prompt listlengths 1 1 | label listlengths 1 1 | messages listlengths 2 142 | tools listlengths 1 1 ⌀ | metadata unknown |
|---|---|---|---|---|
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the integral $\\int \\sqrt{\\cos(x)} \\, dx$."
}
] | [
"\\;\n \\int\\sqrt{\\cos x}\\,dx\n =\\sin x\\;{}_2F_{1}\\!\\Big(\\frac12,\\frac14;\\frac32;\\sin^{2}x\\Big)+C\n =\\frac12\\,B_{\\sin^{2}x}\\!\\Big(\\frac12,\\frac14\\Big)+C\n \\;"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the integral $\\int \\sqrt{\\cos(x)} \\, dx$."
},
{
"role": "assistant",
"reasoning_content": "We need to compute indefinite integral \\int sqrt(cos x) dx. This... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nCalculate the sum:\n\\[ 2\\begin{pmatrix}\n\\cos \\frac{\\pi }{3} & -\\sin \\frac{\\pi }{3} \\\\ \\sin \\frac{\\pi }{3}\n & \\cos \\frac{\\pi }{3}\n\\end{pmatrix}+ 4\\begin{pmat... | [
"\n\\frac{2^{\\,n+1}}{\\sqrt3}\n\\begin{pmatrix}\n\\sin\\frac{(n+1)\\pi}{3}&\\cos\\frac{(n+1)\\pi}{3}\\\\[2pt]\n-\\cos\\frac{(n+1)\\pi}{3}&\\sin\\frac{(n+1)\\pi}{3}\n\\end{pmatrix}\n-\n\\begin{pmatrix}\n1&\\dfrac{1}{\\sqrt3}\\\\[2pt]\n-\\dfrac{1}{\\sqrt3}&1\n\\end{pmatrix}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nCalculate the sum:\n\\[ 2\\begin{pmatrix}\n\\cos \\frac{\\pi }{3} & -\\sin \\frac{\\pi }{3} \\\\ \\sin \\frac{\\pi }{3}\n & \\cos \\frac{\\pi }{3}\n\\end{pmatrix}+ 4\\begin{pmat... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( g \\) be the harmonic mean of the arithmetic mean of the geometric mean of \\( abc \\) and \\( def \\) and the arithmetic mean of \\( ace \\) and \\( bdf \\). Compute \\... | [
"\\displaystyle \ng=\\frac{2\\sqrt{a b c d e f}\\,\\bigl(a c e+b d f\\bigr)}\n{2\\sqrt{a b c d e f}+a c e+b d f}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( g \\) be the harmonic mean of the arithmetic mean of the geometric mean of \\( abc \\) and \\( def \\) and the arithmetic mean of \\( ace \\) and \\( bdf \\). Compute \\... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"idx": 10841,
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn $\\triangle ABC$ with angles $\\alpha, \\beta, \\gamma$, angle $\\gamma$ is trisected, with the trisectors meeting $AB$ at points $D$ and $E$, where $D$ is closer to $A$. If ... | [
"AD:DE:EB\\;=\\;\\sin^{2}\\beta\\;:\\;2\\sin\\alpha\\sin\\beta\\cos\\!\\Bigl(\\frac{\\gamma}{3}\\Bigr)-\\sin^{2}\\beta\\;:\\;\\sin^{2}\\alpha"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn $\\triangle ABC$ with angles $\\alpha, \\beta, \\gamma$, angle $\\gamma$ is trisected, with the trisectors meeting $AB$ at points $D$ and $E$, where $D$ is closer to $A$. If ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSpin a regular coin repeatedly until both heads and tails have appeared $k$ times ($k = 1, 2, \\dots$). Denote by $v_k$ the number of necessary throws. Find the distribution of ... | [
""
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSpin a regular coin repeatedly until both heads and tails have appeared $k$ times ($k = 1, 2, \\dots$). Denote by $v_k$ the number of necessary throws. Find the distribution of ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the integer solutions of $\\frac{x+y}{y+z} = \\frac{x+z}{x+y}$."
}
] | [
"\n\\begin{aligned}\nx&=\\frac{\\varepsilon\\,k\\,pq+ k\\,q^{2}-k\\,p^{2}}{2}\n =\\frac{k\\bigl(\\varepsilon pq+q^{2}-p^{2}\\bigr)}{2},\\\\[2mm]\ny&=\\frac{\\varepsilon\\,k\\,pq+ k\\,p^{2}-k\\,q^{2}}{2}\n =\\frac{k\\bigl(\\varepsilon pq+p^{2}-q^{2}\\bigr)}{2},\\\\[2mm]\nz&=\\frac{k\\bigl(p^{2}+q^{2}-\\varep... | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the integer solutions of $\\frac{x+y}{y+z} = \\frac{x+z}{x+y}$."
},
{
"role": "assistant",
"reasoning_content": "We need integer solutions (x,y,z) presumably in... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 1,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nCalculate the integral $\\mathrm{I}(a) = \\displaystyle \\int\\limits_{0}^{+\\infty} \\dfrac{e^{-ax}}{x^2+1}\\mathrm{d}x$ for $a \\ge 0$."
}
] | [
"\\displaystyle \nI(a)=-\\operatorname{Im}\\!\\bigl[e^{\\,ia}\\,E_{1}(ia)\\bigr]\n =\\operatorname{Im}\\!\\bigl[e^{-\\,ia}\\,E_{1}(-ia)\\bigr] ,\n"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nCalculate the integral $\\mathrm{I}(a) = \\displaystyle \\int\\limits_{0}^{+\\infty} \\dfrac{e^{-ax}}{x^2+1}\\mathrm{d}x$ for $a \\ge 0$."
},
{
"role": "assistant",
... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": 1,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEvaluate the integral $\\int_{0}^{t} \\sqrt{1+t^4} \\, dt$."
}
] | [
"\\displaystyle\n\\int_{0}^{t}\\sqrt{1+x^{4}}\\,dx\n= \\frac{t}{3}\\sqrt{1+t^{4}}\n +\\frac13\\Bigl[\n F\\!\\Bigl(\\arctan\\frac{t^{2}-1}{\\sqrt2\\,t}\\,\\Big|\\,\\frac12\\Bigr)\n +K\\!\\Bigl(\\frac1{\\sqrt2}\\Bigr)\\Bigr] "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEvaluate the integral $\\int_{0}^{t} \\sqrt{1+t^4} \\, dt$."
},
{
"role": "assistant",
"reasoning_content": "We need to evaluate integral ∫_0^t sqrt(1 + t^4) dt. Usu... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDerive a formula for the area of a trapezoid using the lengths of the two bases and the lengths of the two diagonals."
}
] | [
"\\,K\n =\\sqrt{s\\,(s-p)\\,(s-q)\\,\\bigl(s-(a+b)\\bigr)}\\;"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDerive a formula for the area of a trapezoid using the lengths of the two bases and the lengths of the two diagonals."
},
{
"role": "assistant",
"reasoning_content":... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all possible solutions of \\(x\\) and \\(y\\) where \\(x - y^3 = \\text{LCM}(x, y)\\)."
}
] | [
"\n\\begin{cases}\n(x,y)=(n,0) &\\text{with }n\\ge 0,\\\\[4pt]\n(x,y)=(0,-1),\\\\[4pt]\n(x,y)=(-4k^{3},\\,-2k) &\\text{with }k\\in\\mathbb{Z}_{>0}.\n\\end{cases}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all possible solutions of \\(x\\) and \\(y\\) where \\(x - y^3 = \\text{LCM}(x, y)\\)."
},
{
"role": "assistant",
"reasoning_content": "We need to find integer ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all pairs \\((x, p)\\), where \\(p\\) is a prime number and \\(x \\leq 2p\\), such that \\(\\frac{x^{(p-1)}}{(p-1)x + 1}\\) is an integer."
}
] | [
"(x,p)=(-2,2)"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all pairs \\((x, p)\\), where \\(p\\) is a prime number and \\(x \\leq 2p\\), such that \\(\\frac{x^{(p-1)}}{(p-1)x + 1}\\) is an integer."
},
{
"role": "assistant"... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the equation \\(abcx^2 + (a^2 + b^2 + c^2)x + ab + bc + ca = 0\\) for \\(x\\) in the set of rational numbers, where \\(a, b, c \\in \\mathbb{Z}\\)."
}
] | [
"\\;x=-\\dfrac{bc}{\\,b^{2}+c^{2}\\,}\\;"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the equation \\(abcx^2 + (a^2 + b^2 + c^2)x + ab + bc + ca = 0\\) for \\(x\\) in the set of rational numbers, where \\(a, b, c \\in \\mathbb{Z}\\)."
},
{
"role": "... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( P \\) be a point inside triangle \\( ABC \\). Let \\( M, N, Q \\) be the feet of the perpendiculars from \\( P \\) to \\( BC, AC, AB \\) respectively. Find the locus of ... | [
"\\;P\\text{ must lie on the median }B\\!D\\text{ (the line joining }B\n\\text{ to the midpoint of }AC).\\;"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( P \\) be a point inside triangle \\( ABC \\). Let \\( M, N, Q \\) be the feet of the perpendiculars from \\( P \\) to \\( BC, AC, AB \\) respectively. Find the locus of ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \n$$\nP = \\frac{2x_1 \\cdot 2x_2 \\dots 2x_n}{(x_1 + x_2)(x_2 + x_3)\\dots(x_n + x_{n+1})}\n$$\nfor positive real numbers $x_1, \\dots, x_{n+1}$ such that $x_1 \\ge x_2 \\g... | [
"1\\;\\le\\;P\\;<\\;\\sqrt{C}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \n$$\nP = \\frac{2x_1 \\cdot 2x_2 \\dots 2x_n}{(x_1 + x_2)(x_2 + x_3)\\dots(x_n + x_{n+1})}\n$$\nfor positive real numbers $x_1, \\dots, x_{n+1}$ such that $x_1 \\ge x_2 \\g... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 1,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the equation \\( x^4 + 4x = 1 \\)."
}
] | [
"\\,x=-\\frac{\\sqrt2}{2}-\\sqrt{\\sqrt2-\\frac12}\\approx-1.6632519388\\,"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the equation \\( x^4 + 4x = 1 \\)."
},
{
"role": "assistant",
"reasoning_content": "We need to solve x^4 + 4x = 1, i.e., find all real (or complex) solutions? ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nUsing the sequence of prime numbers 2, 3, 5, 7, 11, 13, 17, find the number 19 by performing operations (multiplication and subtraction) involving all these primes."
}
] | [
"((((13-(11-17))-7)-5)\\times3)-2"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nUsing the sequence of prime numbers 2, 3, 5, 7, 11, 13, 17, find the number 19 by performing operations (multiplication and subtraction) involving all these primes."
},
{
... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": ... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven a sequence $\\{ a_{n}\\}$ such that $a_{n+1}=\\frac{p_{1}a_{n}+p_{2}}{p_{3}a_{n}+p_{4}}$ and $a_{1}= p_{0}$. Find the general term of $a_{n}$."
}
] | [
"\\;\na_{n}= \\alpha+\\frac{p_{0}-\\alpha}{1+(n-1)\\displaystyle\\frac{p_{3}\\alpha+p_{4}}{\\alpha}}\n"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven a sequence $\\{ a_{n}\\}$ such that $a_{n+1}=\\frac{p_{1}a_{n}+p_{2}}{p_{3}a_{n}+p_{4}}$ and $a_{1}= p_{0}$. Find the general term of $a_{n}$."
},
{
"role": "assis... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet $\\omega$ be the unit circle and let $a \\in \\omega$. Let $m$ be a point outside $\\omega$. The line segment joining $a$ and $m$ intersects $\\omega$ at another point $n$. ... | [
"\\; n=\\frac{2m-a-m^{2}\\,\\overline a}{|m|^{2}+1-a\\overline m-\\overline a\\,m}\n \\; "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet $\\omega$ be the unit circle and let $a \\in \\omega$. Let $m$ be a point outside $\\omega$. The line segment joining $a$ and $m$ intersects $\\omega$ at another point $n$. ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 1,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the maximum value of the function \n\\[ f(x,y,z) = \\frac{Axy + Byz + Cxz}{x^2 + y^2 + z^2}, \\]\nwhere \\( A, B, C \\in \\mathbb{R} \\) are constants and \\( x, y, z \\in ... | [
"\\displaystyle \\frac12\\,\\lambda_{\\max}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the maximum value of the function \n\\[ f(x,y,z) = \\frac{Axy + Byz + Cxz}{x^2 + y^2 + z^2}, \\]\nwhere \\( A, B, C \\in \\mathbb{R} \\) are constants and \\( x, y, z \\in ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine the highest power of a positive integer $k$ which divides $\\frac{(kn)!}{(n!)^k}$."
}
] | [
"\\,p^{\\,s_{p}(n)}\\;\\text{is the highest power of }p\n \\text{ dividing } \\frac{(pn)!}{(n!)^{p}}."
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine the highest power of a positive integer $k$ which divides $\\frac{(kn)!}{(n!)^k}$."
},
{
"role": "assistant",
"reasoning_content": "We need to determine th... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all pairs of real numbers $(m,n)$ such that $ma^2 + nb^2 > c^2$ for all triangles $ABC$ with sides $BC = a$, $AC = b$, and $AB = c$."
}
] | [
"(m-1)(n-1)\\ge 1"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all pairs of real numbers $(m,n)$ such that $ma^2 + nb^2 > c^2$ for all triangles $ABC$ with sides $BC = a$, $AC = b$, and $AB = c$."
},
{
"role": "assistant",
... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIf $\\min \\frac{ax^2+b}{\\sqrt{x^2+1}}=3$ for all $x \\in \\mathbb{R}$, find the range of $b$ and for every given $b$, find $a$."
}
] | [
" a = \\frac{\\,b-\\sqrt{b^{2}-9}\\,}{2}\\qquad (b>3)"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIf $\\min \\frac{ax^2+b}{\\sqrt{x^2+1}}=3$ for all $x \\in \\mathbb{R}$, find the range of $b$ and for every given $b$, find $a$."
},
{
"role": "assistant",
"reasoni... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": ... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine all triples \\((x, y, z) \\in \\mathbb{Z}^3\\) such that\n\\[ 2^x + (2^n - 1)^y = (2^n + 1)^z \\]\nfor a nonnegative integer \\( n \\)."
}
] | [
""
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine all triples \\((x, y, z) \\in \\mathbb{Z}^3\\) such that\n\\[ 2^x + (2^n - 1)^y = (2^n + 1)^z \\]\nfor a nonnegative integer \\( n \\)."
},
{
"role": "assistan... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all $x \\in \\mathbb{R}$ such that $f_n(x) = \\sin(7^n \\pi x)$ converges and find its limit."
}
] | [
"\\;\n\\Bigl\\{\\,x\\in\\mathbb R:\\;x=\\dfrac{p}{7^{\\,k}}\\text{ with }p\\in\\mathbb Z,\\;k\\in\\mathbb N_{0}\\Bigr\\}\n\\qquad\\text{and}\\qquad\n\\lim_{n\\to\\infty}\\sin\\bigl(7^{n}\\pi x\\bigr)=0\\;"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all $x \\in \\mathbb{R}$ such that $f_n(x) = \\sin(7^n \\pi x)$ converges and find its limit."
},
{
"role": "assistant",
"reasoning_content": "We need to find a... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_low_no_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_medium_with_tool": {
"count": 8,
"pass": 1,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the sequence defined by $u_0 = 3$ and $u_{n+1} = u_n + \\sqrt{u_n^2 - u_n - 2}$, find an expression for $u_n$ in terms of $n$."
}
] | [
"\\,u_n=\\frac{2\\alpha_n^{\\,2}+1}{\\alpha_n^{\\,2}-1}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the sequence defined by $u_0 = 3$ and $u_{n+1} = u_n + \\sqrt{u_n^2 - u_n - 2}$, find an expression for $u_n$ in terms of $n$."
},
{
"role": "assistant",
"reas... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all positive integers \\( x, y, z \\) such that \\( x^y + y^z + z^x = 2(x + y + z) \\)."
}
] | [
"(1,2,3)"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all positive integers \\( x, y, z \\) such that \\( x^y + y^z + z^x = 2(x + y + z) \\)."
},
{
"role": "assistant",
"reasoning_content": "We need to find all pos... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 1,
"accuracy": 0.1429
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet $[n] = \\{1, 2, \\dots, n\\}$ for some natural number $n$. Find the function $f(n)$ that maps $n$ to the maximum value $m$ such that there exists a subset of $[n]$ with $m$ ... | [
"\\,\\sqrt{n}\\le f(n)\\le\\frac{1+\\sqrt{8n-7}}{2}\\,"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet $[n] = \\{1, 2, \\dots, n\\}$ for some natural number $n$. Find the function $f(n)$ that maps $n$ to the maximum value $m$ such that there exists a subset of $[n]$ with $m$ ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven a cyclic quadrilateral \\(ABCD\\), what are the complex coordinates of the center of the circumscribed circle?"
}
] | [
"\\;\no=\n\\frac{\\displaystyle\\det\n\\begin{pmatrix}\na\\overline a & a & 1\\\\[2pt]\nb\\overline b & b & 1\\\\[2pt]\nc\\overline c & c & 1\n\\end{pmatrix}}\n{\\displaystyle\\det\n\\begin{pmatrix}\n\\overline a & a & 1\\\\[2pt]\n\\overline b & b & 1\\\\[2pt]\n\\overline c & c & 1\n\\end{pmatrix}}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven a cyclic quadrilateral \\(ABCD\\), what are the complex coordinates of the center of the circumscribed circle?"
},
{
"role": "assistant",
"reasoning_content": ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor each valid \\( N \\), define a sequence \\( a_N \\) by the number of different ways (up to different sizes of the \\( N \\) squares) to fill a square exactly with \\( N \\) ... | [
"1,0,0,1,0,1,2,5,11,24,53,115,247,531,1155,2535,\\ldots"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor each valid \\( N \\), define a sequence \\( a_N \\) by the number of different ways (up to different sizes of the \\( N \\) squares) to fill a square exactly with \\( N \\) ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGeneralize the situation where every $k^{\\text{th}}$ prisoner is killed in a circle of $N$ prisoners. Which prisoner will be set free?"
}
] | [
"\\,r+1\\,"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGeneralize the situation where every $k^{\\text{th}}$ prisoner is killed in a circle of $N$ prisoners. Which prisoner will be set free?"
},
{
"role": "assistant",
"r... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven that the equation \\(x^2 + p_1x + q_1 = 0\\) has roots \\(x_1\\) and \\(x_2\\), and the equation \\(x^2 + p_2x + q_2 = 0\\) has roots \\(x_1\\) and \\(x_2'\\) (these equat... | [
"\\,x^{2}\n+\\Bigl(p_{1}+p_{2}+\\frac{2\\,(q_{2}-q_{1})}{p_{1}-p_{2}}\\Bigr)x\n+\\frac{q_{1}q_{2}\\,(p_{1}-p_{2})}{\\,q_{1}p_{2}-p_{1}q_{2}\\,}=0\\,"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven that the equation \\(x^2 + p_1x + q_1 = 0\\) has roots \\(x_1\\) and \\(x_2\\), and the equation \\(x^2 + p_2x + q_2 = 0\\) has roots \\(x_1\\) and \\(x_2'\\) (these equat... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind natural numbers \\( p, q, r, s \\) such that \\( \\sqrt{n} < \\frac{p}{q} < \\frac{r}{s} < \\sqrt{n+1} \\) for a given natural number \\( n \\)."
}
] | [
"\n\\begin{aligned}\np&=\\Bigl\\lfloor (n+2)^{2}\\sqrt n\\Bigr\\rfloor+1,\\\\[2mm]\nq&=(n+2)^{2},\\\\[2mm]\nr&=\\Bigl\\lfloor (n+2)^{2}\\sqrt{\\,n+1\\,}\\Bigr\\rfloor,\\\\[2mm]\ns&=(n+2)^{2}.\n\\end{aligned}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind natural numbers \\( p, q, r, s \\) such that \\( \\sqrt{n} < \\frac{p}{q} < \\frac{r}{s} < \\sqrt{n+1} \\) for a given natural number \\( n \\)."
},
{
"role": "assi... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEliminate $\\theta$ from the equations $a = \\csc \\theta + \\tan \\theta$ and $b = \\sec \\theta + \\cot \\theta$."
}
] | [
" \\;4ab\\,(a+b+1)^{2}=(a+b)^{4}+2(a+b)^{3}+4(a+b)+4\\; "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEliminate $\\theta$ from the equations $a = \\csc \\theta + \\tan \\theta$ and $b = \\sec \\theta + \\cot \\theta$."
},
{
"role": "assistant",
"reasoning_content": "... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine all $n \\times n$ tables of nonnegative integers with rows and columns labelled by $0,1,2,...,n-1$ such that for each $0\\le i,j\\le n-1$, the number in the cell $(i,j... | [
"\n\\begin{array}{c}\n\\textbf{The condition “the entry }a_{i,j}\\text{ equals the number of }i\\text{'s in row }j”"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine all $n \\times n$ tables of nonnegative integers with rows and columns labelled by $0,1,2,...,n-1$ such that for each $0\\le i,j\\le n-1$, the number in the cell $(i,j... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all integer values of \\(a\\) and \\(b\\) such that \\( \\cos(a) + \\cos(a+b) + 2\\cos(b) = -2 \\)."
}
] | [
"\\ \\cos\\frac b2\\Bigl(\\,\\cos\\Bigl(a+\\frac b2\\Bigr)+2\\cos\\frac b2\\Bigr)=0\\ "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all integer values of \\(a\\) and \\(b\\) such that \\( \\cos(a) + \\cos(a+b) + 2\\cos(b) = -2 \\)."
},
{
"role": "assistant",
"reasoning_content": "We need int... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( p \\) be a prime of the form \\( 3k+2 \\) where \\( k \\) is a natural number. Find all solutions \\((x, y, z)\\) modulo \\( p \\) to the equation \\( x^3 + y^3 + z^3 \\... | [
"p^{2}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( p \\) be a prime of the form \\( 3k+2 \\) where \\( k \\) is a natural number. Find all solutions \\((x, y, z)\\) modulo \\( p \\) to the equation \\( x^3 + y^3 + z^3 \\... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_medium_with_tool": {
"count": 8,
"pass": 4,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven points \\( A_1 (x_1, y_1, z_1), A_2 (x_2, y_2, z_2), \\ldots, A_n (x_n, y_n, z_n) \\), let \\( P (x, y, z) \\) be the point which minimizes \\( \\sum ( |x - x_i| + |y - y_... | [
"\n\\begin{array}{l}\n\\text{If }n=2m+1\\;(m\\ge2):\\;A_i=(0,i,i),\\;A_{m+i}=(i,0,i),\\;A_{2m+1}=(0,0,0).\\\\[4pt]\n\\text{If }n=2m\\;(m\\ge3):\\;A_i=(0,i,i),\\;A_{m+i}=(i,0,i).\\\\[4pt]\n\\text{Then }P=(0,0,\\lceil m/2\\rceil)\\text{ and }P\\notin\\operatorname{conv}\\{A_i\\}.\n\\end{array}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven points \\( A_1 (x_1, y_1, z_1), A_2 (x_2, y_2, z_2), \\ldots, A_n (x_n, y_n, z_n) \\), let \\( P (x, y, z) \\) be the point which minimizes \\( \\sum ( |x - x_i| + |y - y_... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the greatest and least values of \\(x + y\\) where \\(x\\) and \\(y\\) are real numbers with \\(x \\geq -2\\) and \\(y \\geq -3\\), and satisfy the equation \\(x - 2\\sqrt{... | [
"\\text{Greatest value }=4+2\\sqrt{14}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the greatest and least values of \\(x + y\\) where \\(x\\) and \\(y\\) are real numbers with \\(x \\geq -2\\) and \\(y \\geq -3\\), and satisfy the equation \\(x - 2\\sqrt{... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the sequences $\\{a_n\\}$, $\\{b_n\\}$, $\\{c_n\\}$, and $\\{d_n\\}$ defined by:\n\\[ a_{n+1} = a_n + 2b_n + 3c_n \\]\n\\[ b_{n+1} = a_n + b_n + 3d_n \\]\n\\[ c_{n+1} = a_... | [
"\n\\begin{aligned}\n\\lim_{n\\to\\infty}\\frac{b_n}{a_n}&=\\sqrt{\\frac{q}{\\,p+q+1\\,}},\\\\[4pt]\n\\lim_{n\\to\\infty}\\frac{c_n}{a_n}&=\\sqrt{\\frac{p}{\\,p+q+1\\,}},\\\\[4pt]\n\\lim_{n\\to\\infty}\\frac{d_n}{a_n}&=\\frac{1}{\\sqrt{p+q+1}}.\n\\end{aligned}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the sequences $\\{a_n\\}$, $\\{b_n\\}$, $\\{c_n\\}$, and $\\{d_n\\}$ defined by:\n\\[ a_{n+1} = a_n + 2b_n + 3c_n \\]\n\\[ b_{n+1} = a_n + b_n + 3d_n \\]\n\\[ c_{n+1} = a_... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all possible positive integers \\( (a, b) \\) such that \\( (ab^2 + b + 7) \\) divides \\( (a^2 b + a + b) \\)."
}
] | [
"\\;(a,b)=(11,1),\\;(49,1)\\; "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all possible positive integers \\( (a, b) \\) such that \\( (ab^2 + b + 7) \\) divides \\( (a^2 b + a + b) \\)."
},
{
"role": "assistant",
"reasoning_content": ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 3,
"accuracy": 0.375
},
"reason_medium_with_tool": {
"count": 8,
"pass": 1,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind a hundred-digit natural number with non-zero digits such that it is divisible by the sum of its digits."
}
] | [
"5417571967195422991949535349234328481939555173173189291483452147721165832846678514537332733323892876"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind a hundred-digit natural number with non-zero digits such that it is divisible by the sum of its digits."
},
{
"role": "assistant",
"reasoning_content": "We need... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": ... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSketch a rectangle tiled by nine squares with side lengths $2, 5, 7, 9, 16, 25, 28, 33$, and $36$. Label each square in your sketch by its side length. The squares must fill the... | [
""
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSketch a rectangle tiled by nine squares with side lengths $2, 5, 7, 9, 16, 25, 28, 33$, and $36$. Label each square in your sketch by its side length. The squares must fill the... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all pairs of real numbers $(x, y)$ such that\n$$x^7 + y^7 = x^4 + y^4.$$"
}
] | [
"\\;\n\\begin{aligned}\n&\\text{Either }(x,y)\\in\\{(0,0),(0,1),(1,0),(1,1)\\},\\\\[2mm]\n&\\text{or }(x-1)(y-1)<0\\text{ and }x^{4}(x-1)=-\\,y^{4}(y-1).\n\\end{aligned}\n\\;"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all pairs of real numbers $(x, y)$ such that\n$$x^7 + y^7 = x^4 + y^4.$$"
},
{
"role": "assistant",
"reasoning_content": "We need to solve real solutions to x^7... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 5,
"accuracy": 0.625
},
"reason_low_no_tool": {
"count": 8,
"pass": 5,
"accuracy": 0.625
},
"reason_medium_with_tool": {
"count": 8,
"pass": ... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\(ABCD\\) be a parallelogram with points \\(X, Y, Z\\) lying on \\(AD, AB, BC\\) respectively such that \\(AX = CZ\\). Determine if the inequality \\(2(XY + YZ) \\ge AC + B... | [
"\\,2\\bigl(XY+YZ\\bigr)\\ge AC+BD\\,"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\(ABCD\\) be a parallelogram with points \\(X, Y, Z\\) lying on \\(AD, AB, BC\\) respectively such that \\(AX = CZ\\). Determine if the inequality \\(2(XY + YZ) \\ge AC + B... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_low_no_tool": {
"count": 8,
"pass": 7,
"accuracy": 0.875
},
"reason_medium_with_tool": {
"count": 8,
"pass": 3,... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor what integer $n$ can a $1995 \\times n$ rectangle be cut into pairwise different strips, where a strip is defined as a rectangle of size $1 \\times k$ for any natural number... | [
"\\,n=1\\,"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor what integer $n$ can a $1995 \\times n$ rectangle be cut into pairwise different strips, where a strip is defined as a rectangle of size $1 \\times k$ for any natural number... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the closed form for the summation \\((i+1) \\cdot 10^3 \\sum_{i=0}^v (t - id) - b\\), where \\(v = \\frac{1}{d}(t - b) - 1\\)."
}
] | [
"\\,S=10^{3}\\Bigl[\\frac{t}{2}(v+1)(v+2)-\\frac{d}{3}v(v+1)(v+2)\\Bigr]\n -b\\,(v+1),\\qquad \n v=\\frac{t-b}{d}-1.\\,"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the closed form for the summation \\((i+1) \\cdot 10^3 \\sum_{i=0}^v (t - id) - b\\), where \\(v = \\frac{1}{d}(t - b) - 1\\)."
},
{
"role": "assistant",
"reaso... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the simultaneous Diophantine equations $a^2 + ab + b^2 = c^2$ and $p^2 - pq + q^2 = c^2$, where $\\gcd(a, b, c) = 1$ and $\\gcd(p, q, c) = 1$."
}
] | [
"\n\\begin{aligned}\nc &=u^{2}+uv+v^{2},\\\\\na &=u^{2}-v^{2},\\qquad \nb =2uv+v^{2},\\\\\np &=u^{2}-v^{2},\\qquad \nq =-(2uv+v^{2}),\n\\end{aligned}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the simultaneous Diophantine equations $a^2 + ab + b^2 = c^2$ and $p^2 - pq + q^2 = c^2$, where $\\gcd(a, b, c) = 1$ and $\\gcd(p, q, c) = 1$."
},
{
"role": "assis... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSuppose that three congruent disks with radius $R$ have their centers at the vertices of a given triangle and cover the triangle. Find the smallest value of $R$ for a given tria... | [
"\\displaystyle R_{\\min}= \\frac12\\;\\max\\{AB,\\;BC,\\;CA\\}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSuppose that three congruent disks with radius $R$ have their centers at the vertices of a given triangle and cover the triangle. Find the smallest value of $R$ for a given tria... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 1,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn an acute-angled triangle \\(ABC\\), \\(E\\) and \\(F\\) are the feet of the altitudes from \\(B\\) and \\(C\\) respectively. \\(J\\) is the circumcenter of \\(\\Delta AEF\\),... | [
"BC=\\sqrt2\\,AC"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn an acute-angled triangle \\(ABC\\), \\(E\\) and \\(F\\) are the feet of the altitudes from \\(B\\) and \\(C\\) respectively. \\(J\\) is the circumcenter of \\(\\Delta AEF\\),... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all positive integers \\( x_1, x_2, \\ldots, x_{10} \\) such that:\ni) \\( x_1^2 + x_2^2 + \\cdots + x_{10}^2 = 10x_1 x_2 \\cdots x_{10} \\)\nii) At least one of \\( x_1, x... | [
""
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all positive integers \\( x_1, x_2, \\ldots, x_{10} \\) such that:\ni) \\( x_1^2 + x_2^2 + \\cdots + x_{10}^2 = 10x_1 x_2 \\cdots x_{10} \\)\nii) At least one of \\( x_1, x... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": ... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind a pair of points whose distance is not tripled by the map \\( f(x) = 3x \\mod 1 \\)."
}
] | [
"(x,y)=(0.3,\\,0.5)"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind a pair of points whose distance is not tripled by the map \\( f(x) = 3x \\mod 1 \\)."
},
{
"role": "assistant",
"reasoning_content": "We need to parse the quest... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": 1... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEvaluate the summation for \\(n\\): \\(\\sum_{k=1}^{n} \\frac{2^k}{k}\\)"
}
] | [
"\\displaystyle \n\\sum_{k=1}^{n}\\frac{2^{k}}{k}\n = H_{n}+\\int_{0}^{1}\\frac{(1+x)^{n}-1}{x}\\,dx\n =\\int_{0}^{1}\\frac{1-(2x)^{\\,n}}{1-2x}\\,dx "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEvaluate the summation for \\(n\\): \\(\\sum_{k=1}^{n} \\frac{2^k}{k}\\)"
},
{
"role": "assistant",
"reasoning_content": "We need to evaluate the summation for n: ∑_... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 1,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven a tangential quadrilateral \\(ABCD\\), what are the complex coordinates of the incenter?"
}
] | [
"\\displaystyle \nI=\\frac{a\\,c-b\\,d\\;\\pm\\;\\sqrt{(a-b)(b-c)(c-d)(d-a)}}\n {\\,a-b+c-d\\,}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven a tangential quadrilateral \\(ABCD\\), what are the complex coordinates of the incenter?"
},
{
"role": "assistant",
"reasoning_content": "We need to parse: \"G... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGive an example of an injective polarising function $f: \\mathbb{R}^2 \\to \\mathcal{H}$, where $\\mathcal{H}$ is the set of all lines in the plane and $P \\in f(Q)$ implies $Q ... | [
"\\,f\\big((a,b)\\big)=\\{(x,y)\\in\\mathbb R^{2}\\mid (a-1)x+by=a\\}\\,"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGive an example of an injective polarising function $f: \\mathbb{R}^2 \\to \\mathcal{H}$, where $\\mathcal{H}$ is the set of all lines in the plane and $P \\in f(Q)$ implies $Q ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the system of equations for \\(x\\), \\(y\\), and \\(z\\):\n\n\\[\nx + y + z = a\n\\]\n\\[\nx^2 + y^2 + z^2 = b^2\n\\]\n\\[\nxy = z^2\n\\]\n\nwhere \\(a\\) and \\(b\\) are... | [
" (u^{2}+v^{2}+uv)\\,b^{2} = (u^{2}+v^{2}-uv)\\,a^{2} "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the system of equations for \\(x\\), \\(y\\), and \\(z\\):\n\n\\[\nx + y + z = a\n\\]\n\\[\nx^2 + y^2 + z^2 = b^2\n\\]\n\\[\nxy = z^2\n\\]\n\nwhere \\(a\\) and \\(b\\) are... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the cubic equation \\(x^3 + ax^2 + bx + c = 0\\) by depression, where the equation is not factorable. Provide an example and show the steps to solve it."
}
] | [
"x=-2+\\sqrt[3]{\\frac12+\\sqrt{\\frac{23}{108}}}\n +\\sqrt[3]{\\frac12-\\sqrt{\\frac{23}{108}}}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the cubic equation \\(x^3 + ax^2 + bx + c = 0\\) by depression, where the equation is not factorable. Provide an example and show the steps to solve it."
},
{
"rol... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven \\(0 < a_1 < a_2 < \\cdots < a_n\\) and \\(\\frac{b_1}{a_1} < \\frac{b_2}{a_2} < \\cdots < \\frac{b_n}{a_n}\\), find the minimum value of the function \\(f(x) = \\max \\{|... | [
"\\displaystyle \n\\min_{x\\in\\mathbb R}\\max_{1\\le i\\le n}|a_i x-b_i|\n =\\min_{1\\le i<j\\le n}\\frac{a_i b_j-a_j b_i}{a_i+a_j}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven \\(0 < a_1 < a_2 < \\cdots < a_n\\) and \\(\\frac{b_1}{a_1} < \\frac{b_2}{a_2} < \\cdots < \\frac{b_n}{a_n}\\), find the minimum value of the function \\(f(x) = \\max \\{|... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_low_no_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSuppose \\(a, b, c\\) are positive real numbers such that \\(\\frac{1}{a} + \\frac{1}{b} + \\frac{2}{c} = 1\\). Find the minimum value of \\(ab + bc + ca + \\sqrt{a^2b^2 + b^2c^... | [
"24+8\\sqrt2+16\\Bigl(\\sqrt{\\sqrt2+1}+\\sqrt{\\sqrt2-1}\\Bigr)"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSuppose \\(a, b, c\\) are positive real numbers such that \\(\\frac{1}{a} + \\frac{1}{b} + \\frac{2}{c} = 1\\). Find the minimum value of \\(ab + bc + ca + \\sqrt{a^2b^2 + b^2c^... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve for \\( x \\) in the equation \\( x^2 = 2^x \\)."
}
] | [
"\\( x = 2 \\), \\( x = 4 \\), \\( x = -\\frac{2\\text{LambertW}\\left(\\frac{1}{2}\\ln(2)\\right)}{\\ln(2)} \\)"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve for \\( x \\) in the equation \\( x^2 = 2^x \\)."
},
{
"role": "assistant",
"reasoning_content": "We need to solve x^2 = 2^x. Problem: find all real solutions?... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"idx": 9669,
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIf \\( m, n \\in \\mathbb{Z} \\), then \\( m^2 + 3mn - 4n^2 \\) cannot be."
}
] | [
"5"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIf \\( m, n \\in \\mathbb{Z} \\), then \\( m^2 + 3mn - 4n^2 \\) cannot be."
},
{
"role": "assistant",
"reasoning_content": "We need to parse: \"If m, n ∈ Z, then m^2... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"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 for any \\( x, y, z \\in \\mathbb{R} \\),\n\n\\[ f(x+y) + f(y+z) + f(z+x) \\geqslant 3f(x+2y+3z). \\]"
}
] | [
"f(a)+f(b)+f(c)\\ge 3\\,f(2b+c)"
] | [
{
"role": "user",
"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 for any \\( x, y, z \\in \\mathbb{R} \\),\n\n\\[ f(x+y) + f(y+z) + f(z+x) \\geqslant 3f(x+2y+3z). \\]"
},
... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the sequence defined by \\( n! - (n-1)! + (n-2)! - \\cdots \\pm 1! \\), compute the values \\( A \\) and \\( B \\) such that \\( n! = A + B \\). For \\( n > 3 \\), conject... | [
"\\,n! = A_n + B_n\\, "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the sequence defined by \\( n! - (n-1)! + (n-2)! - \\cdots \\pm 1! \\), compute the values \\( A \\) and \\( B \\) such that \\( n! = A + B \\). For \\( n > 3 \\), conject... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all integer solutions \\((x, y)\\) to the equation \\(x^2 + 3xy + 4006(x + y) + 20032 = 0\\)."
}
] | [
"\\bigl(x,y\\bigr)=\\Bigl(\\frac{d-4006}{3},\n \\frac{31915784-d^{2}-4006d}{9d}\\Bigr)"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all integer solutions \\((x, y)\\) to the equation \\(x^2 + 3xy + 4006(x + y) + 20032 = 0\\)."
},
{
"role": "assistant",
"reasoning_content": "We need integer s... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the equation \\(4xy - y - x = z^2\\) using modulo arithmetic."
}
] | [
"(x,y,z)=(0,0,0)"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the equation \\(4xy - y - x = z^2\\) using modulo arithmetic."
},
{
"role": "assistant",
"reasoning_content": "We need to parse: \"Solve the equation \\(4xy - ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": 1,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven \\( r(R + 3) = \\frac{s^2}{3} - \\frac{Q}{6} \\) where \\( Q = \\sum (a - b)^2 \\), further analyze the relationship and implications."
}
] | [
"\\,\\frac{6}{7}\\le R<1,\\qquad0<r\\le\\frac{3}{7}\\,"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven \\( r(R + 3) = \\frac{s^2}{3} - \\frac{Q}{6} \\) where \\( Q = \\sum (a - b)^2 \\), further analyze the relationship and implications."
},
{
"role": "assistant",
... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nOn an $m \\times n$ board ($m, n \\ge 3$), there is a black token in the lower-left corner and white tokens in the lower-right and upper-left corners. Petryk and Vasyl take turn... | [
"\\text{Petryk can stop Vasyl unless } m=n\\text{ is odd.}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nOn an $m \\times n$ board ($m, n \\ge 3$), there is a black token in the lower-left corner and white tokens in the lower-right and upper-left corners. Petryk and Vasyl take turn... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 7,
"pass": 2,
"accuracy": 0.2857
},
"reason_medium_with_tool": {
"count": 8,
"pass": ... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind integers \\( n \\) and \\( a_i \\) such that \n\\[\n(a_1)! + 2(a_2)! + \\cdots + n(a_n)! = (n+1)[a_{n+1}]!\n\\]"
}
] | [
"\\; n=2\\,q!\\,,\\qquad a_1=a_2=\\dots =a_n=1\\,,\\qquad a_{n+1}=q\\;(q\\in\\mathbb N) \\;"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind integers \\( n \\) and \\( a_i \\) such that \n\\[\n(a_1)! + 2(a_2)! + \\cdots + n(a_n)! = (n+1)[a_{n+1}]!\n\\]"
},
{
"role": "assistant",
"reasoning_content": ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": ... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nUsing a compass and a ruler, split a triangle into two smaller triangles such that the sum of the squares of the sides of each smaller triangle is the same."
}
] | [
"AB^{2}+BD^{2}=AC^{2}+CD^{2}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nUsing a compass and a ruler, split a triangle into two smaller triangles such that the sum of the squares of the sides of each smaller triangle is the same."
},
{
"role"... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nCalculate the product $\\prod_{k=0}^n\\left[1+\\left(\\frac{1+i}{2}\\right)^{2^k} \\right]$."
}
] | [
"\\displaystyle\n\\prod_{k=0}^{n}\\Bigl[1+\\Bigl(\\frac{1+i}{2}\\Bigr)^{2^{k}}\\Bigr]\n=(1+i)\\Bigl[1-\\Bigl(\\frac{i}{2}\\Bigr)^{2^{\\,n}}\\Bigr] "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nCalculate the product $\\prod_{k=0}^n\\left[1+\\left(\\frac{1+i}{2}\\right)^{2^k} \\right]$."
},
{
"role": "assistant",
"reasoning_content": "We need to compute prod... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 2,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the simultaneous Diophantine equations $a^2 + b^2 + c^2 = z^2 + 1$ and $p^2 + q^2 + r^2 = z^2 + 2$."
}
] | [
"a=z,\\qquad b=1,\\qquad c=0,\\qquad \np=z,\\qquad q=1,\\qquad r=1 "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the simultaneous Diophantine equations $a^2 + b^2 + c^2 = z^2 + 1$ and $p^2 + q^2 + r^2 = z^2 + 2$."
},
{
"role": "assistant",
"reasoning_content": "We need to... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 2,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the number of bijective functions \\( f: \\{1, 2, \\ldots, 2n\\} \\to \\{1, 2, \\ldots, 2n\\} \\) such that \\( |f(i) - i| < n \\) for all \\( i = 1, 2, \\ldots, 2n \\)."
... | [
"\\; (2n)!\\displaystyle\\sum_{k=0}^{2n}\\frac{(-1)^k}{k!}\n =\\left\\lfloor\\frac{(2n)!}{e}+\\frac12\\right\\rfloor \\;"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the number of bijective functions \\( f: \\{1, 2, \\ldots, 2n\\} \\to \\{1, 2, \\ldots, 2n\\} \\) such that \\( |f(i) - i| < n \\) for all \\( i = 1, 2, \\ldots, 2n \\)."
... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor \\( f: \\mathbb{R}_+ \\rightarrow (0,1) \\) being an increasing function with \\( f(\\overline{x}) = 0.5 \\), determine if \\( f \\) must be concave or have an inflection po... | [
" \\; f(x)=\\frac12 +c\\bigl(e^{k(x-\\bar x)}-1\\bigr),\\; 0<c<\\tfrac14,\n\\;k>0\\; "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor \\( f: \\mathbb{R}_+ \\rightarrow (0,1) \\) being an increasing function with \\( f(\\overline{x}) = 0.5 \\), determine if \\( f \\) must be concave or have an inflection po... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind \\(a\\) and \\(b\\) such that \\((a^2 + b)(a + b^2) = (a - b)^3\\)."
}
] | [
"(a,b)=\\bigl(-\\tfrac95,-3\\bigr)"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind \\(a\\) and \\(b\\) such that \\((a^2 + b)(a + b^2) = (a - b)^3\\)."
},
{
"role": "assistant",
"reasoning_content": "We need to find a and b such that (a^2 + b)... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor a fixed \\( n \\), find \\( m \\in \\mathbb{N} \\) such that \\( p \\) is maximized, where\n\\[ p = \\sum_{i=n/2-m}^{n/2} \\binom{i}{n-m}. \\]"
}
] | [
"\\,m=\\dfrac{n}{2}\\,"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor a fixed \\( n \\), find \\( m \\in \\mathbb{N} \\) such that \\( p \\) is maximized, where\n\\[ p = \\sum_{i=n/2-m}^{n/2} \\binom{i}{n-m}. \\]"
},
{
"role": "assista... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_medium_with_tool": {
"count": 8,
"pass": 2,... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nUsing five 5s and any mathematical operations, form the number 2023."
}
] | [
"55^{2}-(5+5)^{3}-\\bigl\\lfloor\\sqrt{5}\\bigr\\rfloor"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nUsing five 5s and any mathematical operations, form the number 2023."
},
{
"role": "assistant",
"reasoning_content": "We need to express 2023 using five copies of th... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 1,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn a 6-dimensional space over a field of \\( p \\) elements, how many ways can two four-dimensional subspaces be chosen so that they intersect in a three-dimensional subspace?"
... | [
"\\displaystyle\np\\,(p+1)^{2}\\,(p^{2}+1)\\,(p^{2}-p+1)\\,(p^{2}+p+1)\\,\n (p^{4}+p^{3}+p^{2}+p+1)"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn a 6-dimensional space over a field of \\( p \\) elements, how many ways can two four-dimensional subspaces be chosen so that they intersect in a three-dimensional subspace?"
... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven a triangle \\( ABC \\) with \\( BC = a \\), \\( CA = b \\), \\( AB = c \\), \\( \\angle ABC = 45^\\circ \\), and \\( \\angle ACB = 120^\\circ \\). Point \\( I \\) is taken... | [
"\\,AI=\\dfrac{c\\,(b-2a)}{a}\\,"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven a triangle \\( ABC \\) with \\( BC = a \\), \\( CA = b \\), \\( AB = c \\), \\( \\angle ABC = 45^\\circ \\), and \\( \\angle ACB = 120^\\circ \\). Point \\( I \\) is taken... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind a closed formula for the sum $\\sum_{k=0}^{n} k^2 r^k$ using finite differences."
}
] | [
"\\;\\sum_{k=0}^{n}k^{2}\n =\\frac{n(n+1)(2n+1)}{6}\\;"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind a closed formula for the sum $\\sum_{k=0}^{n} k^2 r^k$ using finite differences."
},
{
"role": "assistant",
"reasoning_content": "We need a closed formula for s... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all sequences $\\{a_i\\}_{i=1}^{\\infty}$ such that:\n\\[ a_n = \\frac{a_{n-1} + a_{n-2}}{\\gcd(a_{n-1}, a_{n-2})} \\]"
}
] | [
"a_1=g u,\\; a_2=g v,\\; a_3=u+v,\\;\n a_n=a_{n-1}+a_{n-2}\\ (n\\ge4)"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all sequences $\\{a_i\\}_{i=1}^{\\infty}$ such that:\n\\[ a_n = \\frac{a_{n-1} + a_{n-2}}{\\gcd(a_{n-1}, a_{n-2})} \\]"
},
{
"role": "assistant",
"reasoning_con... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 3,
"accuracy": 0.375
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the inverse function of \\( f(x) = x^3 + x + \\sqrt{x} \\), if it exists. Specifically, express \\( x \\) in terms of \\( y \\)."
}
] | [
"\\,f^{-1}(y)=u(y)^{2}=y^{2}-2y^{3}+5y^{4}-14y^{5}+42y^{6}\n -138y^{7}+ \\cdots\\,"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the inverse function of \\( f(x) = x^3 + x + \\sqrt{x} \\), if it exists. Specifically, express \\( x \\) in terms of \\( y \\)."
},
{
"role": "assistant",
"rea... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nExpress the sequence \\(\\{a_n\\}_{n=1}^\\infty = \\{1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, \\ldots\\}\\) in closed form, where the sequence is formed by taking one odd integer, ... | [
"\\displaystyle \na_n = \\Big\\lfloor \\frac{4n+1-\\sqrt{8n+1}}{2}\\Big\\rfloor "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nExpress the sequence \\(\\{a_n\\}_{n=1}^\\infty = \\{1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, \\ldots\\}\\) in closed form, where the sequence is formed by taking one odd integer, ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nHow many ways are there to choose $n$ weights from $m$ weights with values $1, 2, 3, \\ldots, m$ such that the sum of the chosen weights is not more than $r$ (with $r \\in \\mat... | [
"(6)"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nHow many ways are there to choose $n$ weights from $m$ weights with values $1, 2, 3, \\ldots, m$ such that the sum of the chosen weights is not more than $r$ (with $r \\in \\mat... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the following system of equations in $\\mathbb{Z}^3$ for integer $A \\neq 0$:\n\\[ x + y^2 + z^3 = A \\]\n\\[ \\frac{1}{x} + \\frac{1}{y^2} + \\frac{1}{z^3} = \\frac{1}{A}... | [
""
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the following system of equations in $\\mathbb{Z}^3$ for integer $A \\neq 0$:\n\\[ x + y^2 + z^3 = A \\]\n\\[ \\frac{1}{x} + \\frac{1}{y^2} + \\frac{1}{z^3} = \\frac{1}{A}... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven an equilateral triangle \\(ABC\\) and an inner point \\(P\\) such that \\(PA:PB:PC = 3:4:5\\), find the point \\(P\\) by geometric construction."
}
] | [
"P\\Bigl(\\,\\frac{9+42\\sqrt3}{193}\\,a\\;,\\;\\frac{150-72\\sqrt3}{193}\\,a\\,\\Bigr)"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven an equilateral triangle \\(ABC\\) and an inner point \\(P\\) such that \\(PA:PB:PC = 3:4:5\\), find the point \\(P\\) by geometric construction."
},
{
"role": "ass... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the lengths of wallpaper pieces needed (3.5, 3.2, 2.1, 2.1, 2.6, 2.7, 1.6, 1.7, 0.8, 0.8) and rolls of wallpaper that are each 10 meters long, find the best combination of... | [
"\n\\begin{array}{l}\n\\text{Roll 1: }3.5+2.6+0.8=6.9\\ \\text{m}\\\\[2mm]\n\\text{Roll 2: }3.2+2.7+0.8=6.7\\ \\text{m}\\\\[2mm]\n\\text{Roll 3: }2.1+2.1+1.7+1.6=7.5\\ \\text{m}\n\\end{array}\n"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the lengths of wallpaper pieces needed (3.5, 3.2, 2.1, 2.1, 2.6, 2.7, 1.6, 1.7, 0.8, 0.8) and rolls of wallpaper that are each 10 meters long, find the best combination of... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThe coefficients in the series \\( T = 2 + 8x + 18x^2 + 37x^3 + \\cdots + a_r x^r + \\cdots \\) satisfy a recurrence relation of the form \\( a_{r+2} + p a_{r+1} + q a_r = 0 \\)... | [
" a_r=\\frac{14}{3}\\,2^{\\,r}-\\frac{8}{3}\\,2^{-r}\\;(r\\ge0). "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThe coefficients in the series \\( T = 2 + 8x + 18x^2 + 37x^3 + \\cdots + a_r x^r + \\cdots \\) satisfy a recurrence relation of the form \\( a_{r+2} + p a_{r+1} + q a_r = 0 \\)... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the sums of all possible pairs of 5 positive numbers \\(x_1, x_2, x_3, x_4, x_5\\), how can one determine the original 5 numbers?"
}
] | [
"\\;\n\\begin{aligned}\nx_{(3)} &= \\frac14\\sum_{k=1}^{10}p_k\\;-\\;p_{1}\\;-\\;p_{10},\\\\[4pt]\n\\{x_{(1)},x_{(2)}\\} &=\\{\\,p_k-x_{(3)}\\mid p_k-x_{(3)}\\text{ form a pair whose sum}=p_{1}\\,\\},\\\\[4pt]\n\\{x_{(4)},x_{(5)}\\} &=\\{\\,p_k-x_{(3)}\\mid p_k-x_{(3)}\\text{ are the remaining two numbers}\\,\\}.\n... | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the sums of all possible pairs of 5 positive numbers \\(x_1, x_2, x_3, x_4, x_5\\), how can one determine the original 5 numbers?"
},
{
"role": "assistant",
"r... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven three parallel lines \\( l, m, n \\) with distances \\( p \\) and \\( q \\) between them respectively, and points \\( A, B, C \\) on \\( l, m, n \\) such that \\( \\triang... | [
"\\frac{\\sqrt3\\,(p+q)}{2\\sqrt{p^{2}+q^{2}-pq}}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven three parallel lines \\( l, m, n \\) with distances \\( p \\) and \\( q \\) between them respectively, and points \\( A, B, C \\) on \\( l, m, n \\) such that \\( \\triang... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 7,
"pass": 2,
"accuracy": 0.2857
},
"reason_medium_with_tool": {
"count": 8,
"pass":... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all polynomials \\( P(x) \\) with real coefficients such that \\( P(x)P(2x^2 - 1) = P(x^2)P(2x - 1) \\)."
}
] | [
"Q(x)\\,Q(2x^{2}-1)=Q(x^{2})\\,Q(2x-1)"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all polynomials \\( P(x) \\) with real coefficients such that \\( P(x)P(2x^2 - 1) = P(x^2)P(2x - 1) \\)."
},
{
"role": "assistant",
"reasoning_content": "We nee... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven positive integers \\( m < n \\) and \\( p < q \\) such that \\( \\gcd(m, n) = 1 \\) and \\( \\gcd(p, q) = 1 \\), and \\(\\frac{m}{n} = \\tan \\alpha\\) and \\(\\frac{p}{q}... | [
"m=\\frac{2q}{\\,\\displaystyle\\bigl\\lfloor\\frac{q}{n}\\bigr\\rfloor+1\\,}-n "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven positive integers \\( m < n \\) and \\( p < q \\) such that \\( \\gcd(m, n) = 1 \\) and \\( \\gcd(p, q) = 1 \\), and \\(\\frac{m}{n} = \\tan \\alpha\\) and \\(\\frac{p}{q}... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nConsider the set \\( S \\subset \\mathbb{Z}^2 \\) such that \\((a,b) \\in S\\) if and only if there exist \\( c, d \\in \\mathbb{Z} \\) such that \\( a + bi = (c + di)^n \\) for... | [
"\\,2^{\\,v_2(n)}\\,"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nConsider the set \\( S \\subset \\mathbb{Z}^2 \\) such that \\((a,b) \\in S\\) if and only if there exist \\( c, d \\in \\mathbb{Z} \\) such that \\( a + bi = (c + di)^n \\) for... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 3,
"accuracy": 0.375
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": ... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind a natural number $N$ that is a multiple of $2009$ and the sum of its (decimal) digits equals $2009$."
}
] | [
"5\\underbrace{99\\ldots9}_{120}6\\underbrace{99\\ldots9}_{102}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind a natural number $N$ that is a multiple of $2009$ and the sum of its (decimal) digits equals $2009$."
},
{
"role": "assistant",
"reasoning_content": "We need a ... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nHow would one compute the radius of the circumcircle of a given triangle when the vertices are provided in barycentric coordinates?"
}
] | [
"\nR\\;=\\;\n\\frac{\\displaystyle\\sqrt{\n\\Bigl(a^{2}(y_{2}z_{3}-y_{3}z_{2})^{2}+b^{2}(y_{3}z_{1}-y_{1}z_{3})^{2}+c^{2}(y_{1}z_{2}-y_{2}z_{1})^{2}\\Bigr)\n\\Bigl(a^{2}(x_{2}z_{3}-x_{3}z_{2})^{2}+b^{2}(x_{3}z_{1}-x_{1}z_{3})^{2}+c^{2}(x_{1}z_{2}-x_{2}z_{1})^{2}\\Bigr)\n\\Bigl(a^{2}(x_{2}y_{3}-x_{3}y_{2})^{2}+b^{2}... | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nHow would one compute the radius of the circumcircle of a given triangle when the vertices are provided in barycentric coordinates?"
},
{
"role": "assistant",
"reaso... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the system of equations:\n\\[ \\frac{x^3}{y} + \\frac{4y^3}{x} = 1 + 2xy \\]\n\\[ 13y^2 - 4x^2 = 1 \\]"
}
] | [
"\n\\begin{aligned}\n(x,y)&=\\Bigl(\\pm\\frac{3\\sqrt{17}+29}{86}\\sqrt{\\frac{43}{5+2\\sqrt{17}}},\n\\;\\pm\\sqrt{\\frac{5+2\\sqrt{17}}{43}}\\Bigr),\\\\[4pt]\n(x,y)&=\\Bigl(\\pm\\frac{17-3\\sqrt{13}}{258}\\sqrt{\\frac{129}{3+2\\sqrt{13}}},\n\\;\\pm\\sqrt{\\frac{3+2\\sqrt{13}}{129}}\\Bigr).\n\\end{aligned}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the system of equations:\n\\[ \\frac{x^3}{y} + \\frac{4y^3}{x} = 1 + 2xy \\]\n\\[ 13y^2 - 4x^2 = 1 \\]"
},
{
"role": "assistant",
"reasoning_content": "We need... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the equation $6x \\cdot \\sin{x} = 1$."
}
] | [
"x\\approx0.414159933"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the equation $6x \\cdot \\sin{x} = 1$."
},
{
"role": "assistant",
"reasoning_content": "We need to solve equation 6 x sin x = 1. So find all real solutions x s... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the equation $(x+y)^2 + 4y + x = 0$ for integers $x$ and $y$."
}
] | [
"\n\\left\\{\\left(\\frac{k^{2}+6k-7}{12},\\frac{1-k^{2}}{12}\\right),\\;\n\\left(\\frac{k^{2}-6k-7}{12},\\frac{1-k^{2}}{12}\\right)\n\\;\\Big|\\;k\\in\\mathbb Z,\\;k\\equiv1,5\\pmod 6\\right\\}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the equation $(x+y)^2 + 4y + x = 0$ for integers $x$ and $y$."
},
{
"role": "assistant",
"reasoning_content": "We need integer solutions (x,y) to equation (x+y... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 1,
"accuracy": 0.1429
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all positive integer solutions \\((x, y)\\) to the equation \\(2x(x+1) = y(y+1)\\)."
}
] | [
"\n\\begin{cases}\nx_{1}=2,\\; y_{1}=3,\\\\[1mm]\nx_{n+1}=3x_{n}+2y_{n}+2,\\\\[1mm]\ny_{n+1}=4x_{n}+3y_{n}+3\\qquad (n\\ge1).\n\\end{cases}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all positive integer solutions \\((x, y)\\) to the equation \\(2x(x+1) = y(y+1)\\)."
},
{
"role": "assistant",
"reasoning_content": "We need to find all positiv... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn a 2-dimensional 30x30 box, a random walker starts from an arbitrary point inside the box. Determine the probability that the walker will exit the box through any of the sides... | [
"\nP_{\\text{exit}}(n)\n \\;=\\;\n 1-\n \\sum_{k=1}^{30}\\sum_{\\ell=1}^{30}\n \\Biggl[\n \\frac{4}{31^{2}}\\,\n \\sin\\!\\Bigl(\\frac{k\\pi\\,i_{0}}{31}\\Bigr)\\,\n \\sin\\!\\Bigl(\\frac{\\ell\\pi\\,j_{0}}{31}\\Bigr)\n \\Biggr]^{\\!2}\n \\Biggl[\n \\frac{\\cos\\!\\bigl(\\frac{k\\pi}{31}\\bigr)\n +\\... | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn a 2-dimensional 30x30 box, a random walker starts from an arbitrary point inside the box. Determine the probability that the walker will exit the box through any of the sides... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 1,
... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( f(x) = x^5 + x^4 + x^3 + x + 1 \\). Let \\( R \\) be the remainder when \\( f(x^{12}) \\) is divided by \\( f(x) \\). Solve the equation \\( 4^x - 2\\sin x - Rx = 0 \\).... | [
"x\\approx0.1848955952\\quad\\text{or}\\quad x\\approx1.6925851507 "
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( f(x) = x^5 + x^4 + x^3 + x + 1 \\). Let \\( R \\) be the remainder when \\( f(x^{12}) \\) is divided by \\( f(x) \\). Solve the equation \\( 4^x - 2\\sin x - Rx = 0 \\).... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0... |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine all integral solutions \\((x, y, z, w)\\) of the simultaneous Diophantine equations:\n\\[ x^2 + y^2 + z^2 = 2w^2 \\]\n\\[ x^4 + y^4 + z^4 = 2w^4 \\]"
}
] | [
"\\text{All integral solutions are given by (4) with }(p,q,w)\\text{ satisfying }p^{2}+pq+q^{2}=w^{2}, \n\\text{ i.e. by (7) and the subsequent substitution.}"
] | [
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine all integral solutions \\((x, y, z, w)\\) of the simultaneous Diophantine equations:\n\\[ x^2 + y^2 + z^2 = 2w^2 \\]\n\\[ x^4 + y^4 + z^4 = 2w^4 \\]"
},
{
"rol... | [
{
"type": "function",
"function": {
"name": "python",
"description": "Call this function to execute Python code in a stateful Jupyter notebook environment. Python will respond with the output of the execution or time out after 20.0 seconds.",
"parameters": {
"type": "object",
... | {
"points": [
1
],
"source": "aops",
"difficulty": {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 7,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 0,
... |
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