id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-caa707e345d13baaafb6a6b84d4a347cf0f039d5e283a1d68d11f648e7939431 | What numbers can be written in the form $x^2 + 2y^2$? | [
"\\text{All positive integers where primes congruent to }5 \\text{ or }7 \\pmod{8} \\text{ have even exponents in their factorization.}"
] | [
"What numbers can be written in the form $x^2 + 2y^2$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-27337a9820769efcb984c39396312f437b8e799cdfca8038f3022265d022fda2 | What numbers divide the numerical expression $1^{1001} + 2^{1001} + \ldots + 1001^{1001}$? | [
"1002 and \\(1001^2\\)"
] | [
"What numbers divide the numerical expression $1^{1001} + 2^{1001} + \\ldots + 1001^{1001}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4a2b3d82991eb85d36845a531ce691883a8dba23763bb75414ca95697541b815 | What numeral is in the 100th decimal place in the decimal representation of $\frac{6}{7}$? | [
"1"
] | [
"What numeral is in the 100th decimal place in the decimal representation of $\\frac{6}{7}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3003ff55e209d9c9fe217e73e384d7074cf0c38812148868d0ac5fe198769dee | What pair of positive real numbers \((x, y)\) satisfies \(x^2y^2 = 144\) and \((x - y)^3 = 64\)? | [
"(6, 2)"
] | [
"What pair of positive real numbers \\((x, y)\\) satisfies \\(x^2y^2 = 144\\) and \\((x - y)^3 = 64\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-0aad968f7d36ad896c8aefce4ddaf6355ac015ceb9cd41161bb4fe7683957d11 | What parametrization of $x$ and $y$ should be chosen to evaluate the line integral $\int_{C} |y| \,ds$ where $C$ is the curve $(x^2 + y^2)^2 = r^2(x^2 - y^2)$? The parametrization $x = r\cos{\theta}$ and $y = r\sin{\theta}$ does not satisfy the curve $C$. | [
"x = \\frac{r \\cos t}{1 + \\sin^2 t}, \\quad y = \\frac{r \\sin t \\cos t}{1 + \\sin^2 t}"
] | [
"What parametrization of $x$ and $y$ should be chosen to evaluate the line integral $\\int_{C} |y| \\,ds$ where $C$ is the curve $(x^2 + y^2)^2 = r^2(x^2 - y^2)$? The parametrization $x = r\\cos{\\theta}$ and $y = r\\sin{\\theta}$ does not satisfy the curve $C$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-0b9dbc3a25be77b1621f2b844be2f6f67d5989619c1d10a71a9b9321fc5b4367 | What partition sums can be obtained from the set $\{1, 2, 3, 4, \dots, n\}$? Given a set $\{1, 2, 3, 4, \dots, n\}$, partition the set into disjoint subsets, multiply the numbers within each subset, and then sum these products. Determine all possible values of these partition sums. | [
"\\left[ \\frac{n(n+1)}{2} - 1, \\, n! + 1 \\right]"
] | [
"What partition sums can be obtained from the set $\\{1, 2, 3, 4, \\dots, n\\}$? \n\nGiven a set $\\{1, 2, 3, 4, \\dots, n\\}$, partition the set into disjoint subsets, multiply the numbers within each subset, and then sum these products. Determine all possible values of these partition sums."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-c585712f3308b121bc5e88fe3e502124a7e77bc4ecf0c6855664c419eb9c5ecc | What percent (or fraction) of the perfect squares starting from $100$ have an even digit in the tens place? | [
"\\(\\frac{4}{5}\\)"
] | [
"What percent (or fraction) of the perfect squares starting from $100$ have an even digit in the tens place?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-7872c89c7a2b5fe7ec52ab6c19c715eaaf26e6b680256767612d6a9ef0f84403 | What percent of $x$ is equal to $40\%$ of $50\%$ of $x$? | [
"20"
] | [
"What percent of $x$ is equal to $40\\%$ of $50\\%$ of $x$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0713f9752a9bed9340b3a7a03ed90fe2fa54f0b4e63fa1fdb8aea6b60c7e138c | What percent of 300 is 550? | [
"\\( 183.\\overline{3} \\% \\)"
] | [
"What percent of 300 is 550?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5524540bd5b7137ec8bf680739cd92b1008ac0e84d2fec8e48b5e0cfc2f94fe3 | What percent of American women are between 66 and 71 inches tall, given that their heights are normally distributed with a mean of 68.5 inches and a standard deviation of 2.5 inches? | [
"68%"
] | [
"What percent of American women are between 66 and 71 inches tall, given that their heights are normally distributed with a mean of 68.5 inches and a standard deviation of 2.5 inches?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-f2185102f056ecd184414913062fd80e435b8d411893f5f516fcf5e22287d159 | What percent of American women are shorter than 61 inches, given that their heights are normally distributed with a mean of 68.5 inches and a standard deviation of 2.5 inches? | [
"0.15%"
] | [
"What percent of American women are shorter than 61 inches, given that their heights are normally distributed with a mean of 68.5 inches and a standard deviation of 2.5 inches?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e88e5a1445b146d2611db354a0508e70e079eb389984d88f98c7c2e2ff8f802a | What percent of American women are taller than 73.5 inches, given that their heights are normally distributed with a mean of 68.5 inches and a standard deviation of 2.5 inches? | [
"2.5%"
] | [
"What percent of American women are taller than 73.5 inches, given that their heights are normally distributed with a mean of 68.5 inches and a standard deviation of 2.5 inches?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dced01e1fe6f2fbead07d81f1d1b0424026ae1ddac03bd1959d7d5dd88243696 | What percent of all applicants had scores higher than 13 on the biological sciences part of the MCAT, given that the scores were approximately Normal with a mean of 9.9 and a standard deviation of 2.6? | [
"11.70%"
] | [
"What percent of all applicants had scores higher than 13 on the biological sciences part of the MCAT, given that the scores were approximately Normal with a mean of 9.9 and a standard deviation of 2.6?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-0293da17bb99bc4be9c8d671e42ffc340509c5a680ef96fb035babc9c9a9e5a6 | What percent of the junior biology majors are ineligible for Bio research, given that 52% have taken statistics, 23% have taken a computer course, and 7% have taken both? | [
"32%"
] | [
"What percent of the junior biology majors are ineligible for Bio research, given that 52% have taken statistics, 23% have taken a computer course, and 7% have taken both?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-994a13610266427777da55971271ef3b7d42afcb114dc4d1f4df83c5da448bae | What percent of the perfect squares in the strings from $10^2$ to $109^2$, $10^2$ to $1009^2$, $10^2$ to $10009^2$, and so on (where each string contains 100, 1000, 10000, etc., numbers) have a tens digit of $1, 3, 5, 7,$ or $9$? | [
"20\\%"
] | [
"What percent of the perfect squares in the strings from $10^2$ to $109^2$, $10^2$ to $1009^2$, $10^2$ to $10009^2$, and so on (where each string contains 100, 1000, 10000, etc., numbers) have a tens digit of $1, 3, 5, 7,$ or $9$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c2bb30b81e0ca0a24ed5363ae7a6b53d96e8e209126ff8eb50643fd536ecdd85 | What percent of the positive integers less than or equal to $100$ have no remainders when divided by $5?$ | [
"20"
] | [
"What percent of the positive integers less than or equal to $100$ have no remainders when divided by $5?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-29096187c9b851f49d9163f0311dc97873dd9aed4336ba15a4048ca63e6601ff | What percent of the prime numbers less than 12 are divisible by 2? | [
"20\\%"
] | [
"What percent of the prime numbers less than 12 are divisible by 2?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c09f57e6ed9ce2a426d905f17ca8c548f8d2cad7a4f6f830773ef0518356fe41 | What percent of those who entered medical school had scores between 9 and 12 on the biological sciences part of the MCAT, given that the mean score for those who entered was 10.4 and the standard deviation was 1.8? | [
"59.52%"
] | [
"What percent of those who entered medical school had scores between 9 and 12 on the biological sciences part of the MCAT, given that the mean score for those who entered was 10.4 and the standard deviation was 1.8?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 11
} | 25 | 0 | 0 |
omr-ws-48573b05a4aa9370d16ab3bf6dd6d81a133ce48e8d1fbba5f026c48fa309397c | What percentage of the U.S. population were prison inmates at the end of 2002, given that there were an estimated 476 prison inmates per 100,000 U.S. residents? | [
"0.476%"
] | [
"What percentage of the U.S. population were prison inmates at the end of 2002, given that there were an estimated 476 prison inmates per 100,000 U.S. residents?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-b67b8901b64cf34be5bf53af4f76db28e65cb08f11e0590995dd739e798689d2 | What percentage of the space taken up by closest-packed spheres is not in the spheres themselves? | [
"\\(25.95\\%\\)"
] | [
"What percentage of the space taken up by closest-packed spheres is not in the spheres themselves?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 31
} | 54 | 0 | 0 |
omr-ws-b406af769f218a0543f62862df95ebcac1e10595bc7e1ee770cfaf7001778a44 | What percentage of the two-letter combinations generated from the letters $A, B, C, D, E$ (without repeating any letter) do not contain a consonant? | [
"10%"
] | [
"What percentage of the two-letter combinations generated from the letters $A, B, C, D, E$ (without repeating any letter) do not contain a consonant?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-422f159c50a3414173144041e947243983f908366ea79921e2c11065dedd1f48 | What perfect-square integer is closest to 273? | [
"289"
] | [
"What perfect-square integer is closest to 273?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-50e60a16b44c6d70875b99bf606f482d33ddeac4153383ca0fc110394ba469e2 | What point within a triangle has the property that the sum of its distances from the sides is a minimum? | [
"\\text{The vertex opposite the longest side}"
] | [
"What point within a triangle has the property that the sum of its distances from the sides is a minimum?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-30e80addf87f91afca5c3c3d0c28a5d71e6e4fd7347113c511546c6398bd21a8 | What polynomial \( P \in \mathbb{R}[x] \) satisfies \( |\sqrt[3]{x} - P(x)| \leq \frac{1}{100} \) for all \( x \in [0,1] \)? | [
"P(x)"
] | [
"What polynomial \\( P \\in \\mathbb{R}[x] \\) satisfies \\( |\\sqrt[3]{x} - P(x)| \\leq \\frac{1}{100} \\) for all \\( x \\in [0,1] \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-fce917c2ffca5f34118eeb3f01678837aefc9c50d742146fb228e6e3eb060abf | What polynomial satisfies the function $f(\sqrt{3}+\sqrt{2})= \sqrt{3}-\sqrt{2}$? | [
"\\( f(x) = -x^3 + 10x \\)"
] | [
"What polynomial satisfies the function $f(\\sqrt{3}+\\sqrt{2})= \\sqrt{3}-\\sqrt{2}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-18e266b0a51be474fcc8b84cf9702244a038ec180b2381525177a35f9086346a | What positive integer $n$ satisfies $\log_{10} (225!) - \log_{10} (223!) = 1 + \log_{10} (n!)$? | [
"7"
] | [
"What positive integer $n$ satisfies $\\log_{10} (225!) - \\log_{10} (223!) = 1 + \\log_{10} (n!)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-f40ec7f853ba7ffa7046e755cd97c784ae7f3becc52c4857901ac1b190ad216d | What positive integer can be divided by two adjacent numbers without leaving a remainder or being a decimal? | [
"$kn(n+1)$"
] | [
"What positive integer can be divided by two adjacent numbers without leaving a remainder or being a decimal?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c49a709ac62a02fb45d9723a2ec7790ee575885f70f942b4711aacfa985f2366 | What positive integer not greater than 100, when input into a machine that halves even integers and adds 3 to odd integers, will make the machine run for as long as possible before it repeats a number? | [
"67"
] | [
"What positive integer not greater than 100, when input into a machine that halves even integers and adds 3 to odd integers, will make the machine run for as long as possible before it repeats a number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9a0652e562b628bb2b2f370320c6ae5da262cff8380fe4fd181d4b16385f4605 | What prime is 4 greater than a perfect square and 7 less than the next perfect square? | [
"29"
] | [
"What prime is 4 greater than a perfect square and 7 less than the next perfect square?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c9ee02a00ff8ca667fd483cecbc64a611a457e28cab757de7cd60b45fa54836a | What principle would allow the transformation from the following integral: \[ \int_{-\infty}^{\infty} \int_{x}^{x+a} f(t) \, dt \, dx, \] to: \[ \int_{-\infty}^{\infty} \left( \int_{k}^{k+a} f(k) \, dx \right) dk? \] Would it be necessary to use Fubini's Theorem? | [
"Fubini's Theorem"
] | [
"What principle would allow the transformation from the following integral:\n\n\\[\n\\int_{-\\infty}^{\\infty} \\int_{x}^{x+a} f(t) \\, dt \\, dx,\n\\]\n\nto:\n\n\\[\n\\int_{-\\infty}^{\\infty} \\left( \\int_{k}^{k+a} f(k) \\, dx \\right) dk?\n\\]\n\nWould it be necessary to use Fubini's Theorem?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-ab614a3a8ce61873ae95af97a684a9ef7d4b5a7ddfc538eab3b15d5f55844766 | What properties should the function $f$ have so that $\lim_{x\to 0} xf(x) = 0$? | [
"\\( f(x) = o\\left(\\frac{1}{x}\\right) \\) as \\( x \\to 0 \\)"
] | [
"What properties should the function $f$ have so that $\\lim_{x\\to 0} xf(x) = 0$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-89da03d8828908501abfbe23734c2e14455dce31531af77203beb10b46c56b71 | What proportion of a circle centered at the origin with radius $ r$ is located below the line $ y = r/2$? | [
"\\(\\frac{3\\sqrt{3} + 8\\pi}{12\\pi}\\)"
] | [
"What proportion of a circle centered at the origin with radius $ r$ is located below the line $ y = r/2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-ac6aa8cf9274df4513456a38e741a15df194dfdabeb682d4df662a18d21d77f4 | What proportion of a normal distribution with mean 42 and standard deviation 8 is between 10 and 42? | [
"0.49997"
] | [
"What proportion of a normal distribution with mean 42 and standard deviation 8 is between 10 and 42?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-c357f3c59b2e7b2f90b71a00c9824af6c54c81b8536dbb424bccab5e6fc3694a | What proportion of a normal distribution with mean 42 and standard deviation 8 is between 44 and 58? | [
"0.37854"
] | [
"What proportion of a normal distribution with mean 42 and standard deviation 8 is between 44 and 58?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-1ffbf009fa18e03c9ff7d289c72bb8ed20f941fc2f07eeb258744b64f4246e63 | What proportion of a normal distribution with mean 42 and standard deviation 8 is less than 34? | [
"0.15866"
] | [
"What proportion of a normal distribution with mean 42 and standard deviation 8 is less than 34?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-b86d3a624295f9516b6d1a8ac2ab7b061d9f5a159f80ccf4edec066da7a1d6ee | What proportion of a normal distribution with mean 42 and standard deviation 8 is less than 36 or greater than 48? | [
"0.45326"
] | [
"What proportion of a normal distribution with mean 42 and standard deviation 8 is less than 36 or greater than 48?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-079e2a842e56b691612da9bb3d32f36b1a419bc7d4615c199c87a29a7340405e | What proportion of powers of two begin with a $7$ when expressed in base $10$? | [
"$\\log_{10}\\frac{8}{7}$ or approximately $0.057991947$"
] | [
"What proportion of powers of two begin with a $7$ when expressed in base $10$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-1aced4a37c5b2acb9dccccc4ff8746162dfab9b8cb53f7d6f56f48f70637347e | What quadrilateral is formed by the points $J(-1, 0)$, $K(7,0)$, $L(5,-4)$, and $M(-5,-4)$? | [
"Trapezoid"
] | [
"What quadrilateral is formed by the points $J(-1, 0)$, $K(7,0)$, $L(5,-4)$, and $M(-5,-4)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-60053408c763f75262e894c06e0214e2de73ab904e40e7efc46942b96501dc22 | What radius of a cylindrical can with a volume of $250\pi$ cm^3 requires the least material? | [
"\\( r = 5 \\sqrt[3]{2} \\)"
] | [
"What radius of a cylindrical can with a volume of $250\\pi$ cm^3 requires the least material?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b17da5e1ff4f31a54ca1bf85bea380e3de21675581686653fa900e4942c8e6d1 | What rational number does the repeating decimal 0.9999... equal? | [
"1"
] | [
"What rational number does the repeating decimal 0.9999... equal?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e6d7dc0d6f02ffbfb4b6623381564287475c41705cf66dd6065c76ded92e344d | What real number $ x $ satisfies the infinite equation: \[ {{e}^{x^{{e}^{{x}^{...}}}}} = 2. \] | [
"\\(\\frac{\\ln(2)}{2}\\)"
] | [
"What real number $ x $ satisfies the infinite equation:\n\\[ {{e}^{x^{{e}^{{x}^{...}}}}} = 2. \\]"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5dc0297c5312ad4edfa89b053eaa4f11406f48377cc5bc39d386989fb1d1f1cd | What real number is equal to the expression $2 + \frac{4}{1 + \frac{4}{2 + \frac{4}{1 + \cdots}}}$, where the $1$s and the $2$s alternate? | [
"4"
] | [
"What real number is equal to the expression $2 + \\frac{4}{1 + \\frac{4}{2 + \\frac{4}{1 + \\cdots}}}$, where the $1$s and the $2$s alternate?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0389b35e145cbf83dc0b46a87afd46a1264768a12e56b9ebbcf7bebbca16c05c | What reduced fraction $\frac{p}{q}$ with $8 \le q \le 99$ is closest to $\frac{3}{7}$? | [
"\\(\\frac{41}{96}\\)"
] | [
"What reduced fraction $\\frac{p}{q}$ with $8 \\le q \\le 99$ is closest to $\\frac{3}{7}$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 62
} | 94 | 0 | 0 |
omr-ws-fa66ac3b569bbacbcec4f050c3adf0ec754e01cc0b0929d7b6db0b5bfecc3c71 | What region \( R \) maximizes the value of the integral \( \iint_R (9 - x^2 - y^2) \, dA \)? | [
"\\( x^2 + y^2 \\leq 9 \\)"
] | [
"What region \\( R \\) maximizes the value of the integral \\( \\iint_R (9 - x^2 - y^2) \\, dA \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-22078a10289ffb36b782a7b23a0a1681e5cee701dcdf35a05373d876c533faaa | What relation exists between the legs and the base of an isosceles triangle such that the perimeter in centimeters equals the area in square centimeters? Are there any special limiting cases? | [
"No integer solutions exist for the sides of the isosceles triangle."
] | [
"What relation exists between the legs and the base of an isosceles triangle such that the perimeter in centimeters equals the area in square centimeters? Are there any special limiting cases?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6215be2e6906c87eb56f3e44c54fc41eb38d5251f87b18f21f4c617d690a0c89 | What relation should there be between the three points represented as complex numbers to show that the triangle formed by these points is isosceles? | [
"|z_1 - z_2| = |z_1 - z_3| \\text{ or } |z_1 - z_2| = |z_2 - z_3| \\text{ or } |z_1 - z_3| = |z_2 - z_3|"
] | [
"What relation should there be between the three points represented as complex numbers to show that the triangle formed by these points is isosceles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dca823b9baf346e26a63a83d9ce8f9f42a6e9fff221a538a1f09be7adb2ccc63 | What relationship do the convergents of $\sqrt{d}$ have with the solutions to the equation $x^2 - d y^2 = 1$ or $x^2 - d y^2 = -1$? | [
"\\text{The convergents of } \\sqrt{d} \\text{ provide solutions to the Pell equations } x^2 - dy^2 = 1 \\text{ or } x^2 - dy^2 = -1."
] | [
"What relationship do the convergents of $\\sqrt{d}$ have with the solutions to the equation $x^2 - d y^2 = 1$ or $x^2 - d y^2 = -1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-4935aa2ae7bbcf1e57761b8dc0bb0fca49ec33c6cafec47c9e8b5ebd0e1cd419 | What relationship must connect the numbers $a_1,a_2,a_3$ for the system $$\left\{\begin{array}{lll} (1+a_1)x+y+z=1,\medskip \\ x+(1+a_2)y+z=1,\medskip \\ x+y+(1+a_3)z=1 \end{array}\right.$$ to be solvable and have a unique solution? | [
"D=a_1a_2+a_2a_3+a_1a_3+a_1a_2a_3\\ne 0"
] | [
"What relationship must connect the numbers $a_1,a_2,a_3$ for the system $$\\left\\{\\begin{array}{lll} (1+a_1)x+y+z=1,\\medskip \\\\ x+(1+a_2)y+z=1,\\medskip \\\\ x+y+(1+a_3)z=1 \\end{array}\\right.$$ to be solvable and have a unique solution?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6ce1cbf76947cd49a857a85827a957c20ac4c532de2e28819f8ca3c77f469c4 | What relationship should be between the positive real numbers $a$ and $b$ such that the sequence $\left(\left( a\sqrt[n]{n} + b \right)^{\frac{n}{\ln n}}\right)_{n\ge 1}$ has a nonzero and finite limit? For such $a$ and $b$, calculate the limit of this sequence. | [
"\\(e^a\\)"
] | [
"What relationship should be between the positive real numbers $a$ and $b$ such that the sequence $\\left(\\left( a\\sqrt[n]{n} + b \\right)^{\\frac{n}{\\ln n}}\\right)_{n\\ge 1}$ has a nonzero and finite limit? For such $a$ and $b$, calculate the limit of this sequence."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-2ddaf4227c0921499c965e41b8da45eefe1c7aeaebd1c547c6aa199670a27de3 | What restriction should be placed on \( a_{i,j} \) such that \[ \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} a_{i,j} = \sum_{j=1}^{\infty} \sum_{i=1}^{\infty} a_{i,j} \] given that \(\sum_{i=1}^{\infty} a_{i,j}\) and \(\sum_{j=1}^{\infty} a_{i,j}\) are convergent for any natural number \( j \) and \( i \), respectively? | [
"The terms are nonnegative or the whole double sum is absolutely convergent."
] | [
"What restriction should be placed on \\( a_{i,j} \\) such that \\[ \\sum_{i=1}^{\\infty} \\sum_{j=1}^{\\infty} a_{i,j} = \\sum_{j=1}^{\\infty} \\sum_{i=1}^{\\infty} a_{i,j} \\] given that \\(\\sum_{i=1}^{\\infty} a_{i,j}\\) and \\(\\sum_{j=1}^{\\infty} a_{i,j}\\) are convergent for any natural number \\( j \\) and... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-6987ec751719ac15de75d14ff844817d109613bb81b3fcd947ff03557499ba33 | What right triangle has a perimeter in centimeters equal to its area in square centimeters? | [
"(8, 6, 10) or (5, 12, 13)"
] | [
"What right triangle has a perimeter in centimeters equal to its area in square centimeters?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 17
} | 26 | 0 | 0 |
omr-ws-f5e3ec5c683dd05ef0bf68c052da68d691e773557bc6b46dcb96a7fe4ef2831c | What score does Terri need on the final exam, which counts as 2/3 of the final grade, to earn an average score of 80, given her previous test scores of 86, 80, 84, and 90? | [
"77.5"
] | [
"What score does Terri need on the final exam, which counts as 2/3 of the final grade, to earn an average score of 80, given her previous test scores of 86, 80, 84, and 90?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-3c5201c27867045f23779fd9c9fb4b77c4e179d90ba1b5f92ce88ffffbc0ef50 | What set of numbers, when added together, equals 2013, such that their product is maximized? | [
"741 numbers, each approximately \\( \\frac{2013}{741} \\approx 2.7166 \\)"
] | [
"What set of numbers, when added together, equals 2013, such that their product is maximized?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 2,
"tir": 19
} | 23 | 0 | 0 |
omr-ws-48af36da750ac92f76fa918977f720c6f158fea9c76fc0581106fe91ecc29f0e | What shape should an additional piece of wall of length M be built to enclose the maximal area when using an existing infinite straight wall? | [
"\\(\\text{Semicircle with radius } \\frac{M}{\\pi}\\)"
] | [
"What shape should an additional piece of wall of length M be built to enclose the maximal area when using an existing infinite straight wall?"
] | [
"c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-17e4818ed31241386301384352068d039c59388a7ab1add6bc42c8be8f11e55b | What should be the height of a truncated cone, whose base circles are inscribed in a hemisphere, to minimize its lateral surface area? | [
"\\dfrac{2\\sqrt{2}}{3}R"
] | [
"What should be the height of a truncated cone, whose base circles are inscribed in a hemisphere, to minimize its lateral surface area?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-83300ccb534fe9f9153c4404a5fd8e42cb841c32cde2c02c594379eee7b6acb4 | What should be the ratio of the edges of a rectangular dart board so that the probability of a random hit being closer to the origin than to the edge is maximized? | [
"1",
"2"
] | [
"What should be the ratio of the edges of a rectangular dart board so that the probability of a random hit being closer to the origin than to the edge is maximized?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-b120661f189a4639310d1bf4597bab004b0846d6f56120a76b337d0e91c5629b | What should be the value of the parameter $ \alpha$ in order that the equation $ \sqrt{(x-\frac{1}{2})^2} + \sqrt{(x - \sin \alpha)^2} = \cos 3 \alpha$ has a single solution? Find this solution. | [
"\\( \\alpha = 30^\\circ \\), \\( x = \\frac{1}{2} \\)"
] | [
"What should be the value of the parameter $ \\alpha$ in order that the equation $ \\sqrt{(x-\\frac{1}{2})^2} + \\sqrt{(x - \\sin \\alpha)^2} = \\cos 3 \\alpha$ has a single solution? Find this solution."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 24
} | 24 | 0 | 0 |
omr-ws-854ad1454de6be8ff3139ac1bc4f22de759f73fcc71b56210b71077c5922e740 | What should the fixed amount be for sums greater than 4 when rolling a pair of dice, so that the expected payment is equivalent to 5 bucks per week? | [
"2"
] | [
"What should the fixed amount be for sums greater than 4 when rolling a pair of dice, so that the expected payment is equivalent to 5 bucks per week?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-01aeb6ad55090232d70ca992e273fcef7188ab5ec1d21fe8de70eee699ec7193 | What should the maximum total area of circles \(a\), \(b\), and \(c\) be so that when positioned in a right triangle, they do not overlap and do not cross outside the triangle, but occupy as much area as possible? | [
"(7 - 4\\sqrt{2}) \\pi r^2"
] | [
"What should the maximum total area of circles \\(a\\), \\(b\\), and \\(c\\) be so that when positioned in a right triangle, they do not overlap and do not cross outside the triangle, but occupy as much area as possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-a38dbd5404c768f144ba865b046fad55d31aef473ed168384cfcbc68ad3b0211 | What should the value of \( q \) be so that the convergence of the iteration formula \( X_{n+1} = X_n + q(AX_n - I) \) is as fast as possible? | [
"\\( q_{opt} = -\\frac{2}{\\lambda_{min} + \\lambda_{max}} \\)"
] | [
"What should the value of \\( q \\) be so that the convergence of the iteration formula \\( X_{n+1} = X_n + q(AX_n - I) \\) is as fast as possible?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 12
} | 34 | 0 | 0 |
omr-ws-7ed7f66f1814edf670f7ead4bf1caf7c5a0fc5d010879df2f8d131fb4bc6ef6c | What should the value of \( x \) be to fit the largest possible circle inside a triangle with side lengths 2, 2, and \( 2x \)? | [
"\\( x = \\sqrt{5} - 1 \\)"
] | [
"What should the value of \\( x \\) be to fit the largest possible circle inside a triangle with side lengths 2, 2, and \\( 2x \\)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-6f508117e8a0ce1949059d8d8a50f45d6eb3a23837e1fcb51700a8389988392f | What sizes of squares with integer sides can be completely covered without overlap by identical tiles consisting of three squares with side $1$ joined together in one $L$ shape? | [
"6k"
] | [
"What sizes of squares with integer sides can be completely covered without overlap by identical tiles consisting of three squares with side $1$ joined together in one $L$ shape?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-bd21d84297ff09f2937d011ba38d198b4794b3a13449d42608d4ad7b93057ff8 | What slope field is generated by the differential equation $\dfrac{dy}{dx}=2x-y$? | [
"\\(\\frac{dy}{dx} = 2x - y\\)"
] | [
"What slope field is generated by the differential equation $\\dfrac{dy}{dx}=2x-y$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-b6abe34f791230e450982e2ce5b4012a60c5c8fccbf85a54b0265bb7df4d139e | What square is the product of four consecutive odd integers? | [
"9"
] | [
"What square is the product of four consecutive odd integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-04d164fd862f6a066e4fbef81ffa0d383585b32a2b3e44118702b8f0792c24e0 | What string represents $2016$ in base $-2$? | [
"1100000100000"
] | [
"What string represents $2016$ in base $-2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 15
} | 37 | 0 | 0 |
omr-ws-126613a3c719de633dc802fd4620cde3c36648784e7aba07afc66ec48a604541 | What subgroups \( H \) of a group \( G \) exist such that for all \( x \in G \) and all \( h \in H \), the operation \( xhx^{-1} = h^{-1} \)? What changes if we restrict \( x \) to various subgroups of \( G \)? | [
"H \\leq C_G(K) \\text{ and } H \\text{ is an elementary abelian 2-group}"
] | [
"What subgroups \\( H \\) of a group \\( G \\) exist such that for all \\( x \\in G \\) and all \\( h \\in H \\), the operation \\( xhx^{-1} = h^{-1} \\)? What changes if we restrict \\( x \\) to various subgroups of \\( G \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-6587d95d0fddc4e4995b92815519f2388aa74b02ee439cda04c38b36438712fa | What subsets of the unit circle can be the set of points at which a power series \( f(z) = \sum c_n z^n \) with radius of convergence 1 converges? | [
"\\text{All } F_\\sigma \\text{ subsets}"
] | [
"What subsets of the unit circle can be the set of points at which a power series \\( f(z) = \\sum c_n z^n \\) with radius of convergence 1 converges?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-497c3b3ea81749889b687e3afb62852dc13a4887bb4dc5f37085f746b9da6c36 | What systems of inequalities would describe the situation where you need to use at least 1 dollar worth of stamps to mail a package, with as many 3 cent stamps as you need and only four 32 cent stamps available? | [
"\\( 3x + 32y \\ge 100, \\, x \\ge 0, \\, 0 \\le y \\le 4 \\)"
] | [
"What systems of inequalities would describe the situation where you need to use at least 1 dollar worth of stamps to mail a package, with as many 3 cent stamps as you need and only four 32 cent stamps available?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 18
} | 40 | 0 | 0 |
omr-ws-6364059506da8c2bd7ad180ae3e05ecd5bd3909a66a325db88aaf8e3a925285b | What three numbers should come next in the sequence: 8, 5, 4, 9, 1, 7, 6, …? | [
"3, 2, 0"
] | [
"What three numbers should come next in the sequence: 8, 5, 4, 9, 1, 7, 6, …?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 6,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-5d604d8ecdcb70a6faa361a9c1bf819e31e3c6fe4d0c8a41875aed8be6669052 | What three-digit integer is equal to the sum of the factorials of its digits? | [
"145"
] | [
"What three-digit integer is equal to the sum of the factorials of its digits?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e726d3b2fa1552852601ff36c6f33983555052f05544bad858ada697bfd028c5 | What three-digit number is formed by the 2nd, 3rd, and 4th digits (counting from the left) of the expansion $3^{2020}$? | [
"094"
] | [
"What three-digit number is formed by the 2nd, 3rd, and 4th digits (counting from the left) of the expansion $3^{2020}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-6532d6782d7861e28556f215acd143a02594210e51f32a6d3bc550bfb6277750 | What three-digit number with units digit 2 and hundreds digit 4 is divisible by 9? | [
"432"
] | [
"What three-digit number with units digit 2 and hundreds digit 4 is divisible by 9?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4bdd547413d4d8ee76c2e02f356a0b4016ab88e2058faae9b2cb5fc37aa1019c | What time is 3 hours and 17 minutes before 2:07 AM? | [
"10:50 PM"
] | [
"What time is 3 hours and 17 minutes before 2:07 AM?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-f1dd0f8729e57373ee48749cda7c73f28d252246ab256d4be1ba890a25e43188 | What topological space is obtained when two copies of $\mathbb{R}$ are glued along $\mathbb{R}-\{0\}$? Is the resulting space Hausdorff? | [
"The resulting space is a model of $\\mathbb{R}$ with two origins and is not Hausdorff."
] | [
"What topological space is obtained when two copies of $\\mathbb{R}$ are glued along $\\mathbb{R}-\\{0\\}$? Is the resulting space Hausdorff?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-d7dbce2709f20414e1a4fa4123c374ad5d780069c665ed1b1daa2e2d398aaf40 | What topological space is obtained when two copies of $\mathbb{R}$ are glued along the origin? Is the resulting space Hausdorff? | [
"The resulting space is the union of $y=x^2$ and $y=-x^2$ in $\\mathbb{R}^2$ and it is Hausdorff."
] | [
"What topological space is obtained when two copies of $\\mathbb{R}$ are glued along the origin? Is the resulting space Hausdorff?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 5,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-ded891349055a3ffc03733170acbac19800759a454f8cb2c18ffd088edd82a6d | What transformation does the equation \( y = (ax)^2 \) represent compared to \( y = x^2 \)? | [
"horizontal compression by a factor of \\( \\frac{1}{a} \\)"
] | [
"What transformation does the equation \\( y = (ax)^2 \\) represent compared to \\( y = x^2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 7
} | 16 | 0 | 0 |
omr-ws-021ea7bd7b6f5022629b7a11835d8ebf72b18b06b8db3f0951c6103d6a455bc7 | What translation maps the parabola \( f(x) = x^2 + 3x + 2 \) to the parabola \( g(x) = x^2 \)? | [
"1.5 units to the right, 0.25 units up"
] | [
"What translation maps the parabola \\( f(x) = x^2 + 3x + 2 \\) to the parabola \\( g(x) = x^2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-e9f0a6042f20e624d48f7139ea1736f721714493f91c8bb2620d458930122223 | What two-digit positive integer is one more than a multiple of 2, 3, 4, 5 and 6? | [
"61"
] | [
"What two-digit positive integer is one more than a multiple of 2, 3, 4, 5 and 6?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fb46e7744549fe3471ed063749fb2dd5978a0f60c72f0fc3da6c988573088aff | What two-place number is smaller than the sum of the squares of its digits by 11 and larger than their doubled product by 5? | [
"15 or 95"
] | [
"What two-place number is smaller than the sum of the squares of its digits by 11 and larger than their doubled product by 5?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31fcbcb44819c11fc6bb2794355495cde8a8a92ae329996693ae5625ab339e6b | What two-variable function represents a plane that contains the line given by the equation \( y = mx + b \)? | [
"\\( y - mx + Az = b \\) or equivalently \\( C(mx - y + b) \\) for any constant \\( C \\)."
] | [
"What two-variable function represents a plane that contains the line given by the equation \\( y = mx + b \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 2,
"tir": 1
} | 8 | 0 | 0 |
omr-ws-cd3085f5f2412546a64d2ba3ced651117721c42e60e27619e1b82ae4c18c70fb | What type of contour should be chosen to integrate $\int_0^\infty \frac{\sin x}{x} \, dx$ using the method of contour integration? | [
"A rectangle in the upper half-plane."
] | [
"What type of contour should be chosen to integrate $\\int_0^\\infty \\frac{\\sin x}{x} \\, dx$ using the method of contour integration?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ec01ff511d8b43253d6f0617c83ad5aa8b5b4a286702b38f40815b7fdbe6e7cd | What type of correlation would you expect to find between the number of players on a soccer team and the number of minutes a player spends in the game rather than on the sideline? | [
"Negative correlation."
] | [
"What type of correlation would you expect to find between the number of players on a soccer team and the number of minutes a player spends in the game rather than on the sideline?"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4b18777f2716b67e1795b3efe8d8c91cf13e5718c443640a8cbd1d4c55c85988 | What type of curve is defined by the polar equation $r(\theta) = 1 - 2 \sin^2 (3\theta)$? | [
"12-petal rose."
] | [
"What type of curve is defined by the polar equation $r(\\theta) = 1 - 2 \\sin^2 (3\\theta)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 5
} | 23 | 0 | 0 |
omr-ws-91b4b4d3683d33d03614d2fba4e23d2036511bd87a789473df3a8962e6e308e8 | What type of curve is the graph of the equation \( x^2 + xy + y^2 = 1 \)? Find at least 6 points on this curve and make a sketch. | [
"\\text{Ellipse}"
] | [
"What type of curve is the graph of the equation \\( x^2 + xy + y^2 = 1 \\)? Find at least 6 points on this curve and make a sketch."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-66c75a16d3d84a233a7d423713139265a6564513efc45532b28b25de68cb0811 | What type of differential system is represented by the following equations? \[ \frac{dx}{x} = \frac{dy}{y} = \frac{dz}{z}; \quad x > 0, y > 0, z > 0 \] Note: The system involves differentials \(dx\), \(dy\), and \(dz\) rather than derivatives like \(x'\), \(y'\), or \(x''\). | [
"\\text{Symmetric system of differential equations}"
] | [
"What type of differential system is represented by the following equations?\n\n\\[\n\\frac{dx}{x} = \\frac{dy}{y} = \\frac{dz}{z}; \\quad x > 0, y > 0, z > 0\n\\]\n\nNote: The system involves differentials \\(dx\\), \\(dy\\), and \\(dz\\) rather than derivatives like \\(x'\\), \\(y'\\), or \\(x''\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-ce5ff0bbc915f7d1e7e5230dc1cf0a8e58b8e60477eb4663e3c3f03fbe9fdff9 | What type of numbers get reversed when they are multiplied by 4? (For example, if a number is represented as "abcd", it should become "dcba" when reversed.) | [
"2178"
] | [
"What type of numbers get reversed when they are multiplied by 4? (For example, if a number is represented as \"abcd\", it should become \"dcba\" when reversed.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7d79f91c7e038d2af84f0fda95a098c130698c317b98c96c6c957470968c3411 | What type of transformation has the same effect as a reflection in the line \( y=7 \) followed by a reflection in the line \( y=3 \)? | [
"Translation by \\( (0, -8) \\)"
] | [
"What type of transformation has the same effect as a reflection in the line \\( y=7 \\) followed by a reflection in the line \\( y=3 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-cc49e083a7047c5b04dbe30156bea4851d4babeebfe22ccceb969a50f9b27bde | What type of transformation has the same effect as a reflection in the line \( y=x \) followed by a reflection in the line \( y=2x+5 \)? | [
"\\text{Rotation about } (-5, -5) \\text{ by } \\arctan\\left(\\frac{3}{4}\\right)"
] | [
"What type of transformation has the same effect as a reflection in the line \\( y=x \\) followed by a reflection in the line \\( y=2x+5 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-14bdee279ff12b2284938cd667ae47ffe83348215db4dfb4ae9b189db72682dc | What type of transformation has the same effect as a translation (4,8) followed by a translation (-2,9)? | [
"Translation (2, 17)"
] | [
"What type of transformation has the same effect as a translation (4,8) followed by a translation (-2,9)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 3,
"tir": 22
} | 40 | 0 | 0 |
omr-ws-338715f5034c8f227ba53f622463e33d27304d1768f47328c2b07a36a7eaa117 | What value can have the largest angle of the triangle, if its sides $a, b, c$ satisfy $$(\sqrt{a + b} + \sqrt{a - b})(\sqrt{a + c} + \sqrt{a - c}) = \sqrt{2}(a + b + c)?$$ | [
"90^\\circ"
] | [
"What value can have the largest angle of the triangle, if its sides $a, b, c$ satisfy $$(\\sqrt{a + b} + \\sqrt{a - b})(\\sqrt{a + c} + \\sqrt{a - c}) = \\sqrt{2}(a + b + c)?$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4a8384bc63696ad94af1229c4485537185398e9f1e11525fe1f44a5325da55b1 | What value must $\frac{F_{n+1}}{F_{n}}$ approach as $n$ grows, assuming that it does approach some value? | [
"\\dfrac{1+\\sqrt{5}}{2}"
] | [
"What value must $\\frac{F_{n+1}}{F_{n}}$ approach as $n$ grows, assuming that it does approach some value?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-790e8b64d99632882ff0e3ecf196bec7be0b952d1f61dec0a6d516e1b095d2be | What value of $\theta$ in the first quadrant satisfies the equation \[\cos 5^\circ = \sin 25^\circ + \sin \theta\]? | [
"35 degrees"
] | [
"What value of $\\theta$ in the first quadrant satisfies the equation \\[\\cos 5^\\circ = \\sin 25^\\circ + \\sin \\theta\\]?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-9f419478850ff9854b9239d86dc605398162ef4a6f1cd3758f1959d15af5497d | What value of $\theta$ results in the maximal travel of the projectile through the air, that is the longest distance as measured along the actual arc of the projectile beginning at the mouth of the cannon and terminating where the projectile hits the ground? Assume the cannon sits at ground level. | [
"\\(\\theta \\approx 56.46^\\circ\\)"
] | [
"What value of $\\theta$ results in the maximal travel of the projectile through the air, that is the longest distance as measured along the actual arc of the projectile beginning at the mouth of the cannon and terminating where the projectile hits the ground? Assume the cannon sits at ground level."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-cf663ab41758821c9f922b22e33c80961e973ee3244461e2af91a67b7cd7917b | What value of $k$ will make $x^2 - 16x + k$ the square of a binomial? | [
"64"
] | [
"What value of $k$ will make $x^2 - 16x + k$ the square of a binomial?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cd3988408e1983cd330491ed140d1f712b263b059bee78c3e0eb5dedc706fc69 | What value of $x$ makes the equation below true: $$2x + 4 = |{-17 + 3}|$$ | [
"5"
] | [
"What value of $x$ makes the equation below true: $$2x + 4 = |{-17 + 3}|$$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ecc7688ac16380764a65c17a25a8bd4653ffa5960052edc4a223a57b023bf56d | What value of $x$ will give the maximum value for $-x^2- 6x + 12$? | [
"-3"
] | [
"What value of $x$ will give the maximum value for $-x^2- 6x + 12$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-172bcfdc3614e96337263eaf4a376feb9ed4a52b72ed37cb1ae2c8ce7ab78c58 | What value of $x$ will give the minimum value for $x^2 + 11x - 5$? | [
"-\\frac{11}{2}"
] | [
"What value of $x$ will give the minimum value for $x^2 + 11x - 5$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-348553c397ac3ce71f4b15878b7b2e83ace5d7972e7c6b94fd9daa19f3d19328 | What value of $x$ will give the minimum value for $x^2- 10x + 24$? | [
"5"
] | [
"What value of $x$ will give the minimum value for $x^2- 10x + 24$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4e2af2e0d7b0e0915e6e07d8bd2af432f30260760f39ec40c8cc9ba6a1452852 | What value of $x$ will give the minimum value of $2x^2 - 12x + 3$? | [
"3"
] | [
"What value of $x$ will give the minimum value of $2x^2 - 12x + 3$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bfdb2d511c8a61d2f098daece6ab7232a42aa242ebf80b9963415a4cc17d0c90 | What value of \( k \) gives the most distinct \( k \)-th power residues modulo 29? Specifically, find a \( k \) such that \( n^k \) is unique for \( n = 0, 1, 2, \ldots, 28 \). | [
"\\( k \\) such that \\( \\gcd(k, 28) = 1 \\)"
] | [
"What value of \\( k \\) gives the most distinct \\( k \\)-th power residues modulo 29? Specifically, find a \\( k \\) such that \\( n^k \\) is unique for \\( n = 0, 1, 2, \\ldots, 28 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
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