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[
{
"role": "user",
"content": "George is planning a dinner party for three other couples, his wife, and himself. He plans to seat the four couples around a circular table for 8, and wants each husband to be seated opposite his wife. How many seating arrangements can he make, if rotations and reflections of e... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00017383a5c19212cf96aff297af8aa643ec16d7:ec13bcb53b2a | |
[
{
"role": "user",
"content": "George is planning a dinner party for three other couples, his wife, and himself. He plans to seat the four couples around a circular table for 8, and wants each husband to be seated opposite his wife. How many seating arrangements can he make, if rotations and reflections of e... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00017383a5c19212cf96aff297af8aa643ec16d7:68915487ea01 | |
[
{
"role": "user",
"content": "George is planning a dinner party for three other couples, his wife, and himself. He plans to seat the four couples around a circular table for 8, and wants each husband to be seated opposite his wife. How many seating arrangements can he make, if rotations and reflections of e... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00017383a5c19212cf96aff297af8aa643ec16d7:a7d76893975c | |
[
{
"role": "user",
"content": "George is planning a dinner party for three other couples, his wife, and himself. He plans to seat the four couples around a circular table for 8, and wants each husband to be seated opposite his wife. How many seating arrangements can he make, if rotations and reflections of e... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00017383a5c19212cf96aff297af8aa643ec16d7:cd454c441852 | |
[
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(x+y)=f(x)f(y)-2x^2y\\]\nfor all real numbers $x$ and $y.$ If $f(1)=3$, find $f(3)$.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, let's tackle this p... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0003c5e28625bb826c995a2b63ca8bdb00909c67:293b6c5e2303 | |
[
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(x+y)=f(x)f(y)-2x^2y\\]\nfor all real numbers $x$ and $y.$ If $f(1)=3$, find $f(3)$.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I have this prob... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0003c5e28625bb826c995a2b63ca8bdb00909c67:36a70c456a74 | |
[
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers. If $z$ is a complex number such that $|z-a|=2$ and $|z-b|=4$, determine $|a-b|$.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, let's see. I need to find... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00051344a88b44580bb289dd5178c0e2ccd3a766:ac85a5fc604a | |
[
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers. If $z$ is a complex number such that $|z-a|=2$ and $|z-b|=4$, determine $|a-b|$.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I need to figure out t... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00051344a88b44580bb289dd5178c0e2ccd3a766:00240dd7e66f | |
[
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers. If $z$ is a complex number such that $|z-a|=2$ and $|z-b|=4$, determine $|a-b|$.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I need to find the val... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00051344a88b44580bb289dd5178c0e2ccd3a766:9fd8ec4656a9 | |
[
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers. If $z$ is a complex number such that $|z-a|=2$ and $|z-b|=4$, determine $|a-b|$.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I have this problem he... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00051344a88b44580bb289dd5178c0e2ccd3a766:da6cd24e58b6 | |
[
{
"role": "user",
"content": "Given two arrays X and Y of n integers each, where X represents the maximum amount of time (in minutes) an electric car battery can charge at a specific charging station, and Y represents the minimum amount of time (in minutes) an electric car can travel after charging at the s... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00068c820b209d9f9c8b7e23d9d7441fca1d46c3:4ab2cbdebc61 | |
[
{
"role": "user",
"content": "Given two arrays X and Y of n integers each, where X represents the maximum amount of time (in minutes) an electric car battery can charge at a specific charging station, and Y represents the minimum amount of time (in minutes) an electric car can travel after charging at the s... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00068c820b209d9f9c8b7e23d9d7441fca1d46c3:8ad99a2bdef1 | |
[
{
"role": "user",
"content": "Given two arrays X and Y of n integers each, where X represents the maximum amount of time (in minutes) an electric car battery can charge at a specific charging station, and Y represents the minimum amount of time (in minutes) an electric car can travel after charging at the s... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00068c820b209d9f9c8b7e23d9d7441fca1d46c3:ba7017f08141 | |
[
{
"role": "user",
"content": "Given two arrays X and Y of n integers each, where X represents the maximum amount of time (in minutes) an electric car battery can charge at a specific charging station, and Y represents the minimum amount of time (in minutes) an electric car can travel after charging at the s... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00068c820b209d9f9c8b7e23d9d7441fca1d46c3:ba048b2e4fd8 | |
[
{
"role": "user",
"content": "In a box, there are white and black marbles, with a total number not exceeding 2000. When three marbles are drawn at random without replacement, the probability that they are either all white or all black is exactly 1/5. What is the largest possible number of white marbles in t... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0007312a425920580af349a47674f400bf997a43:5497c08ffd62 | |
[
{
"role": "user",
"content": "In a box, there are white and black marbles, with a total number not exceeding 2000. When three marbles are drawn at random without replacement, the probability that they are either all white or all black is exactly 1/5. What is the largest possible number of white marbles in t... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0007312a425920580af349a47674f400bf997a43:a51b5ace3880 | |
[
{
"role": "user",
"content": "In a box, there are white and black marbles, with a total number not exceeding 2000. When three marbles are drawn at random without replacement, the probability that they are either all white or all black is exactly 1/5. What is the largest possible number of white marbles in t... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0007312a425920580af349a47674f400bf997a43:a355ecb26253 | |
[
{
"role": "user",
"content": "In a box, there are white and black marbles, with a total number not exceeding 2000. When three marbles are drawn at random without replacement, the probability that they are either all white or all black is exactly 1/5. What is the largest possible number of white marbles in t... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0007312a425920580af349a47674f400bf997a43:96188bf46ceb | |
[
{
"role": "user",
"content": "A cube has a volume of 64 cubic centimeters. Its three edges meet at a corner, forming a solid angle $\\omega$. If $\\omega$ subtends a solid angle of $\\frac{\\pi}{12}$ steradians at the center of a sphere circumscribing the cube, what is the length of a diagonal of the cube?\... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 000818fd835f16ca56bcf35e25a3efcfd07e8e3e:8ac3443846c3 | |
[
{
"role": "user",
"content": "A cube has a volume of 64 cubic centimeters. Its three edges meet at a corner, forming a solid angle $\\omega$. If $\\omega$ subtends a solid angle of $\\frac{\\pi}{12}$ steradians at the center of a sphere circumscribing the cube, what is the length of a diagonal of the cube?\... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 000818fd835f16ca56bcf35e25a3efcfd07e8e3e:2a6ea73450de | |
[
{
"role": "user",
"content": "A cube has a volume of 64 cubic centimeters. Its three edges meet at a corner, forming a solid angle $\\omega$. If $\\omega$ subtends a solid angle of $\\frac{\\pi}{12}$ steradians at the center of a sphere circumscribing the cube, what is the length of a diagonal of the cube?\... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 000818fd835f16ca56bcf35e25a3efcfd07e8e3e:556365c5fcb9 | |
[
{
"role": "user",
"content": "A cube has a volume of 64 cubic centimeters. Its three edges meet at a corner, forming a solid angle $\\omega$. If $\\omega$ subtends a solid angle of $\\frac{\\pi}{12}$ steradians at the center of a sphere circumscribing the cube, what is the length of a diagonal of the cube?\... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 000818fd835f16ca56bcf35e25a3efcfd07e8e3e:926af741045e | |
[
{
"role": "user",
"content": "In a circle of 1000 switches, initially all set to \"off\", a man starts at switch number 1 and flips every 2nd switch to \"on\". Then, starting from the first switch that was left \"off\", he flips every 3rd switch to \"on\". He continues this process, each time flipping every... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0008e95e5e500cc6d5d735a3f9b9e4184159f4b8:849ce6809795 | |
[
{
"role": "user",
"content": "Let $\\mathcal{R}$ be the region in the $xy$-plane defined by the inequalities $2\\le x \\le 3$ and $0\\le y \\le x \\left\\lfloor \\sqrt x \\right\\rfloor$.\n\nFind the smallest positive integer $n$ such that there is no way to cover the entire region $\\mathcal{R}$ with $n$ p... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00094dfc9bef1cbb04c015235332eac7c3c03c21:b8429d15c8d0 | |
[
{
"role": "user",
"content": "Find the $1000^{\\text{th}}$ digit past the decimal point in the decimal expansion of $\\dfrac{17}{27}$, where the expansion is obtained by a long division algorithm with the remainder always being less than or equal to half of the divisor.\n\nLet's think step by step and outpu... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 000ba563853784b097807380d928a5074146bfa9:2eda091d8d0f | |
[
{
"role": "user",
"content": "Find the $1000^{\\text{th}}$ digit past the decimal point in the decimal expansion of $\\dfrac{17}{27}$, where the expansion is obtained by a long division algorithm with the remainder always being less than or equal to half of the divisor.\n\nLet's think step by step and outpu... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 000ba563853784b097807380d928a5074146bfa9:cc18d58806ec | |
[
{
"role": "user",
"content": "Find the $1000^{\\text{th}}$ digit past the decimal point in the decimal expansion of $\\dfrac{17}{27}$, where the expansion is obtained by a long division algorithm with the remainder always being less than or equal to half of the divisor.\n\nLet's think step by step and outpu... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 000ba563853784b097807380d928a5074146bfa9:9c3211e4d9d3 | |
[
{
"role": "user",
"content": "Find the $1000^{\\text{th}}$ digit past the decimal point in the decimal expansion of $\\dfrac{17}{27}$, where the expansion is obtained by a long division algorithm with the remainder always being less than or equal to half of the divisor.\n\nLet's think step by step and outpu... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 000ba563853784b097807380d928a5074146bfa9:802244cb9819 | |
[
{
"role": "user",
"content": "A convex hexagon has interior angles with measures $x$, $2x$, $3x$, $4x$, $5x$, and $6x$ degrees. The sum of the measures of any three consecutive angles is equal to 540 degrees. What is the measure of the smallest angle?\n\nLet's think step by step and output the final answe... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 000dc454c4bd71c377dfad87d6e748e3ee600d91:f369c7e6ce4a | |
[
{
"role": "user",
"content": "In the quadrilateral $ABCD$ the line segments $BE$ and $CF$ are medians of the triangles $ABC$ and $ACD$ respectively. If the area of the quadrilateral $ABCD$ is equal to $80$ square centimeters, the area of $AEBF$ is equal to $32$ square centimeters, and the ratio of $AB$ and ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 000f252c733bb7657dc1cc57e64ca32d9a2fd6f5:f3a7b36df9f4 | |
[
{
"role": "user",
"content": "In the quadrilateral $ABCD$ the line segments $BE$ and $CF$ are medians of the triangles $ABC$ and $ACD$ respectively. If the area of the quadrilateral $ABCD$ is equal to $80$ square centimeters, the area of $AEBF$ is equal to $32$ square centimeters, and the ratio of $AB$ and ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 000f252c733bb7657dc1cc57e64ca32d9a2fd6f5:cca3aed5dc9a | |
[
{
"role": "user",
"content": "Find the largest number of flags that can be placed on a geoboard such that no two flags lie on the same horizontal, vertical, or diagonal line, and no three flags form an isosceles triangle.\n\n[asy]\ndraw((0,0)--(0,6)--(6,6)--(6,0)--cycle,linewidth(1));\nfor(int i=1;i<6;++i)\... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0010425c3e539a859ae83cbd309bc0981e8e0207:a469ccaa89ac | |
[
{
"role": "user",
"content": "Find the largest number of flags that can be placed on a geoboard such that no two flags lie on the same horizontal, vertical, or diagonal line, and no three flags form an isosceles triangle.\n\n[asy]\ndraw((0,0)--(0,6)--(6,6)--(6,0)--cycle,linewidth(1));\nfor(int i=1;i<6;++i)\... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0010425c3e539a859ae83cbd309bc0981e8e0207:34805965242b | |
[
{
"role": "user",
"content": "If $\\cos 45^\\circ \\tan 20^\\circ$ can be written as $\\sin \\theta + \\cos\\theta$ for some acute angle $\\theta,$ what is $\\theta?$ (Give your answer in degrees, not radians.)\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0015af834932c896bb51196bab1f61c46098c4f1:84f9bc81b83f | |
[
{
"role": "user",
"content": "If $\\cos 45^\\circ \\tan 20^\\circ$ can be written as $\\sin \\theta + \\cos\\theta$ for some acute angle $\\theta,$ what is $\\theta?$ (Give your answer in degrees, not radians.)\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0015af834932c896bb51196bab1f61c46098c4f1:6b8fca0829b7 | |
[
{
"role": "user",
"content": "Let $\\mathcal{P}$ be the parabola in the plane determined by the equation $y = x^2.$ Suppose a circle $\\mathcal{C}$ intersects $\\mathcal{P}$ at four distinct points. If three of these points are $(-19,361),$ $(-4,16),$ and $(11,121),$ find the sum of the distances from the... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 001f51694506f5174272dd4aed6e136a640809e5:c7f0807b4062 | |
[
{
"role": "user",
"content": "Let $\\mathcal{P}$ be the parabola in the plane determined by the equation $y = x^2.$ Suppose a circle $\\mathcal{C}$ intersects $\\mathcal{P}$ at four distinct points. If three of these points are $(-19,361),$ $(-4,16),$ and $(11,121),$ find the sum of the distances from the... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 001f51694506f5174272dd4aed6e136a640809e5:3606f697e192 | |
[
{
"role": "user",
"content": "Let $\\mathcal{P}$ be the parabola in the plane determined by the equation $y = x^2.$ Suppose a circle $\\mathcal{C}$ intersects $\\mathcal{P}$ at four distinct points. If three of these points are $(-19,361),$ $(-4,16),$ and $(11,121),$ find the sum of the distances from the... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 001f51694506f5174272dd4aed6e136a640809e5:84869bb20a69 | |
[
{
"role": "user",
"content": "Let $\\mathcal{P}$ be the parabola in the plane determined by the equation $y = x^2.$ Suppose a circle $\\mathcal{C}$ intersects $\\mathcal{P}$ at four distinct points. If three of these points are $(-19,361),$ $(-4,16),$ and $(11,121),$ find the sum of the distances from the... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 001f51694506f5174272dd4aed6e136a640809e5:28f8d2f4a173 | |
[
{
"role": "user",
"content": "A cargo ship can transport containers that weigh up to 2000 kg. A company needs to transport a total of 35000 kg of cargo, but some of the containers are already partially filled with cargo. The available containers have the following weights: 500 kg, 700 kg, and 1000 kg. How m... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0021ec42c931e1aa3501802dffe0f4b39f888cbf:b558ccdb39f5 | |
[
{
"role": "user",
"content": "A cargo ship can transport containers that weigh up to 2000 kg. A company needs to transport a total of 35000 kg of cargo, but some of the containers are already partially filled with cargo. The available containers have the following weights: 500 kg, 700 kg, and 1000 kg. How m... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0021ec42c931e1aa3501802dffe0f4b39f888cbf:573cf278dcda | |
[
{
"role": "user",
"content": "A cargo ship can transport containers that weigh up to 2000 kg. A company needs to transport a total of 35000 kg of cargo, but some of the containers are already partially filled with cargo. The available containers have the following weights: 500 kg, 700 kg, and 1000 kg. How m... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0021ec42c931e1aa3501802dffe0f4b39f888cbf:1099167ddceb | |
[
{
"role": "user",
"content": "A cargo ship can transport containers that weigh up to 2000 kg. A company needs to transport a total of 35000 kg of cargo, but some of the containers are already partially filled with cargo. The available containers have the following weights: 500 kg, 700 kg, and 1000 kg. How m... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0021ec42c931e1aa3501802dffe0f4b39f888cbf:9f228367d978 | |
[
{
"role": "user",
"content": "A polynomial p(x) with integer coefficients has the property that its remainder when divided by x^n is n for all positive integers n. Find the remainder when p(x) is divided by x^6 - 1.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"rol... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0023012a6078b31d6d69876aff560fc5f98f0dc3:0c32e7a2ac9b | |
[
{
"role": "user",
"content": "A polynomial p(x) with integer coefficients has the property that its remainder when divided by x^n is n for all positive integers n. Find the remainder when p(x) is divided by x^6 - 1.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"rol... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0023012a6078b31d6d69876aff560fc5f98f0dc3:2520da4a75fe | |
[
{
"role": "user",
"content": "A polynomial p(x) with integer coefficients has the property that its remainder when divided by x^n is n for all positive integers n. Find the remainder when p(x) is divided by x^6 - 1.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"rol... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0023012a6078b31d6d69876aff560fc5f98f0dc3:f05992ef7547 | |
[
{
"role": "user",
"content": "A polynomial p(x) with integer coefficients has the property that its remainder when divided by x^n is n for all positive integers n. Find the remainder when p(x) is divided by x^6 - 1.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"rol... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0023012a6078b31d6d69876aff560fc5f98f0dc3:003359e30448 | |
[
{
"role": "user",
"content": "Five numbered cards are placed in a row in ascending order. The cards are then rearranged so that the number on each card is either one more or one less than the number on the card that was originally in its position. What is the probability that the new arrangement is in desce... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0024577c38c0d76f8bd0929e1d1a5e012f8f7c7f:8a3bf870e085 | |
[
{
"role": "user",
"content": "Five numbered cards are placed in a row in ascending order. The cards are then rearranged so that the number on each card is either one more or one less than the number on the card that was originally in its position. What is the probability that the new arrangement is in desce... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0024577c38c0d76f8bd0929e1d1a5e012f8f7c7f:68dc3593da00 | |
[
{
"role": "user",
"content": "Five numbered cards are placed in a row in ascending order. The cards are then rearranged so that the number on each card is either one more or one less than the number on the card that was originally in its position. What is the probability that the new arrangement is in desce... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0024577c38c0d76f8bd0929e1d1a5e012f8f7c7f:02cafae7721a | |
[
{
"role": "user",
"content": "Five numbered cards are placed in a row in ascending order. The cards are then rearranged so that the number on each card is either one more or one less than the number on the card that was originally in its position. What is the probability that the new arrangement is in desce... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0024577c38c0d76f8bd0929e1d1a5e012f8f7c7f:f4be953bbf90 | |
[
{
"role": "user",
"content": "In a Cartesian coordinate plane, a circle of radius 5 units is centered at the point (3, 4). A square is inscribed in the circle such that two of its vertices lie on the circle and the other two vertices are on the x-axis. What is the area of the square?\n\nLet's think step by ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0025aa9d5ce85c094d33605d791f960dd119250a:92677eb1639d | |
[
{
"role": "user",
"content": "In a Cartesian coordinate plane, a circle of radius 5 units is centered at the point (3, 4). A square is inscribed in the circle such that two of its vertices lie on the circle and the other two vertices are on the x-axis. What is the area of the square?\n\nLet's think step by ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0025aa9d5ce85c094d33605d791f960dd119250a:036f58ea72da | |
[
{
"role": "user",
"content": "In a Cartesian coordinate plane, a circle of radius 5 units is centered at the point (3, 4). A square is inscribed in the circle such that two of its vertices lie on the circle and the other two vertices are on the x-axis. What is the area of the square?\n\nLet's think step by ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0025aa9d5ce85c094d33605d791f960dd119250a:98394bc6816c | |
[
{
"role": "user",
"content": "In a Cartesian coordinate plane, a circle of radius 5 units is centered at the point (3, 4). A square is inscribed in the circle such that two of its vertices lie on the circle and the other two vertices are on the x-axis. What is the area of the square?\n\nLet's think step by ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0025aa9d5ce85c094d33605d791f960dd119250a:8eabe6e3bc87 | |
[
{
"role": "user",
"content": "Find the value of $c$ for the smallest positive real number $d$ such that all the roots of the polynomial $x^4 - dx^3 + cx^2 - dx + 1$ are real. For this value of $d$, the value of $c$ is unique.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 002fb901f6bf5bb760941953a8b3d9e9d67f2f3c:9273c751c399 | |
[
{
"role": "user",
"content": "Find the value of $c$ for the smallest positive real number $d$ such that all the roots of the polynomial $x^4 - dx^3 + cx^2 - dx + 1$ are real. For this value of $d$, the value of $c$ is unique.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 002fb901f6bf5bb760941953a8b3d9e9d67f2f3c:01a0a94f9f25 | |
[
{
"role": "user",
"content": "Find the value of $c$ for the smallest positive real number $d$ such that all the roots of the polynomial $x^4 - dx^3 + cx^2 - dx + 1$ are real. For this value of $d$, the value of $c$ is unique.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 002fb901f6bf5bb760941953a8b3d9e9d67f2f3c:a164b8fbafe9 | |
[
{
"role": "user",
"content": "Find the value of $c$ for the smallest positive real number $d$ such that all the roots of the polynomial $x^4 - dx^3 + cx^2 - dx + 1$ are real. For this value of $d$, the value of $c$ is unique.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 002fb901f6bf5bb760941953a8b3d9e9d67f2f3c:de8eacf5c5ec | |
[
{
"role": "user",
"content": "The product of four positive integers $w$, $x$, $y$, and $z$ is 1200. If $w-4 = x+4 = 3y-12 = z/4$, what is the value of the sum $w + x + y + z$?\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think>... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 003229d1dac6696f3e6db30921be6883ee8cc6b0:dba96e04cb71 | |
[
{
"role": "user",
"content": "Let $m$ and $n$ be positive integers such that $m^2 - mn + n^2 = 225$. Determine the maximum value of $m + n$.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I have this problem here:... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0033f3163e70aae607dd62fd739506fe024114f6:71f935d6c911 | |
[
{
"role": "user",
"content": "Let $m$ and $n$ be positive integers such that $m^2 - mn + n^2 = 225$. Determine the maximum value of $m + n$.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I have this problem here:... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0033f3163e70aae607dd62fd739506fe024114f6:e6d919235232 | |
[
{
"role": "user",
"content": "Let $m$ and $n$ be positive integers such that $m^2 - mn + n^2 = 225$. Determine the maximum value of $m + n$.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I need to find positive i... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0033f3163e70aae607dd62fd739506fe024114f6:0cb6d10603b0 | |
[
{
"role": "user",
"content": "Let $m$ and $n$ be positive integers such that $m^2 - mn + n^2 = 225$. Determine the maximum value of $m + n$.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I have this problem here:... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0033f3163e70aae607dd62fd739506fe024114f6:a102fc9d27b8 | |
[
{
"role": "user",
"content": "Find the range of the function\n\\[f(x) = \\frac{2x^2 + 1}{|x - 3|}.\\]\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I need to find the range of the function f(x) = (2x² + 1)/|x - 3... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0033f34309f190eb8c21579168b224532d7c10c4:a632befd1637 | |
[
{
"role": "user",
"content": "Find the range of the function\n\\[f(x) = \\frac{2x^2 + 1}{|x - 3|}.\\]\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I have to find the range of the function f(x) = (2x² + 1)/|x - 3... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0033f34309f190eb8c21579168b224532d7c10c4:1cfdb86d431e | |
[
{
"role": "user",
"content": "Find the range of the function\n\\[f(x) = \\frac{2x^2 + 1}{|x - 3|}.\\]\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I need to find the range of the function f(x) = (2x² + 1)/|x - 3... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0033f34309f190eb8c21579168b224532d7c10c4:53b79d62113e | |
[
{
"role": "user",
"content": "Find the range of the function\n\\[f(x) = \\frac{2x^2 + 1}{|x - 3|}.\\]\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I need to find the range of the function \\( f(x) = \\frac{2x^2 ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0033f34309f190eb8c21579168b224532d7c10c4:d9643a9a453c | |
[
{
"role": "user",
"content": "The diagonals of a regular polygon with $n$ sides intersect in a point, dividing each diagonal into two segments. Let $a_n$ be the ratio of the length of the longer segment to the length of the shorter segment for each diagonal. Find the maximum value of $n$ for which $a_n$ is ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0034e7e950655a542b31c38cd33b65a727e7851d:064410b3481e | |
[
{
"role": "user",
"content": "Let $\\{a_i\\}_{i=1}^{2019}$ be a sequence of real numbers satisfying $|a_{i+1} - a_i| \\le 1$ for $i = 1,2,\\ldots,2018.$ Suppose that the polynomial $$P(x) = (x-a_1)(x-a_2)\\cdots(x-a_{2019})$$satisfies $$\\frac{|P(0)|}{|P(1)|} < \\frac{1}{2019}.$$Determine the value of $a_1... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 003d913e21f1f3988075b4f0f313196444b633bd:873bffe70ff6 | |
[
{
"role": "user",
"content": "Let $\\{a_i\\}_{i=1}^{2019}$ be a sequence of real numbers satisfying $|a_{i+1} - a_i| \\le 1$ for $i = 1,2,\\ldots,2018.$ Suppose that the polynomial $$P(x) = (x-a_1)(x-a_2)\\cdots(x-a_{2019})$$satisfies $$\\frac{|P(0)|}{|P(1)|} < \\frac{1}{2019}.$$Determine the value of $a_1... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 003d913e21f1f3988075b4f0f313196444b633bd:f276cc0f059c | |
[
{
"role": "user",
"content": "Let $\\{a_i\\}_{i=1}^{2019}$ be a sequence of real numbers satisfying $|a_{i+1} - a_i| \\le 1$ for $i = 1,2,\\ldots,2018.$ Suppose that the polynomial $$P(x) = (x-a_1)(x-a_2)\\cdots(x-a_{2019})$$satisfies $$\\frac{|P(0)|}{|P(1)|} < \\frac{1}{2019}.$$Determine the value of $a_1... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 003d913e21f1f3988075b4f0f313196444b633bd:4882397da463 | |
[
{
"role": "user",
"content": "Quadrilateral $ABCD$ is inscribed in a circle with $AB$ as a diameter, $AB = 2 \\sqrt{2},$ and $BC = CD = 2.$ Points $E$ and $F$ are on $AB$ and $AD$ respectively so that $BE = AF = x.$ If $EF = CD = 2,$ determine the measure of angle $ABC$ in terms of $x.$\n\nLet's think step ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 003d96cbf4ab9ea6e2795fe3758d1d3f799e3a33:2ef5e4bfab62 | |
[
{
"role": "user",
"content": "Quadrilateral $ABCD$ is inscribed in a circle with $AB$ as a diameter, $AB = 2 \\sqrt{2},$ and $BC = CD = 2.$ Points $E$ and $F$ are on $AB$ and $AD$ respectively so that $BE = AF = x.$ If $EF = CD = 2,$ determine the measure of angle $ABC$ in terms of $x.$\n\nLet's think step ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 003d96cbf4ab9ea6e2795fe3758d1d3f799e3a33:857225b46647 | |
[
{
"role": "user",
"content": "Find the maximum value of\n\\[\\cos^2 2x + 2 \\sin^2 3x,\\]as $x$ varies over all real numbers.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I need to find the maximum value of the ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 003e6a26f1b1ebad3cf77076bd536624a18a7c48:17900d66c1aa | |
[
{
"role": "user",
"content": "Find the maximum value of\n\\[\\cos^2 2x + 2 \\sin^2 3x,\\]as $x$ varies over all real numbers.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I need to find the maximum value of the ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 003e6a26f1b1ebad3cf77076bd536624a18a7c48:988c1773b9a3 | |
[
{
"role": "user",
"content": "Find the maximum value of\n\\[\\cos^2 2x + 2 \\sin^2 3x,\\]as $x$ varies over all real numbers.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I need to find the maximum value of the ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 003e6a26f1b1ebad3cf77076bd536624a18a7c48:3f78ef210983 | |
[
{
"role": "user",
"content": "Find the maximum value of\n\\[\\cos^2 2x + 2 \\sin^2 3x,\\]as $x$ varies over all real numbers.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I need to find the maximum value of the ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 003e6a26f1b1ebad3cf77076bd536624a18a7c48:b64cf16c472b | |
[
{
"role": "user",
"content": "Let $P$ be a point inside the square $ABCD,$ so that $PA = 1,$ $PB = 2,$ and $PC = 3.$ Then\n\\[\\overrightarrow{P} = x \\overrightarrow{A} + y \\overrightarrow{B} + z \\overrightarrow{C} + w \\overrightarrow{D},\\]where $x,$ $y,$ $z,$ and $w$ are constants such that $x + y + ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0040ac5692ebda2ecd04a2bf3fa71b8b474dd01f:408577d7e583 | |
[
{
"role": "user",
"content": "Let $P$ be a point inside the square $ABCD,$ so that $PA = 1,$ $PB = 2,$ and $PC = 3.$ Then\n\\[\\overrightarrow{P} = x \\overrightarrow{A} + y \\overrightarrow{B} + z \\overrightarrow{C} + w \\overrightarrow{D},\\]where $x,$ $y,$ $z,$ and $w$ are constants such that $x + y + ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0040ac5692ebda2ecd04a2bf3fa71b8b474dd01f:189a52dcfaa6 | |
[
{
"role": "user",
"content": "Let $P$ be a point inside the square $ABCD,$ so that $PA = 1,$ $PB = 2,$ and $PC = 3.$ Then\n\\[\\overrightarrow{P} = x \\overrightarrow{A} + y \\overrightarrow{B} + z \\overrightarrow{C} + w \\overrightarrow{D},\\]where $x,$ $y,$ $z,$ and $w$ are constants such that $x + y + ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0040ac5692ebda2ecd04a2bf3fa71b8b474dd01f:9c6aebc2f56f | |
[
{
"role": "user",
"content": "Let $P$ be a point inside the square $ABCD,$ so that $PA = 1,$ $PB = 2,$ and $PC = 3.$ Then\n\\[\\overrightarrow{P} = x \\overrightarrow{A} + y \\overrightarrow{B} + z \\overrightarrow{C} + w \\overrightarrow{D},\\]where $x,$ $y,$ $z,$ and $w$ are constants such that $x + y + ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0040ac5692ebda2ecd04a2bf3fa71b8b474dd01f:bc633eb0c0a2 | |
[
{
"role": "user",
"content": "The vertices of a cube are labeled with the numbers 1 through 8 in such a way that the sum of the numbers on each face is the same. Find the maximum possible value of the sum of the numbers on a single face.\n\n(No further text)\n\nLet's think step by step and output the final ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0042ba38d8dd23dae58d30ada97113e5591c88cc:c6f1bb469a53 | |
[
{
"role": "user",
"content": "The vertices of a cube are labeled with the numbers 1 through 8 in such a way that the sum of the numbers on each face is the same. Find the maximum possible value of the sum of the numbers on a single face.\n\n(No further text)\n\nLet's think step by step and output the final ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0042ba38d8dd23dae58d30ada97113e5591c88cc:fa6f0b44c0bf | |
[
{
"role": "user",
"content": "The vertices of a cube are labeled with the numbers 1 through 8 in such a way that the sum of the numbers on each face is the same. Find the maximum possible value of the sum of the numbers on a single face.\n\n(No further text)\n\nLet's think step by step and output the final ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0042ba38d8dd23dae58d30ada97113e5591c88cc:d12481f5ef8c | |
[
{
"role": "user",
"content": "The vertices of a cube are labeled with the numbers 1 through 8 in such a way that the sum of the numbers on each face is the same. Find the maximum possible value of the sum of the numbers on a single face.\n\n(No further text)\n\nLet's think step by step and output the final ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0042ba38d8dd23dae58d30ada97113e5591c88cc:1bc8ee4401ab | |
[
{
"role": "user",
"content": "In the figure below, $O$ is the circumcenter of triangle $ABC$, and $M$ is the midpoint of $\\overline{BC}$. Let $D$ be the intersection of $\\overline{AO}$ and $\\overline{BM}$. What is the ratio of $OD$ to $OA$?\n\n[asy]\n\nsize (4cm,4cm);\n\npair A,B,C,D,M,O;\n\nB=(0,0);\n... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00440659511f0e12a78d74f1d0e609668b4acceb:da8db1688f0e | |
[
{
"role": "user",
"content": "In the figure below, $O$ is the circumcenter of triangle $ABC$, and $M$ is the midpoint of $\\overline{BC}$. Let $D$ be the intersection of $\\overline{AO}$ and $\\overline{BM}$. What is the ratio of $OD$ to $OA$?\n\n[asy]\n\nsize (4cm,4cm);\n\npair A,B,C,D,M,O;\n\nB=(0,0);\n... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00440659511f0e12a78d74f1d0e609668b4acceb:1175dbcd40a7 | |
[
{
"role": "user",
"content": "In the figure below, $O$ is the circumcenter of triangle $ABC$, and $M$ is the midpoint of $\\overline{BC}$. Let $D$ be the intersection of $\\overline{AO}$ and $\\overline{BM}$. What is the ratio of $OD$ to $OA$?\n\n[asy]\n\nsize (4cm,4cm);\n\npair A,B,C,D,M,O;\n\nB=(0,0);\n... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00440659511f0e12a78d74f1d0e609668b4acceb:27709a9425e1 | |
[
{
"role": "user",
"content": "In the figure below, $O$ is the circumcenter of triangle $ABC$, and $M$ is the midpoint of $\\overline{BC}$. Let $D$ be the intersection of $\\overline{AO}$ and $\\overline{BM}$. What is the ratio of $OD$ to $OA$?\n\n[asy]\n\nsize (4cm,4cm);\n\npair A,B,C,D,M,O;\n\nB=(0,0);\n... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 00440659511f0e12a78d74f1d0e609668b4acceb:864bf1728aba | |
[
{
"role": "user",
"content": "Determine all positive integer pairs (a,b) such that the sum of the last three digits of a^2 + b^2 is 1000.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Alright, let me try to tackle this pr... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 004581d40f2981928dce283ba94dd1e8202bac43:078023c24047 | |
[
{
"role": "user",
"content": "Determine all positive integer pairs (a,b) such that the sum of the last three digits of a^2 + b^2 is 1000.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so I have this problem here: I ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 004581d40f2981928dce283ba94dd1e8202bac43:5081d30ca936 | |
[
{
"role": "user",
"content": "Determine all positive integer pairs (a,b) such that the sum of the last three digits of a^2 + b^2 is 1000.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Alright, so I need to find all positi... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 004581d40f2981928dce283ba94dd1e8202bac43:a3ebbb0752e2 | |
[
{
"role": "user",
"content": "Determine all positive integer pairs (a,b) such that the sum of the last three digits of a^2 + b^2 is 1000.\n\nLet's think step by step and output the final answer within \\boxed{}."
},
{
"role": "assistant",
"content": "<think> Okay, so the problem is asking for al... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 004581d40f2981928dce283ba94dd1e8202bac43:c3b7028a0afe | |
[
{
"role": "user",
"content": "In a rectangular parallelepiped, all edges are colored in red, green, or blue. Each face has a distinct color, and each color appears on exactly two opposite faces. The total length of all edges is $234$ units. Determine the total area of all faces of the parallelepiped.\n\nLet... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0049491ec2b18f26b0e2385ffe05c2d180d23803:cedd2cff602b | |
[
{
"role": "user",
"content": "In a rectangular parallelepiped, all edges are colored in red, green, or blue. Each face has a distinct color, and each color appears on exactly two opposite faces. The total length of all edges is $234$ units. Determine the total area of all faces of the parallelepiped.\n\nLet... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 0049491ec2b18f26b0e2385ffe05c2d180d23803:3140b7badc5c | |
[
{
"role": "user",
"content": "In a group of people, each person can speak one or more of the following three languages: French, Spanish, and English. If 10 people speak French, 11 people speak Spanish, and 12 people speak English, what is the minimum number of people in the group, assuming that there is at ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 004b50f6a851fb246e41cce3e67bfb4b95b5034c:e5b7db33b2cd | |
[
{
"role": "user",
"content": "In a group of people, each person can speak one or more of the following three languages: French, Spanish, and English. If 10 people speak French, 11 people speak Spanish, and 12 people speak English, what is the minimum number of people in the group, assuming that there is at ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 004b50f6a851fb246e41cce3e67bfb4b95b5034c:faa87500a469 | |
[
{
"role": "user",
"content": "In a group of people, each person can speak one or more of the following three languages: French, Spanish, and English. If 10 people speak French, 11 people speak Spanish, and 12 people speak English, what is the minimum number of people in the group, assuming that there is at ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 004b50f6a851fb246e41cce3e67bfb4b95b5034c:f0465ff54e28 | |
[
{
"role": "user",
"content": "In a group of people, each person can speak one or more of the following three languages: French, Spanish, and English. If 10 people speak French, 11 people speak Spanish, and 12 people speak English, what is the minimum number of people in the group, assuming that there is at ... | math | ai2-adapt-dev/openmath-2-math | OpenThoughts3-1.2M | train | 004b50f6a851fb246e41cce3e67bfb4b95b5034c:f7db93c773bc |
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