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-aaf04a6bc41dfb33da3567bfc21edb22b00edf540da4be536ca6de49ad373ae3 | What value of \( p \) maximizes the value of \[ \mathbb{E}\left[\frac{1}{\gamma}\left(1 + r + p(X - r)\right)^{-\gamma}\right] \] where \( X \sim \text{log-normal}(\mu, \sigma^2) \), \( 0 < \gamma < 1 \), and \( r \) is a constant? | [
"0"
] | [
"What value of \\( p \\) maximizes the value of \n\\[\n\\mathbb{E}\\left[\\frac{1}{\\gamma}\\left(1 + r + p(X - r)\\right)^{-\\gamma}\\right]\n\\]\nwhere \\( X \\sim \\text{log-normal}(\\mu, \\sigma^2) \\), \\( 0 < \\gamma < 1 \\), and \\( r \\) is a constant?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-2dc362024d09345bccdbb80fbd0674173db25b746fea3dc5897e70b9a6ea3e9d | What value of \( x \) satisfies the equation? \[ \sqrt{x + \log_{\sqrt{2}}\sqrt[4]{\sqrt{2}\sqrt{\sqrt{2}}}} = \sqrt{\log_{\sqrt{\sqrt{2}}}\sqrt[4]{\sqrt{2\sqrt{2}}}} \] | [
"\\( x = \\frac{3}{8} \\)"
] | [
"What value of \\( x \\) satisfies the equation?\n\\[\n\\sqrt{x + \\log_{\\sqrt{2}}\\sqrt[4]{\\sqrt{2}\\sqrt{\\sqrt{2}}}} = \\sqrt{\\log_{\\sqrt{\\sqrt{2}}}\\sqrt[4]{\\sqrt{2\\sqrt{2}}}}\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 29
} | 42 | 0 | 0 |
omr-ws-601782152d1250b9070e47f078b460f9e8db960bb0d71c1227ad7da2114c710f | What value of the central angle $\alpha$ will yield the greatest possible volume of the cone formed by cutting out a sector from a circle and joining the edges? | [
"\\(\\frac{2\\pi (3-\\sqrt{6})}{3}\\)"
] | [
"What value of the central angle $\\alpha$ will yield the greatest possible volume of the cone formed by cutting out a sector from a circle and joining the edges?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-abe64b9eefbc85ff102ee2de2d24547b636ab84fca6d19d6bcc64ff40412c330 | What values can be made using combinations of postage stamps with denominations of $5p$ and $7p$? | [
"All integers greater than 23."
] | [
"What values can be made using combinations of postage stamps with denominations of $5p$ and $7p$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 23
} | 42 | 0 | 0 |
omr-ws-c4ee631d89765c3e98dd4d55ddd5b29abd8c0850884d5436858a79b650ef7284 | What values can the ratio \( v = \frac{a}{b} \) take for a tetrahedron where two edges have length \( a \) and the other four edges have length \( b \), with \( a \) and \( b \) being any positive real numbers? | [
"\\( 0 < v < 2 \\)"
] | [
"What values can the ratio \\( v = \\frac{a}{b} \\) take for a tetrahedron where two edges have length \\( a \\) and the other four edges have length \\( b \\), with \\( a \\) and \\( b \\) being any positive real numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-ca4f151568fe80ce7b8c8b8b4422af1f0430abc4a8839fa15628c6ce89cbd70d | What values of \( x \) satisfy the inequality \(\log_{x^2 - 3}27 \ge 3\)? | [
"\\( (-\\sqrt{6}, -2) \\cup (-2, -\\sqrt{3}) \\cup (\\sqrt{3}, 2) \\cup (2, \\sqrt{6}) \\)"
] | [
"What values of \\( x \\) satisfy the inequality \\(\\log_{x^2 - 3}27 \\ge 3\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7345d35e335564deb761ba593f8d2a94f18e00196e8166f48e0d52fd7c0b7c7f | What was Ticked-Off Todd's average speed in miles per hour if he walked continuously from 10:12 AM to approximately 7:32 PM at a steady pace of 740 meters every 16 minutes? | [
"1.7243 mph"
] | [
"What was Ticked-Off Todd's average speed in miles per hour if he walked continuously from 10:12 AM to approximately 7:32 PM at a steady pace of 740 meters every 16 minutes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-01066e0f07f4a86e25f48214d48c767d774590d20227387a2bb0600c738da10d | What was the last palindromic date in abbreviated form (MM-DD-YY) prior to October 1, 2001? | [
"09-09-99"
] | [
"What was the last palindromic date in abbreviated form (MM-DD-YY) prior to October 1, 2001?"
] | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-36c6a8ad8d0bb78584a172fbed72fdfd6bb593ec641f19e8bbec966c73c2fa7c | What was the last palindromic date in complete form (MM-DD-YYYY) prior to October 1, 2001? | [
"09-09-2001"
] | [
"What was the last palindromic date in complete form (MM-DD-YYYY) prior to October 1, 2001?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-46d14755291f83802b0f76167dcfe9a77034df5bc273fbb8f462376bc7a144c1 | What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class, given the following data? - 1 day: 1 student - 2 days: 2 students - 3 days: 1.5 students - 4 days: 3.5 students - 5 days: 4.5 students - 6 days: 2 students - 7 days: 1.5 students | [
"4.36"
] | [
"What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class, given the following data?\n\n- 1 day: 1 student\n- 2 days: 2 students\n- 3 days: 1.5 students\n- 4 days: 3.5 students\n- 5 days: 4.5 students\n- 6 days: 2 students\n- 7 days: 1.... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-61254afa83f40a360dd4883d633d6669f48a7b9e4fa8665ed72e8d7e0834974a | What was the original cost of an item that has a book value of $6344 after 7 years if the reducing balance depreciation rate is 18% per year? | [
"25451.64"
] | [
"What was the original cost of an item that has a book value of $6344 after 7 years if the reducing balance depreciation rate is 18% per year?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af7c13fa4dc172875f8ec848a6d13c77a57909521d299e60e1ea441edc5288f5 | What was the reading on a gas meter with four dials, given that the reading is the largest possible such that all four pointers on the meter were exactly parallel? | [
"9696"
] | [
"What was the reading on a gas meter with four dials, given that the reading is the largest possible such that all four pointers on the meter were exactly parallel?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 3
} | 23 | 0 | 0 |
omr-ws-b0f500ba29015055662b82a9136bcfc9f37931829d2df74374e2fcf652264629 | What weights bracket the middle 50% of the population, given that the weight of a medium apple follows a normal distribution with a mean of 200 grams and a standard deviation of 12 grams? | [
"(191.91, 208.09)"
] | [
"What weights bracket the middle 50% of the population, given that the weight of a medium apple follows a normal distribution with a mean of 200 grams and a standard deviation of 12 grams?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-46efd6fac7a5e20ce8681543ff1feb06ad1afce4814c9b75131134e80a913618 | What will be the remainder when \(23!\) is divided by 47? | [
"46"
] | [
"What will be the remainder when \\(23!\\) is divided by 47?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-40a02558246f2f215e59b78eb42a684c4436a28530941b2f5d5a0fd9c11743b4 | What will be the time 8640002017 seconds after 5 o'clock in the afternoon, using a 12-hour clock time convention? | [
"5:33:37 PM"
] | [
"What will be the time 8640002017 seconds after 5 o'clock in the afternoon, using a 12-hour clock time convention?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-3c7524017ec919b8e7211282f06d1f9d9c9f3fd636821bbbbbd9203a70cfe755 | What would be the amount of compound interest on $6,000 invested for two years at 9%, compounded semiannually? Round your answer to the nearest dollar. | [
"1155"
] | [
"What would be the amount of compound interest on $6,000 invested for two years at 9%, compounded semiannually? Round your answer to the nearest dollar."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5d0b0e7b2de84b414b7b22ada8e5d77df24bd9412112c99ba76cf0e1e74db6d3 | What would happen if the set wasn't compact for the series $\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}\sin\left(1+\frac{x}{n}\right)$? | [
"The series does not converge uniformly on $\\mathbb{R}$."
] | [
"What would happen if the set wasn't compact for the series $\\sum_{n=1}^\\infty \\frac{(-1)^n}{\\sqrt{n}}\\sin\\left(1+\\frac{x}{n}\\right)$?"
] | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-1c4c721514292d4cbc4e232541c0558a0ed7677e986711958bfaff5f06ce7a34 | What would the Bernoulli distribution Ber (q) be reduced to if $q = 0$ or $q = 1$? | [
"When \\( q = 0 \\), Ber(q) reduces to \\( P(X = 0) = 1 \\). When \\( q = 1 \\), Ber(q) reduces to \\( P(X = 1) = 1 \\)."
] | [
"What would the Bernoulli distribution Ber (q) be reduced to if $q = 0$ or $q = 1$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-bc7f9091b2c4639425bd36a6b543e986d3d938ed972eeb7d0ae3e4f3698ed76c | What would the bearing from vector A (67 degrees west of south) to vector B (1 degree west of north) be? | [
"33"
] | [
"What would the bearing from vector A (67 degrees west of south) to vector B (1 degree west of north) be?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-d1c0afd2158b303fd1d0622baa60d05aa616440fc7704a24b71feb867eb52bea | What would the graph of $y=\{x\}$ look like, where $\{x\}$ denotes the fractional part of $x$? | [
"The graph of \\( y = \\{x\\} \\) consists of line segments that repeat every integer interval, starting from \\( y = 0 \\) and increasing linearly to \\( y = 1 \\) before resetting at the next integer."
] | [
"What would the graph of $y=\\{x\\}$ look like, where $\\{x\\}$ denotes the fractional part of $x$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 5
} | 25 | 0 | 0 |
omr-ws-3911fa4316c4b3cdd84611779d2b5d1f3d9074b5a87cf5b7536e19c2fa0e4bb1 | What would the graph of the equation \( y = 1 + \sqrt{-x^2 + 6x - 8} \) look like? | [
"The graph is the upper half of a circle centered at \\((3,1)\\) with radius \\(1\\)."
] | [
"What would the graph of the equation \\( y = 1 + \\sqrt{-x^2 + 6x - 8} \\) look like?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-1e63cae12bb128ecc71377a2d9d18db81535b973da94cd705919fe99cb0fcade | What would the phase curve look like for the system of differential equations given by: \[ x' = x \] \[ y' = 2x + 3y \] | [
"\\( y = -x + Cx^3 \\)"
] | [
"What would the phase curve look like for the system of differential equations given by:\n\\[ x' = x \\]\n\\[ y' = 2x + 3y \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-6605a7acf13f559272d92bf94b460045537f06b00b4d2da3873006220a978e2d | What's the coefficient of the $m^4n^4$ term in the expansion of $(m+n)^8$? | [
"70"
] | [
"What's the coefficient of the $m^4n^4$ term in the expansion of $(m+n)^8$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-74f50b5ae447226e3ae7e33bcf9d9afe0b2ca80250d5db32b96601e7c63f428b | What's the coefficient of the $x^2y^4$ term in the expansion of $(x+y)^6$? | [
"15"
] | [
"What's the coefficient of the $x^2y^4$ term in the expansion of $(x+y)^6$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b3650a1afe586c58cfe4b6fb5167b64916edc7db2be74d2485713d91680154fe | What's the expected number of consecutively numbered balls being pulled out consecutively? In other words, how many times should we expect to hear Ball $n$ then Ball $n \pm 1$ if we pull all 75 balls? | [
"\\dfrac{148}{75}"
] | [
"What's the expected number of consecutively numbered balls being pulled out consecutively? In other words, how many times should we expect to hear Ball $n$ then Ball $n \\pm 1$ if we pull all 75 balls?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c4154d3194e84b73abcb43e7c1704cfa1cdaf9fe93962362e473960111975d0e | What’s the maximum number of Friday the 13th’s that can occur in a year? | [
"3"
] | [
"What’s the maximum number of Friday the 13th’s that can occur in a year?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b5257153fe5f4c1ef76d78f276e142e475a73cf06d0bcb171377aef659fce373 | What’s the maximum number of circles of radius $4$ that fit into a $24 \times 15$ rectangle without overlap? | [
"5"
] | [
"What’s the maximum number of circles of radius $4$ that fit into a $24 \\times 15$ rectangle without overlap?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b5afed05442926728983c5de544f5c7e04ec3a07496503d9c6fcdbc2cc09b18e | When $ (1+ x)^{38} $ is expanded in ascending powers of $ x $, $ N_{1} $ of the coefficients leave a remainder of $ 1 $ when divided by $ 3 $, while $ N_{2} $ of the coefficients leave a remainder of $ 2 $ when divided by $ 3 $. Find $ N_{1}-N_{2} $. | [
"4"
] | [
"When $ (1+ x)^{38} $ is expanded in ascending powers of $ x $, $ N_{1} $ of the coefficients leave a remainder of $ 1 $ when divided by $ 3 $, while $ N_{2} $ of the coefficients leave a remainder of $ 2 $ when divided by $ 3 $. Find $ N_{1}-N_{2} $."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e022fc442f1989e792b2cc51f41c3507da8dccabedc478b0b258af2d4e94af14 | When $(1+x)^{38}$ is expanded in ascending powers of $x$, let $N_1$ be the number of coefficients that leave a remainder of $1$ when divided by $3$, and let $N_2$ be the number of coefficients that leave a remainder of $2$ when divided by $3$. Find $N_1 - N_2$. | [
"4"
] | [
"When $(1+x)^{38}$ is expanded in ascending powers of $x$, let $N_1$ be the number of coefficients that leave a remainder of $1$ when divided by $3$, and let $N_2$ be the number of coefficients that leave a remainder of $2$ when divided by $3$. Find $N_1 - N_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-44d1c2b6f4e2ff7c3d160181f2e493eafaa10653e5100cb61e8fdb513238b029 | When $(2x^{2}-3x+1)(x+1)$ is divided by $x^{2}+4$, find the quotient and residue. | [
"\\(2x - 1\\) and \\(5 - 9x\\)"
] | [
"When $(2x^{2}-3x+1)(x+1)$ is divided by $x^{2}+4$, find the quotient and residue."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ebc44442b9de4101879ae9a659c93158b87d995f56f7ba839791f32cad8c2a3 | When $(3x+4y)^{75} \cdot (15x-y)^{75}$ is expanded, the term with the largest coefficient can be expressed as $a \cdot x^{n} \cdot y^{p}$. Find the remainder when $a+n+p$ is divided by $100$. | [
"50"
] | [
"When $(3x+4y)^{75} \\cdot (15x-y)^{75}$ is expanded, the term with the largest coefficient can be expressed as $a \\cdot x^{n} \\cdot y^{p}$. Find the remainder when $a+n+p$ is divided by $100$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-1a8efb588b590ac5ae5940986ad5006df0c7171b58668af2368a6cd7d1464b0c | When $(a + b)(b + c)(c + a) \ne 0$, consider the set of equations: \[ \frac{a^2}{a + b} = \frac{a^2}{a + c} + 31, \] \[ \frac{b^2}{b + c} = \frac{b^2}{a + b} + x, \] \[ \frac{c^2}{a + c} = \frac{c^2}{b + c} + y. \] If $|x|$ and $|y|$ are both perfect squares, what is the value of $x \cdot y$? | [
"-57600"
] | [
"When $(a + b)(b + c)(c + a) \\ne 0$, consider the set of equations:\n\\[\n\\frac{a^2}{a + b} = \\frac{a^2}{a + c} + 31,\n\\]\n\\[\n\\frac{b^2}{b + c} = \\frac{b^2}{a + b} + x,\n\\]\n\\[\n\\frac{c^2}{a + c} = \\frac{c^2}{b + c} + y.\n\\]\nIf $|x|$ and $|y|$ are both perfect squares, what is the value of $x \\cdot y... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-b719a6cb3b902f5d67cffecf68964c14e21c888ec0a5c960dd877f92b3f20483 | When $(a-b+c-d)^{24}$ is expanded and like terms are combined, the result contains exactly $p$ terms with positive coefficients. What is the sum of the digits of $p$? | [
"14"
] | [
"When $(a-b+c-d)^{24}$ is expanded and like terms are combined, the result contains exactly $p$ terms with positive coefficients. What is the sum of the digits of $p$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2b6718f0d901d008788b38e276e4230447e34c849f4d714d499ba64049d0180e | When $(x+1)^{m}$ is multiplied out, the number of coefficients that are not divisible by $2$, $3$, and $5$ are $64$, $16$, and $8$ respectively. Find the value of $m$. | [
"\\(63\\)"
] | [
"When $(x+1)^{m}$ is multiplied out, the number of coefficients that are not divisible by $2$, $3$, and $5$ are $64$, $16$, and $8$ respectively. Find the value of $m$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72355865525ccdb5b2ec74a0820e0f506b185da412d9dabd33f803c64084190a | When $(x+1)^{m}$ is multiplied out, we get 64 odd coefficients, 16 coefficients not divisible by 3, and 8 coefficients not divisible by 5. Find the value of $m$. | [
"1000"
] | [
"When $(x+1)^{m}$ is multiplied out, we get 64 odd coefficients, 16 coefficients not divisible by 3, and 8 coefficients not divisible by 5. Find the value of $m$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-2b817a750d5ae2da7c3cf10daff90342f188f8625556ad91f30833cb942d9a95 | When $(x+2y-z)^8$ is expanded, what is the coefficient of $x^3y^2z^3$? | [
"-2240"
] | [
"When $(x+2y-z)^8$ is expanded, what is the coefficient of $x^3y^2z^3$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-abe062ec276e14542fc31c121fe9e0b492fd9d34c6d6028139611b765390b4f4 | When $(x^3 + x + 1)^{24}$ is expanded and like terms are combined, what is the sum of the exponents of the terms with an odd coefficient? | [
"240"
] | [
"When $(x^3 + x + 1)^{24}$ is expanded and like terms are combined, what is the sum of the exponents of the terms with an odd coefficient?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-5577032d5ee89802e13c1267883a1cdd6664c06474acb2f44f843b9468753485 | When $0.42\overline{153}$ is expressed as a fraction in the form $\frac{x}{99900}$, what is the value of $x$? | [
"42111"
] | [
"When $0.42\\overline{153}$ is expressed as a fraction in the form $\\frac{x}{99900}$, what is the value of $x$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fde8dd1228289941f9288bed02487bc4b76023673926e8a8ffa573242f4d29a9 | When $0.\overline{36}$ is expressed as a common fraction in lowest terms, what is the sum of the numerator and denominator? | [
"15"
] | [
"When $0.\\overline{36}$ is expressed as a common fraction in lowest terms, what is the sum of the numerator and denominator?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e9c9e556c59c93a93c930b82cb4c024489a89b874287adf82486c17f0729e5a | When $1 + 7 + 7^2 + \cdots + 7^{2004}$ is divided by $1000$, a remainder of $N$ is obtained. Determine the value of $N$. | [
"801"
] | [
"When $1 + 7 + 7^2 + \\cdots + 7^{2004}$ is divided by $1000$, a remainder of $N$ is obtained. Determine the value of $N$.\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-51a669c3b5c9816ba24a45ee7e59f11681089e83068993eda05705cf3205a692 | When $1 - i \sqrt{3}$ is converted to the exponential form $re^{i \theta}$, what is $\theta$? | [
"\\frac{5\\pi}{3}"
] | [
"When $1 - i \\sqrt{3}$ is converted to the exponential form $re^{i \\theta}$, what is $\\theta$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-700d8ee58429e11c8043ec7b304abaa1a1fb91b31bfe8a0be6f00c312c16aaa5 | When $1000^{100}$ is expanded out, the result is $1$ followed by how many zeros? | [
"300"
] | [
"When $1000^{100}$ is expanded out, the result is $1$ followed by how many zeros?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-40e2c3a97937bd2020193caf8a8bccd0efe0718d5e0b549c4d6395c8a2f37c24 | When $10^{93} - 93$ is expressed as a single whole number, what is the sum of the digits? | [
"826"
] | [
"When $10^{93} - 93$ is expressed as a single whole number, what is the sum of the digits?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-c8a0ef5c5d8b61d123cc89d7af585c386e501d8818ad30ca721cd25bb91a8c8d | When $11^4$ is written out in base 10, the sum of its digits is $16=2^4$. What is the largest base $b$ such that the base-$b$ digits of $11^4$ do not add up to $2^4$? (Note: here, $11^4$ in base $b$ means that the base-$b$ number $11$ is raised to the fourth power.) | [
"6"
] | [
"When $11^4$ is written out in base 10, the sum of its digits is $16=2^4$. What is the largest base $b$ such that the base-$b$ digits of $11^4$ do not add up to $2^4$? (Note: here, $11^4$ in base $b$ means that the base-$b$ number $11$ is raised to the fourth power.)"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5ff7fe1d9498fa3012cde3ce7cc7f2e374e7f3dec8f65f87081d9aac8bea0ad0 | When $1216$ is multiplied by $k$, where $k$ is a positive integer, the resulting product has exactly $61$ divisors, not including one and the product itself. Determine the smallest value of $k$ for which this is true. | [
"171"
] | [
"When $1216$ is multiplied by $k$, where $k$ is a positive integer, the resulting product has exactly $61$ divisors, not including one and the product itself. Determine the smallest value of $k$ for which this is true."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 23
} | 46 | 0 | 0 |
omr-ws-c11d0ac0b1768d36b286c62f11f121b21743fba7d7b11dbe0177d7bb5068746d | When $200^6$ is converted to base 4, how many zeroes does it have at the end? | [
"9"
] | [
"When $200^6$ is converted to base 4, how many zeroes does it have at the end?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-297d3d19949953f5a792beb446ae1fbafa76c0d2da6c93b4d537f9ad64d2cd41 | When $2025^{2026} - 2025$ is divided by $2025^2 + 2026$, what is the remainder? | [
"0"
] | [
"When $2025^{2026} - 2025$ is divided by $2025^2 + 2026$, what is the remainder?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-273eee389262fc983d576aca6cd2ca4645a18339e6739817c14b3067e00f940e | When $2n-1$ consecutive integers are added, the result is the square of the $n$th integer. If the smallest of these integers is $2015$, what is the largest integer? | [
"6043"
] | [
"When $2n-1$ consecutive integers are added, the result is the square of the $n$th integer. If the smallest of these integers is $2015$, what is the largest integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-df8ec80ef412347acb2e38ec076e821ee59fc87c79754a2b01cd4799253da110 | When $3z^3-4z^2-14z+3$ is divided by $3z+5$, the quotient is $z^2-3z+\frac{1}{3}$. What is the remainder? | [
"\\frac{4}{3}"
] | [
"When $3z^3-4z^2-14z+3$ is divided by $3z+5$, the quotient is $z^2-3z+\\frac{1}{3}$. What is the remainder?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3a704d8c4f8e14d38f4ff90a5b8e4154497b816c93b7fc34ad3fb0253316ba01 | When $4444^{4444}$ is written in decimal notation, the sum of its digits is $A$. Let $B$ be the sum of the digits of $A$. Find the sum of the digits of $B$. | [
"7"
] | [
"When $4444^{4444}$ is written in decimal notation, the sum of its digits is $A$. Let $B$ be the sum of the digits of $A$. Find the sum of the digits of $B$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ac775183ae4d992926fbf67711b55c6d6502432abba204ff8484251777f946f1 | When $555_{10}$ is expressed in this base, it has 4 digits, in the form ABAB, where A and B are two different digits. What base is it? | [
"6"
] | [
"When $555_{10}$ is expressed in this base, it has 4 digits, in the form ABAB, where A and B are two different digits. What base is it?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61154381340316619923f0ba375a9a42607bd1ee8cc19512fb2491b8fce19f4c | When $907$ is divided by two $2$-digit numbers $N$ and $N+2$, the remainder obtained in both cases is $x$. Find the value of $x$ for the maximum possible value of $N$. | [
"8"
] | [
"When $907$ is divided by two $2$-digit numbers $N$ and $N+2$, the remainder obtained in both cases is $x$. Find the value of $x$ for the maximum possible value of $N$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 3
} | 25 | 0 | 0 |
omr-ws-11d7cd2ad80f38fca332525402ff420e0671691d4d3080600318095ded581f84 | When $N$ standard 6-sided dice are rolled, the probability of obtaining a sum of 2019 is greater than zero and is the same as the probability of obtaining a sum of $S$. What is the smallest possible value of $S$? | [
"340"
] | [
"When $N$ standard 6-sided dice are rolled, the probability of obtaining a sum of 2019 is greater than zero and is the same as the probability of obtaining a sum of $S$. What is the smallest possible value of $S$?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 32
} | 63 | 0 | 0 |
omr-ws-d7b6e5dd798baff7a839bc395032ada2a50610bf22e3f789eae758f37f349e08 | When $P(x)$ is divided by $x(x-1)$, the remainder is $x+1$. When $P(x)$ is divided by $x^2-1$, the remainder is $3x-1$. Find all possible polynomials $P(x)$. | [
"\\( P(x) = (x^3 - x) \\cdot Q(x) - 2x^2 + 3x + 1 \\) for any polynomial \\( Q(x) \\)."
] | [
"When $P(x)$ is divided by $x(x-1)$, the remainder is $x+1$. When $P(x)$ is divided by $x^2-1$, the remainder is $3x-1$. Find all possible polynomials $P(x)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 4
} | 22 | 0 | 0 |
omr-ws-f57f7ea512468d8661caefe1a24deec30da6970c014203b1f8bd6e0be8393d24 | When $P(x)$ is divided by $x-1$, the remainder is $1$, and when divided by $(x-2)(x-3)$, the remainder is $5$. Find the remainder when $P(x)$ is divided by $(x-1)(x-2)(x-3)$. | [
"-2x^2 + 10x - 7"
] | [
"When $P(x)$ is divided by $x-1$, the remainder is $1$, and when divided by $(x-2)(x-3)$, the remainder is $5$. Find the remainder when $P(x)$ is divided by $(x-1)(x-2)(x-3)$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6bc0c6c0cca3ac9187ff82d36c3e9ca66d2be9f03453957aef288f50b3ad4519 | When $P(x)$ is divided by $x-2012$, the remainder is $2102$. If the remainder when $P(x)$ is divided by $(x-2012)^2$ is $r(x)$, what is $r(2777)$? | [
"2102"
] | [
"When $P(x)$ is divided by $x-2012$, the remainder is $2102$. If the remainder when $P(x)$ is divided by $(x-2012)^2$ is $r(x)$, what is $r(2777)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fe86e467ee1ce6de2097f1b3d0748d9e284def038299a4bbcd2641032a2369b3 | When $X_\alpha$ is non-connected, how many connected components does $X_\alpha$ have, where $X_\alpha = D(-v,3) \cup D(v,3) \cup \{(x, \alpha x) : x \in \mathbb{R}\}$, with $v = (5,0)$? | [
"3"
] | [
"When $X_\\alpha$ is non-connected, how many connected components does $X_\\alpha$ have, where $X_\\alpha = D(-v,3) \\cup D(v,3) \\cup \\{(x, \\alpha x) : x \\in \\mathbb{R}\\}$, with $v = (5,0)$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6af0ab96bdfea2a94304941700a1883cd4eb98df54a7e9395a6f7a6ce5dccc40 | When $\frac{1}{1111}$ is expressed as a decimal, what is the sum of the first 40 digits after the decimal point? | [
"90"
] | [
"When $\\frac{1}{1111}$ is expressed as a decimal, what is the sum of the first 40 digits after the decimal point?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-526fcbad7849b9ec466e2b107f72e637f8ac5d41fe1d11550e5bbfdf2f23d1dd | When $\frac{3}{1250}$ is written as a decimal, how many zeros are there between the decimal point and the first non-zero digit? | [
"2"
] | [
"When $\\frac{3}{1250}$ is written as a decimal, how many zeros are there between the decimal point and the first non-zero digit?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-efc3d9d43cd7fa73f714c2d62ea61d8038527fe3d12052c4cdfaba90920e29b6 | When $\left(x^2 + x + y^2 + y\right)^{10}$ is expanded and like terms are combined, what is the number of distinct terms in the resulting expression? | [
"166"
] | [
"When $\\left(x^2 + x + y^2 + y\\right)^{10}$ is expanded and like terms are combined, what is the number of distinct terms in the resulting expression?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 4,
"tir": 13
} | 21 | 0 | 0 |
omr-ws-3f1bfc5c7fb1d70d13a8ec0f03a9115333b7f2ef52f9a02d1f50a2f1cb76c930 | When $\log_{25}{125} = x$ is expressed in the form of $a^b = c$, what is the value of $a+bc$, in terms of $x$? | [
"\\( 25 + 125x \\)"
] | [
"When $\\log_{25}{125} = x$ is expressed in the form of $a^b = c$, what is the value of $a+bc$, in terms of $x$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-890381141cea3cf6202f3bf875715097a1b9dc9ae62f3569bf7a0fa724f829e6 | When $\sqrt[3]{2700}$ is simplified, the result is $a\sqrt[3]{b}$, where $a$ and $b$ are positive integers and $b$ is as small as possible. What is $a+b$? | [
"103"
] | [
"When $\\sqrt[3]{2700}$ is simplified, the result is $a\\sqrt[3]{b}$, where $a$ and $b$ are positive integers and $b$ is as small as possible. What is $a+b$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b42ecd491cb9673924c135cca5941117c017604f6308cb0d38dff757f0aeabf3 | When $\sqrt[4]{2^7\cdot3^3}$ is fully simplified, the result is $a\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$? | [
"218"
] | [
"When $\\sqrt[4]{2^7\\cdot3^3}$ is fully simplified, the result is $a\\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e1b93b01ebe15c865e25ed77e42933b5f8da9afe617218ee1e0ad528da1e0508 | When $f(x) = ax^3 - 6x^2 + bx - 5$ is divided by $x - 1,$ the remainder is $-5.$ When $f(x)$ is divided by $x + 2,$ the remainder is $-53.$ Find the ordered pair $(a,b).$ | [
"(2,4)"
] | [
"When $f(x) = ax^3 - 6x^2 + bx - 5$ is divided by $x - 1,$ the remainder is $-5.$ When $f(x)$ is divided by $x + 2,$ the remainder is $-53.$ Find the ordered pair $(a,b).$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-85ea6426153b5989e58c851cf9dee8b142fe1db8ee139c28eac0465d21090650 | When $f(x)$ is divided by $x+1$, the remainder is $6$. When $f(x)$ is divided by $2x-1$, the remainder is $3$. Find the remainder when $f(x)$ is divided by $2x^{2}+x+1$. | [
"-2x + 4"
] | [
"When $f(x)$ is divided by $x+1$, the remainder is $6$. When $f(x)$ is divided by $2x-1$, the remainder is $3$. Find the remainder when $f(x)$ is divided by $2x^{2}+x+1$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3438415c4c7e12920c2d6db0d29eb289108275fb8af80ce6d0e5a3a922948af1 | When $k$ is an even positive integer, find $$\lim_{k\to\infty} \frac{\mathbb{E}[\tau]}{f(k)}$$ where $\tau$ is the first time every vertex in the 2-3 tree $T_k$ has been visited by a simple random walk starting from the root $o$. | [
"2"
] | [
"When $k$ is an even positive integer, find \n\n$$\\lim_{k\\to\\infty} \\frac{\\mathbb{E}[\\tau]}{f(k)}$$\n\nwhere $\\tau$ is the first time every vertex in the 2-3 tree $T_k$ has been visited by a simple random walk starting from the root $o$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-a2bbaba2105283189c4df25a16b01b10e0a289e166c510bf297f207e4a8bb80b | When $n \geq 3$, calculate the sum of the capacities of all odd subsets of $S_{n}$. | [
"\\( n(n+1) \\cdot 2^{n-3} \\)"
] | [
"When $n \\geq 3$, calculate the sum of the capacities of all odd subsets of $S_{n}$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-c461464b8c2ba11b5e81281740abf18264ff08336d94664684351d6cd79bea8f | When $n$ dice are thrown at one time, find the probability that the sum of the dice is $n+3$. | [
"\\(\\frac{\\binom{n+2}{n}}{6^n}\\)"
] | [
"When $n$ dice are thrown at one time, find the probability that the sum of the dice is $n+3$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 3
} | 17 | 0 | 0 |
omr-ws-d37936a789ae26a305ade45f850031bfd2f28c3e00005cdbcc6ebeb6b8b5c52b | When $n$ dice are thrown at one time, find the probability that the sum of the numbers on the dice is $n+2.$ | [
"\\(\\frac{n^2 + n}{2 \\cdot 6^n}\\)"
] | [
"When $n$ dice are thrown at one time, find the probability that the sum of the numbers on the dice is $n+2.$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-755f0d8aba52ce036d9ccee9256dac8a9ba548d4e7e9b9fc5fce013211527b5b | When $n$ different dice are thrown, which sums of the numbers on the dice are the most probable? | [
"\\( \\frac{7n}{2} \\) if \\( n \\) is even, and \\( \\left\\lfloor \\frac{7n}{2} \\right\\rfloor \\) and \\( \\left\\lfloor \\frac{7n}{2} \\right\\rfloor + 1 \\) if \\( n \\) is odd."
] | [
"When $n$ different dice are thrown, which sums of the numbers on the dice are the most probable?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-3765b1f5f630c12b69bddedf9d662524c68827215e1c91ef0af4f3274fd60654 | When $n$ is divided by $d$, it leaves a remainder of $9$; when the square of $n$ is divided by $d$, it leaves a remainder of $1$. Find the number of possible values of $d$. | [
"5"
] | [
"When $n$ is divided by $d$, it leaves a remainder of $9$; when the square of $n$ is divided by $d$, it leaves a remainder of $1$. Find the number of possible values of $d$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-2a9c52e0e51371924257417fd48de9c2d053e38b25960af37c2fea48e71e3380 | When $n$ is divided by 3, the remainder is 2. What is the remainder when $5n$ is divided by 3? | [
"1"
] | [
"When $n$ is divided by 3, the remainder is 2. What is the remainder when $5n$ is divided by 3?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d4ebaf27d1a2b2ddcee1860797b8fe23259047a43f04bedd8b083016ed6fc16a | When $n$ is divided by 6, a remainder of 1 is given. What is the remainder when $n+2010$ is divided by 6? | [
"1"
] | [
"When $n$ is divided by 6, a remainder of 1 is given. What is the remainder when $n+2010$ is divided by 6?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9b468278b198b50f18645b4d04384161aeec95e4b0078e7743e74a1ee8c55ea | When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. The smallest possible value of $S$ is? | [
"337"
] | [
"When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. The smallest possible value of $S$ is?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-746232fec84bb4f4820e4ec4515472281e4065b624bf26343bf310bf39045e20 | When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. What is the smallest possible value of $S$? | [
"337"
] | [
"When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. What is the smallest possible value of $S$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-39d23c8f55980c21e9434625a7eb259d7d09e24556c01b3841969ce6b3e83b86 | When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 2019 is greater than zero and is the same as the probability of obtaining a sum of $S$. What is the smallest possible value of $S$? | [
"340"
] | [
"When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 2019 is greater than zero and is the same as the probability of obtaining a sum of $S$. What is the smallest possible value of $S$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-0d43eb9b2ff099ba5618a1c59d921502874543a82d8f80dfcf7dafd72f43dedd | When $n$ standard six-sided dice are rolled, the probability of obtaining a sum of $1994$ is greater than zero and is the same as the probability of obtaining a sum of $S$. What is the smallest possible value of $S$? | [
"337"
] | [
"When $n$ standard six-sided dice are rolled, the probability of obtaining a sum of $1994$ is greater than zero and is the same as the probability of obtaining a sum of $S$. What is the smallest possible value of $S$?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 35
} | 82 | 0 | 0 |
omr-ws-d350821fbdd23701362755c5657ec5292397a96b66489d2dd665f4828f7893c4 | When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of 936 possible values. What is $n$? | [
"11"
] | [
"When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of 936 possible values. What is $n$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-fcd71dd373f66940d21fef2bd4c4448f3bc1d0a0c4a7c487ca8808c93ffe9022 | When $p(x) = Ax^5 + Bx^3 + Cx + 4$ is divided by $x - 3,$ the remainder is 11. Find the remainder when $p(x)$ is divided by $x + 3.$ | [
"-3"
] | [
"When $p(x) = Ax^5 + Bx^3 + Cx + 4$ is divided by $x - 3,$ the remainder is 11. Find the remainder when $p(x)$ is divided by $x + 3.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f32452c0a7e900412c3b86148b44fac638eb1df0ea7dc50febaba03509855dd4 | When $t$ varies in the range $\frac{\pi}{4} \leq t \leq \frac{\pi}{2}$, find the area $S$ of the figure formed by moving the line segment $PQ$, where $P$ is the point with coordinates $(f(t), g(t))$ and $Q$ is the point of tangency of the line $L$ with the unit circle. | [
"\\(\\frac{7\\pi^3}{192}\\)"
] | [
"When $t$ varies in the range $\\frac{\\pi}{4} \\leq t \\leq \\frac{\\pi}{2}$, find the area $S$ of the figure formed by moving the line segment $PQ$, where $P$ is the point with coordinates $(f(t), g(t))$ and $Q$ is the point of tangency of the line $L$ with the unit circle."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-37ef11930c2eb2f69d02e0c87a5b43c9abfb8767b4031d1ca5a0972781d5e60e | When $u$ satisfies $a \leq u \leq b$, let $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ be the points of intersection of the parabolas $C_1: y=-x^2+1$ and $C_2: y=(x-u)^2+u$. Express $2|x_1y_2 - x_2y_1|$ in terms of $u$. | [
"\\((u^2 + u + 1) \\sqrt{2 - 2u - u^2}\\)"
] | [
"When $u$ satisfies $a \\leq u \\leq b$, let $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ be the points of intersection of the parabolas $C_1: y=-x^2+1$ and $C_2: y=(x-u)^2+u$. Express $2|x_1y_2 - x_2y_1|$ in terms of $u$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-9beaea333826a079261ba48c5455f67a9b172de0a051c28eead418792a000d9f | When $x$ and $y$ are positive integers, let $B(x, y) = \int_{0}^{1} t^{x-1}(1-t)^{y-1}\ dt$. Find the value of $B(5, 7) + B(7, 5)$. | [
"\\dfrac{1}{1155}"
] | [
"When $x$ and $y$ are positive integers, let $B(x, y) = \\int_{0}^{1} t^{x-1}(1-t)^{y-1}\\ dt$. Find the value of $B(5, 7) + B(7, 5)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-252aff49a6d9574afa0258df42c589159fba0818165a568f30edded9f795b8e2 | When $x$ and $y$ are positive real numbers and $x + y = 2$, find the minimum real value of $x^2y^2(x^2 + y^2)$. | [
"0"
] | [
"When $x$ and $y$ are positive real numbers and $x + y = 2$, find the minimum real value of $x^2y^2(x^2 + y^2)$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2f42ef5c94eae634784a7d72fceec7d7b8ac72d55a5a50d3e5a9eebeecc10259 | When $x$ is divided by each of $4$, $5$, and $6$, remainders of $3$, $4$, and $5$ (respectively) are obtained. What is the smallest possible positive integer value of $x$? | [
"59"
] | [
"When $x$ is divided by each of $4$, $5$, and $6$, remainders of $3$, $4$, and $5$ (respectively) are obtained. What is the smallest possible positive integer value of $x$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4724ab1c8b62994eb1064bfdfc591ebb2832722c339bf9decf8a0cde1d97228b | When 1000 is divided by some positive integer \( x \), the remainder is 16. What is the sum of the smallest and largest value of \( x \)? | [
"1008"
] | [
"When 1000 is divided by some positive integer \\( x \\), the remainder is 16. What is the sum of the smallest and largest value of \\( x \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-7858017e63b67178e19efa90f9e5d2de5319b417d8587ecaa8fae8e69fa6399b | When 14 apple trees are planted per acre in a certain orchard, the average yield per tree is 480 apples per year. For each additional tree planted per acre, the annual yield per tree decreases by 100 apples. How many additional trees should be planted in the orchard so that the maximum number of apples will be produced... | [
"0"
] | [
"When 14 apple trees are planted per acre in a certain orchard, the average yield per tree is 480 apples per year. For each additional tree planted per acre, the annual yield per tree decreases by 100 apples. How many additional trees should be planted in the orchard so that the maximum number of apples will be pro... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2b762656fd0ca4bc51c60a8d7fd41deadabe1a6bd6a048102ca888a7d5343f46 | When 15 is appended to a list of integers, the mean is increased by 2. When 1 is appended to the enlarged list, the mean of the enlarged list is decreased by 1. How many integers were in the original list? | [
"4"
] | [
"When 15 is appended to a list of integers, the mean is increased by 2. When 1 is appended to the enlarged list, the mean of the enlarged list is decreased by 1. How many integers were in the original list?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 62
} | 62 | 0 | 0 |
omr-ws-7d846bed76ed3e6af5b9af7152fadcd2d00b4ec680312e0898c8d183560a3474 | When 2010 is divided by a two-digit positive integer \( n \), the remainder is 10. How many different possible values of \( n \) are there? | [
"7"
] | [
"When 2010 is divided by a two-digit positive integer \\( n \\), the remainder is 10. How many different possible values of \\( n \\) are there?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06589fe933f50e5bcc061e070da927557a5ebc6374dc1d730842f8199fb2a3bb | When 21105 is divided by 666, the remainder is 27. What is the remainder when 21970 is divided by 666? | [
"226"
] | [
"When 21105 is divided by 666, the remainder is 27. What is the remainder when 21970 is divided by 666?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ac775f03e55d14d81b743ee9f20f901bbd2999eeb7b367e414607bcf3be8ec45 | When 3 times a number is subtracted from 8, and the result is multiplied by the sum of twice the number and \(\frac{16}{3}\), the result is 12. Compute the square of the number. | [
"\\(\\frac{46}{9}\\)"
] | [
"When 3 times a number is subtracted from 8, and the result is multiplied by the sum of twice the number and \\(\\frac{16}{3}\\), the result is 12. Compute the square of the number."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-059677ca7135a8e5384160d7325d1b8dcddfe209f521c7c8acd53cc33767948d | When 31513 and 38364 are divided by a certain three-digit number, the remainders are equal. Find the remainders. | [
"79, 474, 191"
] | [
"When 31513 and 38364 are divided by a certain three-digit number, the remainders are equal. Find the remainders."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe18d1343eeeec0e156031f5c6ffd974002876472d08b40503f5b63c7a276a2d | When 480 people are surveyed using a method where a dice is rolled and if it shows 6, the box marked "yes" is ticked, otherwise they answer if they have lied to their boss about being sick (with yes or no), and 143 people tick the "yes" box, estimate the proportion of people who have lied to their boss about being sick... | [
"15.8%"
] | [
"When 480 people are surveyed using a method where a dice is rolled and if it shows 6, the box marked \"yes\" is ticked, otherwise they answer if they have lied to their boss about being sick (with yes or no), and 143 people tick the \"yes\" box, estimate the proportion of people who have lied to their boss about b... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-29ae2f428e221dbdf228f619fbb107a7af43b770295b5d5732b59256a7d0cabc | When 6 darts are thrown onto a circular dartboard of radius 10 units, find the minimum possible distance $d$ such that there will always be at least 2 darts which are at most $d$ units apart. | [
"10"
] | [
"When 6 darts are thrown onto a circular dartboard of radius 10 units, find the minimum possible distance $d$ such that there will always be at least 2 darts which are at most $d$ units apart."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-698292defaf4810a2d947851b2192bc1adea411c96b263959001ed837d413e9e | When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as $\frac{n}{6^7}$, where $n$ is a positive integer. What is $n$? | [
"84"
] | [
"When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as $\\frac{n}{6^7}$, where $n$ is a positive integer. What is $n$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cf238f4e87dd6f63c7c6520339c934c76c7d4a3b58ef14d316183cc9287b68c3 | When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as \(\frac{n}{6^7}\), where \(n\) is a positive integer. What is \(n\)? | [
"84"
] | [
"When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as \\(\\frac{n}{6^7}\\), where \\(n\\) is a positive integer. What is \\(n\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 4
} | 36 | 0 | 0 |
omr-ws-e736827b6b4d0b02ab9b6ace9e4e7dd546ce9b7ffaf564ccedc96c70f854b2f5 | When 9 letters, $AAAABBCCC$, are aligned randomly, what is the probability that a symmetrical alignment, such as $AABCCCBAA$, is generated? | [
"\\(\\frac{1}{105}\\)"
] | [
"When 9 letters, $AAAABBCCC$, are aligned randomly, what is the probability that a symmetrical alignment, such as $AABCCCBAA$, is generated?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 26
} | 74 | 0 | 0 |
omr-ws-31a8d9d4574ec126ff03aa955e5cac19349f86391705b432bd0bb6ec4a23fe4e | When Alicia was twice as old as Bert, she was as old as Carly was when Bert was twice as old as Carly. Right now, Alicia is 73 years old, and Carly is 34 years old. How old is Bert right now? | [
"60"
] | [
"When Alicia was twice as old as Bert, she was as old as Carly was when Bert was twice as old as Carly. Right now, Alicia is 73 years old, and Carly is 34 years old. How old is Bert right now?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-a4e2d749ea63be62f0e145a8bff370ac8205fa919fa4525f7f0e83205063b8b7 | When Anne entered the room, the mean age increased by 4 years. When Beth then entered the room, the mean age increased by a further 3 years. Anne and Beth were the same age. How many people were in the room before Anne entered? | [
"6"
] | [
"When Anne entered the room, the mean age increased by 4 years. When Beth then entered the room, the mean age increased by a further 3 years. Anne and Beth were the same age. How many people were in the room before Anne entered?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3d9999de905b32e70246960962147d09c827bf1ac212210f18f945fcc521e044 | When B first says that he cannot tell what the two numbers are, A receives a large amount of information. But when A first says that he cannot tell what the two numbers are, B already knows that A cannot tell what the two numbers are. What good does it do B to listen to A? | [
"The iterative process of their statements allows B to narrow down the possibilities until he can determine the numbers."
] | [
"When B first says that he cannot tell what the two numbers are, A receives a large amount of information. But when A first says that he cannot tell what the two numbers are, B already knows that A cannot tell what the two numbers are. What good does it do B to listen to A?"
] | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-c3e668dd8473a7e6574accf9c1eb8347db3de8e145d5d9416114d0b1cfd3755d | When Ben works at twice his normal rate and Sammy works at his normal rate, they can finish a project together in 6 hours. When Ben works at his normal rate and Sammy works as three times his normal rate, they can finish the same project together in 4 hours. How many hours does it take Ben and Sammy to finish that proj... | [
"\\dfrac{60}{7}"
] | [
"When Ben works at twice his normal rate and Sammy works at his normal rate, they can finish a project together in 6 hours. When Ben works at his normal rate and Sammy works as three times his normal rate, they can finish the same project together in 4 hours. How many hours does it take Ben and Sammy to finish that... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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