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-1acb6f7a347a97facd8d672d2920a7f922e02a7124de428908c213be099d1dd1 | All numbers $1, 2, \ldots, 100$ are arbitrarily placed on each cell of a $10 \times 10$ square grid. Each cell contains exactly one number, and all numbers are distinct. For every configuration of numbers, there is always a row or a column where the difference between the largest and smallest number is at least $k$. Fi... | [
"50"
] | [
"All numbers $1, 2, \\ldots, 100$ are arbitrarily placed on each cell of a $10 \\times 10$ square grid. Each cell contains exactly one number, and all numbers are distinct. For every configuration of numbers, there is always a row or a column where the difference between the largest and smallest number is at least ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 22,
"genselect": 6
} | 28 | 0 | 0 |
omr-ws-ac303f7a846fdfbe037228f6fad7b1163f321c43f366a733acbadb564d721329 | All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Marcy could have? | [
"4"
] | [
"All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Marcy could have?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-652b9c430e0ac1dd284811c528e6761850f87427b62637da96918ee758f9413f | All of the interior angles of octagon EFGHIJKL are 135 degrees. EF=GH=IJ=KL=2 and FG=HI=JK=LE=6. A) Find the perimeter of the octagon. B) Find the area of the octagon. C) Find EI. | [
"Perimeter = 32, Area = 36, EI = 2√19"
] | [
"All of the interior angles of octagon EFGHIJKL are 135 degrees. EF=GH=IJ=KL=2 and FG=HI=JK=LE=6.\nA) Find the perimeter of the octagon.\nB) Find the area of the octagon.\nC) Find EI."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1faf2a700525caea9220be92d914a1671414d89536924d4d2343330c3c3c8e3b | All of the roots of the polynomial $x^3 + ax^2 + bx + c$ are positive integers greater than $2$, and the coefficients satisfy $a + b + c + 1 = -2009$. Find the value of $a$. | [
"-58"
] | [
"All of the roots of the polynomial $x^3 + ax^2 + bx + c$ are positive integers greater than $2$, and the coefficients satisfy $a + b + c + 1 = -2009$. Find the value of $a$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-1f14b6490baa7b2ad02d69709ed424994c39a0b8c4d536b2f883f58fdd82efad | All perfect squares and all perfect squares multiplied by two are written in a row in increasing order. Let \( f(n) \) be the \( n \)-th number in this sequence. Find the largest integer \( n \) such that all the numbers \[ f(n), f(2n), f(3n), \dots, f(10n^2) \] are perfect squares. | [
"no_valid_answer_found"
] | [
"All perfect squares and all perfect squares multiplied by two are written in a row in increasing order. Let \\( f(n) \\) be the \\( n \\)-th number in this sequence. Find the largest integer \\( n \\) such that all the numbers\n\\[ f(n), f(2n), f(3n), \\dots, f(10n^2) \\]\nare perfect squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f780ce91e40477aa89c516388307f93ab0f123bb3459c880ca853069683e2e34 | All points in a 100 x 100 grid are colored in one of four colors: red, green, blue, and yellow, such that there are 25 points of each color in each row and each column. Let $N$ be the number of ways to choose two rows and two columns such that their four intersection points are all different colors. Calculate the remai... | [
"no_valid_answer_found"
] | [
"All points in a 100 x 100 grid are colored in one of four colors: red, green, blue, and yellow, such that there are 25 points of each color in each row and each column. Let $N$ be the number of ways to choose two rows and two columns such that their four intersection points are all different colors. Calculate the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-76be1b098587ec9bacbeceb9d878bd6f0ca7006feaeb213ff2d62e4cddf98ba6 | All points in the plane are colored using $n$ colors. In each line, there are points of no more than two colors. What is the maximum number of colors that can be used? | [
"3"
] | [
"All points in the plane are colored using $n$ colors. In each line, there are points of no more than two colors. What is the maximum number of colors that can be used?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-24008af9ffc0a1194347fdc89614988d415affa9bc42685e0d7b1abc3abac095 | All points in the plane are colored using three colors: red, blue, and green, with each color used more than once. Let \( f(d) \) be the minimum number of distinct distances between points of different colors that must exist for any such coloring. Find \( f(10) \). | [
"9"
] | [
"All points in the plane are colored using three colors: red, blue, and green, with each color used more than once. Let \\( f(d) \\) be the minimum number of distinct distances between points of different colors that must exist for any such coloring. Find \\( f(10) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-c971a1a04ac70c19af0c10085a4a892bdd2ce46af874c8d6cebfbb7c22de63a9 | All points on the circumference of a circle are painted either red, blue, or green. Let $N$ be the number of ways to choose three points of the same color that form an isosceles triangle. Find the minimum possible value of $N$ if there are at least three points of each color on the circumference. | [
"0"
] | [
"All points on the circumference of a circle are painted either red, blue, or green. Let $N$ be the number of ways to choose three points of the same color that form an isosceles triangle. Find the minimum possible value of $N$ if there are at least three points of each color on the circumference."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-345b666c7a5d16455a85f16ed441bd36866246549a363cbdae5db0ab12784e6a | All points with integer coordinates in the \(xy\)-plane are colored using three colors: red, blue, and green, with each color being used at least once. The point \((0,0)\) is red and the point \((0,1)\) is blue. Find the minimum possible area of a right-angled triangle whose vertices are points with integer coordinates... | [
"\\dfrac{1}{2}"
] | [
"All points with integer coordinates in the \\(xy\\)-plane are colored using three colors: red, blue, and green, with each color being used at least once. The point \\((0,0)\\) is red and the point \\((0,1)\\) is blue. Find the minimum possible area of a right-angled triangle whose vertices are points with integer ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18b791747e5d9b3dc98ba2a3b801c80869776207cb734d7c89d20edbbd964d17 | All positive integers $1, 2, \ldots, 121$ are arranged in the squares of an $11 \times 11$ table. Dima found the product of the numbers in each row, and Sasha found the product of the numbers in each column. If it is possible for them to get the same set of $11$ numbers, calculate the product of these $11$ numbers. If ... | [
"121!"
] | [
"All positive integers $1, 2, \\ldots, 121$ are arranged in the squares of an $11 \\times 11$ table. Dima found the product of the numbers in each row, and Sasha found the product of the numbers in each column. If it is possible for them to get the same set of $11$ numbers, calculate the product of these $11$ numbe... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b20f1a56700aec741c4a3f845262ae176e5c613fc5336dea57422bfd4c08655 | All positive integers 1, 2, ..., 121 are arranged in the squares of an 11x11 table. Dima calculates the product of numbers in each row, and Sasha calculates the product of the numbers in each column. If it is possible for Dima and Sasha to obtain the same set of 11 numbers, find the product of the numbers in the first ... | [
"0"
] | [
"All positive integers 1, 2, ..., 121 are arranged in the squares of an 11x11 table. Dima calculates the product of numbers in each row, and Sasha calculates the product of the numbers in each column. If it is possible for Dima and Sasha to obtain the same set of 11 numbers, find the product of the numbers in the f... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-fd793766c34a7b6a76b52f4dfc6e4858df74f0dea8d02d3e931d7e633901f1ec | All positive integers are colored either red or green such that: - There are equally many red as green integers. - The sum of three (not necessarily distinct) red integers is red. - The sum of three (not necessarily distinct) green integers is green. Find all colorings that satisfy these conditions. | [
"\\text{All even numbers are one color and all odd numbers are the other color.}"
] | [
"All positive integers are colored either red or green such that:\n- There are equally many red as green integers.\n- The sum of three (not necessarily distinct) red integers is red.\n- The sum of three (not necessarily distinct) green integers is green.\nFind all colorings that satisfy these conditions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-214ac0f56f6bdd71de4f4423359f2a57541815b61680253f8171f486e53c32be | All positive integers are distributed among two disjoint sets \( N_1 \) and \( N_2 \) such that no difference of two numbers belonging to the same set is a prime greater than 100. Find all such distributions. | [
"All the odd positive integers are in \\( N_1 \\) and all the even positive integers are in \\( N_2 \\), or all the odd positive integers are in \\( N_2 \\) and all the even positive integers are in \\( N_1 \\)."
] | [
"All positive integers are distributed among two disjoint sets \\( N_1 \\) and \\( N_2 \\) such that no difference of two numbers belonging to the same set is a prime greater than 100. Find all such distributions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 6,
"tir": 16
} | 69 | 0 | 0 |
omr-ws-69fc2149176d7756c3c3520eae6d737a3fa75069afd5da3560a177312689f33c | All positive integers relatively prime to $2015$ are written in increasing order. Let the twentieth number be $p$. The value of \[\dfrac{2015}{p}-1\] can be expressed as $\dfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Compute $a+b$. | [
"2015"
] | [
"All positive integers relatively prime to $2015$ are written in increasing order. Let the twentieth number be $p$. The value of \n\\[\\dfrac{2015}{p}-1\\]\ncan be expressed as $\\dfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Compute $a+b$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d8618abf04f1817a77bf413842f11e0b55f82a2a29c7f66557b8ce94b0c664ad | All positive integers whose digits add up to 11 are listed in increasing order: $29, 38, 47, ...$. What is the eleventh number in that list? | [
"137"
] | [
"All positive integers whose digits add up to 11 are listed in increasing order: $29, 38, 47, ...$. What is the eleventh number in that list?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2e4009b411cf72ea5c7878de631a50f7231d75ecce5242692bd1122669494838 | All positive integers with a digit sum of 2020 are written in increasing order. Which number is in the 230th position? | [
"699997\\underbrace{999\\ldots999}_{219}"
] | [
"All positive integers with a digit sum of 2020 are written in increasing order. Which number is in the 230th position?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-82e61fbded0e667a1c17de68486d91767250578bfcf0fd30ff8dff501b19eabf | All possible 2000-gons are inscribed in a circle of perimeter \( p \). Let \( X = \{ x \mid \text{an inscribed polygon has a side of length } x \} \). What is the greatest member of \( X \)? | [
"\\(\\frac{p}{\\pi}\\)"
] | [
"All possible 2000-gons are inscribed in a circle of perimeter \\( p \\). Let \\( X = \\{ x \\mid \\text{an inscribed polygon has a side of length } x \\} \\). What is the greatest member of \\( X \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-39bbcb620d0e088e63a25fc8f0ddd23c97f33da37cd7b6f37211a732fd48b06f | All possible 4-digit numbers are created using the digits 5, 6, 7, 8 and then sorted from smallest to largest. Similarly, all possible 4-digit numbers are created using the digits 3, 4, 5, 6 and then sorted from smallest to largest. The first number of the second type is subtracted from the first number of the first ty... | [
"568832"
] | [
"All possible 4-digit numbers are created using the digits 5, 6, 7, 8 and then sorted from smallest to largest. Similarly, all possible 4-digit numbers are created using the digits 3, 4, 5, 6 and then sorted from smallest to largest. The first number of the second type is subtracted from the first number of the fir... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 25
} | 26 | 0 | 0 |
omr-ws-9684559547f92cb161cf8a5f01ae9fb81918bc4eb48ffd84238fe367acf64967 | All possible 6-digit numbers, in each of which the digits occur in non-increasing order (from left to right, e.g., 877550), are written as a sequence in increasing order. Find the 2005-th number in this sequence. | [
"864110"
] | [
"All possible 6-digit numbers, in each of which the digits occur in non-increasing order (from left to right, e.g., 877550), are written as a sequence in increasing order. Find the 2005-th number in this sequence."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-54b34e27ad7b4e4fe1d1d1862410446233f0fc1f877798923a944ba0060ab38c | All possible rearrangements of the letters 'AAGMMR' are placed in alphabetical order. On this list of 'words,' in what place is the word 'MAGMAR'? | [
"99"
] | [
"All possible rearrangements of the letters 'AAGMMR' are placed in alphabetical order. On this list of 'words,' in what place is the word 'MAGMAR'?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-32fb89f8e401f07198c8c24415ed40101c9f46dd0592c2974cc19444ca04cecb | All possible sequences of four consecutive positive integers less than 200 are created. A sequence is called orderly if the smallest member is divisible by 2, the next smallest member is divisible by 3, the next smallest is divisible by 4, and the largest is divisible by 5. What is the probability that a randomly-selec... | [
"\\dfrac{1}{49}"
] | [
"All possible sequences of four consecutive positive integers less than 200 are created. A sequence is called orderly if the smallest member is divisible by 2, the next smallest member is divisible by 3, the next smallest is divisible by 4, and the largest is divisible by 5. What is the probability that a randomly-... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-518b07478440937aec70b3f8ea70c7f1bec0d5de340f13a76f9eacaf73b867ff | All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each sequence, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values. | [
"100"
] | [
"All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each sequence, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-816437e8a79cca63abeffe1fa0abee6acffaa353d19b648040d992f9cf9d4ec6 | All possible straight lines joining the vertices of a cube with mid-points of its edges are drawn. At how many points inside the cube do two or more of these lines meet? | [
"12"
] | [
"All possible straight lines joining the vertices of a cube with mid-points of its edges are drawn. At how many points inside the cube do two or more of these lines meet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-505b20d7b12d6ccc85f07bdef766a6c3991acbbf4ff7a6cf096438e627b7b12a | All possible values of \( \frac{y}{x} \) are listed out in a row (where \( x \) and \( y \) are nonzero numbers which can be imaginary) such that \( y^4 = 5x^2y^2 + 69x^4 \). Each of the \( r_i \) are summed, and two \( r_i \) are randomly selected from the list. The two numbers are multiplied and inserted back into th... | [
"\\dfrac{1}{6}"
] | [
"All possible values of \\( \\frac{y}{x} \\) are listed out in a row (where \\( x \\) and \\( y \\) are nonzero numbers which can be imaginary) such that \\( y^4 = 5x^2y^2 + 69x^4 \\). Each of the \\( r_i \\) are summed, and two \\( r_i \\) are randomly selected from the list. The two numbers are multiplied and ins... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-b8ca75cef1b4df6b3562c2e790bda377818112c11c8033e899d7f61fcb8e75a2 | All possible vertical lines $x = k$ and horizontal lines $y = m$ are drawn on the coordinate plane, where $k$ and $m$ are integers. A red line with the equation $19x + 96y = c$ is also drawn. Let $M$ be the number of segments of different lengths formed on the red line when intersecting with the black lines. What value... | [
"2"
] | [
"All possible vertical lines $x = k$ and horizontal lines $y = m$ are drawn on the coordinate plane, where $k$ and $m$ are integers. A red line with the equation $19x + 96y = c$ is also drawn. Let $M$ be the number of segments of different lengths formed on the red line when intersecting with the black lines. What ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-4d3220bca3178df4f1e155cb065a56daae6275f02896b3477ab30f9a4bf6659d | All products produced in a manufacturing facility are screened. 97% of the defective products are identified. 8% of the products that are not defective are also identified as defective by the screening process. If 1 out of every 1500 products are defective, what is the probability that a product identified by the scree... | [
"0.008"
] | [
"All products produced in a manufacturing facility are screened. 97% of the defective products are identified. 8% of the products that are not defective are also identified as defective by the screening process. If 1 out of every 1500 products are defective, what is the probability that a product identified by the ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-14abdf61ff172f38cf53b0209c086ad5abe9bed843b694d9515ccf2bb33d7011 | All roots of $$\prod^{25}_{n=1} \prod^{2n}_{k=0}(-1)^k \cdot x^k = 0$$ are written in the form $r(\cos \phi + i\sin \phi)$ for $i^2 = -1$, $r > 0$, and $0 \le \phi < 2\pi$. What is the smallest positive value of $\phi$ in radians? | [
"\\dfrac{\\pi}{51}"
] | [
"All roots of $$\\prod^{25}_{n=1} \\prod^{2n}_{k=0}(-1)^k \\cdot x^k = 0$$ are written in the form $r(\\cos \\phi + i\\sin \\phi)$ for $i^2 = -1$, $r > 0$, and $0 \\le \\phi < 2\\pi$. What is the smallest positive value of $\\phi$ in radians?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-677d7ce675f33f47af812a95c8bdebbaba280988da43a227df3cbaebc16a1335 | All roots of \( ax^3 - x^2 + bx - 1 = 0 \) are real and positive where \( a, b \) are both non-zero reals. Find the minimum value of \( S = \frac{5a^2 - 3ab + 2}{a^2(b - a)} \). | [
"12\\sqrt{3}"
] | [
"All roots of \\( ax^3 - x^2 + bx - 1 = 0 \\) are real and positive where \\( a, b \\) are both non-zero reals. Find the minimum value of \\( S = \\frac{5a^2 - 3ab + 2}{a^2(b - a)} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-cffc610965ccf377f232fb1d225635886db273c3afe0755301ce7f3be2c0a084 | All roots of the polynomial \( f = x^4 + ax^3 + bx^2 + cx + d \) are non-positive integers, and \( a + b + c + d = 2010 \). Find the value of \( d \). | [
"0"
] | [
"All roots of the polynomial \\( f = x^4 + ax^3 + bx^2 + cx + d \\) are non-positive integers, and \\( a + b + c + d = 2010 \\). Find the value of \\( d \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-abca845d698e8eacb0ab7414eb525576c93a82a25ceea4bf03ce9fab1952f5fd | All segments joining $n$ points (no three of which are collinear) are colored in one of $k$ colors. What is the smallest $k$ for which there always exists a closed polygonal line with vertices at some of the $n$ points, whose sides are all of the same color? | [
"\\( k = \\left\\lceil \\frac{n}{2} \\right\\rceil - 1 \\)"
] | [
"All segments joining $n$ points (no three of which are collinear) are colored in one of $k$ colors. What is the smallest $k$ for which there always exists a closed polygonal line with vertices at some of the $n$ points, whose sides are all of the same color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 8,
"tir": 4
} | 33 | 0 | 0 |
omr-ws-ccd1de323035e5feec2e8e20afc78bb51a92dc6081b34ac508aa92872cf845b9 | All shirts sell at the same price, and all caps sell at a different price than the shirts (but the same as other caps). Bob pays \$104 for 6 shirts and 11 caps. Joe pays an integral amount of dollars for 1 shirt and 2 caps. Janie buys 1 shirt and 1 cap. Do you even need to know anything about Joe? | [
"14"
] | [
"All shirts sell at the same price, and all caps sell at a different price than the shirts (but the same as other caps). Bob pays \\$104 for 6 shirts and 11 caps. Joe pays an integral amount of dollars for 1 shirt and 2 caps. Janie buys 1 shirt and 1 cap. Do you even need to know anything about Joe?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 1,
"tir": 4
} | 12 | 0 | 0 |
omr-ws-7c0ecc76a87abc81640fabac3cb6efa124d53c68862c49dbd841ef017c82f7ea | All shirts sell at the same price, and all caps sell at a different price than the shirts (but the same as other caps). Bob pays \$104 for 6 shirts and 11 caps. Joe pays an integral amount of dollars for 1 shirt and 2 caps. Janie buys 1 shirt and 1 cap. How much does Janie pay? | [
"14"
] | [
"All shirts sell at the same price, and all caps sell at a different price than the shirts (but the same as other caps). Bob pays \\$104 for 6 shirts and 11 caps. Joe pays an integral amount of dollars for 1 shirt and 2 caps. Janie buys 1 shirt and 1 cap. How much does Janie pay?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-0f84544983afaa917a9250103411b9fb907dc7cb5d5b255c5be1a51d66116a8f | All sides and diagonals of a 25-gon are drawn either red or white. Calculate the minimum number of monochromatic triangles, having all three sides in the same color and all three vertices from the vertices of the 25-gon, that can be guaranteed to exist. | [
"506"
] | [
"All sides and diagonals of a 25-gon are drawn either red or white. Calculate the minimum number of monochromatic triangles, having all three sides in the same color and all three vertices from the vertices of the 25-gon, that can be guaranteed to exist."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-2e1d5c4da70bc98db98117a218a4e88eadf7caa4928c6df9f5d8d0ac31f2a36a | All sides of a convex hexagon $ABCDEF$ are $1$. Let $M$ and $m$ be the maximum and minimum possible values of the three diagonals $AD$, $BE$, and $CF$. Find the range of $M$ and $m$. | [
"[\\sqrt{3}, 2]"
] | [
"All sides of a convex hexagon $ABCDEF$ are $1$. Let $M$ and $m$ be the maximum and minimum possible values of the three diagonals $AD$, $BE$, and $CF$. Find the range of $M$ and $m$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-5eac98de0c278175fd584087c80952117caed381de9470b47f95bc9174410962 | All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A = \angle B = 90^\circ$. What is the degree measure of $\angle E$? | [
"150^\\circ"
] | [
"All sides of the convex pentagon $ABCDE$ are of equal length, and $\\angle A = \\angle B = 90^\\circ$. What is the degree measure of $\\angle E$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b696895605e9804b0ee2717eb9c84301036d2f7f6cabc594f90a1b4299f60366 | All six sides of a convex hexagon \(A_1A_2A_3A_4A_5A_6\) are colored red. Each of the diagonals is either blue or red. Find the number of such colorings such that each triangle \(A_iA_jA_k\) (where \(i, j, k\) are distinct) has at least one red side. | [
"32"
] | [
"All six sides of a convex hexagon \\(A_1A_2A_3A_4A_5A_6\\) are colored red. Each of the diagonals is either blue or red. Find the number of such colorings such that each triangle \\(A_iA_jA_k\\) (where \\(i, j, k\\) are distinct) has at least one red side."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2d84298df2fdafcf390dd6b7e3554bee7eb061ea237c194ccd61c708123f2412 | All six-digit natural numbers from $100000$ to $999999$ are written in ascending order without spaces. What is the largest value of $k$ for which the same $k$-digit number can be found in at least two different places in this string? | [
"5"
] | [
"All six-digit natural numbers from $100000$ to $999999$ are written in ascending order without spaces. What is the largest value of $k$ for which the same $k$-digit number can be found in at least two different places in this string?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-227f2a9cd7817d755274a0268404b7985f81b4df94c7d7b9b288267b3ea2690d | All squares of a board of size \( (n+1) \times (n-1) \) are painted with three colors such that for any two different columns and any two different rows, the 4 squares at their intersections do not all have the same color. Find the greatest possible value of \( n \). | [
"5"
] | [
"All squares of a board of size \\( (n+1) \\times (n-1) \\) are painted with three colors such that for any two different columns and any two different rows, the 4 squares at their intersections do not all have the same color. Find the greatest possible value of \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-c56456013cc5838b8fe4c929e5a415df007b198aa2c42f0bbc83dde4f5316f07 | All the angles of a hexagon are 120 degrees and four consecutive sides have lengths of 5, 7, 4, and 6 units. Find the sum of the lengths of the other two sides. | [
"11"
] | [
"All the angles of a hexagon are 120 degrees and four consecutive sides have lengths of 5, 7, 4, and 6 units. Find the sum of the lengths of the other two sides."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6f710efc3c807276b97d36328609a543781332cf6790218efd98d86c1d0a69bc | All the cells on a $100 \times 100$ board are painted in various colors such that no pair of squares with a common side or vertex can have the same color. Determine the minimum number of colors with which it is possible to make such a coloring. | [
"4"
] | [
"All the cells on a $100 \\times 100$ board are painted in various colors such that no pair of squares with a common side or vertex can have the same color. Determine the minimum number of colors with which it is possible to make such a coloring."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3a554556818546930b973fa7f3b1dfeafa4b5fee21efd69212c53e87fadc86e7 | All the complex roots of $(z + 1)^5 = 32z^5,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. | [
"\\frac{2}{3}"
] | [
"All the complex roots of $(z + 1)^5 = 32z^5,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e6f833dafbc209e9eaee8a540308e8228b0c81ccc37051a54ad4320c9dbf980a | All the digits of a 15-digit number are either p or c. p shows up 3 more times than c does, and the average of the digits is c - p. What is p + c? | [
"11"
] | [
"All the digits of a 15-digit number are either p or c. p shows up 3 more times than c does, and the average of the digits is c - p. What is p + c?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-5632eeb17abb9fe3c4dff366362614e52d137da8ded31e3adeec11144c7954c7 | All the digits of a number are different, the first digit is not zero, and the sum of the digits is 36. There are $ N \times 7! $ such numbers. What is the value of $ N $? | [
"107"
] | [
"All the digits of a number are different, the first digit is not zero, and the sum of the digits is 36. There are $ N \\times 7! $ such numbers. What is the value of $ N $?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-0142b9175d28391f83492c05986934f4fb46f44cf8233609d833d1327b82b110 | All the digits of the positive integer N are either 0 or 1. The remainder after dividing N by 37 is 18. What is the smallest number of times that the digit 1 can appear in N? | [
"5"
] | [
"All the digits of the positive integer N are either 0 or 1. The remainder after dividing N by 37 is 18. What is the smallest number of times that the digit 1 can appear in N?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-e6bb25178453774336e7402eae5271e2f0678206928dd83f7906a1c20c21ff6f | All the divisors of \(360^{10}\) are written on a board. At a move, we can take two numbers, neither of which is divisible by the other, and replace them with their greatest common divisor and lowest common multiple. At some point, we will no longer be able to perform new operations. How many different numbers will be ... | [
"61"
] | [
"All the divisors of \\(360^{10}\\) are written on a board. At a move, we can take two numbers, neither of which is divisible by the other, and replace them with their greatest common divisor and lowest common multiple. At some point, we will no longer be able to perform new operations. How many different numbers w... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-26cd3cc4e1afb906a15c8fb8fd3094e04cd92db0c379be64a677c9486759781e | All the divisors of \(8 \cdot 10^6\) are written on a board. At a move, we can take two numbers, neither of which is divisible by the other, and replace them with their greatest common divisor and lowest common multiple. At some point, we will no longer be able to perform new operations. How many different numbers will... | [
"17"
] | [
"All the divisors of \\(8 \\cdot 10^6\\) are written on a board. At a move, we can take two numbers, neither of which is divisible by the other, and replace them with their greatest common divisor and lowest common multiple. At some point, we will no longer be able to perform new operations. How many different numb... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-62033a9206c7eb3c44aae85405faa582add6eb181c94ac9650e73c2c6cf83f17 | All the edges of a regular prism $ABCA_1B_1C_1$ are $a$ long. We consider line segments with their endpoints lying on the diagonals $BC_1$ and $CA_1$ of the lateral faces, the segments being parallel to the plane $ABB_1A_1$. (a) One of the segments is drawn through the point $M$ of the diagonal $BC_1$ such that $|BM|:|... | [
"(a) $\\frac{a\\sqrt 5}{3}$; (b) $\\frac{a\\sqrt 5}{5}$"
] | [
"All the edges of a regular prism $ABCA_1B_1C_1$ are $a$ long. We consider line segments with their endpoints lying on the diagonals $BC_1$ and $CA_1$ of the lateral faces, the segments being parallel to the plane $ABB_1A_1$. (a) One of the segments is drawn through the point $M$ of the diagonal $BC_1$ such that $|... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-333622830b25e9cf256b212c07f1c7171dc7272eae0ae44477dd6277d5ff3552 | All the edges of the triangular pyramid $SABC$ are equal. A point $M$ is taken on the edge $SA$ such that $| SM | = | MA |$. A point $N$ is taken on the edge $SB$ such that $| SN | = \frac{1}{3} | SB |$. A plane is drawn through the points $M$ and $N$ which is parallel to the median $AD$ of the base $ABC$. Find the rat... | [
"$\\frac{1}{6}$"
] | [
"All the edges of the triangular pyramid $SABC$ are equal. A point $M$ is taken on the edge $SA$ such that $| SM | = | MA |$. A point $N$ is taken on the edge $SB$ such that $| SN | = \\frac{1}{3} | SB |$. A plane is drawn through the points $M$ and $N$ which is parallel to the median $AD$ of the base $ABC$. Find ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1577177f2dcfa5fa15e458c220be426c2772ecfdd6c2845475614b5018fe0f92 | All the edges of the triangular pyramid $SABC$ are equal. A point $M$ taken on the edge $SA$ is such that $| SM | = | MA |$, a point $N$ taken on the edge $SB$ is such that $3 | SN | = | SB |$. A plane drawn through the points $M$ and $N$ is parallel to the median $AD$ of the base $ABC$. Find the ratio of the volume of... | [
"$\\frac{1}{6}$"
] | [
"All the edges of the triangular pyramid $SABC$ are equal. A point $M$ taken on the edge $SA$ is such that $| SM | = | MA |$, a point $N$ taken on the edge $SB$ is such that $3 | SN | = | SB |$. A plane drawn through the points $M$ and $N$ is parallel to the median $AD$ of the base $ABC$. Find the ratio of the volu... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a0b97d97e9cf0eba8e778d38db5dc4f46dd06c59e5da615ee8fab8459cd6869b | All the faces of a parallelepiped are equal rhombuses with sides $a$ and acute angles $\alpha $. Determine the volume of the parallelepiped. | [
"$V=2a^3 \\sin\\frac{\\alpha }{2}\\sqrt {\\sin\\frac{3\\alpha }{2}\\sin\\frac{\\alpha }{2}}$"
] | [
"All the faces of a parallelepiped are equal rhombuses with sides $a$ and acute angles $\\alpha $. Determine the volume of the parallelepiped."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f05ac38df0724c5efdeee4f712989f1c106b519cee84160c13c96c304589c9ea | All the faces of a triangular pyramid are isosceles right triangles, except the base, which is an equilateral triangle. The slant height of the pyramid is \( L \). Find the volume of the pyramid in terms of \( L \). | [
"\\(\\frac{\\sqrt{6} L^3}{36}\\)"
] | [
"All the faces of a triangular pyramid are isosceles right triangles, except the base, which is an equilateral triangle. The slant height of the pyramid is \\( L \\). Find the volume of the pyramid in terms of \\( L \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b8934331bf21d0338cd499f984e9947a22c48d8b29dc18be340c8d48d50222d4 | All the faces of the triangular pyramid are equal isosceles triangles and the altitude of the pyramid coincides with the altitude of one of its lateral faces. Find the volume of the pyramid if the distance between the larger opposite edges is unity. | [
"$\\frac{2}{3}$"
] | [
"All the faces of the triangular pyramid are equal isosceles triangles and the altitude of the pyramid coincides with the altitude of one of its lateral faces. Find the volume of the pyramid if the distance between the larger opposite edges is unity."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48bcd62e5be30b52cdbed0feebcf45e3b59651a4632d73f9d142a455d9570faf | All the harvesting combines on a certain farm can take in the harvest in 24 hours. However, according to the work plan, only one combine was in operation during the first hour, two during the second hour, three during the third hour, and so on, until all combines were in operation. All machines worked together for only... | [
"25 harvesting combines"
] | [
"All the harvesting combines on a certain farm can take in the harvest in 24 hours. However, according to the work plan, only one combine was in operation during the first hour, two during the second hour, three during the third hour, and so on, until all combines were in operation. All machines worked together for... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b287cf057aad5f70449766ddaba3323ac025ffc9189d1117c2d361d15566a2f | All the integer multiples of 9 are written as a sequence of the form $$9;\ 18;\ 27;\ 36;\ 45;\ 54;\ 63;\ 72;\ 81;\ 90;\ 99;\ 108;\ 117; \ldots \eqno(*)$$ and for each of these numbers the sum of its digits is found: $$9;\ 9;\ 9;\ 9;\ 9;\ 9;\ 9;\ 9;\ 9;\ 9;\ 18;\ 9;\ 9; \ldots \eqno(**)$$ In what place in sequence $(**)... | [
"For the first time the number 81 occurs in the 111111111th place. The number 27 is consecutively repeated 4 times earlier than the number 36 first occurs"
] | [
"All the integer multiples of 9 are written as a sequence of the form $$9;\\ 18;\\ 27;\\ 36;\\ 45;\\ 54;\\ 63;\\ 72;\\ 81;\\ 90;\\ 99;\\ 108;\\ 117; \\ldots \\eqno(*)$$ and for each of these numbers the sum of its digits is found: $$9;\\ 9;\\ 9;\\ 9;\\ 9;\\ 9;\\ 9;\\ 9;\\ 9;\\ 9;\\ 18;\\ 9;\\ 9; \\ldots \\eqno(**)$... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b8516c76482be01be898e2edb322a6652130b21050fa5407022585695156a7ed | All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3 \times 3$ array of squares, one number in each square, such that if two numbers are consecutive, they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center? | [
"7"
] | [
"All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3 \\times 3$ array of squares, one number in each square, such that if two numbers are consecutive, they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 1
} | 21 | 0 | 0 |
omr-ws-70b03008bfdb88c0e1272b2f7dc3ab42f267c230becc08731fc72a8a7ad15087 | All the numbers $1, 2, \ldots, 9$ are written in the cells of a $3 \times 3$ table (one number per cell). In each move, an arbitrary $2 \times 2$ square of the table can be chosen, and all four numbers in that square can be either decreased by 1 or increased by 1. After some number of such moves, all numbers in the tab... | [
"\\( a = 1, 5, 9 \\)"
] | [
"All the numbers $1, 2, \\ldots, 9$ are written in the cells of a $3 \\times 3$ table (one number per cell). In each move, an arbitrary $2 \\times 2$ square of the table can be chosen, and all four numbers in that square can be either decreased by 1 or increased by 1. After some number of such moves, all numbers in... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-6e42cc1b01c3e100fea9eba8cc236f77ef06ee39d581ec66016e6b1c15622855 | All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of th... | [
"729"
] | [
"All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-54e86dd09a4a9ec4a9bd7398ee5e80341ebe13e7c0174714006d1a1cb582129f | All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the ... | [
"729"
] | [
"All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-70faf949a051e9be6e9f7ae1359143d53ab42d25fbe9dd0e75fbf193b1cea076 | All the permutations of the letters a, b, c, d, e are written down and arranged in alphabetical order. What is the position of the arrangement "debac"? | [
"95"
] | [
"All the permutations of the letters a, b, c, d, e are written down and arranged in alphabetical order. What is the position of the arrangement \"debac\"?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-90b3f995d18199cb50a7a00b64bf4c6b6f28a202cc277288dc3551fcbaecaf77 | All the points in the plane are colored using three colors. Consider the smallest possible area of a triangle with vertices of the same color such that either it is isosceles or its angles are in geometric progression. Find the maximum possible value of this area if the distance between any two points of the same color... | [
"\\dfrac{\\sqrt{3}}{4}"
] | [
"All the points in the plane are colored using three colors. Consider the smallest possible area of a triangle with vertices of the same color such that either it is isosceles or its angles are in geometric progression. Find the maximum possible value of this area if the distance between any two points of the same ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-ce89a95d889566f91aade412d28fe0e5eae47b70f59e43eb3a86b6f602d75252 | All the positive integers from 1 to 1000 are written on the blackboard in some order, and there is a collection of cards each containing 10 numbers. If there is a card with numbers $1 \le a_1 < a_2 < \ldots < a_{10} \le 1000$ in the collection, then it is allowed to arrange in increasing order the numbers at places $a_... | [
"111"
] | [
"All the positive integers from 1 to 1000 are written on the blackboard in some order, and there is a collection of cards each containing 10 numbers. If there is a card with numbers $1 \\le a_1 < a_2 < \\ldots < a_{10} \\le 1000$ in the collection, then it is allowed to arrange in increasing order the numbers at pl... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-08f52f8c6283646afcff1079f744934a6c6b8c2f4317240871a41f9a13896514 | All the positive integers from 1 to 1000 are written on the blackboard in some order. There is a collection of cards, each containing 10 numbers. If a card has numbers \(1 \le a_1 < a_2 < \ldots < a_{10} \le 1000\), it is allowed to arrange the numbers at positions \(a_1, a_2, \ldots, a_{10}\) in increasing order. What... | [
"111"
] | [
"All the positive integers from 1 to 1000 are written on the blackboard in some order. There is a collection of cards, each containing 10 numbers. If a card has numbers \\(1 \\le a_1 < a_2 < \\ldots < a_{10} \\le 1000\\), it is allowed to arrange the numbers at positions \\(a_1, a_2, \\ldots, a_{10}\\) in increasin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-b1ae23e7402f42d0ea63abc2878315c3958cb67dc17be9bef8137551129103ae | All the positive integers that are coprime to 2021 are grouped in increasing order, such that the nth group contains \(2n - 1\) numbers. Hence the first three groups are \(\{1\}\), \(\{2, 3, 4\}\), \(\{5, 6, 7, 8, 9\}\). Suppose that 2022 belongs to the \(k\)th group. Find \(k\). | [
"43"
] | [
"All the positive integers that are coprime to 2021 are grouped in increasing order, such that the nth group contains \\(2n - 1\\) numbers. Hence the first three groups are \\(\\{1\\}\\), \\(\\{2, 3, 4\\}\\), \\(\\{5, 6, 7, 8, 9\\}\\). Suppose that 2022 belongs to the \\(k\\)th group. Find \\(k\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-43c1218d18df139293900003f593dde9d4ba00754a66b16d2b2dc81dd5d7c9ac | All the rationals are coloured with $n$ colours such that if rationals $a$ and $b$ are colored with different colours, then $\frac{a+b}{2}$ is coloured with a colour different from both $a$ and $b$. Let $f(n)$ be the number of distinct colours used when $n=3$. Calculate $f(3)$. | [
"3"
] | [
"All the rationals are coloured with $n$ colours such that if rationals $a$ and $b$ are colored with different colours, then $\\frac{a+b}{2}$ is coloured with a colour different from both $a$ and $b$. Let $f(n)$ be the number of distinct colours used when $n=3$. Calculate $f(3)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e0bedba99a6894482ca46a8a9be7353c6ff98a9067e4008002cef2b957a31afa | All the roots of polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$? | [
"-88"
] | [
"All the roots of polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 2
} | 50 | 0 | 0 |
omr-ws-6a8c012d8761c36cf815f1fdba923e798cacf1d14a2745104de6fa1a03737ae7 | All the roots of the equation $x^6 - 6x^5 + 15x^4 - ax^3 + 15x^2 - bx + 1 = 0$ are positive. Find the values of $a$ and $b$. | [
"a = 20, b = 6"
] | [
"All the roots of the equation $x^6 - 6x^5 + 15x^4 - ax^3 + 15x^2 - bx + 1 = 0$ are positive. Find the values of $a$ and $b$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-a348bcf267c0a195b5a1f97e0a018ca4326abf1b95e46abeddfae2ad6d67ecfc | All the roots of the polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers, possibly repeated. What is the value of $B$? | [
"-88"
] | [
"All the roots of the polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers, possibly repeated. What is the value of $B$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-41ce6b8510f92c9eb0f6fa3fc3fbb88364bc0336c3fca0a2adc03e067bed12fa | All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$? | [
"\\(-30\\)"
] | [
"All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5cabb987247943dfae9780a708d8af62c7587ecb7397e3aa06d0521e39569e22 | All the roots of the polynomial \( z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16 \) are positive integers, possibly repeated. What is the value of \( B \)? | [
"-88"
] | [
"All the roots of the polynomial \\( z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16 \\) are positive integers, possibly repeated. What is the value of \\( B \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-d6f4c255fa8bc5081b0b6e46b7f9b40f06ea6eb46a6169d6ea91fa0ba5c1da10 | All the roots of the polynomial \( z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16 \) are positive integers. What is the value of \( B \)? | [
"-88"
] | [
"All the roots of the polynomial \\( z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16 \\) are positive integers. What is the value of \\( B \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-8dd8e83eab4c17af9cb2d58134439dd4402158120287688f30a15e3a7c9bd6bc | All the roots of the polynomial \(z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16\) are positive integers, possibly repeated. What is the value of \(B\)? | [
"-88"
] | [
"All the roots of the polynomial \\(z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16\\) are positive integers, possibly repeated. What is the value of \\(B\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ef17a8103a574f515b9a6f12add376e8b185cf55296595c50b715ecef7e0be94 | All the squares of a board of $(n+1) \times (n-1)$ squares are painted with three colors such that, for any two different columns and any two different rows, the 4 squares in their intersections do not all have the same color. Find the greatest possible value of $n$. | [
"10"
] | [
"All the squares of a board of $(n+1) \\times (n-1)$ squares are painted with three colors such that, for any two different columns and any two different rows, the 4 squares in their intersections do not all have the same color. Find the greatest possible value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-2a4b5d297fb604ef10084212a1d62fc1123227805433d004a70c55b2b8e4aa29 | All the three face angles of a trihedral angle are acute and one of them is equal to $\alpha $. The dihedral angles whose edges are the sides of this face angle are equal to $\beta $ and $\gamma $ respectively. Find the other two face angles. | [
"\\tan \\gamma _1=\\frac{\\sin \\alpha \\tan \\beta \\csc \\gamma } {1+\\cos \\alpha \\tan \\beta \\cot \\gamma } and \\tan \\beta _1=\\frac{\\sin \\alpha \\tan \\gamma \\csc \\beta } {1+\\cos \\alpha \\tan \\gamma \\cot \\beta }"
] | [
"All the three face angles of a trihedral angle are acute and one of them is equal to $\\alpha $. The dihedral angles whose edges are the sides of this face angle are equal to $\\beta $ and $\\gamma $ respectively. Find the other two face angles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e042c6f30e22a69799d0e7ad5cc4ea70e4af33089a9b4a76f158f5ec930e01f | All three vertices of an equilateral triangle are on the parabola \( y = x^2 \), and one of its sides has a slope of 2. The x-coordinates of the three vertices have a sum of \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. What is the value of \( m + n \)? | [
"14"
] | [
"All three vertices of an equilateral triangle are on the parabola \\( y = x^2 \\), and one of its sides has a slope of 2. The x-coordinates of the three vertices have a sum of \\( \\frac{m}{n} \\), where \\( m \\) and \\( n \\) are relatively prime positive integers. What is the value of \\( m + n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-42db18ba0247be31dd1d3200ad5c1aa72f83f4f0701e459372226d537ce33fcf | All vertices and centers of all faces were marked on a cube ($14$ points in total). It turned out that the distance from any of these points to a certain plane takes only two different values, the smaller of which is equal to $1$. Find the length of the edge of the cube. | [
"4"
] | [
"All vertices and centers of all faces were marked on a cube ($14$ points in total). It turned out that the distance from any of these points to a certain plane takes only two different values, the smaller of which is equal to $1$. Find the length of the edge of the cube."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-ab784cdd8f7ea669d32a044a5d2ae171efcfbaf110bb4f9fc77739dce4c72199 | All vertices of a $789$-gon are marked red, and $615$ more red points are marked inside it. No three red points lie on a straight line. The polygon is divided into triangles such that all red points, and only red points, are their vertices. How many triangles are there? | [
"2017"
] | [
"All vertices of a $789$-gon are marked red, and $615$ more red points are marked inside it. No three red points lie on a straight line. The polygon is divided into triangles such that all red points, and only red points, are their vertices. How many triangles are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-9577611ffb108e46da8da4da54b25a7879b87922d77bff7b78700b5c987448d5 | All vertices of a convex polygon lie at the vertices of the square grid, no edge length exceeds $4$ (the edge length of a square is $1$). What is the largest number of sides such a polygon can have? | [
"32"
] | [
"All vertices of a convex polygon lie at the vertices of the square grid, no edge length exceeds $4$ (the edge length of a square is $1$). What is the largest number of sides such a polygon can have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a7349282f0ba8df24a55dd3df20d7cabc22884d33f1f81ca80e9ca2f9ce73c13 | All vertices of a pyramid lie in the facet planes of a cube but not on the lines including its edges, and each facet plane contains at least one vertex. What is the maximum possible number of the vertices of the pyramid? | [
"6"
] | [
"All vertices of a pyramid lie in the facet planes of a cube but not on the lines including its edges, and each facet plane contains at least one vertex. What is the maximum possible number of the vertices of the pyramid?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-de725d20fba8e0ce182b8fbce164d55c8dead2cdcded148fd9e578de51610731 | All vertices of a pyramid lie on the facets of a cube but not on its edges, and each facet contains at least one vertex. What is the maximum possible number of the vertices of the pyramid? | [
"6"
] | [
"All vertices of a pyramid lie on the facets of a cube but not on its edges, and each facet contains at least one vertex. What is the maximum possible number of the vertices of the pyramid?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-40b38e0eea26077302adc7bf0d011fc7bf25316d3dc4f87368ef89ec94925b33 | All vertices of a regular $2016$-gon are initially white. What is the least number of them that can be repainted black so that there is no right-angled triangle with white vertices? | [
"1008"
] | [
"All vertices of a regular $2016$-gon are initially white. What is the least number of them that can be repainted black so that there is no right-angled triangle with white vertices?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7dbcbbc18125226d48df0705b632fdb9535052ef821403665cf6ba3358cef0fa | Allan and Zarwa are playing a game by tossing a coin. Allan wins as soon as a head is tossed, and Zarwa wins if two tails are tossed. What is the probability that Allan wins? | [
"\\(\\frac{3}{4}\\)"
] | [
"Allan and Zarwa are playing a game by tossing a coin. Allan wins as soon as a head is tossed, and Zarwa wins if two tails are tossed. What is the probability that Allan wins?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-baad19f85ccd5ebc177d17a3bfcfd6327181d2e9e8ba87f1a45a6dbca0b237d4 | Allan and Zarwa are playing a game where they toss a coin. Allan wins as soon as a head is tossed, and Zarwa wins if two tails are tossed in a row. What is the probability that Allan wins? | [
"\\(\\frac{3}{4}\\)"
] | [
"Allan and Zarwa are playing a game where they toss a coin. Allan wins as soon as a head is tossed, and Zarwa wins if two tails are tossed in a row. What is the probability that Allan wins?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-0030fb8396ac7e66082144927b7447f77a8aab439e373c84b77dc18f1bc8ce3e | Allen Iverson starts at one vertex of a tetrahedron and in each move he walks along a random edge to an adjacent vertex. What is the probability that after 60 moves, Allen Iverson will be at the same vertex he started at? | [
"\\dfrac{3^{59} + 1}{4 \\cdot 3^{59}}"
] | [
"Allen Iverson starts at one vertex of a tetrahedron and in each move he walks along a random edge to an adjacent vertex. What is the probability that after 60 moves, Allen Iverson will be at the same vertex he started at?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ef808026ebcbfef77f016c92eb7c03b526101237f4827ea50275b4fdd2b5288 | Allen and Alan play a game. A nonconstant polynomial \( P(x,y) \) with real coefficients and a positive integer \( d \) greater than the degree of \( P \) are known to both. Alan thinks of a polynomial \( Q(x,y) \) with real coefficients and degree at most \( d \) and keeps it secret. Allen can make queries of the form... | [
"\\(\\binom{d+2}{2}\\)"
] | [
"Allen and Alan play a game. A nonconstant polynomial \\( P(x,y) \\) with real coefficients and a positive integer \\( d \\) greater than the degree of \\( P \\) are known to both. Alan thinks of a polynomial \\( Q(x,y) \\) with real coefficients and degree at most \\( d \\) and keeps it secret. Allen can make quer... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 4,
"tir": 1
} | 14 | 0 | 0 |
omr-ws-af616d3b461bcbd64fd2931cd2be6f72255691b5fdc70ec94758e999465635f3 | Allen and Ben are painting a fence. The ratio of the amount of work Allen does to the amount of work Ben does is $3:5$. If the fence requires a total of $240$ square feet to be painted, how many square feet does Ben paint? | [
"150"
] | [
"Allen and Ben are painting a fence. The ratio of the amount of work Allen does to the amount of work Ben does is $3:5$. If the fence requires a total of $240$ square feet to be painted, how many square feet does Ben paint?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-431b8630008efb375f965afff6211b663002f1777604645b647c965002d1ab68 | Allen and Bethany each arrive at a party at a random time between 1:00 and 2:00. Each stays for 15 minutes, then leaves. What is the probability that Allen and Bethany see each other at the party? | [
"\\frac{7}{16}"
] | [
"Allen and Bethany each arrive at a party at a random time between 1:00 and 2:00. Each stays for 15 minutes, then leaves. What is the probability that Allen and Bethany see each other at the party?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-03e2f68b25b875d34081cde7753769d7c677ca7af1790799a8bdd2f95b1fe360 | Allen randomly draws 6 cards. Find the ratio of the probability that he sees exactly 1 spade in his hand to the probability that he sees no spades. | [
"\\dfrac{39}{17}"
] | [
"Allen randomly draws 6 cards. Find the ratio of the probability that he sees exactly 1 spade in his hand to the probability that he sees no spades."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-65ba3a6295424d42a2caf8e713a0901307c36cba50574b95e2516c2d22dc7ecf | Allen takes a $1 \times 69420$ grid where initially every cell contains the number $1$. In each move, he selects 69 consecutive cells such that the leftmost cell contains $1$ and multiplies each of those numbers by $-1$. After a series of moves, Allen stops when no more moves can be made. Let $N$ be the number of cells... | [
"414"
] | [
"Allen takes a $1 \\times 69420$ grid where initially every cell contains the number $1$. In each move, he selects 69 consecutive cells such that the leftmost cell contains $1$ and multiplies each of those numbers by $-1$. After a series of moves, Allen stops when no more moves can be made. Let $N$ be the number of... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-03414e1da902a4e3e5bf2cf1490786f142f5ba182fded069aa0644eda0dd9133 | Allen the ant starts on a vertex of a cube. Each move, he randomly goes to one of the 3 adjacent vertices with equal probability. What is the expected number of moves it takes for Allen to return to his starting vertex? | [
"\\( \\frac{6}{5} \\)"
] | [
"Allen the ant starts on a vertex of a cube. Each move, he randomly goes to one of the 3 adjacent vertices with equal probability. What is the expected number of moves it takes for Allen to return to his starting vertex?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19229f87a3e1387351960457f8ffb2487cc772afa2a0089cf34736e896fcf4d3 | Alli rolls a standard $6$-sided die twice. What is the probability of rolling integers that differ by $2$ on her first two rolls? Express your answer as a common fraction. | [
"\\dfrac{2}{9}"
] | [
"Alli rolls a standard $6$-sided die twice. What is the probability of rolling integers that differ by $2$ on her first two rolls? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-00de1954ba4d980773df4146c54d28abd19f8a0d126911ca86a32dcf9a4af449 | Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $f(n)$ points, where \[f(n) = \left\{ \begin{array}{cl} 6 & \text{ if }n\text{ is a multiple of 2 and 3}, \\ 2 & \text{ if }n\text{ is only a multiple of 2}, \\ 0 & \text{ if }n\text{ is not a multiple of 2}.... | [
"32"
] | [
"Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $f(n)$ points, where \\[f(n) = \\left\\{\n\\begin{array}{cl} 6 & \\text{ if }n\\text{ is a multiple of 2 and 3}, \\\\\n2 & \\text{ if }n\\text{ is only a multiple of 2}, \\\\\n0 & \\text{ if }n\\text{ i... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b0682f25ed7d97262ff1d24ad91c041359c870cdabf30cfcbe215c047f787e6b | Allison, Brent, and Cody want to meet for lunch at Moe's between 1 and 2 pm. Allison will wait for 5 minutes before leaving, Brent will wait for 10 minutes before leaving, and Cody will wait for 15 minutes before leaving. What is the probability that they all meet up for lunch? | [
"\\dfrac{689}{10368}"
] | [
"Allison, Brent, and Cody want to meet for lunch at Moe's between 1 and 2 pm. Allison will wait for 5 minutes before leaving, Brent will wait for 10 minutes before leaving, and Cody will wait for 15 minutes before leaving. What is the probability that they all meet up for lunch?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-8136f67ae1b29cff2ad091447f1b017a1122866bde1db3d6860d70c58bc284ac | Allison, Brian and Noah each have a 6-sided cube. All of the faces on Allison's cube have a 5. The faces on Brian's cube are numbered 1, 2, 3, 4, 5 and 6. Three of the faces on Noah's cube have a 2 and three of the faces have a 6. All three cubes are rolled. What is the probability that Allison's roll is greater than e... | [
"\\frac{1}{3}"
] | [
"Allison, Brian and Noah each have a 6-sided cube. All of the faces on Allison's cube have a 5. The faces on Brian's cube are numbered 1, 2, 3, 4, 5 and 6. Three of the faces on Noah's cube have a 2 and three of the faces have a 6. All three cubes are rolled. What is the probability that Allison's roll is greater t... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c6f776d868552521426ae4c2ba7a44d3a23a19a7beba2229a7a532ae52acc8d5 | Ally, Bella, Aria, Barbara, Ava, Brianna, and Cindy are playing a game, with turns taken in the order listed above. Ally starts with 1 muffin and has the choice to either "double it and give it to the next person" or "square it and give it to the next person". Ally then passes the resulting amount to Bella, and this pr... | [
"1895"
] | [
"Ally, Bella, Aria, Barbara, Ava, Brianna, and Cindy are playing a game, with turns taken in the order listed above. Ally starts with 1 muffin and has the choice to either \"double it and give it to the next person\" or \"square it and give it to the next person\". Ally then passes the resulting amount to Bella, an... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 14
} | 32 | 0 | 0 |
omr-ws-623ba48a82b14d4e73c0d9796eb1a8e4fcb315d60b24aa6a3f1dadf57da49a34 | Along a circle are written $n$ numbers, each being $+1$ or $-1$. Our aim is to find the product of all $n$ numbers. In one question, we can find the product of $k$ successive numbers $a_i \ldots a_{i+k-1}$. Here, $a_{n+1} = a_1$ and so on. How many questions $q(n,k)$ are necessary to find the product of all $n$ numbers... | [
"\\gcd(n, k)"
] | [
"Along a circle are written $n$ numbers, each being $+1$ or $-1$. Our aim is to find the product of all $n$ numbers. In one question, we can find the product of $k$ successive numbers $a_i \\ldots a_{i+k-1}$. Here, $a_{n+1} = a_1$ and so on. How many questions $q(n,k)$ are necessary to find the product of all $n$ n... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-6c1bfb0422ed3a09c844bf4905ea88ebe41e255c544607254895bfb472a469a7 | Along a circle with 100 positions forming a regular 100-gon, B writes an integer in each position. A and B take turns making moves: A can add 1 to two diametrically opposed positions, and B can add 1 to two consecutive positions. A's goal is to maximize the number of odd numbers. What is the maximum number of odd numbe... | [
"50"
] | [
"Along a circle with 100 positions forming a regular 100-gon, B writes an integer in each position. A and B take turns making moves: A can add 1 to two diametrically opposed positions, and B can add 1 to two consecutive positions. A's goal is to maximize the number of odd numbers. What is the maximum number of odd ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b4b6df344e57b63562f5fa712fc60fa41a7597628d86777ced435c5a96a2cff | Along a circle, 10 iron weights have been placed. Between every two weights, there is a brass ball. The mass of each ball is equal to the difference of the masses of its neighboring weights. If the masses of the iron weights are given by the sequence \(a_1, a_2, \ldots, a_{10}\), find the sum of all possible values of ... | [
"0"
] | [
"Along a circle, 10 iron weights have been placed. Between every two weights, there is a brass ball. The mass of each ball is equal to the difference of the masses of its neighboring weights. If the masses of the iron weights are given by the sequence \\(a_1, a_2, \\ldots, a_{10}\\), find the sum of all possible va... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e37d90f54f678aef624c7b58df5ec75e3cb1e5709a124cd8bcff55fae8d58870 | Along a circle, there are 2n numbers whose sum is positive. For each number, define the 'right sum' as the sum of the number and the n-1 numbers to its right, and the 'left sum' as the sum of the number and the n-1 numbers to its left. If n = 5, find the minimum number of integers among these 2n numbers that must have ... | [
"1"
] | [
"Along a circle, there are 2n numbers whose sum is positive. For each number, define the 'right sum' as the sum of the number and the n-1 numbers to its right, and the 'left sum' as the sum of the number and the n-1 numbers to its left. If n = 5, find the minimum number of integers among these 2n numbers that must ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-320f72c7eb4e6eaf6379f78e6dc0d761be3f742c9d33dde18d0dc2187a6bce53 | Along a circle-shaped path are 100 boys and 100 girls. The distance between two points on the path is defined as the length of the smaller arc through which it is possible to get from one point to the other. Calculate the maximum possible difference between the sum of distances between pairs of the same gender and the ... | [
"9900"
] | [
"Along a circle-shaped path are 100 boys and 100 girls. The distance between two points on the path is defined as the length of the smaller arc through which it is possible to get from one point to the other. Calculate the maximum possible difference between the sum of distances between pairs of the same gender and... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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