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-b597a6c6f5f1f926daaa679425da045daaad6d5bc98874bd7166be61badd2c06 | What is the asymptotic behavior of the expected value of the largest prime factor of \( n \)? Specifically, consider the expression \[ \frac{1}{2k+1}\sum_{|m-n| \le k}E(m) \] as \( n, k \to \infty \) and \( n >> k \), where \( E(m) \) denotes the largest prime factor of \( m \). | [
"\\frac{\\pi^2}{12} \\frac{n}{\\log n}"
] | [
"What is the asymptotic behavior of the expected value of the largest prime factor of \\( n \\)? Specifically, consider the expression\n\\[ \\frac{1}{2k+1}\\sum_{|m-n| \\le k}E(m) \\]\nas \\( n, k \\to \\infty \\) and \\( n >> k \\), where \\( E(m) \\) denotes the largest prime factor of \\( m \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-9b430c03b7ff43537bc13028f244e508b26240ee55de58fee2a913a201e44612 | What is the asymptotic behavior of the series $\sum_{n>0}u^{n^2}$ as $u$ approaches 1 from below? | [
"\\(\\sqrt{\\frac{\\pi}{4(1-u)}}\\)"
] | [
"What is the asymptotic behavior of the series $\\sum_{n>0}u^{n^2}$ as $u$ approaches 1 from below?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-ea487e069c985e2db419c7f45b68ad897df22486372c604d15e5cdd9b582c5eb | What is the asymptotic behavior of the sum \(\sum_{r=0}^{n}\cos\left(\frac{\sqrt{r}}{n}\right)\) as \(n\) approaches \(\infty\)? | [
"\\(n + \\frac{3}{4} - \\frac{17}{72}n^{-1} + \\frac{59}{2880}n^{-2} + O(n^{-3})\\)"
] | [
"What is the asymptotic behavior of the sum \\(\\sum_{r=0}^{n}\\cos\\left(\\frac{\\sqrt{r}}{n}\\right)\\) as \\(n\\) approaches \\(\\infty\\)?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 18
} | 20 | 0 | 0 |
omr-ws-48992ad0811266c89bbe60938c1d8b7e4384f9ae5dfd563dd984a11f3f971b91 | What is the asymptotic complexity (big theta notation) of $f(n, p)$, where $f(n, p)$ is the least positive integer value of $k$ such that the probability that some pair of the $k$ people in the room share the same birthday is at least $p$, assuming a year has $n$ days? | [
"\\(\\Theta(\\sqrt{n})\\)"
] | [
"What is the asymptotic complexity (big theta notation) of $f(n, p)$, where $f(n, p)$ is the least positive integer value of $k$ such that the probability that some pair of the $k$ people in the room share the same birthday is at least $p$, assuming a year has $n$ days?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 4,
"tir": 9
} | 22 | 0 | 0 |
omr-ws-2de2d36505847082f9ddd49ad0960faf9ac13c2aab10f92f456ed16f0674aaad | What is the asymptotic density of integers \( n \) such that the three consecutive numbers \( 4n + 1 \), \( 4n + 2 \), and \( 4n + 3 \) are all square-free? If this density is not zero, does it imply that there are infinitely many such integers \( n \)? | [
"\\text{Yes}"
] | [
"What is the asymptotic density of integers \\( n \\) such that the three consecutive numbers \\( 4n + 1 \\), \\( 4n + 2 \\), and \\( 4n + 3 \\) are all square-free? If this density is not zero, does it imply that there are infinitely many such integers \\( n \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-14cf256a4c6f42e9cacff192cb2c2854ff2fe4923db3a2af22c7b2e9b941a453 | What is the asymptotic limit of \( \frac{S(n)}{n^2} \) as \( n \) approaches infinity, where \( S(n) = 1^{\mu(1)} + 2^{\mu(2)} + 3^{\mu(3)} + \cdots + n^{\mu(n)} \) and \( \mu(n) \) is the Möbius function of \( n \)? | [
"\\dfrac{3}{2\\pi^2}"
] | [
"What is the asymptotic limit of \\( \\frac{S(n)}{n^2} \\) as \\( n \\) approaches infinity, where \\( S(n) = 1^{\\mu(1)} + 2^{\\mu(2)} + 3^{\\mu(3)} + \\cdots + n^{\\mu(n)} \\) and \\( \\mu(n) \\) is the Möbius function of \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4e5390dec68e14fd77f9416d1b0ce13320ec9a7958bd4b953d8aa22953b29888 | What is the asymptotic probability that the number of copies of $K_4$ in $G(n, \sqrt{n})$ is greater than $n$? | [
"0"
] | [
"What is the asymptotic probability that the number of copies of $K_4$ in $G(n, \\sqrt{n})$ is greater than $n$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-159bc24d78348fa930c49ee52bb20c30da3c04412b843379e211e2047e5c0304 | What is the asymptotic upper bound on the divisor function (number of divisors of an integer)? | [
"\\( O(n^\\epsilon) \\) for any \\( \\epsilon > 0 \\)"
] | [
"What is the asymptotic upper bound on the divisor function (number of divisors of an integer)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 2,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-acf394e16e9c27751101872b3998cf4e89a40925e8f73a106b4b3851b20860d0 | What is the average area of a triangle given that its circumradius is 1? | [
"\\(\\frac{11\\pi}{64}\\)"
] | [
"What is the average area of a triangle given that its circumradius is 1?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c097ce188d01ece93ec2e845dc898ca6ef99cfe4bb4b2582f8b78a5dcd52a604 | What is the average area of triangle ABC, where the points A(1,a), B(2,b), and C(3,c) are defined with a, b, and c ranging over the set {0,1,2,3}? (If A, B, and C are collinear, consider the area to be 0.) | [
"\\(\\frac{3}{4}\\)"
] | [
"What is the average area of triangle ABC, where the points A(1,a), B(2,b), and C(3,c) are defined with a, b, and c ranging over the set {0,1,2,3}? (If A, B, and C are collinear, consider the area to be 0.)"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f35a280c253859e1c7333d32bf0e4ab57832cc18b8ddb6f6d7db61fe2403876e | What is the average area of triangle ABC, where the points A(1,a), B(2,b), and C(3,c) are defined with a, b, and c ranging over the set {0,1,2,3}? Count collinear points as a triangle with area 0. | [
"\\dfrac{9}{8}"
] | [
"What is the average area of triangle ABC, where the points A(1,a), B(2,b), and C(3,c) are defined with a, b, and c ranging over the set {0,1,2,3}? Count collinear points as a triangle with area 0."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e876c69c1861d8cf5264e04da34ab45b42d6ce15844b3e3262b22d13d40c102 | What is the average book width, in centimeters, of five books with the following widths: $6$, $\frac{1}{2}$, $1$, $2.5$, and $10$? | [
"4"
] | [
"What is the average book width, in centimeters, of five books with the following widths: $6$, $\\frac{1}{2}$, $1$, $2.5$, and $10$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-44453a99ec88eb21957263cfcfb4cb0b34b60401f98ae2f9d3418aa155dc41f8 | What is the average distance between 8 random points in a sector of a circle that covers 1/3rd of the circle's area? | [
"\\dfrac{128R}{45\\pi}"
] | [
"What is the average distance between 8 random points in a sector of a circle that covers 1/3rd of the circle's area?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-c90b59970ba9951783a0fd78f1c48fcc97c3d569cc4e67375294480ca48beda4 | What is the average distance between two points in a unit square? | [
"\\dfrac{2 + \\sqrt{2} + 5 \\ln(1 + \\sqrt{2})}{15}"
] | [
"What is the average distance between two points in a unit square?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e6366b680a971ad36e47707ecbb8e0729bcb303aefb00acd48127427c2a39d16 | What is the average distance between two points on a square? No other problems are identified in the post. | [
"\\dfrac{2 + \\sqrt{2} + 5 \\ln(1 + \\sqrt{2})}{15}"
] | [
"What is the average distance between two points on a square?\n\nNo other problems are identified in the post."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-92c23c680cb81c969236d96f707dbc00330fd63b913d25fcc9a8bc7363aab09d | What is the average distance between two randomly chosen points in a square with a side length of $1$? | [
"\\( \\frac{2+\\sqrt{2}+5\\ln(1+\\sqrt{2})}{15} \\approx 0.5214 \\)"
] | [
"What is the average distance between two randomly chosen points in a square with a side length of $1$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-0178d09b062334dd4896558bb4606ff63249bdaf93ed48485a2a8ebfbf3f7ae9 | What is the average distance between two randomly chosen points in a unit square? | [
"\\(\\frac{1}{15}(2 + \\sqrt{2} + 5\\log(1+\\sqrt{2}))\\)"
] | [
"What is the average distance between two randomly chosen points in a unit square?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-bf774b225dd67f41c3bdb6ae44889bad0e7b3b6c382a0f9127ce269e2245af12 | What is the average head length of a permutation of the sequence $[1, 1, 2, 3, \dotsc, n]$? | [
"\\dfrac{n-1}{3}"
] | [
"What is the average head length of a permutation of the sequence $[1, 1, 2, 3, \\dotsc, n]$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-f5074feb57f033947ec876989790db5f3376400279b84153ab6ee7ca0f72132b | What is the average head length of a sequence of length $n$ consisting of the numbers $\{1, 2, 3, 4\}$? | [
"\\dfrac{4}{3}\\left(1 - \\left(\\dfrac{1}{4}\\right)^n\\right)"
] | [
"What is the average head length of a sequence of length $n$ consisting of the numbers $\\{1, 2, 3, 4\\}$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a83c645bbcd1eef6e8306ff0c2e479a820574dbcf72893e8c2541e2cfbb1c67b | What is the average length of the longest increasing subsequence of a random permutation of $1, 2, \cdots, n$? | [
"\\(2\\sqrt{n}\\)"
] | [
"What is the average length of the longest increasing subsequence of a random permutation of $1, 2, \\cdots, n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-1093c263bf22d7689a2857823a922d835ccca8725e48a13322c468b568cf5a43 | What is the average number of cards drawn from a shuffled deck until an ace is drawn? | [
"\\(\\frac{53}{5}\\)"
] | [
"What is the average number of cards drawn from a shuffled deck until an ace is drawn?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-183213b227e23c4506022eee5309ef4b1a0ce098230022bd499627d97428175f | What is the average number of coin tosses required to obtain 2 heads in a row? | [
"6"
] | [
"What is the average number of coin tosses required to obtain 2 heads in a row?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a429da2ce98a2a5ddc62867a2876ff720fbf3f35abbd0f074be474d124af300b | What is the average number of independent sets of size $k$ in graphs with $V$ vertices and $E$ edges? | [
"\\(\\frac{\\binom{V}{k} \\binom{\\binom{V}{2} - \\binom{k}{2}}{E}}{\\binom{\\binom{V}{2}}{E}}\\)"
] | [
"What is the average number of independent sets of size $k$ in graphs with $V$ vertices and $E$ edges?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 16
} | 63 | 0 | 0 |
omr-ws-37fb5aca6d894f02afda47b1e5cf86076d7b3ca5e82f8720c2d1fa64fa27be72 | What is the average number of numbers you need to pick before repeating when picking out of \( n \) numbers? | [
"\\sum_{k=0}^n \\frac{n!}{n^k (n - k)!}"
] | [
"What is the average number of numbers you need to pick before repeating when picking out of \\( n \\) numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cbdbcb088620db6d0d0174347fefca341e663884167bd14d2bdc166cb4b6e1af | What is the average number of pairs of consecutive integers in a randomly selected subset of 5 distinct integers chosen from the set $\{ 1, 2, 3, \ldots, 30\}$? | [
"\\(\\frac{2}{3}\\)"
] | [
"What is the average number of pairs of consecutive integers in a randomly selected subset of 5 distinct integers chosen from the set $\\{ 1, 2, 3, \\ldots, 30\\}$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-2186e3792e893005e51c1939e829edd0044a3045b59a7a5d7639a58bcea74fc9 | What is the average number of runs in an n-digit integer where adjacent digits are always different and the leading digit can be 0? A run is defined as a maximal string of digits that are either strictly increasing or strictly decreasing. | [
"\\dfrac{19n - 11}{27}"
] | [
"What is the average number of runs in an n-digit integer where adjacent digits are always different and the leading digit can be 0? A run is defined as a maximal string of digits that are either strictly increasing or strictly decreasing."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-edd6a129fe2ab57283abf92933a04bd3013dfb44a53e64d619d03059da884890 | What is the average number of times a coin should be tossed such that the number of heads and tails are the same? | [
"Infinite"
] | [
"What is the average number of times a coin should be tossed such that the number of heads and tails are the same?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-7bfcd10f8d1a99c4758ca565ebde4c40b0a2488663bb7d38f628a67a0e6e352d | What is the average number of times one should toss a coin to get \( n \) heads in a row, where \( n \) is any positive integer? | [
"2^{n+1} - 2"
] | [
"What is the average number of times one should toss a coin to get \\( n \\) heads in a row, where \\( n \\) is any positive integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed7203c3a648b5a7b82d9129d1af32e41c520a9c1c25793ebcae12c7f61b7131 | What is the average number of tosses required to get two heads in a row when a coin is repeatedly thrown? | [
"6"
] | [
"What is the average number of tosses required to get two heads in a row when a coin is repeatedly thrown?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e3f28989cc9cbfba981b1f28787c381888c49610242916b8063225fc916590f | What is the average of all 3-digit even numbers with distinct digits formed by choosing from the digits 1, 3, 4, 6? | [
"335"
] | [
"What is the average of all 3-digit even numbers with distinct digits formed by choosing from the digits 1, 3, 4, 6?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-56b2af346c43878d1d7e5f93c98f9ab735315076b7bc36baf1b91522214e432e | What is the average of all 4-digit numbers with distinct digits formed by choosing from the digits 1, 3, 4, 6? | [
"3888.5"
] | [
"What is the average of all 4-digit numbers with distinct digits formed by choosing from the digits 1, 3, 4, 6?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-2c481fc423c63c371fefd5f8aaa14386e8245e7b4df30cd47d94705cc17bf994 | What is the average of all different (m+n)-digit numbers where each consists of the digit m m times and the digit n n times? | [
"\\dfrac{(m^2 + n^2)(10^{m+n} - 1)}{9(m + n)}"
] | [
"What is the average of all different (m+n)-digit numbers where each consists of the digit m m times and the digit n n times?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bee9d35df8d4f40cdf679615dddd3665328626e6a20727a0f55624ed7a276047 | What is the average of all different 9-digit numbers where each consists of the digit 5 five times and the digit 4 four times? | [
"506172839"
] | [
"What is the average of all different 9-digit numbers where each consists of the digit 5 five times and the digit 4 four times?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-8cb26824bb7b7dd077d075ed93d0e517bee4d26f553cec68ebf65ac4d8701a79 | What is the average of all integer divisors of $222$? | [
"0"
] | [
"What is the average of all integer divisors of $222$?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-3dbf697f2dc98b1393ee2ee379995914ac90e4c0a8d4a7ae9b02867e2bda568d | What is the average of all the integer values of $M$ such that $\frac{M}{56}$ is strictly between $\frac{3}{7}$ and $\frac{1}{4}$? | [
"19"
] | [
"What is the average of all the integer values of $M$ such that $\\frac{M}{56}$ is strictly between $\\frac{3}{7}$ and $\\frac{1}{4}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e766ffec327fe179c0ce5a33e386d288cce00e62cb6bcc511e3f93742c7be9c | What is the average of all two-digit integers that satisfy the condition that the number is equal to the sum of the ten's digit and the unit's digit and the product of the ten's digit and the unit's digit? | [
"59"
] | [
"What is the average of all two-digit integers that satisfy the condition that the number is equal to the sum of the ten's digit and the unit's digit and the product of the ten's digit and the unit's digit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-c43db35f2213202a72df6a76c1b78b000cf52515a78e388c598f0f3ce04a455c | What is the average of the elements occurring in the $r$-subsets of $\{1, 2, 3, \ldots, n\}$? Note: $r$-subsets refer to elements taken $r$ at a time, and the elements are arranged in ascending order from left to right. | [
"\\(\\frac{n+1}{2}\\)"
] | [
"What is the average of the elements occurring in the $r$-subsets of $\\{1, 2, 3, \\ldots, n\\}$? Note: $r$-subsets refer to elements taken $r$ at a time, and the elements are arranged in ascending order from left to right."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 23
} | 46 | 0 | 0 |
omr-ws-55475ccec7cf4659d4f4186bc0777b3ee97ca1c2a1f2ed4d47a059e43347df98 | What is the average squared distance between two randomly and uniformly selected points inside the circle \( x^2 + y^2 \le 1 \)? (Note: The explicit use of densities is NOT allowed.) | [
"1"
] | [
"What is the average squared distance between two randomly and uniformly selected points inside the circle \\( x^2 + y^2 \\le 1 \\)? \n\n(Note: The explicit use of densities is NOT allowed.)"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-274c0379cb56d327ac1f232b0575f15b28dca37e3d15955c6330d2b0414c991c | What is the average temperature of the surface of a spherical planet with center $(0,0,0)$ and radius $20$, where the temperature at any point $(x,y,z)$ on the surface is given by $T(x,y,z) = (x + y)^2 + (y - z)^2$ degrees? | [
"\\dfrac{1600}{3}"
] | [
"What is the average temperature of the surface of a spherical planet with center $(0,0,0)$ and radius $20$, where the temperature at any point $(x,y,z)$ on the surface is given by $T(x,y,z) = (x + y)^2 + (y - z)^2$ degrees?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-06d1a2907821dcca5af04bdaa9f12a80785519dcf63ae568d8d14121d9d9233d | What is the base $2$ representation of $84_{10}$? | [
"1010100_2"
] | [
"What is the base $2$ representation of $84_{10}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6641585ca43211e1579df28584bad35ca98d6ffad6492b5b5a795bbb547a4c9d | What is the base 4 representation of the base 2 number $11011000_2$? | [
"3120_4"
] | [
"What is the base 4 representation of the base 2 number $11011000_2$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-84609f62adb1dbd5b7da0298bc4724e2ba9094445b463cda4831a551c2f2a169 | What is the base five product of the numbers $121_{5}$ and $11_{5}$? | [
"1331"
] | [
"What is the base five product of the numbers $121_{5}$ and $11_{5}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eb276df89483971d7e0a17ac35b7b4fe986de20cc917f893d990933d97bd6f80 | What is the base four equivalent of $123_{10}$? | [
"1323_{4}"
] | [
"What is the base four equivalent of $123_{10}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c3d9d7fd8787608f6125e48117eb67dbf3b6bf6b15e025d88ff5e08cb53090b8 | What is the base ten equivalent of $12345_{6}$? | [
"1865"
] | [
"What is the base ten equivalent of $12345_{6}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-303f769b6d3a5df64aaf6ffe01235425f26457b4a07066dccf0b95784f3e45f5 | What is the base ten equivalent of $54321_6$? | [
"7465"
] | [
"What is the base ten equivalent of $54321_6$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-52aaad5f25cd971b68de19b5fc0865174b5d23afceee6e74b264dbe93948c1c2 | What is the base-10 integer 515 when expressed in base 6? | [
"2215_6"
] | [
"What is the base-10 integer 515 when expressed in base 6?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8f8058c24f04a6c5b87bf30f85a6e391dff91929e7bc30f86668ddbc7499fe9a | What is the base-10 sum of all positive integers such that, when expressed in binary, they have 7 digits and no two consecutive 1’s? | [
"963"
] | [
"What is the base-10 sum of all positive integers such that, when expressed in binary, they have 7 digits and no two consecutive 1’s?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-6e1fff028c93415538d41492aaee4cd5116ad420d4a93c5e07d495030263a1d1 | What is the base-ten number represented by the base-eight number 31? | [
"25"
] | [
"What is the base-ten number represented by the base-eight number 31?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c0f86c2b29d6ef4cabb8384921711f7ee6686d5a1d3536537f138e4e65d4f9da | What is the behavior of the fractional part of $(1+\sqrt{2})^k$ as $k \to \infty$? | [
"\\text{The fractional part approaches } 0 \\text{ for odd } k \\text{ and } 1 \\text{ for even } k \\text{ as } k \\to \\infty."
] | [
"What is the behavior of the fractional part of $(1+\\sqrt{2})^k$ as $k \\to \\infty$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-7f0b8189a18be833f6116cde22330277bea1c87c469534743e112d6ce5fa32c0 | What is the best approximation for $\pi(x)$ in the form $\pi(x+a) \approx \frac{x}{\ln(bx)}$ for real numbers $a$ and $b$? | [
"a = 0 \\text{ and } b = \\frac{1}{e}"
] | [
"What is the best approximation for $\\pi(x)$ in the form $\\pi(x+a) \\approx \\frac{x}{\\ln(bx)}$ for real numbers $a$ and $b$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-1c6d66671a9bc7df2bd12757cf202d32fc4e2b864b3facaa42ead20c5456a9ad | What is the best bound to compare the number of distinct prime factors of \( m! \) and \( n! \), where \( m \) and \( n \) are natural numbers? | [
"\\omega(m!) \\geq \\omega(n!) \\text{ if and only if } m \\geq n"
] | [
"What is the best bound to compare the number of distinct prime factors of \\( m! \\) and \\( n! \\), where \\( m \\) and \\( n \\) are natural numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-44c2e51519df70bb034c9e04412ea0c404fc44123062dbec28fa9945f7685878 | What is the best constant $ k_n $ such that for any real numbers $ a_1, a_2, \ldots, a_n $, the inequality $ (a_1-a_2)^2 + (a_2-a_3)^2 + \cdots + (a_n-a_1)^2 \geq k_n \sum_{i<j} (a_i-a_j)^2 $ holds? | [
"\\dfrac{4}{n} \\sin^2\\left( \\dfrac{\\pi}{n} \\right)"
] | [
"What is the best constant $ k_n $ such that for any real numbers $ a_1, a_2, \\ldots, a_n $, the inequality $ (a_1-a_2)^2 + (a_2-a_3)^2 + \\cdots + (a_n-a_1)^2 \\geq k_n \\sum_{i<j} (a_i-a_j)^2 $ holds?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-09d3e39c3bbe75861c24f4d9da22a87292e165d49078c4c2695175a43b40143b | What is the best constant \( c \) for which the inequality \( x^2 + y^2 + xy + 1 \geq c(x + y) \) holds? | [
"1"
] | [
"What is the best constant \\( c \\) for which the inequality \\( x^2 + y^2 + xy + 1 \\geq c(x + y) \\) holds?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cc3a196bffb61024a6bd6c6278793f9037bf51846d0b8fa6978d3d3227bac59a | What is the best general method to solve the system of equations given by: \[ a + b + c = p \] \[ ab + bc + ac = m \] \[ abc = n \] | [
"Solve the polynomial \\(x^3 - px^2 + mx - n = 0\\) to find the roots \\(a, b, c\\)."
] | [
"What is the best general method to solve the system of equations given by:\n\\[ a + b + c = p \\]\n\\[ ab + bc + ac = m \\]\n\\[ abc = n \\]"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-c71f0fb040764aae335f0d1f9f22c3b2cfe310ff63fd0744e234e116e860e5d2 | What is the best positive real number \( k \) such that for any \( a, b, c > 0 \), the following inequality holds: \[ \sum_{\text{cyc}} \frac{a^2}{b^2} + k \ge (3 + k) \frac{a^2 + b^2 + c^2}{ab + bc + ca} \]? | [
"6.263970365"
] | [
"What is the best positive real number \\( k \\) such that for any \\( a, b, c > 0 \\), the following inequality holds:\n\\[ \\sum_{\\text{cyc}} \\frac{a^2}{b^2} + k \\ge (3 + k) \\frac{a^2 + b^2 + c^2}{ab + bc + ca} \\]?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-e76acaa06e08d3d2a25ef9d7e1111e0404818b09fdb0129cb9042d29d56f6398 | What is the best strategy for John to win at least 3 dollars when flipping a coin 3 times, and what is the probability of winning with that strategy? | [
"\\dfrac{1}{8}"
] | [
"What is the best strategy for John to win at least 3 dollars when flipping a coin 3 times, and what is the probability of winning with that strategy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-67429198c3964fe5aa3a9eeef1a108e2ec5b25c4c218282ba3f420ff7b4c5662 | What is the best upper bound for the expression $\sqrt{x(y+z)}+\sqrt{y(z+x)}+\sqrt{z(x+y)}$? | [
"\\( \\sqrt{6(xy+yz+zx)} \\)"
] | [
"What is the best upper bound for the expression $\\sqrt{x(y+z)}+\\sqrt{y(z+x)}+\\sqrt{z(x+y)}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-e9626496290898d472055955c0e2f82c282cb9ecfdf0575e2a77a200af6b033d | What is the biggest natural number \( n \) such that the inequality \( a^4 - na + 8 > 0 \) holds for all real values of \( a \)? | [
"8"
] | [
"What is the biggest natural number \\( n \\) such that the inequality \\( a^4 - na + 8 > 0 \\) holds for all real values of \\( a \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-066e006ea5b792f754414b7fd3ea3d7fed24cd32199c19c024f5f76123b01bd0 | What is the biggest prime factor of the number $13^4 + 16^5 - 172^2$? | [
"1321"
] | [
"What is the biggest prime factor of the number $13^4 + 16^5 - 172^2$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eec86a9e51f903ca6eabacde7b1663d7eafc68f50cf6ed9a67149b65ca1a9b80 | What is the biggest shadow that a cube of side length 1 can have, with the sun at its peak? Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane. | [
"\\sqrt{3}"
] | [
"What is the biggest shadow that a cube of side length 1 can have, with the sun at its peak? Note: \"The biggest shadow of a figure with the sun at its peak\" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-bc293e3a2e8b4034c62743589b9460096063e460d36e5358fe9afe459bde6885 | What is the binary (base 2) representation of the base 10 fraction $\frac{1}{5}$? | [
"0.\\overline{0011}"
] | [
"What is the binary (base 2) representation of the base 10 fraction $\\frac{1}{5}$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ad4b3db72d1b4d016d0fd63a9e62242f7b3c0817952572c7b3178a282e729e5d | What is the binary representation of $ \frac{1}{7} $? | [
"\\( 0.001001_2 \\)"
] | [
"What is the binary representation of $ \\frac{1}{7} $?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-4af7500557dd427aa37155a559c1c9096a96939ecf28859ca4621a7572d24a40 | What is the bounce number for the initial state $[(x, y); \overrightarrow{(p, q)}]$ where $p, q$ are relatively prime positive integers, assuming the striker never hits a corner? | [
"p + q - 2"
] | [
"What is the bounce number for the initial state $[(x, y); \\overrightarrow{(p, q)}]$ where $p, q$ are relatively prime positive integers, assuming the striker never hits a corner?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-5465c08cdd067ad73afd943f58e099cd77f4cf49d6128fda5bfc94f61cc83dd9 | What is the cardinality of $\mathcal{F}_\sigma$, the set of all countable unions of elements of $\mathcal{F}$? | [
"$2^c$"
] | [
"What is the cardinality of $\\mathcal{F}_\\sigma$, the set of all countable unions of elements of $\\mathcal{F}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 3,
"tir": 13
} | 26 | 0 | 0 |
omr-ws-8c36648c0ee142ddb4c9b9a6f6df0196322b155dcc7b32ed87070cbdf13b06c0 | What is the cardinality of Lebesgue non-measurable sets, assuming the axiom of choice? | [
"2^{\\aleph_0}"
] | [
"What is the cardinality of Lebesgue non-measurable sets, assuming the axiom of choice?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-dce86633542fa83960181674bd9b86fae5e3c439f584454bd0bf5b00f7c72022 | What is the cardinality of all dense subsets of $\mathbb{R}$, where a dense subset $S \subseteq \mathbb{R}$ satisfies $\bar{S} = \mathbb{R}$? | [
"$card(P(\\mathbb{R}))$"
] | [
"What is the cardinality of all dense subsets of $\\mathbb{R}$, where a dense subset $S \\subseteq \\mathbb{R}$ satisfies $\\bar{S} = \\mathbb{R}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-c1d750be390773e194ba676cb7da760dd1efcd58f84ffefcf9b2ea491b7994d2 | What is the cardinality of the family $\mathcal{F}$ of all closed subsets of the Euclidean plane $\mathbb{R}^2$? | [
"$\\mathfrak{c}$ or $2^{\\aleph_0}$"
] | [
"What is the cardinality of the family $\\mathcal{F}$ of all closed subsets of the Euclidean plane $\\mathbb{R}^2$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-067234e1c1abb89de124d229061269d849b67a080d9262304804b6b91d418a89 | What is the cardinality of the set $\{\varphi(1), \varphi(2), \ldots, \varphi(n)\}$? | [
"A082695(n)"
] | [
"What is the cardinality of the set $\\{\\varphi(1), \\varphi(2), \\ldots, \\varphi(n)\\}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-5eec5d778f24130ca116dd16bd52585e30e3677c60227c903d2b2888d73d0ec8 | What is the cardinality of the set $\{x \mid x^d = 1\}$ where $d$ is a divisor of $2^{nm} - 1$? | [
"d"
] | [
"What is the cardinality of the set $\\{x \\mid x^d = 1\\}$ where $d$ is a divisor of $2^{nm} - 1$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-15345ad8a8e17cdf0410fceac7145a2f7f143d2abbd9f8d17d8b0c7bf8c0be15 | What is the cardinality of the set of all functions $f: \mathbb{R} \to \mathbb{R}$ which satisfy $f(x+y) = f(x) + f(y)$ for all $x$ and $y$? | [
"\\( 2^{|\\mathbb{R}|} \\)"
] | [
"What is the cardinality of the set of all functions $f: \\mathbb{R} \\to \\mathbb{R}$ which satisfy $f(x+y) = f(x) + f(y)$ for all $x$ and $y$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-2bac89f11bc0a7ca3c09193fb54341074a16254f9c30bd9a1edd5da7c59efc82 | What is the cardinality of the set of automorphisms of the complex field $\mathbf{C}$? | [
"$2^c$"
] | [
"What is the cardinality of the set of automorphisms of the complex field $\\mathbf{C}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-479cb4bcc8422cc84d4108ea4fc828834e1a0a3e63d4d753674a8217ceb00205 | What is the cardinality of the set of one-variable functions that are everywhere differentiable on $(-\infty, \infty)$? | [
"\\mathfrak{c}"
] | [
"What is the cardinality of the set of one-variable functions that are everywhere differentiable on $(-\\infty, \\infty)$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a51b88c0c85160109ed8301693c50c2a8ddd0794ec1ce92bc9746162b124a984 | What is the center of the circle passing through the point $(6,0)$ and tangent to the circle $x^{2}+y^{2}=4$ at $(0,2)$. | [
"\\((0, -8)\\)"
] | [
"What is the center of the circle passing through the point $(6,0)$ and tangent to the circle $x^{2}+y^{2}=4$ at $(0,2)$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b045712e511fe34ed6454fa3a51f4e3175df57dd43f70525c5e3ea21ad2fa3b | What is the center of the conic section given by the polar equation \( r = \frac{2}{3 + 5 \cos(\theta)} \) when expressed in Cartesian form? | [
"\\(\\left(\\frac{5}{8}, 0\\right)\\)"
] | [
"What is the center of the conic section given by the polar equation \\( r = \\frac{2}{3 + 5 \\cos(\\theta)} \\) when expressed in Cartesian form?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-33097ef811c6418d2dd3eb57ef53ea10784926295f25a237658afdae6fb51f4e | What is the central angle of a hemisphere? | [
"180^\\circ"
] | [
"What is the central angle of a hemisphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-85305c2260f7d241b96eb1b7042692bc266207bab0638de4941226b907ba1504 | What is the centralizer of the element $(12)$ in $S_{5}$? | [
"\\(C_2 \\times S_3\\)"
] | [
"What is the centralizer of the element $(12)$ in $S_{5}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-424b92ce3f309620baf436ed77e7119afda4549f5d42e18157e8983b5f12b3ca | What is the centralizer of the element $(12)(34)(56)$ in the alternating group $A_6$? | [
"\\(S_4\\)"
] | [
"What is the centralizer of the element $(12)(34)(56)$ in the alternating group $A_6$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 10
} | 27 | 0 | 0 |
omr-ws-37bdf1e976f6fcef4252d372d018828e3c67342d0b9a29cb135bfe27ac9f6ed8 | What is the centre and radius of the circle represented by $\left| \frac{z+1}{z-2} \right| = 2$? | [
"Centre $(3,0)$, Radius $2$"
] | [
"What is the centre and radius of the circle represented by $\\left| \\frac{z+1}{z-2} \\right| = 2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 27
} | 28 | 0 | 0 |
omr-ws-021fecfb3bb30d620ec0c2b094697ba1f3ab467c3d9ae3d1a47b0956c100b5df | What is the characteristic of the commutative ring \( R = F_2[t] / t^3 F_2[t] \)? | [
"2"
] | [
"What is the characteristic of the commutative ring \\( R = F_2[t] / t^3 F_2[t] \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4bd633281aa8137e648841a4b01f57fe2c8e43c622d38dd40c142b7d1b3b78e4 | What is the characterization of the midpoint of the altitude and the foot of the altitude (from point A) in Barycentric coordinates? | [
"\\left(\\frac{1}{2}, \\frac{a^2 + b^2 - c^2}{4a^2}, \\frac{a^2 + c^2 - b^2}{4a^2}\\right)"
] | [
"What is the characterization of the midpoint of the altitude and the foot of the altitude (from point A) in Barycentric coordinates?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-16889a46f729d989fb309adb67c528b4d73d71a31fa506b2c1153caab2bd4fa3 | What is the circumcenter of the triangle $\triangle{I_{a}I_{b}I_{c}}$ in relation to the triangle $\triangle{ABC}$? | [
"The reflection of the incenter over the circumcenter."
] | [
"What is the circumcenter of the triangle $\\triangle{I_{a}I_{b}I_{c}}$ in relation to the triangle $\\triangle{ABC}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-b48f2599b898f63f202fd72c18a9c1cd668e473a2e1de29ecbb31b4981584d27 | What is the class equation of the alternating group A4? | [
"\\( 1 + 3 + 8 \\)"
] | [
"What is the class equation of the alternating group A4?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-57f9e19136ec4a7806dc3a16046f5ee0de1dca24438a86bf80b5f1006e1d567b | What is the class equation of the direct product $K_4 \times Q_8$, where $K_4$ is the Klein four-group and $Q_8$ is the quaternion group? | [
"\\(32=1+1+1+1+1+1+1+1+2+2+2+2+2+2+2+2+2+2+2+2\\)"
] | [
"What is the class equation of the direct product $K_4 \\times Q_8$, where $K_4$ is the Klein four-group and $Q_8$ is the quaternion group?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 13
} | 15 | 0 | 0 |
omr-ws-7d2f98fb795021dcda63727fa71fbbd3701853fdebca02f8f2015bdcafe9b486 | What is the closed form expression for the sum $1^k + 2^k + \cdots + n^k$? | [
"\\(\\sum_{k=1}^n k^p = \\frac{1}{p+1} \\sum_{j=0}^p \\binom{p+1}{j} B_j n^{p+1-j}\\)"
] | [
"What is the closed form expression for the sum $1^k + 2^k + \\cdots + n^k$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-70e5dd491200b6292936e685150738f8d8d752a701e014cb516201f56541e187 | What is the closed form for $\text{det}(A+xB)$ where $A$ and $B$ are $3 \times 3$ matrices? | [
"\\det(A) + x \\left( \\sum_{i=1}^3 \\det(A^{(i)}) \\right) + x^2 \\left( \\sum_{1 \\leq i < j \\leq 3} \\det(A^{(i,j)}) \\right) + x^3 \\det(B)"
] | [
"What is the closed form for $\\text{det}(A+xB)$ where $A$ and $B$ are $3 \\times 3$ matrices?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-33f20c483e2b428d6f21bf84f21462021125f1e2498c9396ff4a80f8b2b07e80 | What is the closed form for $\zeta(2k)$, where $k$ is an integer? | [
"\\(\\zeta(2k) = \\frac{1}{2} (2\\pi)^{2k} \\frac{B_k}{(2k)!}\\)"
] | [
"What is the closed form for $\\zeta(2k)$, where $k$ is an integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c46ee8dfd436d0bbba28227dba845809f1c512906779e0871d5494892d049c9e | What is the closed form of the sum \(1^1 + 2^2 + 3^3 + \cdots + n^n\)? | [
"\\sum_{k=1}^n k^k"
] | [
"What is the closed form of the sum \\(1^1 + 2^2 + 3^3 + \\cdots + n^n\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a332a15b32c2424e684c4711d83b26df34e21688c7e5059ced855ce0ac98e3ee | What is the closed formula for \( \sum_{n=0}^{\infty} n^k x^n \)? | [
"\\frac{x A_k(x)}{(1 - x)^{k + 1}}"
] | [
"What is the closed formula for \\( \\sum_{n=0}^{\\infty} n^k x^n \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-96edb9eb52326b48953ccf9e16ba4572d05585579c7320bee7525ac793d276fe | What is the closed-form solution of $\displaystyle\sum_{n=0}^\infty {3n \choose n}x^n$? | [
"\\frac{2 \\cos\\left( \\frac{1}{3} \\arccos\\left(1 - \\frac{27}{2}x\\right) \\right)}{\\sqrt{4 - 27x}}"
] | [
"What is the closed-form solution of $\\displaystyle\\sum_{n=0}^\\infty {3n \\choose n}x^n$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-7e667ba82c21afc9867e22a08a315546d5038a129da8fbd50ff32da529fad6d5 | What is the closest average interplanetary speed in miles per hour of NASA’s Perseverance Rover, which traveled $292{,}526{,}838$ miles over a period of approximately $6.5$ months? | [
"60,000"
] | [
"What is the closest average interplanetary speed in miles per hour of NASA’s Perseverance Rover, which traveled $292{,}526{,}838$ miles over a period of approximately $6.5$ months?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 25
} | 38 | 0 | 0 |
omr-ws-92271d1fbc19092cc3b1e320863783a02332160b7a70809028f4f8160d0b429b | What is the closest integer to $1000 \times \sum_{n=3}^{10000} \frac{1}{n^2 - 4}$? | [
"521"
] | [
"What is the closest integer to $1000 \\times \\sum_{n=3}^{10000} \\frac{1}{n^2 - 4}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-ec9209d2f2e0f902bfb31972617056f64b0d15e5b0cf5bf486a0564916a2bdd6 | What is the closest predigic number to 2007? Can you prove that it is the closest? A predigic number is one that occurs in the sequence of digits formed by the numbers less than it. For example, 920 is predigic because in the sequence 123456789101112...918919, 920 occurs: 12345...181[b]920[/b]2122...918919. | [
"2010"
] | [
"What is the closest predigic number to 2007? Can you prove that it is the closest?\n\nA predigic number is one that occurs in the sequence of digits formed by the numbers less than it. For example, 920 is predigic because in the sequence \n123456789101112...918919,\n920 occurs:\n12345...181[b]920[/b]2122...918919.... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-3e0e880d981e4a488010ad906c68630d702e1c06589e8d6b14d25ecbbc9770c3 | What is the closest straight line approximation (ax + b) of the function \( f(x) = \sin{2x} \) on the interval \([- \pi, \pi]\)? | [
"\\( y = \\frac{-3}{2\\pi^2}x \\)"
] | [
"What is the closest straight line approximation (ax + b) of the function \\( f(x) = \\sin{2x} \\) on the interval \\([- \\pi, \\pi]\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-f8ad033da92c0448775418270f2696eb12d828a21ac2c20a080b0c807361c8c5 | What is the closure of $\{M^2 : M \in GL_n(\mathbb{R})\}$ in $GL_n(\mathbb{R})$? | [
"GL_n^+(\\mathbb{R})"
] | [
"What is the closure of $\\{M^2 : M \\in GL_n(\\mathbb{R})\\}$ in $GL_n(\\mathbb{R})$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-3276d64fb3df9bac0a649f81d59a020b85d481d4f28b0a09d5fcecba0b5c019e | What is the closure of $\{M^2 : M \in M_n(\mathbb{R})\}$ in $M_n(\mathbb{R})$? | [
"\\(F\\) (where \\(F\\) is the set of real matrices whose Jordan decomposition satisfies property (P))"
] | [
"What is the closure of $\\{M^2 : M \\in M_n(\\mathbb{R})\\}$ in $M_n(\\mathbb{R})$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-813ce16266d319dd47b401da0868079ba241962397f2226b644a8c37e2bf86e2 | What is the coefficient of $(a^{p}b^{q}c^{r}d^{s})$ in $((abcd)^{n}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})^{n})$? | [
"\\(\\frac{n!}{(n-p)!(n-q)!(n-r)!(n-s)!}\\)"
] | [
"What is the coefficient of $(a^{p}b^{q}c^{r}d^{s})$ in $((abcd)^{n}(\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}+\\frac{1}{d})^{n})$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6e3de0fc17a615b616f2b4f9db97885ddd144702f8a8c6f9d166dbb9a6b03e98 | What is the coefficient of $\frac{1}{x}$ in the expansion of $(x+1)^n \left(x + \frac{1}{x}\right)^n$? | [
"$\\sum_{k=0}^{\\left\\lfloor\\frac{n-1}2\\right\\rfloor}\\binom n{2k+1}\\binom nk$ or $\\sum_{k=1}^n\\binom nk\\binom{n+2k}{k-1}(-2)^{n-k}$"
] | [
"What is the coefficient of $\\frac{1}{x}$ in the expansion of $(x+1)^n \\left(x + \\frac{1}{x}\\right)^n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 19
} | 19 | 0 | 0 |
omr-ws-48dd8643f517d85c0e69794757c9b52db36d35e4383b172f4238f7b423dfacae | What is the coefficient of $a^2b^2$ in $(a+b)^4\left(c+\dfrac{1}{c}\right)^6$? | [
"120"
] | [
"What is the coefficient of $a^2b^2$ in $(a+b)^4\\left(c+\\dfrac{1}{c}\\right)^6$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-10599121b18a7cb5022d49466a7039c15e0797e52ebdf2f9bb62662c1d3bd949 | What is the coefficient of $wx^{2}y^{3}z^{4}$ in the expansion of $(8w+\frac{1}{3}x+\frac{1}{2}y+z)^{10}$? | [
"1400"
] | [
"What is the coefficient of $wx^{2}y^{3}z^{4}$ in the expansion of $(8w+\\frac{1}{3}x+\\frac{1}{2}y+z)^{10}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-1ab254842d4fda6d0831938f036bc5487a614e6e5862d2dffe7193fcdfb875ef | What is the coefficient of $x^2$ when $-5x^3 - 5x^2 - 7x + 1$ is multiplied by $-x^2 - 6x + 1$ and the like terms are combined? | [
"36"
] | [
"What is the coefficient of $x^2$ when $-5x^3 - 5x^2 - 7x + 1$ is multiplied by $-x^2 - 6x + 1$ and the like terms are combined?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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