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-b81c025cfb0745750dc37276641284e51a20a71a98fd0bd6fc841f02d69649b0 | When the mean, median, and mode of the list $10, 2, 5, 2, 4, 2, x$ are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$? | [
"20"
] | [
"When the mean, median, and mode of the list $10, 2, 5, 2, 4, 2, x$ are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 2
} | 22 | 0 | 0 |
omr-ws-a274f7165fbbdc32c1641a666f3711740467227b2e34eb6e259f52e0b87eb075 | When the mean, median, and mode of the list $10, 2, 5, 2, 4, 2, x$ are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all real possible values of $x$? | [
"20"
] | [
"When the mean, median, and mode of the list $10, 2, 5, 2, 4, 2, x$ are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all real possible values of $x$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-56b2ebff7e7cc00b86c83b5fb42b32bb683d94734a9687ffc69c53f751139d89 | When the mean, median, and mode of the list 10, 2, 5, 2, 4, 2, x are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of x? | [
"20"
] | [
"When the mean, median, and mode of the list 10, 2, 5, 2, 4, 2, x are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of x?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-010c259b7263ba2654b7237566d67a85e70180c29e0690a516e9379c9572c357 | When the mean, median, and mode of the list \(10, 2, 5, 2, 4, 2, x\) are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of \(x\)? | [
"20"
] | [
"When the mean, median, and mode of the list \\(10, 2, 5, 2, 4, 2, x\\) are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of \\(x\\)?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 63,
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} | 71 | 0 | 0 |
omr-ws-ad84b25297a3c87259fbb2d2d0ad2a96c5dbf920c3338d30a41d2870e8646838 | When the mean, median, and mode of the list \[ 10, 2, 5, 2, 4, 2, x \] are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of \( x \)? | [
"20"
] | [
"When the mean, median, and mode of the list \\[ 10, 2, 5, 2, 4, 2, x \\] are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of \\( x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-5886708c22feee847a68ec03d989f2be71d4498755dc787ef60913500de9ada8 | When the mean, median, and mode of the list \[10, 2, 5, 2, 4, 2, x\] are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of \( x \)? | [
"20"
] | [
"When the mean, median, and mode of the list \\[10, 2, 5, 2, 4, 2, x\\] are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of \\( x \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4563fb83753773c0390d4c45c945f38a2548295e0b81a51b83d7b6d8432490d4 | When the natural numbers are written one after another in an increasing way, you get an infinite succession of digits $123456789101112\ldots$. Denote $A_k$ as the number formed by the first $k$ digits of this sequence. For a given positive integer $n$, find the smallest positive integer $m$ such that: (i) $n$ divides $... | [
"168"
] | [
"When the natural numbers are written one after another in an increasing way, you get an infinite succession of digits $123456789101112\\ldots$. Denote $A_k$ as the number formed by the first $k$ digits of this sequence. For a given positive integer $n$, find the smallest positive integer $m$ such that:\n(i) $n$ di... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-65b425bb7460d909f522ffae8868af82ef648ed13a93413b9880fe6943b547e2 | When the number $10!$ is written in the base 12 system, how many zeros does it end with? | [
"4"
] | [
"When the number $10!$ is written in the base 12 system, how many zeros does it end with?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-2a503ed859f679ab406e95030ee0b077c109e6afa4ceadaf018130110177b082 | When the number $25$ is added to a list of numbers with total sum $S$, the average of all the numbers increases by one. What is the sum of the digits of the greatest possible value of $S$? | [
"9"
] | [
"When the number $25$ is added to a list of numbers with total sum $S$, the average of all the numbers increases by one. What is the sum of the digits of the greatest possible value of $S$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-b0f3b69f7f6a855112cbe3de7e0c3964a1f119afb449a040eb6749f49b5e25ca | When the number $n$ is written in base $b$, its representation is the two-digit number $AB$ where $A = b-2$ and $B = 2$. What is the representation of $n$ in base $b-1$? | [
"\\( 101_{b-1} \\)"
] | [
"When the number $n$ is written in base $b$, its representation is the two-digit number $AB$ where $A = b-2$ and $B = 2$. What is the representation of $n$ in base $b-1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-178d498fee30b7f8d2db1eff73fdf389d23974580128dd206204cf50f3476330 | When the number $n$ is written in base $b$, its representation is the two-digit number $AB$ where $A=b-2$ and $B=2$. What is the representation of $n$ in base $(b-1)$? | [
"\\( 101 \\)"
] | [
"When the number $n$ is written in base $b$, its representation is the two-digit number $AB$ where $A=b-2$ and $B=2$. What is the representation of $n$ in base $(b-1)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-45beeab48d4271b8419e00a5d377c6cd040abbacd19a0a753bebfe4968ceeafe | When the number 210 is written as the sum of the greatest possible number of consecutive positive integers, what is the largest of these integers? | [
"20"
] | [
"When the number 210 is written as the sum of the greatest possible number of consecutive positive integers, what is the largest of these integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-79315f20f8a0c4313ec1f966d3073baa1f94c264122b432601c46010571bd979 | When the number \( n \) is written in base \( b \), its representation is the two-digit number \( AB \) where \( A = b - 2 \) and \( B = 2 \). What is the representation of \( n \) in base \( (b - 1) \)? | [
"\\( 101_{(b - 1)} \\)"
] | [
"When the number \\( n \\) is written in base \\( b \\), its representation is the two-digit number \\( AB \\) where \\( A = b - 2 \\) and \\( B = 2 \\). What is the representation of \\( n \\) in base \\( (b - 1) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-f2c6068464a762f56a49ef876ea208c0b86c975b4d9efe274ab307c9a641c9ac | When the number \( n \) is written in base \( b \), its representation is the two-digit number \( AB \) where \( A = b - 2 \) and \( B = 2 \). What is the representation of \( n \) in base \( b - 1 \)? | [
"101_{b-1}"
] | [
"When the number \\( n \\) is written in base \\( b \\), its representation is the two-digit number \\( AB \\) where \\( A = b - 2 \\) and \\( B = 2 \\). What is the representation of \\( n \\) in base \\( b - 1 \\)?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 5
} | 36 | 0 | 0 |
omr-ws-8d996cf43b2c3ea5d74ceb3299ff52c01755f9576076600da7cc19bb61fab439 | When the number \( n \) is written in base \( b \), its representation is the two-digit number \( AB \) where \( A = b-2 \) and \( B = 2 \). What is the representation of \( n \) in base \( (b-1) \)? | [
"\\((b-1)1\\)",
"\\(101_{b-1}\\)"
] | [
"When the number \\( n \\) is written in base \\( b \\), its representation is the two-digit number \\( AB \\) where \\( A = b-2 \\) and \\( B = 2 \\). What is the representation of \\( n \\) in base \\( (b-1) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 6
} | 38 | 0 | 0 |
omr-ws-e74fc0ebce77b7c31433caef7792ba5dd3983b2267acc6d1146faef0d0ee8578 | When the number \( n \) is written in base \( b \), its representation is the two-digit number \( AB \) where \( A = b-2 \) and \( B = 2 \). What is the representation of \( n \) in base \( b-1 \)? | [
"101",
"\\( 101_{b-1} \\)"
] | [
"When the number \\( n \\) is written in base \\( b \\), its representation is the two-digit number \\( AB \\) where \\( A = b-2 \\) and \\( B = 2 \\). What is the representation of \\( n \\) in base \\( b-1 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 63,
"tir": 9
} | 72 | 0 | 0 |
omr-ws-cd6ed5a60669f8caf329266548577a7709aad8f3642b76ab06215764b79fb45e | When the numbers 19779 and 17997 are divided by a certain three-digit number, they leave the same remainder. Find the largest such divisor and the remainder. | [
"\\(891\\) and the remainder is \\(67\\)."
] | [
"When the numbers 19779 and 17997 are divided by a certain three-digit number, they leave the same remainder. Find the largest such divisor and the remainder."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7192c2a47c790a4b0cb3a4fd9a2585f90db95af3e7884f55ef61e497131972d5 | When the numbers 272758 and 273437 are divided by a two-digit number \( N \), the remainders are 13 and 17, respectively. Find the sum of the digits of \( N \). | [
"9"
] | [
"When the numbers 272758 and 273437 are divided by a two-digit number \\( N \\), the remainders are 13 and 17, respectively. Find the sum of the digits of \\( N \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-91c5e1d67d296aa097273182e2f9a4efb0454731aac1512974c9ab838ea9aa2f | When the parameter \( a = \frac{(x + \frac{1}{x} + 2)^2}{x + \frac{1}{x} - 2} \) is a positive integer, and \( x = \frac{b + \sqrt{b - \sqrt{b}}}{b - \sqrt{b - \sqrt{b}}} \) (with \( b \) a positive integer), find the values of \( a \) and \( b \). | [
"2"
] | [
"When the parameter \\( a = \\frac{(x + \\frac{1}{x} + 2)^2}{x + \\frac{1}{x} - 2} \\) is a positive integer, and \\( x = \\frac{b + \\sqrt{b - \\sqrt{b}}}{b - \\sqrt{b - \\sqrt{b}}} \\) (with \\( b \\) a positive integer), find the values of \\( a \\) and \\( b \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0884e4d708c44d2bc026e55de3542a49690daa123907d640e008fae9b134d28a | When the permutations of the letters in $MMATHS$ are ordered alphabetically starting from $AHMMST$, what is the position $n$ of the permutation $MMATHS$? | [
"173"
] | [
"When the permutations of the letters in $MMATHS$ are ordered alphabetically starting from $AHMMST$, what is the position $n$ of the permutation $MMATHS$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-9bea8f9139ec613caa9347ac6f414e57bc0970b8ce6968261db0de998026471f | When the point \( P \) moves on the line \( x + y = 2 \), find the locus of the point \( Q \). | [
"The locus of \\( Q \\) is a circle centered at \\( \\left( \\frac{1}{4}, \\frac{1}{4} \\right) \\) with radius \\( \\frac{\\sqrt{2}}{4} \\)."
] | [
"When the point \\( P \\) moves on the line \\( x + y = 2 \\), find the locus of the point \\( Q \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-bdc0bd0f8c8011b836959240effe4ba0de51a610d749ffadeb46eb1d0ec0304a | When the point \( P(X, Y) \) moves in the region \( |X| + |Y| \leq 2 \), find the area of the domain of \( Q(X-Y, XY) \). | [
"4"
] | [
"When the point \\( P(X, Y) \\) moves in the region \\( |X| + |Y| \\leq 2 \\), find the area of the domain of \\( Q(X-Y, XY) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 26,
"genselect": 7
} | 33 | 0 | 0 |
omr-ws-6195bfd2cd72e6bed1a625bad96b18769cdcc73609bba5307920620a11a9c287 | When the polynomial $(1+x+x^2)^{50}$ is expanded, determine the sum of the coefficients of the $x^0, x^2, x^4, \ldots, x^{100}$ terms. | [
"\\(\\frac{3^{50} + 1}{2}\\)"
] | [
"When the polynomial $(1+x+x^2)^{50}$ is expanded, determine the sum of the coefficients of the $x^0, x^2, x^4, \\ldots, x^{100}$ terms."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-8f97e9aee15f7ef463fc070b93a81b474235280626d79437410432938c02df60 | When the polynomial $[(w+x+y+z)(w+x+y-z)(w+x-y+z)(-w+x+y+z)]^{2013}$ is expanded and simplified, how many terms are there? | [
"\\dbinom{4029}{3}"
] | [
"When the polynomial $[(w+x+y+z)(w+x+y-z)(w+x-y+z)(-w+x+y+z)]^{2013}$ is expanded and simplified, how many terms are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-0dceb6369761141a80944bf6c3f33a2ff8faf70916326427829bb90bc6d51888 | When the polynomial $x^4 - 6x^3 + 16x^ 2 - 25x + 10$ is divided by $x^2 - 2x + k,$ the remainder is $x + a.$ Enter the ordered pair $(k,a).$ | [
"(5,-5)"
] | [
"When the polynomial $x^4 - 6x^3 + 16x^ 2 - 25x + 10$ is divided by $x^2 - 2x + k,$ the remainder is $x + a.$ Enter the ordered pair $(k,a).$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6f9d8868aaa3be4a9aad985a58d03260a725dea7856d1c4025598f5482d0df57 | When the polynomial $x^{1951}-1$ is divided by $x^4+x^3+2x^2+x+1$ we obtain a quotient and a remainder. Find the coefficient in $x^{14}$ in the quotient. | [
"Use the fact that the polynomial $x^4+x^3+2x^2+x+1$ is a divisor of the binomial $x^{12}-1$. {\\it Answer:} $-1$"
] | [
"When the polynomial $x^{1951}-1$ is divided by $x^4+x^3+2x^2+x+1$ we obtain a quotient and a remainder. Find the coefficient in $x^{14}$ in the quotient."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6399e720cdcb0efd7ba169671cc547d4dc31148e43760b3e408391a8eec65de7 | When the polynomial \( x^n + x^{n-1} + \cdots + x + 1 \) is divided by \( x^2 + x + 1 \), let the remainder be \( ax + b \). Find the value of \( a^2 + b^2 \) when \( n = 100 \). | [
"2"
] | [
"When the polynomial \\( x^n + x^{n-1} + \\cdots + x + 1 \\) is divided by \\( x^2 + x + 1 \\), let the remainder be \\( ax + b \\). Find the value of \\( a^2 + b^2 \\) when \\( n = 100 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c9a6692d3ce508b6a117942d2163c5c5a85a7eb4778abdd4a8d0c832012417b5 | When the polynomial \( x^n + x^{n-1} + \cdots + x + 1 \) is divided by \( x^2 - x + 1 \), let the remainder be \( ax + b \). Find the value of \( a^2 + b^2 \) when \( n = 10 \). | [
"2"
] | [
"When the polynomial \\( x^n + x^{n-1} + \\cdots + x + 1 \\) is divided by \\( x^2 - x + 1 \\), let the remainder be \\( ax + b \\). Find the value of \\( a^2 + b^2 \\) when \\( n = 10 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-5ba683c21ff2f87e574d9d1d64d12b490f08baab622b8ac05aa4ba60db7a99b8 | When the polynomial \(x^{1951} - 1\) is divided by \(P(x) = x^4 + x^3 + 2x^2 + x + 1\), a quotient and a remainder are obtained. What is the coefficient of \(x^{14}\) in the quotient? | [
"-1"
] | [
"When the polynomial \\(x^{1951} - 1\\) is divided by \\(P(x) = x^4 + x^3 + 2x^2 + x + 1\\), a quotient and a remainder are obtained. What is the coefficient of \\(x^{14}\\) in the quotient?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 2
} | 20 | 0 | 0 |
omr-ws-a41a4ff363398a0a5d72fbdbb95a299ebe39e83e7330f6d87bb40a57037682b6 | When the price of a certain commodity is \( p \) dollars per unit, the manufacturer is willing to supply \( x \) hundred units, where \( 110p^2 - 4x^3 = 4000 \). How fast is the supply changing when the price is 9 dollars per unit and is increasing at a rate of 1 dollar per month? | [
"\\(\\frac{1980}{12 \\cdot 1227.5^{2/3}}\\)"
] | [
"When the price of a certain commodity is \\( p \\) dollars per unit, the manufacturer is willing to supply \\( x \\) hundred units, where \\( 110p^2 - 4x^3 = 4000 \\). How fast is the supply changing when the price is 9 dollars per unit and is increasing at a rate of 1 dollar per month?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-58464798db48e2ccd3ed70ef2c6fb437e93bba2c0e9c54d081bea92f9ff3c8c6 | When the product $(3x+2y+1)(x+4y+5)$ is expanded, what is the sum of the coefficients of the terms which contain a nonzero power of $y$? | [
"36"
] | [
"When the product $(3x+2y+1)(x+4y+5)$ is expanded, what is the sum of the coefficients of the terms which contain a nonzero power of $y$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3bd85bdf08bef9d35b7e64e40b5fec9482cfe44ab227ac81836e5711ee4c4851 | When the product $(x+2)(x+3)(x+4)(x+5)(x+6)(x+7)(x+8)(x+9)(x+10)$ is expanded, it can be written as $a_9x^9+ a_8x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$. Find the value of $a_1+a_3+a_5+a_7+a_9$. | [
"19776960"
] | [
"When the product $(x+2)(x+3)(x+4)(x+5)(x+6)(x+7)(x+8)(x+9)(x+10)$ is expanded, it can be written as $a_9x^9+ a_8x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$. Find the value of $a_1+a_3+a_5+a_7+a_9$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b85d9982c673f5740119b330e7e2c69e6ca43e1c4341cc0c1f07d2d7b7f1b6a7 | When the product $1 \cdot 3 \cdot 5 \cdot 7 \cdot \ldots \cdot 99$ is written as a number, what is its second last digit? | [
"7"
] | [
"When the product $1 \\cdot 3 \\cdot 5 \\cdot 7 \\cdot \\ldots \\cdot 99$ is written as a number, what is its second last digit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-2c0f2d5e13a5dec4423bc0c4fe3bf08da663629522712be6633bb468a8c3e246 | When the product \((2021x^{2021} + 2020x^{2020} + \cdots + 3x^3 + 2x^2 + x)(x^{2021} - x^{2020} + \cdots + x^3 - x^2 + x - 1)\) is expanded and simplified, what is the coefficient of \(x^{2021}\)? | [
"-1011"
] | [
"When the product \\((2021x^{2021} + 2020x^{2020} + \\cdots + 3x^3 + 2x^2 + x)(x^{2021} - x^{2020} + \\cdots + x^3 - x^2 + x - 1)\\) is expanded and simplified, what is the coefficient of \\(x^{2021}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b2e59b6c8d9375f2ab0e214ed4975d1289851611c7e964ca23bb7481a2801abd | When the product \((x + 5)(x + 10)(2x^2 + 3)(x^3 + 6x + 16)(x^3 + 6x + 16)(x + 9)(x + 18)^3\) is expanded, how many of the coefficients are odd? | [
"2"
] | [
"When the product \\((x + 5)(x + 10)(2x^2 + 3)(x^3 + 6x + 16)(x^3 + 6x + 16)(x + 9)(x + 18)^3\\) is expanded, how many of the coefficients are odd?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1b09c36966d53e404cc2a504e9872defbb06f4d979978f8f8a019efbf5c29e48 | When the radius of the circular ripple is $20$ cm, how fast is the area enclosed by the ripple increasing? | [
"\\( 400\\pi \\)"
] | [
"When the radius of the circular ripple is $20$ cm, how fast is the area enclosed by the ripple increasing?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 4
} | 6 | 0 | 0 |
omr-ws-4615d16951da81d035daa71a0f0f9930aea1195ddfaa397e65e497965ce18b72 | When the repeating decimal $0.\overline{12}$ is expressed as a common fraction in lowest terms, what is the sum of its numerator and denominator? | [
"37"
] | [
"When the repeating decimal $0.\\overline{12}$ is expressed as a common fraction in lowest terms, what is the sum of its numerator and denominator?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e6fd1bce5610f4e0970e1cf5b18fbc79777b6d372e248d9819436bb4968c8850 | When the repeating decimal 0.2006006006... is expressed as a fraction in simplest form, what is the sum of the numerator and the denominator? | [
"1999"
] | [
"When the repeating decimal 0.2006006006... is expressed as a fraction in simplest form, what is the sum of the numerator and the denominator?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-076eda96f7dfd3744ca54e03dcb093ca82738822e9c0e8f3138f6bb2a73a0158 | When the roots of the polynomial \( P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdots (x-10)^{10} \) are removed from the number line, what remains is the union of 11 disjoint open intervals. On how many of these intervals is \( P(x) \) positive? | [
"6"
] | [
"When the roots of the polynomial \\( P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \\cdots (x-10)^{10} \\) are removed from the number line, what remains is the union of 11 disjoint open intervals. On how many of these intervals is \\( P(x) \\) positive?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0532a83eccb7f70643c484fa14143e514c05563317d0e1db1da6520b7080a65e | When the roots of the polynomial \( P(x) = \prod_{i=1}^{10}(x-i)^{i} \) are removed from the real number line, what remains is the union of 11 disjoint open intervals. On how many of those intervals is \( P(x) \) positive? | [
"6"
] | [
"When the roots of the polynomial \\( P(x) = \\prod_{i=1}^{10}(x-i)^{i} \\) are removed from the real number line, what remains is the union of 11 disjoint open intervals. On how many of those intervals is \\( P(x) \\) positive?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-64ad9bc3ee79a80dd33a811d8ddb3f4d453c95668f9f07489aaffdf51c9dcb85 | When the roots of the polynomial \[ P(x) = (x-1)^1(x-2)^2(x-3)^3 \dots (x-10)^{10} \] are removed from the real number line, what remains is the union of $11$ disjoint open intervals. On how many of these intervals is $P(x)$ positive? | [
"6"
] | [
"When the roots of the polynomial\n\\[ P(x) = (x-1)^1(x-2)^2(x-3)^3 \\dots (x-10)^{10} \\]\nare removed from the real number line, what remains is the union of $11$ disjoint open intervals. On how many of these intervals is $P(x)$ positive?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-14d22e75bb923a7d5869da3923fc5968963601aa1cfaac191e97ec6ef0b61159 | When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of five consecutive positive integers all of which are nonprime? | [
"29"
] | [
"When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of five consecutive positive integers all of which are nonprime?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-17038c75477023a00d83c97a3bb8baeefca3eb932b058c65e85cc5523783ff0a | When the square of three times a positive integer is decreased by the integer, the result is $2010$. What is the integer? | [
"15"
] | [
"When the square of three times a positive integer is decreased by the integer, the result is $2010$. What is the integer?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-04918da5a73e307dc596f1090e938768f7b7d35ed173cd053c83571bda56d616 | When the square root of $x$ is cubed, the answer is 64. What is the value of $x$? | [
"16"
] | [
"When the square root of $x$ is cubed, the answer is 64. What is the value of $x$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7968ed5cbf7d2f22f5f473bb9e2e1e9b76100174811783238b296c6a777d8c64 | When the transformed line \( l \) is tangent to the circle \( C: x^2 + y^2 = 1 \), express the point of tangency in terms of \( p \) and \( q \). | [
"\\(\\left(\\frac{2-p}{q}, \\frac{1}{q}\\right)\\)"
] | [
"When the transformed line \\( l \\) is tangent to the circle \\( C: x^2 + y^2 = 1 \\), express the point of tangency in terms of \\( p \\) and \\( q \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d2bc0532c6d5d4cf64cf22b7c4edaf1d8dd5c777898316796fd0e5e4cc1c8728 | When the triangular piece of paper \(ABC\) is folded along the dotted line \(AF\), point \(B\) lands on side \(AC\). We know \(AB\) is 1 unit and \(AC\) is 2 units. If the measure of angle \(B\) is \(90^\circ\), what is the measure of angle \(AFC\)? | [
"\\(120^\\circ\\)"
] | [
"When the triangular piece of paper \\(ABC\\) is folded along the dotted line \\(AF\\), point \\(B\\) lands on side \\(AC\\). We know \\(AB\\) is 1 unit and \\(AC\\) is 2 units. If the measure of angle \\(B\\) is \\(90^\\circ\\), what is the measure of angle \\(AFC\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-110b9ba8c52a482eab48e3e1b55b9e30f4e2f02d87c27ea4c7c65c92a4689b37 | When the two-digit integer $MM$, with equal digits, is multiplied by the one-digit integer $M$, the result is the three-digit integer $NPM$. What is the greatest possible value of $NPM$? | [
"396"
] | [
"When the two-digit integer $MM$, with equal digits, is multiplied by the one-digit integer $M$, the result is the three-digit integer $NPM$. What is the greatest possible value of $NPM$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-20c63f8e946826e1e320bb654c3ec6c4b90f840cdd75626d33d7eebcc732ad2c | When the value of $y$ is doubled and then this increased value is divided by 5, the result is 10. What is the value of $y$? | [
"25"
] | [
"When the value of $y$ is doubled and then this increased value is divided by 5, the result is 10. What is the value of $y$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cbf5c736ce5d4b13a3495d85d3e01efad6a1fc7bc4dde5bcf4d07b71e20d80a4 | When the vectors $\begin{pmatrix} -5 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ are both projected onto the same vector $\mathbf{v},$ the result is $\mathbf{p}$ in both cases. Find $\mathbf{p}.$ | [
"\\begin{pmatrix} -34/53 \\\\ 119/53 \\end{pmatrix}"
] | [
"When the vectors $\\begin{pmatrix} -5 \\\\ 1 \\end{pmatrix}$ and $\\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix}$ are both projected onto the same vector $\\mathbf{v},$ the result is $\\mathbf{p}$ in both cases. Find $\\mathbf{p}.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a738f7beea531e0d242399ab35772abfd28bf231bd0bf1e99088e7bb3118bb43 | When the vectors $\begin{pmatrix} 4 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} -1 \\ 3 \end{pmatrix}$ are both projected onto the same vector $\mathbf{v},$ the result is $\mathbf{p}$ in both cases. Find $\mathbf{p}.$ | [
"\\begin{pmatrix} 26/29 \\\\ 65/29 \\end{pmatrix}"
] | [
"When the vectors $\\begin{pmatrix} 4 \\\\ 1 \\end{pmatrix}$ and $\\begin{pmatrix} -1 \\\\ 3 \\end{pmatrix}$ are both projected onto the same vector $\\mathbf{v},$ the result is $\\mathbf{p}$ in both cases. Find $\\mathbf{p}.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-575bf5c52d5942e76a701c2a1052610f10beba56e76b70b825a06c916f477f4a | When the vectors $\mathbf{a} = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 0 \\ 3 \\ 0 \end{pmatrix}$ are both projected onto the same vector $\mathbf{v},$ the result is $\mathbf{p}$ in both cases. Furthermore, the vectors $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{p}$ are collinear. Fin... | [
"\\begin{pmatrix} 4/7 \\\\ 5/7 \\\\ 8/7 \\end{pmatrix}"
] | [
"When the vectors $\\mathbf{a} = \\begin{pmatrix} 1 \\\\ -1 \\\\ 2 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 0 \\\\ 3 \\\\ 0 \\end{pmatrix}$ are both projected onto the same vector $\\mathbf{v},$ the result is $\\mathbf{p}$ in both cases. Furthermore, the vectors $\\mathbf{a},$ $\\mathbf{b},$ and $\\math... | [
"MATH_training_set"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-edb083574cd0679286c0d3923bfc3e6794c40160abb70aa94811fd830048491b | When there are $3$ cuts, what is the maximum number of pie slices that can be created? | [
"6"
] | [
"When there are $3$ cuts, what is the maximum number of pie slices that can be created?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93d2ab2c6e63d3b6d7e1948541d42bc422a9d59a41f245a62e45c3bf8dbcacef | When there exists a triangle with side lengths \( \sqrt{a}, \sqrt{b}, \sqrt{c} \) and one of the angles is 60 degrees, what is the maximum value \( M \) of \( \frac{bc}{a} \)? | [
"12"
] | [
"When there exists a triangle with side lengths \\( \\sqrt{a}, \\sqrt{b}, \\sqrt{c} \\) and one of the angles is 60 degrees, what is the maximum value \\( M \\) of \\( \\frac{bc}{a} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b785366569a4254bc97d4c4c28714d57c81d01402b7675c3d7f2077079f495d | When three numbers are added two at a time the sums are 29, 46, and 53. What is the sum of all three numbers? | [
"64"
] | [
"When three numbers are added two at a time the sums are 29, 46, and 53. What is the sum of all three numbers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f7ca6d9265021c5e264a0fe699ea93738d912511e01dd8dbe51a56c4066cbd99 | When three points are randomly selected on the circumference of a circle, what is the probability that all three points lie within $90^{\circ}$ of one another? | [
"\\(\\frac{3}{16}\\)"
] | [
"When three points are randomly selected on the circumference of a circle, what is the probability that all three points lie within $90^{\\circ}$ of one another?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-713f8ce5860bee11b103b0aad11f1568690a4e88631763b1c7a406d6a3ba6719 | When three positive integers $a$, $b$, and $c$ are multiplied together, their product is 100. Suppose $a \leq b \leq c$. In how many ways can the numbers be chosen? | [
"8"
] | [
"When three positive integers $a$, $b$, and $c$ are multiplied together, their product is 100. Suppose $a \\leq b \\leq c$. In how many ways can the numbers be chosen?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c82b408f5d1090acd2b4e8240b6b4b7a592e210c459f2d6efb01443c43a0a3f7 | When three positive integers are divided by $12$, the remainders are $7,$ $9,$ and $10,$ respectively. When the sum of the three integers is divided by $12$, what is the remainder? | [
"2"
] | [
"When three positive integers are divided by $12$, the remainders are $7,$ $9,$ and $10,$ respectively.\n\nWhen the sum of the three integers is divided by $12$, what is the remainder?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-da9cc42ed8f2c064c7f427243394a225e869b50f7b1977209b09a64cd74a20d5 | When three positive integers are divided by $24$, the remainders are $10,$ $4,$ and $12,$ respectively. When the sum of the three integers is divided by $24$, what is the remainder? | [
"2"
] | [
"When three positive integers are divided by $24$, the remainders are $10,$ $4,$ and $12,$ respectively.\n\nWhen the sum of the three integers is divided by $24$, what is the remainder?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2ba7d836651bdd8032c5d60171b8fa7b6d105b10b93f3d13ce0beab0e4b06dd5 | When three positive integers are divided by $47$, the remainders are $25$, $20$, and $3$, respectively. When the sum of the three integers is divided by $47$, what is the remainder? | [
"1"
] | [
"When three positive integers are divided by $47$, the remainders are $25$, $20$, and $3$, respectively.\n\nWhen the sum of the three integers is divided by $47$, what is the remainder?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9712b0418dba9e7b9fd77bde2ea33e53cc9b41352fb1e76493feb16708edbec1 | When three spherical balls, each of radius 10 cm, are placed in a hemispherical dish, the top of the balls are all exactly level with the top of the dish. What is the radius, in centimeters, of the dish? | [
"\\( 10 + \\frac{10\\sqrt{21}}{3} \\)"
] | [
"When three spherical balls, each of radius 10 cm, are placed in a hemispherical dish, the top of the balls are all exactly level with the top of the dish. What is the radius, in centimeters, of the dish?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 15
} | 32 | 0 | 0 |
omr-ws-716f14078d914ea2933a1e85ed1a904d3b2446c51111a1474d440bd963616c23 | When three spherical balls, each of radius 10 cm, are placed in a hemispherical dish, the tops of the balls are all exactly level with the top of the dish. What is the radius, in centimetres, of the dish? | [
"\\( 10 \\left(1 + \\frac{\\sqrt{21}}{3}\\right) \\)"
] | [
"When three spherical balls, each of radius 10 cm, are placed in a hemispherical dish, the tops of the balls are all exactly level with the top of the dish. What is the radius, in centimetres, of the dish?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 4,
"tir": 17
} | 23 | 0 | 0 |
omr-ws-5ab8f2f8b01ee33cdf127935953aef2221ebc2484f5328b7c1638f150384c276 | When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-1)(b-1)(c-1) \neq 0$$ | [
"\\frac{125}{216}"
] | [
"When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-1)(b-1)(c-1) \\neq 0$$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e91b176c2baec17f11f27ff085dbadd2e22d39f06b51af42e771f8c08493339c | When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $abc = 1$. | [
"\\frac1{216}"
] | [
"When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $abc = 1$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ce67ed75e63d2f21b3637d756fbd1d5148efd042a858d2908937e0a21b39e87 | When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $abc = 180$. | [
"\\frac1{72}"
] | [
"When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $abc = 180$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e60decd1d2229944eb29caecf064cb6db0d1a9e5cc3303581702e897ae638024 | When throwing \( n \) balls into \( n \) buckets, each ball independently and with uniform distribution, let \( X_i \) be the number of balls in the \( i \)-th bucket. Define \( M_n = \max_{1 \leq i \leq n} X_i \). Calculate the smallest constant \( C \) such that the probability \( P(M_n \leq C \frac{\log n}{\log \log... | [
"2"
] | [
"When throwing \\( n \\) balls into \\( n \\) buckets, each ball independently and with uniform distribution, let \\( X_i \\) be the number of balls in the \\( i \\)-th bucket. Define \\( M_n = \\max_{1 \\leq i \\leq n} X_i \\). Calculate the smallest constant \\( C \\) such that the probability \\( P(M_n \\leq C \... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-34d1107e77a653aa9ab5cb78ebed7c5525e4f0c89014f2ae86db03e42fbc579e | When tossing an unbiased coin a large number of times, let $E_{HTH}$ be the expected number of times the sequence HTH appears, and let $E_{HTT}$ be the expected number of times the sequence HTT appears. Calculate the value of $\frac{E_{HTH}}{E_{HTT}}$. | [
"1"
] | [
"When tossing an unbiased coin a large number of times, let $E_{HTH}$ be the expected number of times the sequence HTH appears, and let $E_{HTT}$ be the expected number of times the sequence HTT appears. Calculate the value of $\\frac{E_{HTH}}{E_{HTT}}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-fefbd8044167948c837706b46ad9e80623277d2ea562f66e138f0338f85060cd | When two cards are drawn from a standard 52-card deck, what is the probability that they are both black or both jacks (or both black jacks)? | [
"\\(\\frac{55}{221}\\)"
] | [
"When two cards are drawn from a standard 52-card deck, what is the probability that they are both black or both jacks (or both black jacks)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-326ece444ea071eae6905575857b4630e3842488ba3e1a73bb79a39e98f7e54d | When two circles of radius 32 with centers $O_1$ and $O_2$ intersect, they divide the line segment $O_1O_2$ into three equal parts. Find the radius of the circle which internally touches the two given circles and the segment $O_1O_2$. | [
"7"
] | [
"When two circles of radius 32 with centers $O_1$ and $O_2$ intersect, they divide the line segment $O_1O_2$ into three equal parts. Find the radius of the circle which internally touches the two given circles and the segment $O_1O_2$."
] | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 8,
"tir": 9
} | 53 | 0 | 0 |
omr-ws-185c22a659e3e685e771bf66a73c18a28648581285dde1078c64a92106902f34 | When two fair 6-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that the two digit number $ab$ (where $a$ and $b$ are digits) and $a$ and $b$ are all divisible by 3? | [
"\\frac19"
] | [
"When two fair 6-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that the two digit number $ab$ (where $a$ and $b$ are digits) and $a$ and $b$ are all divisible by 3?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36430e9d07233df807504a5319466492fdaae2b4b74319d6b81a90b5ea7423d9 | When two real numbers are selected between 5 and 8, what is the probability that they differ by less than one? | [
"\\(\\frac{5}{9}\\)"
] | [
"When two real numbers are selected between 5 and 8, what is the probability that they differ by less than one?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-7e242f741667193bcc4c5fea1054dcee7d846525733b7b0becef11ba8dce4bd8 | When two resistors in a circuit are connected in parallel, one of resistance $m$ and one of resistance $n$, the equivalent resistance of the circuit $R$ is given by $\frac{1}{R}= \frac{1}{m}+ \frac{1}{n}$. How many ordered pairs $(m, n)$ of positive integers are such that the equivalent resistance of a circuit with two... | [
"9"
] | [
"When two resistors in a circuit are connected in parallel, one of resistance $m$ and one of resistance $n$, the equivalent resistance of the circuit $R$ is given by $\\frac{1}{R}= \\frac{1}{m}+ \\frac{1}{n}$. How many ordered pairs $(m, n)$ of positive integers are such that the equivalent resistance of a circuit ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1614073399bd3e8a238f53687c8868665594d08fcd5c5295434aa6c24a6f2d41 | When two traditional dice are rolled, what is the probability that the product of the numbers showing on the two dice is odd and prime? | [
"\\(\\frac{2}{9}\\)"
] | [
"When two traditional dice are rolled, what is the probability that the product of the numbers showing on the two dice is odd and prime?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cdfb4e423b2d6220e78e94f8b7e5c68ed8e59cd0693d0553f81832380724e288 | When water is pumped out of a full hemisphere with radius \(0.3048 \times r\) meters, what is the depth of the water when \(1/5\) of the work is done? | [
"0.1405r"
] | [
"When water is pumped out of a full hemisphere with radius \\(0.3048 \\times r\\) meters, what is the depth of the water when \\(1/5\\) of the work is done?"
] | [
"aops_c4_high_school_math",
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-790d35aea4a31ec67855e20c1b71c40f13ebb5e2cc6464df2ef8b6afb79dce62 | When we cut a rope into two pieces, we say that the cut is special if both pieces have different lengths. We cut a chord of length $2008$ into two pieces with integer lengths and write those lengths on the board. We continue cutting one of the pieces into two new pieces with integer lengths and writing those lengths on... | [
"5"
] | [
"When we cut a rope into two pieces, we say that the cut is special if both pieces have different lengths. We cut a chord of length $2008$ into two pieces with integer lengths and write those lengths on the board. We continue cutting one of the pieces into two new pieces with integer lengths and writing those lengt... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-7315b392e0af27e3ffc49eb67df76ef7058e54283d6490bc8afdfe93b247d9b8 | When we cut a rope into two pieces, we say that the cut is special if both pieces have different lengths. We cut a chord of length \(2008\) into two pieces with integer lengths and we write those lengths on the board. Afterwards, we cut one of the pieces into two new pieces with integer lengths and we write those lengt... | [
"5839"
] | [
"When we cut a rope into two pieces, we say that the cut is special if both pieces have different lengths. We cut a chord of length \\(2008\\) into two pieces with integer lengths and we write those lengths on the board. Afterwards, we cut one of the pieces into two new pieces with integer lengths and we write thos... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-22f5773a40c3ec7cfebefbebe667a4a46657bf74bfa3a2de4ec205ad6129e72f | When we say that Ray is climbing up the stairs $m$ at a time, we mean that he starts on the floor (step $0$) then jumps to step $m$ and then to $2m$ and so on until the number of steps to the top is less than $m$. Ray climbs up a flight of stairs of $n$ steps in two ways. When he does it $4$ steps at a time, there are ... | [
"27"
] | [
"When we say that Ray is climbing up the stairs $m$ at a time, we mean that he starts on the floor (step $0$) then jumps to step $m$ and then to $2m$ and so on until the number of steps to the top is less than $m$. Ray climbs up a flight of stairs of $n$ steps in two ways. When he does it $4$ steps at a time, there... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a2105aecc8c3799be73932a18250e43440b82c43ea0acb11dd795150e113b05b | When will the next date with no repetition of digits be? Explain your answer. | [
"17/06/2345"
] | [
"When will the next date with no repetition of digits be? Explain your answer."
] | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-c3160461f99f0e9ea94f96578b94c7511a6634292cc9211be9060dc08dddd1b6 | When will the object be 15 meters above the ground, given the distance function \( s = -4.9t^2 + 20t \)? | [
"\\( t = \\frac{20 \\pm \\sqrt{106}}{9.8} \\)"
] | [
"When will the object be 15 meters above the ground, given the distance function \\( s = -4.9t^2 + 20t \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 4,
"tir": 13
} | 20 | 0 | 0 |
omr-ws-5990bf976eb8ce82d8e9f2be17b189b5e2b2209fb9dc455c45e089a4eebf6588 | When writing all the integers from 1 to 2007 inclusive, how many 0's are used? | [
"506"
] | [
"When writing all the integers from 1 to 2007 inclusive, how many 0's are used?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-339113fba84019ca832e81fe35ed9d152d40b73666b61ae046deb71bdc96aff2 | When writing all the integers from 1 to 2007, inclusive, how many 0's are used? | [
"506"
] | [
"When writing all the integers from 1 to 2007, inclusive, how many 0's are used?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d4583c546cc4d4e7c46bbee33c97e724797e6f51fbfa2caa233cd3cc2ad2fd9c | When writing consecutive Arabic numbers starting from 1 up to \( n \), a total of 3289 Arabic numerals are used. What is the value of \( n \)? | [
"1099"
] | [
"When writing consecutive Arabic numbers starting from 1 up to \\( n \\), a total of 3289 Arabic numerals are used. What is the value of \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4e491fa7e52e5896987185aca3138b5f2381c79b7dee07a2c7f4644d8412543c | When written as a decimal number, \(2005^{2005}\) has \(D\) digits and leading digit \(L\). Find \(D + L\). | [
"6626"
] | [
"When written as a decimal number, \\(2005^{2005}\\) has \\(D\\) digits and leading digit \\(L\\). Find \\(D + L\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 30
} | 48 | 0 | 0 |
omr-ws-34ad5e733cf737e5dca896db665e22921eef61e854f7c92bd1b0bd1ccbdb463f | When written as a decimal, the fraction \( \frac{m}{n} \) (with \( m < n \)) contains the consecutive digits 2, 5, 1. Find the smallest possible value of \( n \). | [
"127"
] | [
"When written as a decimal, the fraction \\( \\frac{m}{n} \\) (with \\( m < n \\)) contains the consecutive digits 2, 5, 1. Find the smallest possible value of \\( n \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-72d0c9e1ceb75dde1710b4cfdc5abacdb53af6191d3afe5a746d65763b1addde | When written as a decimal, the fraction \( \frac{m}{n} \), where \( m < n \), contains the consecutive digits 2, 5, 1 (in that order). Find the smallest possible \( n \). | [
"127"
] | [
"When written as a decimal, the fraction \\( \\frac{m}{n} \\), where \\( m < n \\), contains the consecutive digits 2, 5, 1 (in that order). Find the smallest possible \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-f6432284c1ba2cbc322ed950498655db5104856d1ae105284c920f2f3cada5fc | When written in base $4$, the digits of positive integer $n$ are nonzero and sum to $10$. When written in base $2$, the digits of $n$ sum to $7$. Find the number of possible values of $n$. | [
"129"
] | [
"When written in base $4$, the digits of positive integer $n$ are nonzero and sum to $10$. When written in base $2$, the digits of $n$ sum to $7$. Find the number of possible values of $n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-0509b00cf5c97deb1074edd7c28db2f9f7414bda21756261011639560bfbadf4 | When you simplify $\sqrt[3]{24a^4b^6c^{11}}$, what is the sum of the exponents of the variables that are outside the radical? | [
"6"
] | [
"When you simplify $\\sqrt[3]{24a^4b^6c^{11}}$, what is the sum of the exponents of the variables that are outside the radical?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e95012e017f986c5a891cbb3ec1bf9763ea0388673c0abf9d022c7055707716 | Where can we differentiate the function \( g(x) = |x + 1| \cdot \{x\} \cdot [x + 2] \), where \( |x| \) is the absolute value of \( x \), \( [x] \) is the integer part of \( x \), and \( \{x\} = x - [x] \) is the fractional part of \( x \)? | [
"The function \\( g(x) \\) is differentiable for all non-integer values of \\( x \\)."
] | [
"Where can we differentiate the function \\( g(x) = |x + 1| \\cdot \\{x\\} \\cdot [x + 2] \\), where \\( |x| \\) is the absolute value of \\( x \\), \\( [x] \\) is the integer part of \\( x \\), and \\( \\{x\\} = x - [x] \\) is the fractional part of \\( x \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-9f3bf97df99790ab1d483c5e7639f7e0c994ab82c76737e6c2000fb48511ae0d | Where do the graphs of the functions \( f(x) = \frac{1}{x-2} \) and its inverse \( f^{-1}(x) \) intersect? | [
"(1 - \\sqrt{2}, 1 - \\sqrt{2})"
] | [
"Where do the graphs of the functions \\( f(x) = \\frac{1}{x-2} \\) and its inverse \\( f^{-1}(x) \\) intersect?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-231a6aebeda36db473aed0220eaff22dec5dc888efe49c7eef1e02d6ce1cedc5 | Where does the circumcenter \( O \) of triangle \( ABC \) map after inverting over point \( C \) with radius 1? | [
"O'"
] | [
"Where does the circumcenter \\( O \\) of triangle \\( ABC \\) map after inverting over point \\( C \\) with radius 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-28d5c78654969698e4fe0b7d4b28cc88642726e435a672418c2a694a3748a18d | Where does the curve $C$ with the equation $y=\frac{(x+1)(x-3)}{x(x-2)}$ intersect the x-axis? | [
"\\( x = -1 \\) and \\( x = 3 \\)"
] | [
"Where does the curve $C$ with the equation $y=\\frac{(x+1)(x-3)}{x(x-2)}$ intersect the x-axis?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 26
} | 37 | 0 | 0 |
omr-ws-18e2b1cfe4efdbaefd08b9cabf5e0782ec6424fedf652c514c1ee152e2067738 | Where in the interval $[0, 2\pi]$ does the function $y = x - \cos^2(x)$ have a local minimum? | [
"0"
] | [
"Where in the interval $[0, 2\\pi]$ does the function $y = x - \\cos^2(x)$ have a local minimum?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-05e0d8114a25d9b3f92915081ec784cf0378969ca977cdcdfd3c7cd7921fb721 | Where is the function $\log(1-z^2)$ complex analytic? | [
"$\\log(1-z^2)$ is complex analytic in $\\mathbb{C} \\setminus \\{z \\in \\mathbb{C} : |z| \\geq 1 \\text{ and } z \\in \\mathbb{R} \\cup i\\mathbb{R}\\}$."
] | [
"Where is the function $\\log(1-z^2)$ complex analytic?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-edc1f4ae3f0a893d17ac31b7acac5338f89ef105d5f91b0879839473735af22b | Where is the function \( f(x) = \sum_{n \geq 1} \frac{\sin(n^2 x)}{n^2} \) derivable? | [
"The function \\( f(x) = \\sum_{n \\geq 1} \\frac{\\sin(n^2 x)}{n^2} \\) is differentiable everywhere except at points \\( x = \\pi \\cdot \\frac{2p+1}{2q+1} \\) where \\( p \\) and \\( q \\) are integers."
] | [
"Where is the function \\( f(x) = \\sum_{n \\geq 1} \\frac{\\sin(n^2 x)}{n^2} \\) derivable?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-62b913bb7c8b5631f4bf9bff0c7f819f42699304706fd9c8c48fcc83352eafc8 | Where is the function \( f(x, y) = |x| (|x - y|)^{1/2} \) not differentiable? | [
"\\( x = 0 \\) and \\( x = y \\)"
] | [
"Where is the function \\( f(x, y) = |x| (|x - y|)^{1/2} \\) not differentiable?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 12
} | 17 | 0 | 0 |
omr-ws-5c1328ef8f7f854bc0b0c01cb68f093776d2b9531b4087e08307783b609c0dc8 | Where is the series $\sum_{n=0}^{\infty}z^{n}e^{-izn}$ analytic? | [
"$re^y < 1$"
] | [
"Where is the series $\\sum_{n=0}^{\\infty}z^{n}e^{-izn}$ analytic?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 15
} | 35 | 0 | 0 |
omr-ws-d0d805bcd244f0c29f0b21f48af3b83812282636d1d45b9bcb626836e30e7ba6 | Where must a point $P$ be placed on a circle $C$ with center $O$ so that the angle $\angle OPQ$ is the largest possible, given that $Q$ is an inner point of the circle different from $O$? | [
"P \\text{ is the point on } C \\text{ such that } PQ \\perp OQ"
] | [
"Where must a point $P$ be placed on a circle $C$ with center $O$ so that the angle $\\angle OPQ$ is the largest possible, given that $Q$ is an inner point of the circle different from $O$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c2ce3a769e8efb6ef5ce6e708dd1a74eb0ea7015a0ecda144914925068798acb | Where on the interval \([0,1]\) is the function \( f \) continuous, given that \( f(x) = 0 \) if \( x \) is irrational and \( f(x) = \frac{1}{q} \) if \( x \) can be expressed in the form \( \frac{p}{q} \) where \( p \) and \( q \) are integers, \( \frac{p}{q} \) is in lowest terms, and \( q > 0 \)? | [
"The function \\( f \\) is continuous at all irrational numbers in the interval \\([0,1]\\)."
] | [
"Where on the interval \\([0,1]\\) is the function \\( f \\) continuous, given that \\( f(x) = 0 \\) if \\( x \\) is irrational and \\( f(x) = \\frac{1}{q} \\) if \\( x \\) can be expressed in the form \\( \\frac{p}{q} \\) where \\( p \\) and \\( q \\) are integers, \\( \\frac{p}{q} \\) is in lowest terms, and \\( ... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-1491af35e855e5e9e68d0904949f6e6c7dfd27d53b3e0ac576f89f4244a9e6af | Where should a bridge be built on a river so that a person going from house $A$ to house $B$, using the bridge to cross the river, may do so by the shortest path? Assume that the banks of the river are straight and parallel, and the bridge must be perpendicular to the banks. | [
"\\text{The bridge should be built where the line from } A \\text{ to the reflection of } B \\text{ across the river intersects the bank.}"
] | [
"Where should a bridge be built on a river so that a person going from house $A$ to house $B$, using the bridge to cross the river, may do so by the shortest path? Assume that the banks of the river are straight and parallel, and the bridge must be perpendicular to the banks."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6dcfda041860b68eebce19072b24a87e56538a5f792b65bce440df9355d3dd33 | Where should a bus station \( C \) be placed on a line such that the sum \( AC + BC \) is minimized, given that two cities \( A \) and \( B \) are on the same side of the line? | [
"The bus station \\( C \\) should be placed at the intersection of the line and the segment \\( AB' \\), where \\( B' \\) is the reflection of \\( B \\) across the line. \n\nSince the problem asks for a specific location and not a numerical value, the answer is described in terms of the geometric construction."
] | [
"Where should a bus station \\( C \\) be placed on a line such that the sum \\( AC + BC \\) is minimized, given that two cities \\( A \\) and \\( B \\) are on the same side of the line?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-d7a12b357a77634df153d30b9cce3474cfcb3a805f3a00b2499b12cfb91ed19a | Where should a school be built in a triangle $ABC$ with villages at the vertices containing $10$, $20$, and $30$ schoolchildren respectively, so that the total distance traveled by the schoolchildren is minimized? | [
"C"
] | [
"Where should a school be built in a triangle $ABC$ with villages at the vertices containing $10$, $20$, and $30$ schoolchildren respectively, so that the total distance traveled by the schoolchildren is minimized?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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