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-4984ac11096b27ca1992961ac3f9325a32bdbe1a0d753f81c7cde07f759230b5 | What is the expansion of \((x+y+z)^n\) for a general \(n\)? Additionally, determine the number of distinct terms in the expansion and the sum of the coefficients. | [
"Number of distinct terms = \\(\\binom{n+2}{2}\\), Sum of coefficients = \\(3^n\\)"
] | [
"What is the expansion of \\((x+y+z)^n\\) for a general \\(n\\)? Additionally, determine the number of distinct terms in the expansion and the sum of the coefficients."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 21
} | 21 | 0 | 0 |
omr-ws-bec06b14f9644d5cd92543f937af9c420662e3274d41e9ef0fcd8c377ac71d54 | What is the expectation of a real-valued random variable with density \( f(x) = \frac{1}{2x^2} \) for \( |x| \geq 1 \) and \( f(x) = 0 \) otherwise? | [
"\\text{Does not exist}"
] | [
"What is the expectation of a real-valued random variable with density \\( f(x) = \\frac{1}{2x^2} \\) for \\( |x| \\geq 1 \\) and \\( f(x) = 0 \\) otherwise?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-342456b8206d1dfa91d8b73f82b7c4ffedec2436eefaa298d5f49486608622c8 | What is the expectation of the number of real roots of the polynomial \( P(X) = a_0 + a_1 X + \ldots + a_n X^n \), where \( a_0(t), a_1(t), \ldots, a_n(t) \) are independent random variables with normal distribution? | [
"\\dfrac{2}{\\pi} \\ln n"
] | [
"What is the expectation of the number of real roots of the polynomial \\( P(X) = a_0 + a_1 X + \\ldots + a_n X^n \\), where \\( a_0(t), a_1(t), \\ldots, a_n(t) \\) are independent random variables with normal distribution?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-d7cf0eee957137efe6e483ddb5579c8a6fd8b12dd337c39e2849428e9690fff3 | What is the expected area of a 'triangle' drawn on a sphere, assuming the surface area of the sphere is 1? | [
"\\(\\frac{1}{8}\\)"
] | [
"What is the expected area of a 'triangle' drawn on a sphere, assuming the surface area of the sphere is 1?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-cddb66e9c24fbcee60285b348f8c633908a9c756bab7c3f71ffec52059a5bb96 | What is the expected area of the $n$-gon formed by $n$ randomly chosen points on the circumference of a unit circle? | [
"\\(\\sum_{i=0}^{\\left\\lfloor\\frac{n-3}2\\right\\rfloor}\\frac{(-1)^in!}{(n-2i-2)!2^{2i+2}\\pi^{2i+1}}\\)"
] | [
"What is the expected area of the $n$-gon formed by $n$ randomly chosen points on the circumference of a unit circle?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-21771153a46dd16d6259be66b1f2d181d197ad1c7c1bfd86b12663bb6f4c9707 | What is the expected area of the quadrilateral formed by four randomly chosen points on the circumference of a unit circle? | [
"\\(\\frac{3}{\\pi}\\)"
] | [
"What is the expected area of the quadrilateral formed by four randomly chosen points on the circumference of a unit circle?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 9
} | 19 | 0 | 0 |
omr-ws-e6728057d89ffc2f20aef3f0d27aa7a89ef1d8a4aa8e80831c58e39a064b2901 | What is the expected area of the triangle formed by three randomly chosen points on the circumference of a unit circle? | [
"\\(\\frac{3}{2\\pi}\\)"
] | [
"What is the expected area of the triangle formed by three randomly chosen points on the circumference of a unit circle?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 1,
"tir": 16
} | 26 | 0 | 0 |
omr-ws-4bf7203c7449966fb4ce7d0183028070baacd6e37bc19aa5ed3d262f381ee930 | What is the expected average distance between two sites inside a convex polygon with \( n \) vertices? | [
"\\frac{2}{3} \\cdot \\frac{\\text{perimeter}}{n}"
] | [
"What is the expected average distance between two sites inside a convex polygon with \\( n \\) vertices?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1779853aab67296bb2fd54cfd01e6cb93cbdb63bf3f8df5d2de664aaf62e5210 | What is the expected distance between two randomly chosen points on the unit sphere? | [
"\\(\\frac{4}{3}\\)"
] | [
"What is the expected distance between two randomly chosen points on the unit sphere?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-e78172334fd68d3133adae469f9dfd6596afac8e21ec81a7bfa5c6596a1a7da8 | What is the expected length of time until a head is tossed with a biased coin? | [
"\\(\\frac{1}{p}\\)"
] | [
"What is the expected length of time until a head is tossed with a biased coin?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-4107041a61ffab68e553a763a7ca89fcb99751eca98303d2c957433acd962533 | What is the expected length of time until two consecutive heads are tossed with a biased coin? | [
"\\(\\frac{2}{p}\\)"
] | [
"What is the expected length of time until two consecutive heads are tossed with a biased coin?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e30dc51b05de5df563273a207d402b3e80b2ca4079191cbb05832611e3f0040 | What is the expected magnitude of the dot product of two vectors randomly chosen from the d-dimensional unit sphere? | [
"\\(\\frac{\\Gamma\\left(\\frac{d}{2}\\right)}{\\sqrt{\\pi} \\, \\Gamma\\left(\\frac{d+1}{2}\\right)}\\)"
] | [
"What is the expected magnitude of the dot product of two vectors randomly chosen from the d-dimensional unit sphere?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 8,
"tir": 6
} | 18 | 0 | 0 |
omr-ws-da1dc55165751d8db97e111e693b568ddc7a713b98cd9e62b43021a7e52ccc19 | What is the expected magnitude of the sum of two unit vectors in the Cartesian plane if the directions of both vectors are randomly chosen from the interval $[-\pi, \pi]$? | [
"\\dfrac{4}{\\pi}"
] | [
"What is the expected magnitude of the sum of two unit vectors in the Cartesian plane if the directions of both vectors are randomly chosen from the interval $[-\\pi, \\pi]$?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 59
} | 59 | 0 | 0 |
omr-ws-f27719700d768e921522403be110e574db024f488ab02f43c0f8fe2c2ce2b698 | What is the expected number of ball tosses before at least one of the bins contains two balls, given that there are \( n \) balls and \( b \) bins, with each ball equally likely to end up in any bin? | [
"\\sum_{k=0}^{b} \\frac{b!}{(b - k)! \\, b^k}"
] | [
"What is the expected number of ball tosses before at least one of the bins contains two balls, given that there are \\( n \\) balls and \\( b \\) bins, with each ball equally likely to end up in any bin?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ceed809b0227450fd63ee0ed05691f5958fd21fb431ecc41cd98c5d3f48378a5 | What is the expected number of characters you need to type, selecting uniformly at random and independently from A to Z uppercase, to get the sequence "AB"? | [
"676"
] | [
"What is the expected number of characters you need to type, selecting uniformly at random and independently from A to Z uppercase, to get the sequence \"AB\"?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-3f8ad71c6070b5f36d782ce386d9d580514093e162d727c211f4ff59a406add7 | What is the expected number of coin flips Joe needs to get 10 heads in a row? | [
"2046"
] | [
"What is the expected number of coin flips Joe needs to get 10 heads in a row?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-562f9a69eb861a09ef28e9e04468d8a65609f705ff9c027e2dfbddbf0ef7b357 | What is the expected number of coin flips before getting 3 consecutive heads? | [
"14"
] | [
"What is the expected number of coin flips before getting 3 consecutive heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-dde7db801cb4cc974f02f87ac6feba2be9f0262dea5055181ace378edb30f2db | What is the expected number of coin tosses it takes to observe Head followed by two consecutive Tails? (Use the inductive approach) | [
"8"
] | [
"What is the expected number of coin tosses it takes to observe Head followed by two consecutive Tails? (Use the inductive approach)"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ec61caa7e14a387a3bf24165958accebf9a30c5344435b460ea69d244414cd28 | What is the expected number of dice rolls for player A to roll until a sequence of $1-2$ appears? | [
"36"
] | [
"What is the expected number of dice rolls for player A to roll until a sequence of $1-2$ appears?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-57fa5168b1121cb08d04480cbc9802e2a9167fc55e338fe054bbc375e8148b9f | What is the expected number of dice rolls for player B to roll until a sequence of $1-1$ appears? | [
"42"
] | [
"What is the expected number of dice rolls for player B to roll until a sequence of $1-1$ appears?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-fc4cf50134e75d71cd08957ad7ca6d9489201b2e8a889e1bddee1aeb699ce88b | What is the expected number of flips it will take to get a HT sequence? | [
"4"
] | [
"What is the expected number of flips it will take to get a HT sequence?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3b07a18d6fee1538f866e6e34ad628f3cdc49c4cfe14c833a8093ac04a6356dd | What is the expected number of flips to get the sequence "HTHH"? | [
"18"
] | [
"What is the expected number of flips to get the sequence \"HTHH\"?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-737b5428fcf0f4a48e0f2156f98dc57a4c89d6da11b88978e608b6061a0c4f79 | What is the expected number of flips until the first time the last 2008 flips are $HHTTTTHH \cdots HHTTTTHH$ when flipping a fair coin repeatedly? | [
"$2^{2008} + 2^8 + 2^2 + 2$"
] | [
"What is the expected number of flips until the first time the last 2008 flips are $HHTTTTHH \\cdots HHTTTTHH$ when flipping a fair coin repeatedly?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bf475820f2bc4b3b31236caf5489f7c15fe1c807e09387662859da9386576226 | What is the expected number of flips until we see the sequence HTH, where H represents a head and T represents a tail, and we count the beginning of the sequence in the number of tosses? | [
"10"
] | [
"What is the expected number of flips until we see the sequence HTH, where H represents a head and T represents a tail, and we count the beginning of the sequence in the number of tosses?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-6e7bbe6f93a0681f121213feffef2f42b60f4477b24818af83fe00bddb4721c8 | What is the expected number of instances, \( n \), required for a computer to generate each number from 1 to \( k \) at least once? | [
"\\( k \\cdot H_k \\)",
"k \\left(1 + \\frac{1}{2} + \\frac{1}{3} + \\cdots + \\frac{1}{k}\\right)"
] | [
"What is the expected number of instances, \\( n \\), required for a computer to generate each number from 1 to \\( k \\) at least once?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 25
} | 57 | 0 | 0 |
omr-ws-1949d176faf404f48c6389c0fa529c4be28c25261489fc9df20250e92e40cd2a | What is the expected number of letters printed until the sequence "BL" appears for the first time, given that each letter is chosen uniformly at random from the uppercase letters A to Z? | [
"\\( 26^2 \\)"
] | [
"What is the expected number of letters printed until the sequence \"BL\" appears for the first time, given that each letter is chosen uniformly at random from the uppercase letters A to Z?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-ea897c1c98046b8f7f1698926372a2e94a086c79b49ef8aefff992793275878f | What is the expected number of loops when tying the ends of n strings together until only loops remain? | [
"\\( \\sum_{k=1}^n \\frac{1}{2k-1} \\)"
] | [
"What is the expected number of loops when tying the ends of n strings together until only loops remain?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-a40180133aa995cbd8d23b401900d71a2029cce48b081f466cb5ef1c3a31592e | What is the expected number of minutes between two consecutive palindromic times on a 12-hour digital clock, where a palindromic time is of the form a:bc:ba or ab:cc:ba? | [
"\\(\\frac{40}{31}\\)"
] | [
"What is the expected number of minutes between two consecutive palindromic times on a 12-hour digital clock, where a palindromic time is of the form a:bc:ba or ab:cc:ba?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-98964a1d89ecb33213e1df9e31ff4281b5b63c527e03196ac86fdebeab4f98a0 | What is the expected number of moves it will take for an ant starting at the origin of a number line to return to the origin for the first time, given that at each step, the ant walks either forward or backward with equal probability? | [
"\\infty"
] | [
"What is the expected number of moves it will take for an ant starting at the origin of a number line to return to the origin for the first time, given that at each step, the ant walks either forward or backward with equal probability?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b870cd18cf04937f0cf3f656e1731e635760d53ef7b171734b35f254d1cb2c47 | What is the expected number of moves needed to turn a board of nine Xs into one of nine Os, given that each move randomly changes one of the nine positions from X to O or O to X with equal probability for each position? | [
"\\(\\frac{21248}{35}\\)"
] | [
"What is the expected number of moves needed to turn a board of nine Xs into one of nine Os, given that each move randomly changes one of the nine positions from X to O or O to X with equal probability for each position?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-80515582bb97a0c0edf407886953fd4eee32a8719465728f0eb9af974ff4a127 | What is the expected number of pairs of adjacent cards which are both black when the cards of a standard 52-card deck are dealt out in a circle? Express your answer as a common fraction. | [
"\\(\\frac{650}{51}\\)"
] | [
"What is the expected number of pairs of adjacent cards which are both black when the cards of a standard 52-card deck are dealt out in a circle? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-17aadbe5ba13d3229fb6f63f4339489b1f7e464d38443c04d9036ef34a02d2f4 | What is the expected number of persons visible to the person at the ticket window if \( n \) persons, all of different heights, are standing in a queue facing the window and each permutation of the \( n \) persons is equally likely? | [
"\\( H_n \\)"
] | [
"What is the expected number of persons visible to the person at the ticket window if \\( n \\) persons, all of different heights, are standing in a queue facing the window and each permutation of the \\( n \\) persons is equally likely?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 18
} | 35 | 0 | 0 |
omr-ws-ef0aad3b77a937e34306f55f1e4c56919d8a9dd2a8bd4a926bf79b768c1f678d | What is the expected number of presses required to press all n buttons at least once, given that each press is a random button uniformly distributed? | [
"\\(\\sum_{m=1}^{n} \\frac{n}{m}\\) or \\(n \\left( \\frac{1}{1} + \\frac{1}{2} + \\cdots + \\frac{1}{n} \\right)\\)"
] | [
"What is the expected number of presses required to press all n buttons at least once, given that each press is a random button uniformly distributed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-c9f01b5d4d9b84ab3d310875902845805e5501369869682336fe9fbb29399ae0 | What is the expected number of random integers that must be selected to guarantee having 19 different random integers? | [
"19 \\left(1 + \\frac{1}{2} + \\frac{1}{3} + \\cdots + \\frac{1}{19}\\right)"
] | [
"What is the expected number of random integers that must be selected to guarantee having 19 different random integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-b5b14ffe7573f3031674c635e84d06fa34deb9947b1d2aa67dbef992382caca0 | What is the expected number of random numbers between 0 and 1 (inclusive) that need to be generated and summed until the sum first exceeds 1? | [
"\\( e \\)"
] | [
"What is the expected number of random numbers between 0 and 1 (inclusive) that need to be generated and summed until the sum first exceeds 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-1f18f43a4909770461cbc7f75014a9486407b3c8f98bf946d382e1db455ef1da | What is the expected number of rolls it will take to get all six values of a six-sided die face-up at least once? | [
"\\dfrac{147}{10}"
] | [
"What is the expected number of rolls it will take to get all six values of a six-sided die face-up at least once?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3078999069f3493faa9b23ad2ef52825686dadf9706ad838550ad45491234e24 | What is the expected number of rolls needed for the sum of rolls of a fair 6-sided die to be greater than or equal to 700? Any answer within .0001 of the correct answer will be accepted. | [
"200.5"
] | [
"What is the expected number of rolls needed for the sum of rolls of a fair 6-sided die to be greater than or equal to 700? Any answer within .0001 of the correct answer will be accepted."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 4,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-5bcf53f4e8045883915aab102f49013427539be69f3c2234bd3785d04a37c41a | What is the expected number of rolls you roll on a $6$-sided die until you roll a $3$ followed by a multiple of $2$? | [
"12"
] | [
"What is the expected number of rolls you roll on a $6$-sided die until you roll a $3$ followed by a multiple of $2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-913bd761655339bd953fc0dc27fe7d8e258f225492656c1aabc16ee56dc66e16 | What is the expected number of seconds it takes to correctly guess a four-digit password, where each digit can be from 0 to 9, if it takes 2 seconds to type in a guess and there is a delay of \( n \) seconds after the \( n \)-th guess before another guess can be made? | [
"16676667.5"
] | [
"What is the expected number of seconds it takes to correctly guess a four-digit password, where each digit can be from 0 to 9, if it takes 2 seconds to type in a guess and there is a delay of \\( n \\) seconds after the \\( n \\)-th guess before another guess can be made?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 23
} | 40 | 0 | 0 |
omr-ws-4a29bcf63ad97059cb7c2d6e6e685cb323c9e59eb253115322d3fa034328dda7 | What is the expected number of sets in a selection of $n$ cards, where each card is uniquely defined by four properties: color (red, blue, green), number (1, 2, 3), shape (circle, triangle, square), and shading (filled, partially shaded, empty), and $n \in \mathbb{N}, n \geq 3$? | [
"\\dfrac{n(n-1)(n-2)}{474}"
] | [
"What is the expected number of sets in a selection of $n$ cards, where each card is uniquely defined by four properties: color (red, blue, green), number (1, 2, 3), shape (circle, triangle, square), and shading (filled, partially shaded, empty), and $n \\in \\mathbb{N}, n \\geq 3$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4ff6b885c2849a4cd07283ca8eed09d20bacd6a3f7d285b678c2abee3c62b465 | What is the expected number of sets in a selection of 16 cards, where each card is uniquely defined by four properties: color (red, blue, green), number (1, 2, 3), shape (circle, triangle, square), and shading (filled, partially shaded, empty)? | [
"\\dfrac{560}{79}"
] | [
"What is the expected number of sets in a selection of 16 cards, where each card is uniquely defined by four properties: color (red, blue, green), number (1, 2, 3), shape (circle, triangle, square), and shading (filled, partially shaded, empty)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f948a65d6d51ede4aa50beeacfbd971d7a2bba57ae25465d20a47534c291090 | What is the expected number of spins Bob has to make to get all seven figures from a gochapon machine, where each spin randomly dispenses one of seven different figures with equal probability? | [
"\\dfrac{363}{20}"
] | [
"What is the expected number of spins Bob has to make to get all seven figures from a gochapon machine, where each spin randomly dispenses one of seven different figures with equal probability?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-95697cb0dd873184980e972b02b58b5f72e2bbea42552475b999cd24f2ab9edf | What is the expected number of spins in a roulette wheel before you see any dozen NOT appear 17 times in a row? | [
"1995.1"
] | [
"What is the expected number of spins in a roulette wheel before you see any dozen NOT appear 17 times in a row?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-19f7a3c3236b4be3144011102163d2c63ea7ba72200df1e771fe366b12ff9e9f | What is the expected number of steps a person at the origin would have to take to get his $x$ coordinate to be $n>0$, given that he moves up, right, down, and left with probabilities $a, b, c, d$, which satisfy $a+b+c+d = 1$, and starts at the origin? | [
"\\(\\frac{1}{b-d}\\)"
] | [
"What is the expected number of steps a person at the origin would have to take to get his $x$ coordinate to be $n>0$, given that he moves up, right, down, and left with probabilities $a, b, c, d$, which satisfy $a+b+c+d = 1$, and starts at the origin?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-c41e740c110ff58bbf5aecfddac8f4357a1ba9f4f2aad5e5db6890fe3945ba9d | What is the expected number of students who get their lucky pencil back when each student picks a pencil at random from a box containing all the pencils? | [
"1"
] | [
"What is the expected number of students who get their lucky pencil back when each student picks a pencil at random from a box containing all the pencils?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-05f620dab262f7c40fe85ee88bc30ee6be710171d3cf5d7be36605242fb9edcc | What is the expected number of throws of an $N$-sided die so that each number is rolled at least once? | [
"\\( N \\sum_{i=1}^{N} \\frac{1}{i} \\)"
] | [
"What is the expected number of throws of an $N$-sided die so that each number is rolled at least once?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-899e8bfc03bcf4f81b360cb4d78bae1665fc5f583dc185d2ffe97d0c181690e2 | What is the expected number of times a dice must be rolled until three sixes appear in a row? | [
"258"
] | [
"What is the expected number of times a dice must be rolled until three sixes appear in a row?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-a96e0f2d9cd424794b7bd2b8af91b9bf863c3be8aadfa30c6db18db035534bb6 | What is the expected number of times a fair die must be thrown until all scores appear at least once? | [
"\\dfrac{147}{10}"
] | [
"What is the expected number of times a fair die must be thrown until all scores appear at least once?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-caad6a620faacf150b7d1df96303d4f7530744ebdd3c84ab82f624028d2f78a8 | What is the expected number of times you will have to spin the wheel before you are able to get 5 copies of each of the 10 different cards, given that each spin results in one of the 10 cards randomly? | [
"50H_{10}"
] | [
"What is the expected number of times you will have to spin the wheel before you are able to get 5 copies of each of the 10 different cards, given that each spin results in one of the 10 cards randomly?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-ba872ba6faa0a0afb5c29d9b95f4839444cad7d20a9ed1a4630fe1612a157a39 | What is the expected number of tosses to get two $6$'s (not necessarily simultaneously) if we are tossing two six-sided dice at the same time? | [
"\\dfrac{756}{121}"
] | [
"What is the expected number of tosses to get two $6$'s (not necessarily simultaneously) if we are tossing two six-sided dice at the same time?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f84b3abc704b1ad9a1d66ab94ad5d938dd9f00fe880b83a040292d468c48a7e6 | What is the expected number of turns in a randomly chosen 9-step path from (0,0) to (5,4)? A turn is any point where the path changes direction (right to up or up to right). | [
"\\dfrac{40}{9}"
] | [
"What is the expected number of turns in a randomly chosen 9-step path from (0,0) to (5,4)? A turn is any point where the path changes direction (right to up or up to right)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7acfd5bb92db15e0e2555a1719b535d0c534fb8bc348ea0a3cc2f3c91f8f8487 | What is the expected number of turns needed for all five letters to be pulled (i.e., for there to have been a B-, I-, N-, G-, and O-ball pulled out)? | [
"\\dfrac{137}{12}"
] | [
"What is the expected number of turns needed for all five letters to be pulled (i.e., for there to have been a B-, I-, N-, G-, and O-ball pulled out)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3abeabd2cdde06ecc8ab9601b3adc5afc090790043ac49768de14ce8670a6ff2 | What is the expected number of turns needed for the first person to get BINGO given that there are $n$ people playing? | [
"\\frac{41.4}{n}"
] | [
"What is the expected number of turns needed for the first person to get BINGO given that there are $n$ people playing?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-678411e80ccaadd19419f5031603ef03da9fdb4d0a2a41c8bac213a0f6fd4382 | What is the expected size of a random subset of the set $\{1, 2, 3, \ldots, n\}$? | [
"$\\frac{n}{2}$"
] | [
"What is the expected size of a random subset of the set $\\{1, 2, 3, \\ldots, n\\}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-5d96b3cae72b8ce9040ae4732aad789fca794ed209acfd38a52803a2e5935c66 | What is the expected time to get \( n \) heads in a row when flipping a fair coin? Consider the recursive relationship \( E_n = \frac{1}{2}(1 + E_{n-1}) + \frac{1}{2}(1 + E_n) \) and explain why it does not lead to the correct answer of \( 2^{n+1} - 2 \). | [
"\\( 2^{n+1} - 2 \\)"
] | [
"What is the expected time to get \\( n \\) heads in a row when flipping a fair coin? Consider the recursive relationship \\( E_n = \\frac{1}{2}(1 + E_{n-1}) + \\frac{1}{2}(1 + E_n) \\) and explain why it does not lead to the correct answer of \\( 2^{n+1} - 2 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3a8c13ea13d55649a04c98ee4357dfdfd36ce0fd081d870a5cef2f3f21486310 | What is the expected value of $PA^2 + PB^2 + PC^2 + PD^2$ for a point $P$ chosen inside a unit square $ABCD$? | [
"\\(\\frac{8}{3}\\)"
] | [
"What is the expected value of $PA^2 + PB^2 + PC^2 + PD^2$ for a point $P$ chosen inside a unit square $ABCD$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-908be8fc26e848def2c7889ae0770b30ca7c97e6c3506e0511cd90870ecc47d0 | What is the expected value of $x^2 + y^2 + z^2$ for random points chosen in the plane $2x + 3y + z = 2$? | [
"\\dfrac{16}{7}"
] | [
"What is the expected value of $x^2 + y^2 + z^2$ for random points chosen in the plane $2x + 3y + z = 2$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-7782d4477b084a4989f1535acbcbfd83e60969e297fe2788507392222bdd7aab | What is the expected value of \(\sin x\) where \(x\) ranges from 0 to 180 degrees? | [
"\\(\\frac{2}{\\pi}\\)"
] | [
"What is the expected value of \\(\\sin x\\) where \\(x\\) ranges from 0 to 180 degrees?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-db853fedeaaeb2c3db24b8d43a087e663e330dbe60bcf4d9196e6bb5d59b4365 | What is the expected value of \(\sin x\) where \(x\) ranges from 0 to 90 degrees? | [
"\\(\\frac{2}{\\pi}\\)"
] | [
"What is the expected value of \\(\\sin x\\) where \\(x\\) ranges from 0 to 90 degrees?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-705cd1442fb6af022155bfbf5ed3e8842d3cfaaab64b5adf51e972733592ceda | What is the expected value of a die with infinitely many sides labeled 1, 2, 3, ...? | [
"undefined."
] | [
"What is the expected value of a die with infinitely many sides labeled 1, 2, 3, ...?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-30dc260136e0f4015ee4a699afbe965349e827876e51964928402888b4d50dfe | What is the expected value of draws (without replacement) needed to pick a blue marble out of a bag if there are x blue marbles out of a total of y marbles in the bag? | [
"\\dfrac{y + 1}{x + 1}"
] | [
"What is the expected value of draws (without replacement) needed to pick a blue marble out of a bag if there are x blue marbles out of a total of y marbles in the bag?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0eb0f41e69ed52aa4e7fa201eb24d77beab9a7d996dc2d9b842657f993f4c74d | What is the expected value of rolling a 6 when 4 dice are rolled? | [
"\\(\\frac{2}{3}\\)"
] | [
"What is the expected value of rolling a 6 when 4 dice are rolled?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-19eb3b4b96d380724cc5cb4ba01c6a04918f6c685c0b14ba5ff4ca637a414455 | What is the expected value of the $k^\text{th}$ largest number when $n$ numbers are selected uniformly and randomly from the interval $[0,1]$? | [
"$\\frac{n+1-k}{n+1}$"
] | [
"What is the expected value of the $k^\\text{th}$ largest number when $n$ numbers are selected uniformly and randomly from the interval $[0,1]$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-45329f92390d77442e456193fa027b7788b79697618ab39b4233c4f0a44e7af5 | What is the expected value of the area of a random triangle in a unit square? | [
"\\(\\frac{11}{144}\\)"
] | [
"What is the expected value of the area of a random triangle in a unit square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-3970659c285d58d58f73999e4891d6ea09024540a27097b0b05aab181a98a891 | What is the expected value of the area of the convex n-gon formed by any n distinct points on the unit circle? | [
"\\frac{n}{2} \\sin\\left(\\frac{2\\pi}{n}\\right)"
] | [
"What is the expected value of the area of the convex n-gon formed by any n distinct points on the unit circle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-029956646a2ec785948cfd3d87a6c3403a6ab9bbc50423e266c2508e0d680f20 | What is the expected value of the area of the triangle formed by any 3 distinct points on the unit circle? | [
"\\dfrac{3}{2\\pi}"
] | [
"What is the expected value of the area of the triangle formed by any 3 distinct points on the unit circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-e2389ab61e348a636348b3c8c71a1e99c53936128e6735c40206fc361e6b11fd | What is the expected value of the game if played optimally, where you roll a regular six-sided die and accumulate the sum of all rolls, but lose everything if the sum becomes a perfect square? | [
"6"
] | [
"What is the expected value of the game if played optimally, where you roll a regular six-sided die and accumulate the sum of all rolls, but lose everything if the sum becomes a perfect square?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-6df2a1565d8614d4c7f362a99839453a4490b345709e57b2d0b0de259524ed85 | What is the expected value of the length of a chord in a circle with radius $1$? | [
"\\dfrac{4}{\\pi}"
] | [
"What is the expected value of the length of a chord in a circle with radius $1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e00f06e052995d3dc51b99fc82c6e5698f131c64e8c9170229a42722e1af13e | What is the expected value of the longest consecutive sequence of any letter in a string 1000 letters long, where each letter is A, B, C, or D? | [
"6"
] | [
"What is the expected value of the longest consecutive sequence of any letter in a string 1000 letters long, where each letter is A, B, C, or D?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-a42a4aac12b3ec88c243d9f9b67cb4a61a07adae88b244452fe62da215ef044c | What is the expected value of the maximum score obtained when taking a multiple-choice test with 10 questions, each having 4 choices, twice? | [
"3.2561"
] | [
"What is the expected value of the maximum score obtained when taking a multiple-choice test with 10 questions, each having 4 choices, twice?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 8,
"tir": 21
} | 32 | 0 | 0 |
omr-ws-1b55458a1fb4b050c692d64436a485e47c54ac7378f6d34e50dd77189407623c | What is the expected value of the number of cards before the second ace is turned over in a standard deck of 52 cards that is shuffled well and turned over in order? | [
"\\dfrac{106}{5}"
] | [
"What is the expected value of the number of cards before the second ace is turned over in a standard deck of 52 cards that is shuffled well and turned over in order?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-3f519531216ee651ed337d1090e9f86b7b4c8b7026ae8c70a788f7dfbb90215c | What is the expected value of the number of pairs of adjacent red cards in a shuffled deck of 52 cards? | [
"$\\frac{26 \\cdot 25}{51}$"
] | [
"What is the expected value of the number of pairs of adjacent red cards in a shuffled deck of 52 cards?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-47674bfa44ad011b89c2457578ff710cfc1f6c653cd67ba06ec4e2ef67557c2e | What is the expected value of the number of pieces a circular cake will have after it has been cut n times? | [
"\\( 1 + \\frac{1}{3} \\binom{n}{2} \\)"
] | [
"What is the expected value of the number of pieces a circular cake will have after it has been cut n times?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-712e78e65492a883dc546597e9c05b9f3aed92e44166e678fba4cb9dd1c24700 | What is the expected value of the number of rolls needed to roll any number 3 times in a row with a fair 6-sided die? | [
"43"
] | [
"What is the expected value of the number of rolls needed to roll any number 3 times in a row with a fair 6-sided die?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-43474120fbe5cf3334c3dfd20a4cd124e33d9d5a494cab9182ee2e91508a8d2b | What is the expected value of the number of roots of a 6th degree polynomial with real coefficients that are greater than or equal to 0 and less than or equal to 10? | [
"0"
] | [
"What is the expected value of the number of roots of a 6th degree polynomial with real coefficients that are greater than or equal to 0 and less than or equal to 10?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-427fbd05492e5e54449074901b18fc36a8f6401f0aa7e45338ab79a23b86a4ad | What is the expected value of the number of roots of a 6th degree polynomial with real coefficients, with no other restrictions? | [
"6"
] | [
"What is the expected value of the number of roots of a 6th degree polynomial with real coefficients, with no other restrictions?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6c9b3617545bba1cff7c314138bb848d6f4a50932ce630f4591540c84c053fa6 | What is the expected value of the number of tosses required to get all numbers from 1 to N on an N-sided die? | [
"\\(NH_{N}\\)"
] | [
"What is the expected value of the number of tosses required to get all numbers from 1 to N on an N-sided die?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-cbcc3d567cc20cbd3b989a2ac84db5429b99dbe7dcfc042a78093be7e1150a4b | What is the expected value of the number of tosses that occurs when Jason stops on two consecutive tosses of 3 when he tosses a die repeatedly until he gets a 6 or two consecutive tosses of 3? | [
"\\dfrac{21}{4}"
] | [
"What is the expected value of the number of tosses that occurs when Jason stops on two consecutive tosses of 3 when he tosses a die repeatedly until he gets a 6 or two consecutive tosses of 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f8a1623ccefea36dc82c84ce767acff69246a0c39be915bbde679f14f0e96dc2 | What is the expected value of the roll of a fair octahedral die? (An octahedral die has 8 faces, numbered from 1 to 8.) Express your answer as a decimal. | [
"4.5"
] | [
"What is the expected value of the roll of a fair octahedral die? (An octahedral die has 8 faces, numbered from 1 to 8.) Express your answer as a decimal."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-60af17c6905d8ef079b2a0422f4f3fe2233997543cf598ad9668033c6915de72 | What is the expected value of the roll of a standard 6-sided die? | [
"3.5"
] | [
"What is the expected value of the roll of a standard 6-sided die?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b70ebd4b5dd0e494b8c89db2a0ad43c047feedeb4f16c6230675c40bb563c18 | What is the expected value of the second largest of 5 random numbers uniformly distributed between 0 and 1? | [
"\\dfrac{2}{3}"
] | [
"What is the expected value of the second largest of 5 random numbers uniformly distributed between 0 and 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a980eeb09e1245e34dd095197f8ad1e366b6a0ff2eb95ae7b5b96f9941b54043 | What is the expected value of the square of the number of flips Alex makes, given that he continuously flips a fair coin and stops when he gets his first head? | [
"6"
] | [
"What is the expected value of the square of the number of flips Alex makes, given that he continuously flips a fair coin and stops when he gets his first head?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c93b317f1f4dec9f50276daa1da4060cb1a493be016436faa2e29fd97fb88823 | What is the expected value of the volume of a tetrahedron inscribed in another regular tetrahedron with side length 1? | [
"\\dfrac{\\sqrt{2}}{324}"
] | [
"What is the expected value of the volume of a tetrahedron inscribed in another regular tetrahedron with side length 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-dd4e80c9b8362d164240e424e2b5da6675e8c072588b177edba9bd33bf4345c0 | What is the expected value of your earnings, starting with $d$ dollars, if you can click a button 10 times, where each click has a $\frac{1}{3}$ probability of adding 20 dollars, a $\frac{1}{3}$ probability of deducting 20 dollars, and a $\frac{1}{3}$ probability of doubling your current amount? | [
"\\(\\frac{10d}{3}\\)"
] | [
"What is the expected value of your earnings, starting with $d$ dollars, if you can click a button 10 times, where each click has a $\\frac{1}{3}$ probability of adding 20 dollars, a $\\frac{1}{3}$ probability of deducting 20 dollars, and a $\\frac{1}{3}$ probability of doubling your current amount?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f145208cc85e4d8e6ad371c34947dff44472c9bfa6902ba4afa1133c59497a1a | What is the expected value of your earnings, starting with 20 dollars, if you can click a button 10 times, where each click has a $\frac{1}{3}$ probability of adding 20 dollars, a $\frac{1}{3}$ probability of deducting 20 dollars, and a $\frac{1}{3}$ probability of doubling your current amount? | [
"\\( 20 \\left(\\frac{4}{3}\\right)^{10} \\) or approximately \\( 335.54432 \\)"
] | [
"What is the expected value of your earnings, starting with 20 dollars, if you can click a button 10 times, where each click has a $\\frac{1}{3}$ probability of adding 20 dollars, a $\\frac{1}{3}$ probability of deducting 20 dollars, and a $\\frac{1}{3}$ probability of doubling your current amount?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 2
} | 24 | 0 | 0 |
omr-ws-555907d3143d5063bf200a19fbc27f85bca81ccf79e3f60876ac7140579ca8df | What is the expected value when a 1 dollar lottery ticket is bought in which the purchaser wins exactly 10 million dollars if the ticket contains the six winning numbers chosen from the set {1, 2, 3, ..., 50} and the purchaser wins nothing otherwise? | [
"\\(-0.37\\)"
] | [
"What is the expected value when a 1 dollar lottery ticket is bought in which the purchaser wins exactly 10 million dollars if the ticket contains the six winning numbers chosen from the set {1, 2, 3, ..., 50} and the purchaser wins nothing otherwise?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-43ac7fa4024f758b394e8d229a7d8f2c7232aba10b332707e9b97a0a67b7e2fe | What is the expected variance of a uniform random sample of size 10 from the first 10000 positive integers without replacement? | [
"\\dfrac{25002500}{3}"
] | [
"What is the expected variance of a uniform random sample of size 10 from the first 10000 positive integers without replacement?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-a26b681601d2f908dac59b3a68eada030371cd59b813af9ca5d6e6ad54995758 | What is the expected waiting time before a flight delay, given that the odds in favor of a Pythag-Air-US airlines flight being on time are 3:1? | [
"4"
] | [
"What is the expected waiting time before a flight delay, given that the odds in favor of a Pythag-Air-US airlines flight being on time are 3:1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-54e345e12fb1d9c58ab443e602a2cd46391c99d43c6cd407cda717d8c5fc1fcb | What is the expected waiting time for all six faces of a die to show up when rolling repeatedly? | [
"13.7"
] | [
"What is the expected waiting time for all six faces of a die to show up when rolling repeatedly?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4321f411b57b82b7af7397f6586f58150ceb6e9e82bebe2c36b44acfaa1a1f8a | What is the explicit bijection between the interval $(0,1)$ and the set of real numbers $\mathbb{R}$? | [
"\\( x \\mapsto \\tan \\left(\\pi \\cdot \\left(x-\\frac{1}{2}\\right)\\right) \\)"
] | [
"What is the explicit bijection between the interval $(0,1)$ and the set of real numbers $\\mathbb{R}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 3,
"tir": 22
} | 36 | 0 | 0 |
omr-ws-ed56f68d2ff59d63c55838951770cfdfdb14ff160ef15f0d584fce7222d72d68 | What is the exponent of 2 in the prime factorization of the LCM of the first 12 even fabulous numbers, where a fabulous number is defined as a counting number that is a square, a cube, a 4th power, a 5th power, a 6th power, a 7th power, an 8th power, a 9th power, a 10th power, an 11th power, and a 12th power? | [
"110880"
] | [
"What is the exponent of 2 in the prime factorization of the LCM of the first 12 even fabulous numbers, where a fabulous number is defined as a counting number that is a square, a cube, a 4th power, a 5th power, a 6th power, a 7th power, an 8th power, a 9th power, a 10th power, an 11th power, and a 12th power?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-57c4efd7dbe24f68f89c28c076fe523898fade45e21294758a1f944d4702d9c0 | What is the expression for the number of tetrahedra if we have n layers in a 3-dimensional stack, given that the first layer has 1 tetrahedron, the second layer has 5 tetrahedra, and so on? | [
"\\dfrac{n(2n^2 + 1)}{3}"
] | [
"What is the expression for the number of tetrahedra if we have n layers in a 3-dimensional stack, given that the first layer has 1 tetrahedron, the second layer has 5 tetrahedra, and so on?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-547901e3b55342ebed980e27d36def54f827fd062ae604623bba89c99320193c | What is the expression for the repeated differentiation $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\ldots\right)\right)\right)$? | [
"\\( c \\cdot e^x \\)"
] | [
"What is the expression for the repeated differentiation $\\dfrac{\\mathrm{d}}{\\mathrm{d}x}\\left(\\dfrac{\\mathrm{d}}{\\mathrm{d}x}\\left(\\dfrac{\\mathrm{d}}{\\mathrm{d}x}\\left(\\ldots\\right)\\right)\\right)$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aeb512ae41ae5e5a2785269063cf92424f3a62203268e10ffff4aed080807bca | What is the factored form of the quadratic expression \(2y^2 - 6y + 18\)? | [
"2(y^2 - 3y + 9)"
] | [
"What is the factored form of the quadratic expression \\(2y^2 - 6y + 18\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b16a48ecd17c69dd95415ac6da279d777af7dc897cc74b8abb42dc96e879b68b | What is the factorization of the expression \(x^5 - y^5\)? | [
"\\(x^5 - y^5 = (x - y)(x^4 + x^3y + x^2y^2 + xy^3 + y^4)\\)"
] | [
"What is the factorization of the expression \\(x^5 - y^5\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-d1252b6ab2e48cb663ed02cada4bea4f0bf83063db359e43269003968b74911d | What is the fastest way to count the numbers in the list: 3, 7, 11, 14, ..., 2007? | [
"502"
] | [
"What is the fastest way to count the numbers in the list: 3, 7, 11, 14, ..., 2007?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-767b848040b4ee53fc88426aaa1e1b9bbc9bcba8f378e9b600ef43a3d7550a4b | What is the fewest number of bits necessary to completely determine a four-bit code, if up to two bits in any code may be switched? | [
"11"
] | [
"What is the fewest number of bits necessary to completely determine a four-bit code, if up to two bits in any code may be switched?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5027097ab18f125fc77ea53af2b01ae5a4049cf122baeba2e46b22e077236088 | What is the fewest number of colors needed to uniquely color any set of regions on a torus such that no two touching regions have the same color? (Corners don’t count) | [
"7"
] | [
"What is the fewest number of colors needed to uniquely color any set of regions on a torus such that no two touching regions have the same color? (Corners don’t count)"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c481d33445bf465b7d76d0460c42c703dd963628db7393acd5bfc3f2933a3f74 | What is the fewest number of cuts needed to separate a wooden 3x3x3 cube into 27 unit cubes if you're allowed to move blocks of cubes about before cutting? | [
"6"
] | [
"What is the fewest number of cuts needed to separate a wooden 3x3x3 cube into 27 unit cubes if you're allowed to move blocks of cubes about before cutting?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-080a8e34c41d18646b65881036a21fa03508418fa70368e59307b7855cc934b6 | What is the fewest number of moves needed to land in the square immediately to the right of the starting square, given that on each turn the marker can be moved 11, 18, or 25 spaces to either the right or left? | [
"6"
] | [
"What is the fewest number of moves needed to land in the square immediately to the right of the starting square, given that on each turn the marker can be moved 11, 18, or 25 spaces to either the right or left?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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