3z = x \] | (x,y,z)=\bigl(2\cos\theta,\;2\cos3\theta,\;2\cos9\theta\bigr) | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all real solutions to the system:\n\\[ x^3 - 3x = y \\]\n\\[ y^3 - 3y = z \\]\n\\[ z^3 - 3z = x \\]",
"name": "",
"reasoning_content": "",
"role": "user",
"tool_call_id": "",
... | {
"reason_high_no_tool": {
"accuracy": 0.25,
"count": 8,
"pass": 2
},
"reason_high_with_tool": {
"accuracy": 0.25,
"count": 8,
"pass": 2
},
"reason_low_no_tool": {
"accuracy": 0.625,
"count": 8,
"pass": 5
},
"reason_low_with_tool": {
"accuracy": 0.125,
"count": ... | cc-by-4.0 | [
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] | 743b7297-896c-44e4-1a29-a5b0de45d250 | [
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},
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}
] | ||||
What is the total number of squares that can be drawn on an \( m \times n \) chessboard? | \;
\begin{cases}
\displaystyle\frac{m(m+1)(3n-m+1)}{6}, & m\le n,\\[6pt]
\displaystyle\frac{n(n+1)(3m-n+1)}{6}, & n<m.
\end{cases}
| true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nWhat is the total number of squares that can be drawn on an \\( m \\times n \\) chessboard?",
"name": "",
"reasoning_content": "",
"role": "user",
"tool_call_id": "",
"tool_call... | {
"reason_high_no_tool": {
"accuracy": 0.5,
"count": 8,
"pass": 4
},
"reason_high_with_tool": {
"accuracy": 0.75,
"count": 8,
"pass": 6
},
"reason_low_no_tool": {
"accuracy": 0.375,
"count": 8,
"pass": 3
},
"reason_low_with_tool": {
"accuracy": 0.25,
"count": 8,... | cc-by-4.0 | [
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] | bf8daf9b-faa3-6b0c-a8ec-85a504a12694 | [
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}
},
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}
] | ||||
Find all functions \( f, g : \mathbb{R}^+ \to \mathbb{R}^+ \) such that \( f \) is injective and:
\[ f(g(x) + y) = f(x) + g(y) \]
for all positive numbers \( x \) and \( y \). | \;f(x)=x+A,\qquad g(x)=x+B\quad (A,B\ge0)\; | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all functions \\( f, g : \\mathbb{R}^+ \\to \\mathbb{R}^+ \\) such that \\( f \\) is injective and:\n\\[ f(g(x) + y) = f(x) + g(y) \\]\nfor all positive numbers \\( x \\) and \\( y \\).",
... | {
"reason_high_no_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_high_with_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_low_no_tool": {
"accuracy": 0.625,
"count": 8,
"pass": 5
},
"reason_low_with_tool": {
"accuracy": 0.75,
"count": 8,
... | cc-by-4.0 | [
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] | 575d318d-70a4-1463-3c7b-0eed031c73a8 | [
{
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},
"type": ""
}
] | ||||
Let an integer quadrilateral be a quadrilateral whose sides are all of integer length, whose diagonals form $4$ segments of integer length and whose area is also an integer. Find the area of an integer quadrilateral with at least $3$ sides of length $13$. | 120 | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet an integer quadrilateral be a quadrilateral whose sides are all of integer length, whose diagonals form $4$ segments of integer length and whose area is also an integer. Find the area of an int... | {
"reason_high_no_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_high_with_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_low_no_tool": {
"accuracy": 0.75,
"count": 8,
"pass": 6
},
"reason_low_with_tool": {
"accuracy": 0.5,
"count": 8,
"p... | cc-by-4.0 | [
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},
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] | ||||
Given the function \( f(x) = \frac{2x}{1 + x^2} \) and positive numbers \( x_k > 0 \) for \( k = 1, 2, \ldots, n \), determine the minimum and maximum values of the expression \( \sum_{k=1}^{n} f^{-1}(x_k) \). | \;0\le\displaystyle\sum_{k=1}^{n}f^{-1}(x_{k})\le n\; | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the function \\( f(x) = \\frac{2x}{1 + x^2} \\) and positive numbers \\( x_k > 0 \\) for \\( k = 1, 2, \\ldots, n \\), determine the minimum and maximum values of the expression \\( \\sum_{k=... | {
"reason_high_no_tool": {
"accuracy": 0.625,
"count": 8,
"pass": 5
},
"reason_high_with_tool": {
"accuracy": 0.75,
"count": 8,
"pass": 6
},
"reason_low_no_tool": {
"accuracy": 0.375,
"count": 8,
"pass": 3
},
"reason_low_with_tool": {
"accuracy": 0.25,
"count": ... | cc-by-4.0 | [
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] | ||||
Given a quadrilateral \(ABCD\) with midpoints \(M\) and \(N\) of sides \(AB\) and \(CD\) respectively, and given the lengths \(AB = a\), \(BC = b\), \(CD = c\), \(DA = d\), and \(MN = l\), construct the quadrilateral \(ABCD\). | \text{The quadrilateral }ABCD\text{ is obtained by the steps 1–6 above.} | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven a quadrilateral \\(ABCD\\) with midpoints \\(M\\) and \\(N\\) of sides \\(AB\\) and \\(CD\\) respectively, and given the lengths \\(AB = a\\), \\(BC = b\\), \\(CD = c\\), \\(DA = d\\), and \\... | {
"reason_high_no_tool": {
"accuracy": 0.875,
"count": 8,
"pass": 7
},
"reason_high_with_tool": {
"accuracy": 0.75,
"count": 8,
"pass": 6
},
"reason_low_no_tool": {
"accuracy": 0.625,
"count": 8,
"pass": 5
},
"reason_low_with_tool": {
"accuracy": 0.5,
"count": 8... | cc-by-4.0 | [
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] | ||||
Find all positive integers \( a \) and \( b \) for which
\[ \left\lfloor \frac{a^2}{b} \right\rfloor + \left\lfloor \frac{b^2}{a} \right\rfloor = \left\lfloor \frac{a^2 + b^2}{ab} \right\rfloor + ab. \]
Here, \( \lfloor x \rfloor \) denotes the greatest integer not exceeding \( x \). | \;(a,b)\in \mathbb Z_{>0}^2\; \text{such that }\; b=a^{2}+1\; \text{or}\; a=b^{2}+1\; | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all positive integers \\( a \\) and \\( b \\) for which \n\\[ \\left\\lfloor \\frac{a^2}{b} \\right\\rfloor + \\left\\lfloor \\frac{b^2}{a} \\right\\rfloor = \\left\\lfloor \\frac{a^2 + b^2}{a... | {
"reason_high_no_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_high_with_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_low_no_tool": {
"accuracy": 0.5,
"count": 8,
"pass": 4
},
"reason_low_with_tool": {
"accuracy": 1,
"count": 8,
"pass... | cc-by-4.0 | [
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] | 78ada361-90c9-94da-cb8e-5c4a34ccc827 | [
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] | ||||
Given \( x, y, z > 0 \) such that \( 2x + 4y + 7z = 2xyz \), find the minimum of \( S = x + y + z \). | \displaystyle \frac{15}{2} | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven \\( x, y, z > 0 \\) such that \\( 2x + 4y + 7z = 2xyz \\), find the minimum of \\( S = x + y + z \\).",
"name": "",
"reasoning_content": "",
"role": "user",
"tool_call_id": ""... | {
"reason_high_no_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_high_with_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_low_no_tool": {
"accuracy": 0.7143,
"count": 7,
"pass": 5
},
"reason_low_with_tool": {
"accuracy": 0.875,
"count": 8,
... | cc-by-4.0 | [
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] | ||||
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(f(x) + yz) = x^3 + f(y)f(z) \) for all real numbers \( x, y, z \). | \text{No such function exists.} | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that \\( f(f(x) + yz) = x^3 + f(y)f(z) \\) for all real numbers \\( x, y, z \\).",
"name": "",
"reasoning_content": "",
"... | {
"reason_high_no_tool": {
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"count": 8,
"pass": 8
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"count": 7,
"pass": 5
},
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"accuracy": 0.75,
"count": 8,
... | cc-by-4.0 | [
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] | ||||
In triangle \(ABC\), the altitude, angle bisector, and median from \(C\) divide angle \(C\) into four equal angles. Find angle \(B\). | 22.5^{\circ} | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn triangle \\(ABC\\), the altitude, angle bisector, and median from \\(C\\) divide angle \\(C\\) into four equal angles. Find angle \\(B\\).",
"name": "",
"reasoning_content": "",
"rol... | {
"reason_high_no_tool": {
"accuracy": 0.875,
"count": 8,
"pass": 7
},
"reason_high_with_tool": {
"accuracy": 0.875,
"count": 8,
"pass": 7
},
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"accuracy": 0.125,
"count": 8,
"pass": 1
},
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"count":... | cc-by-4.0 | [
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},
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}
] | ||||
For what values of parameter \( m \) does the equation \( x^2 + (m+6)|x| + 2m + 9 = 0 \) have two distinct solutions? | \,m<-\dfrac92\, | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor what values of parameter \\( m \\) does the equation \\( x^2 + (m+6)|x| + 2m + 9 = 0 \\) have two distinct solutions?",
"name": "",
"reasoning_content": "",
"role": "user",
"too... | {
"reason_high_no_tool": {
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"count": 8,
"pass": 8
},
"reason_high_with_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
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"count": 7,
"pass": 4
},
"reason_low_with_tool": {
"accuracy": 0.875,
"count": 8,
... | cc-by-4.0 | [
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] | 0c712016-04a3-66cf-8e2b-a5651f7fa213 | [
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] | ||||
Determine the minimum value of $\frac{IA^3+IB^3+IC^3}{AB^3+BC^3+CA^3}$, where $ABC$ is any triangle, and $I$ is the incenter. | \displaystyle\frac{\sqrt3}{9} | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine the minimum value of $\\frac{IA^3+IB^3+IC^3}{AB^3+BC^3+CA^3}$, where $ABC$ is any triangle, and $I$ is the incenter.",
"name": "",
"reasoning_content": "",
"role": "user",
... | {
"reason_high_no_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_high_with_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_low_no_tool": {
"accuracy": 0.5714,
"count": 7,
"pass": 4
},
"reason_low_with_tool": {
"accuracy": 1,
"count": 8,
"p... | cc-by-4.0 | [
"nano_v3"
] | 43437e23-8117-c361-63ce-30492fa9fe77 | [
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},
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}
] | ||||
Given hexagon $ABCDEF$ with $AB=FA=14$ and $BC=CD=DE=EF=4$, determine the length of $AD$. | 16 | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven hexagon $ABCDEF$ with $AB=FA=14$ and $BC=CD=DE=EF=4$, determine the length of $AD$.",
"name": "",
"reasoning_content": "",
"role": "user",
"tool_call_id": "",
"tool_calls"... | {
"reason_high_no_tool": {
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"count": 8,
"pass": 8
},
"reason_high_with_tool": {
"accuracy": 1,
"count": 8,
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},
"reason_low_no_tool": {
"accuracy": 0.4286,
"count": 7,
"pass": 3
},
"reason_low_with_tool": {
"accuracy": 0,
"count": 8,
"p... | cc-by-4.0 | [
"nano_v3"
] | 912d9802-1751-901b-377b-390c208d613a | [
{
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},
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] | ||||
On a plane, consider a triangle \(ABC\), a circle \(\Gamma\), a point \(P \in \Gamma\), and a point \(Q\). The perpendicular line from \(P\) to \(BC\) intersects \(AQ\) at \(A'\), and similarly define \(B'\) and \(C'\). Determine the locus of the centroid of triangle \(A'B'C'\) as \(P\) moves along \(\Gamma\). | \text{The locus of the centroid of }A'B'C' \text{ is an ellipse (degenerating to a line in special cases).} | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nOn a plane, consider a triangle \\(ABC\\), a circle \\(\\Gamma\\), a point \\(P \\in \\Gamma\\), and a point \\(Q\\). The perpendicular line from \\(P\\) to \\(BC\\) intersects \\(AQ\\) at \\(A'\\)... | {
"reason_high_no_tool": {
"accuracy": 0.875,
"count": 8,
"pass": 7
},
"reason_high_with_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_low_no_tool": {
"accuracy": 0.75,
"count": 8,
"pass": 6
},
"reason_low_with_tool": {
"accuracy": 0.75,
"count": 8,
... | cc-by-4.0 | [
"nano_v3"
] | ab531d06-81de-2956-f18f-76a304a23954 | [
{
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] | ||||
Let $\triangle ABC$ have side lengths $AB=30$, $BC=32$, and $AC=34$. Point $X$ lies in the interior of $\overline{BC}$, and points $I_1$ and $I_2$ are the incenters of $\triangle ABX$ and $\triangle ACX$, respectively. Find the minimum possible area of $\triangle AI_1I_2$ as $X$ varies along $\overline{BC}$. | 126 | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet $\\triangle ABC$ have side lengths $AB=30$, $BC=32$, and $AC=34$. Point $X$ lies in the interior of $\\overline{BC}$, and points $I_1$ and $I_2$ are the incenters of $\\triangle ABX$ and $\\tri... | {
"reason_high_no_tool": {
"accuracy": 0.875,
"count": 8,
"pass": 7
},
"reason_high_with_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_low_no_tool": {
"accuracy": 0.5,
"count": 8,
"pass": 4
},
"reason_low_with_tool": {
"accuracy": 0.875,
"count": 8,
... | cc-by-4.0 | [
"nano_v3"
] | 53d6d554-4e83-a3f5-bc04-21938fa9795a | [
{
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] | ||||
Given $a, b, m, n \in \mathbb{N}$, solve the system of equations:
\[ a^2 + b^2 = m^2 - n^2 \]
\[ ab = 2mn \] | \;
\begin{aligned}
& (a,b,m,n)=(0,t,t,0),\;(t,0,t,0)\;(t\in\mathbb N),\\
&\text{or }(a,b,m,n)=(0,0,s,s)\;(s\in\mathbb N).
\end{aligned}
\; | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven $a, b, m, n \\in \\mathbb{N}$, solve the system of equations:\n\\[ a^2 + b^2 = m^2 - n^2 \\]\n\\[ ab = 2mn \\]",
"name": "",
"reasoning_content": "",
"role": "user",
"tool_cal... | {
"reason_high_no_tool": {
"accuracy": 0.125,
"count": 8,
"pass": 1
},
"reason_high_with_tool": {
"accuracy": 0.375,
"count": 8,
"pass": 3
},
"reason_low_no_tool": {
"accuracy": 0.125,
"count": 8,
"pass": 1
},
"reason_low_with_tool": {
"accuracy": 0.25,
"count":... | cc-by-4.0 | [
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] | 081ebec4-5c0d-d1b1-2bcf-8f977e9d69cd | [
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}
] | ||||
For all integers \( n \) and \( d \) where \( d \) is a divisor of \( n \), do there exist integers \( y, z, t, s, a \) such that \( z \geq 2 \) and
\[ y^z \cdot n^t \cdot d^s + 1 = (n+1)^a \]
Additionally, you can use more than one divisor \( d \) on the left-hand side. | \text{Yes – such integers always exist.} | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor all integers \\( n \\) and \\( d \\) where \\( d \\) is a divisor of \\( n \\), do there exist integers \\( y, z, t, s, a \\) such that \\( z \\geq 2 \\) and\n\\[ y^z \\cdot n^t \\cdot d^s + 1 ... | {
"reason_high_no_tool": {
"accuracy": 0.875,
"count": 8,
"pass": 7
},
"reason_high_with_tool": {
"accuracy": 0.75,
"count": 8,
"pass": 6
},
"reason_low_no_tool": {
"accuracy": 0.5,
"count": 8,
"pass": 4
},
"reason_low_with_tool": {
"accuracy": 0.875,
"count": 8... | cc-by-4.0 | [
"nano_v3"
] | 4bc4e34b-2dfd-e213-2703-a4a3db3b18c6 | [
{
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"name": "",
"parameters": {
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"code": {
"description": "",
"type": ""
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},
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}
] | ||||
Find (if they exist) natural numbers \( a_1, a_2, \ldots, a_{19} \) such that every natural number \( \leq 999999 \) can be expressed as the sum of \( a_{i_1}, \ldots, a_{i_k} \) with \( k \leq 19 \) and \( i_1, \ldots, i_k \leq 19 \). | \text{No such natural numbers exist.} | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind (if they exist) natural numbers \\( a_1, a_2, \\ldots, a_{19} \\) such that every natural number \\( \\leq 999999 \\) can be expressed as the sum of \\( a_{i_1}, \\ldots, a_{i_k} \\) with \\( ... | {
"reason_high_no_tool": {
"accuracy": 0.625,
"count": 8,
"pass": 5
},
"reason_high_with_tool": {
"accuracy": 0,
"count": 8,
"pass": 0
},
"reason_low_no_tool": {
"accuracy": 0.75,
"count": 8,
"pass": 6
},
"reason_low_with_tool": {
"accuracy": 0.5,
"count": 8,
... | cc-by-4.0 | [
"nano_v3"
] | 191dc97f-c14e-3876-3b78-0e14c669a2bd | [
{
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"name": "",
"parameters": {
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"code": {
"description": "",
"type": ""
}
},
"required": [
""
],
"type": ""
}
},
"type": ""
}
] | ||||
What are the properties of a convex hexagon such that each diagonal cuts off a triangle whose area is not less than $\frac{1}{6}$ of the area of the hexagon? | \text{Affine image of a regular hexagon (centrally symmetric with opposite sides parallel and of equal length).} | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nWhat are the properties of a convex hexagon such that each diagonal cuts off a triangle whose area is not less than $\\frac{1}{6}$ of the area of the hexagon?",
"name": "",
"reasoning_conte... | {
"reason_high_no_tool": {
"accuracy": 1,
"count": 8,
"pass": 8
},
"reason_high_with_tool": {
"accuracy": 0.875,
"count": 8,
"pass": 7
},
"reason_low_no_tool": {
"accuracy": 0.5,
"count": 8,
"pass": 4
},
"reason_low_with_tool": {
"accuracy": 0.75,
"count": 8,
... | cc-by-4.0 | [
"nano_v3"
] | a7209302-bbdf-4758-29a4-f796d1424f02 | [
{
"function": {
"description": "",
"name": "",
"parameters": {
"properties": {
"code": {
"description": "",
"type": ""
}
},
"required": [
""
],
"type": ""
}
},
"type": ""
}
] | ||||
Solve the equation \(\frac{\sqrt{3}-1}{\sin x} + \frac{\sqrt{3}+1}{\cos x} = 4\sqrt{2}\). | \displaystyle
x=\frac{\pi}{12}+2\pi n\quad\text{or}\quad
x=\frac{11\pi}{36}+\frac{2\pi}{3}\,n,\qquad n\in\mathbb Z | true | aops | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the equation \\(\\frac{\\sqrt{3}-1}{\\sin x} + \\frac{\\sqrt{3}+1}{\\cos x} = 4\\sqrt{2}\\).",
"name": "",
"reasoning_content": "",
"role": "user",
"tool_call_id": "",
"to... | {
"reason_high_no_tool": {
"accuracy": 0.875,
"count": 8,
"pass": 7
},
"reason_high_with_tool": {
"accuracy": 0.875,
"count": 8,
"pass": 7
},
"reason_low_no_tool": {
"accuracy": 0.75,
"count": 8,
"pass": 6
},
"reason_low_with_tool": {
"accuracy": 0.75,
"count": ... | cc-by-4.0 | [
"nano_v3"
] | e02b36b7-28d3-dcd0-42c4-d077b08e11ac | [
{
"function": {
"description": "",
"name": "",
"parameters": {
"properties": {
"code": {
"description": "",
"type": ""
}
},
"required": [
""
],
"type": ""
}
},
"type": ""
}
] |
End of preview. Expand in Data Studio
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