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-e2745792befa37f69d326ef121c95f986b014571c27dde823c4e062ab746161b | Two positive integers \( n \) and \( k \) are given, where \( n \ge 2 \) and \( k \le 2n \). There is a circle with \( 2n \) points equally spaced on it. Player B draws \( n \) chords on the circle, joining the \( 2n \) points, such that no two chords share any endpoint. Player A picks exactly \( k \) points out of the... | [
"38"
] | [
"Two positive integers \\( n \\) and \\( k \\) are given, where \\( n \\ge 2 \\) and \\( k \\le 2n \\). There is a circle with \\( 2n \\) points equally spaced on it. Player B draws \\( n \\) chords on the circle, joining the \\( 2n \\) points, such that no two chords share any endpoint. Player A picks exactly \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-50992d10cd685208f90768455d2590ae4c3b90629c8ce1c3e44ed02e77c1f7c5 | Two positive integers \( p \) and \( q \) satisfy the property that \( p^2 + q^4 \) is exactly 5 more than a multiple of 8. How many ordered pairs \( (p, q) \) are there such that \( p \leq q < 100 \)? | [
"625"
] | [
"Two positive integers \\( p \\) and \\( q \\) satisfy the property that \\( p^2 + q^4 \\) is exactly 5 more than a multiple of 8. How many ordered pairs \\( (p, q) \\) are there such that \\( p \\leq q < 100 \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-12d150421af2ef66ced19e9f21e69acce50ce65045d48087b8f21a71f2be8dde | Two positive integers \(a\) and \(b\) are chosen randomly, uniformly, and independently from the set of positive integers less than or equal to 1000. What is the expected value of the number of quadrants through which the graph \(x^a + y^b = 1\) will pass? | [
"3.75"
] | [
"Two positive integers \\(a\\) and \\(b\\) are chosen randomly, uniformly, and independently from the set of positive integers less than or equal to 1000. What is the expected value of the number of quadrants through which the graph \\(x^a + y^b = 1\\) will pass?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-9625c2132bf99099b71f802b12089095ef5026256c31c854e3f76e44d310d110 | Two positive integers \(a\) and \(b\) are chosen such that \(1 \leq a \leq b < 50\). The sum \(S = a + b\) is given to mathematician \(S\), and the sum of squares \(Q = a^2 + b^2\) is given to mathematician \(Q\). Both mathematicians are aware of the constraints and the information given to each other. The conversation... | [
"a = 8, b = 9"
] | [
"Two positive integers \\(a\\) and \\(b\\) are chosen such that \\(1 \\leq a \\leq b < 50\\). The sum \\(S = a + b\\) is given to mathematician \\(S\\), and the sum of squares \\(Q = a^2 + b^2\\) is given to mathematician \\(Q\\). Both mathematicians are aware of the constraints and the information given to each ot... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4403bac8e5e05e0cced5fb7254180c34ec1a7090ecf43bffd971b4a65516ce92 | Two positive integers \(a\) and \(b\) are such that \(a\) is equal to the sum of three positive divisors of \(b\) different from \(b\), and \(b\) is equal to the sum of three positive divisors of \(a\) different from \(a\). Find the smallest possible value of \(a + b\). | [
"12"
] | [
"Two positive integers \\(a\\) and \\(b\\) are such that \\(a\\) is equal to the sum of three positive divisors of \\(b\\) different from \\(b\\), and \\(b\\) is equal to the sum of three positive divisors of \\(a\\) different from \\(a\\). Find the smallest possible value of \\(a + b\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-d363bd3059943dc0051bc3d689ae926e84111c88de0f8b06e231270e6f3cf24e | Two positive integers are chosen. The sum is revealed to logician A, and the sum of squares is revealed to logician B. Both A and B are given this information and the information contained in the conversation below. The conversation between A and B goes as follows. B starts: B: "I can’t tell what they are." A: "I can’t... | [
"8 and 9"
] | [
"Two positive integers are chosen. The sum is revealed to logician A, and the sum of squares is revealed to logician B. Both A and B are given this information and the information contained in the conversation below. The conversation between A and B goes as follows. B starts:\n\nB: \"I can’t tell what they are.\"\n... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-d09b430fd327ff3ee298324ce0ee064f9c3e5963381bd85088bd42c2b71cc816 | Two positive integers are chosen. The sum is revealed to logician A, and the sum of squares is revealed to logician B. Both A and B are given this information and the information contained in the conversation below. The conversation between A and B goes as follows: B: "I can't tell what the two numbers are." A: "I can'... | [
"13"
] | [
"Two positive integers are chosen. The sum is revealed to logician A, and the sum of squares is revealed to logician B. Both A and B are given this information and the information contained in the conversation below. The conversation between A and B goes as follows:\n\nB: \"I can't tell what the two numbers are.\"\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-0794965d7945b19891a1eec23cfc3c7378d2eaea9b4fbcdee906c7b1506bc7a2 | Two positive integers are chosen. The sum is revealed to logician A, and the sum of squares is revealed to logician B. Both A and B are given this information and the information contained in this sentence. The conversation between A and B goes as follows. B starts: B: "I can’t tell what they are." A: "I can’t tell wha... | [
"13"
] | [
"Two positive integers are chosen. The sum is revealed to logician A, and the sum of squares is revealed to logician B. Both A and B are given this information and the information contained in this sentence. The conversation between A and B goes as follows. B starts:\n\nB: \"I can’t tell what they are.\"\nA: \"I ca... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-45610b5119671ae3b19f4e147442f3b099afc64523f65c459e1399deede33cbf | Two positive integers are chosen. The sum is revealed to logician A, and the sum of squares is revealed to logician B. Both A and B are given this information and the information contained in this sentence. The conversation between A and B goes as follows: B starts B: "I can't tell what the two numbers are." A: "I can'... | [
"13"
] | [
"Two positive integers are chosen. The sum is revealed to logician A, and the sum of squares is revealed to logician B. Both A and B are given this information and the information contained in this sentence. The conversation between A and B goes as follows: B starts\nB: \"I can't tell what the two numbers are.\"\nA... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-fdfd8a8f5637a98afe350d970ddf5c1a4bc34ca6bba039e9ef8be487575a7a96 | Two positive integers are chosen. The sum is revealed to logician A, and the sum of squares is revealed to logician B. Both A and B are given this information and the information contained in this sentence. The conversation between A and B goes as follows: B starts B: ` I can't tell what they are.' A: ` I can't tell wh... | [
"4 \\text{ and } 13"
] | [
"Two positive integers are chosen. The sum is revealed to logician A, and the sum of squares is revealed to logician B. Both A and B are given this information and the information contained in this sentence. The conversation between A and B goes as follows: B starts\nB: ` I can't tell what they are.'\nA: ` I can't ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-c154c778bc56b212780fae7f6221e9a17a6a4930a5657160dd44290b31885cd0 | Two positive integers are chosen. The sum is revealed to logician A, and the sum of squares is revealed to logician B. Both A and B are given this information and the information contained in this sentence. The conversation between A and B goes as follows: B: "I can't tell what the two numbers are." A: "I can't tell wh... | [
"4 \\text{ and } 13"
] | [
"Two positive integers are chosen. The sum is revealed to logician A, and the sum of squares is revealed to logician B. Both A and B are given this information and the information contained in this sentence. The conversation between A and B goes as follows:\nB: \"I can't tell what the two numbers are.\"\nA: \"I can... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-a78f75e30b14d54ad9792db98430f59f50a6ef3bdc99652fcccc364fb2e2cd81 | Two positive integers are chosen. The sum is revealed to logician A, and the sum of the squares is revealed to logician B. Both A and B are given this information and the information contained in the conversation below. The conversation between A and B goes as follows: B starts. B: "I can't tell what the two numbers ar... | [
"13"
] | [
"Two positive integers are chosen. The sum is revealed to logician A, and the sum of the squares is revealed to logician B. Both A and B are given this information and the information contained in the conversation below. The conversation between A and B goes as follows: B starts.\n\nB: \"I can't tell what the two n... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-8c38e43af611e4c7929680ed9730d4f017b13799adc01a1b74f4a66090a09d8c | Two positive integers are written on a board, with the larger being 2018. If the numbers on the board are \(a\) and \(b\), you can replace one of the numbers with \(\frac{a+b}{2}\) only if \(a+b\) is even. What is the maximum number of steps you can perform? | [
"10"
] | [
"Two positive integers are written on a board, with the larger being 2018. If the numbers on the board are \\(a\\) and \\(b\\), you can replace one of the numbers with \\(\\frac{a+b}{2}\\) only if \\(a+b\\) is even. What is the maximum number of steps you can perform?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-3822d647ac50ffab1387a27c2a143cb189ec3eb924a1d3447177f8289008dd25 | Two positive integers are written on the blackboard. Initially, one of them is $2000$ and the other is smaller than $2000$. If the arithmetic mean $m$ of the two numbers on the blackboard is an integer, the following operation is allowed: one of the two numbers is erased and replaced by $m$. After performing the operat... | [
"2000"
] | [
"Two positive integers are written on the blackboard. Initially, one of them is $2000$ and the other is smaller than $2000$. If the arithmetic mean $m$ of the two numbers on the blackboard is an integer, the following operation is allowed: one of the two numbers is erased and replaced by $m$. After performing the o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-1c1a5c339ddfbf0ebfe0fa57f1b994cabdc204eb7bcbcbca9255e82d9f0306ba | Two positive integers are written on the blackboard. Mary records in her notebook the square of the smaller number and replaces the larger number on the blackboard by the difference of the two numbers. She repeats this process until one of the numbers on the blackboard becomes zero. What will be the sum of the numbers ... | [
"\\( nm \\)"
] | [
"Two positive integers are written on the blackboard. Mary records in her notebook the square of the smaller number and replaces the larger number on the blackboard by the difference of the two numbers. She repeats this process until one of the numbers on the blackboard becomes zero. What will be the sum of the num... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-3a4501ca9509e681141174c86a214aa8c4f70e0c3e682b5f15007c5877f31f51 | Two positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? | [
"156"
] | [
"Two positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers?"
] | [
"MATH_training_set",
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 36
} | 68 | 0 | 0 |
omr-ws-f4667f2c444d14ac0be01de3ea84c35ba05239da34b6a9948362502cfb15e6f5 | Two positive integers differ by 6 and their product is 135. What is the larger integer? | [
"15"
] | [
"Two positive integers differ by 6 and their product is 135. What is the larger integer?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9699608c4ee42a9853b5d40ed79ebab370d9e6a2f0cc619795fafa47663c8b80 | Two positive integers have a difference of 10. The smaller number is a multiple of 3, and the larger number is a multiple of 7. The sum of all the prime factors of the two numbers is 17. What is the sum of the two numbers? | [
"130"
] | [
"Two positive integers have a difference of 10. The smaller number is a multiple of 3, and the larger number is a multiple of 7. The sum of all the prime factors of the two numbers is 17. What is the sum of the two numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-51ea7252c8358e2649539a81200637c1157544f3e04f0db7ef8fc66d8a0174fc | Two positive integers, $a$ and $b$ are chosen randomly, uniformly, and independently from the set of positive integers less than $1000$, $\{1, 2,...,1000\}$. What is the expected value of the number of quadrants through which the graph $x^a + y^b = 1$ will pass? | [
"\\dfrac{15}{4}"
] | [
"Two positive integers, $a$ and $b$ are chosen randomly, uniformly, and independently from the set of positive integers less than $1000$, $\\{1, 2,...,1000\\}$. What is the expected value of the number of quadrants through which the graph $x^a + y^b = 1$ will pass?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-49d1a6ac1d5cbdd61e6729ba8c7efbb8d6f7f965c9185ac6322aaf727ffebb71 | Two positive integers, both greater than 1 and less than 50, are selected. The sum of these numbers is given to Sanjay (S), and the product is given to Pratik (P). Sanjay tells Pratik, "You can never guess the sum S." Later, Pratik calls Sanjay and says, "Now I know the value of S." Finally, Sanjay calls Pratik and say... | [
"4 and 13"
] | [
"Two positive integers, both greater than 1 and less than 50, are selected. The sum of these numbers is given to Sanjay (S), and the product is given to Pratik (P). Sanjay tells Pratik, \"You can never guess the sum S.\" Later, Pratik calls Sanjay and says, \"Now I know the value of S.\" Finally, Sanjay calls Prati... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 7
} | 13 | 0 | 0 |
omr-ws-555edc7ae205d175fe7bd4db468345a86349b2c5a612a3831bab90be5257958a | Two positive numbers $x$ and $y$ are taken at random, neither of them exceeding unity. What is the probability of their sum not exceeding unity if the sum of their squares exceeds 1/4? | [
"$\\frac{1}{2}-\\frac{\\pi }{16}$"
] | [
"Two positive numbers $x$ and $y$ are taken at random, neither of them exceeding unity. What is the probability of their sum not exceeding unity if the sum of their squares exceeds 1/4?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-922a18ee83ac27c8a141bf0b3328667c071cd4ddb2a91617eb414223f70c48b2 | Two positive numbers $x$ and $y$ each of which does not exceed two are taken at random. Find the probability that $xy\le 1$, $y/x\le 2$. | [
"$\\frac{1+3\\ln 2}{8}$"
] | [
"Two positive numbers $x$ and $y$ each of which does not exceed two are taken at random. Find the probability that $xy\\le 1$, $y/x\\le 2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5cc6a5840891a17cbee3f08dac3f6423b826ef9be77dbc5e38e1956212941358 | Two positive numbers exist such that their product, their sum, and their difference of squares are equal. Find the product of the two numbers. | [
"\\(2 + \\sqrt5\\)"
] | [
"Two positive numbers exist such that their product, their sum, and their difference of squares are equal. Find the product of the two numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-3630b02c8fca052ab5608889abfdd33e3ffe225ac0bb2d3dd74fed860ade4f72 | Two positive real numbers \( x \) and \( y \) satisfy \( x^3 + y^3 = 1 \). Calculate the minimum value of the expression \( x^2 + xy + y^2 - x - y \). | [
"0"
] | [
"Two positive real numbers \\( x \\) and \\( y \\) satisfy \\( x^3 + y^3 = 1 \\). Calculate the minimum value of the expression \\( x^2 + xy + y^2 - x - y \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-94f37fde400a4c82ccc03ea37dd4a75b437a9ea36b73dbac9f5f6a17147eb125 | Two positive real numbers have geometric mean $\sqrt{3}$ and harmonic mean $\frac{3}{2}.$ Enter the two numbers, separated by commas. | [
"1,3"
] | [
"Two positive real numbers have geometric mean $\\sqrt{3}$ and harmonic mean $\\frac{3}{2}.$ Enter the two numbers, separated by commas."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-223a9d3a9615864fd8cacf230458c2085bdc972e13088ed1e074e2b945fe3935 | Two positive sequences $\{x_n\}$ and $\{y_n\}$ are defined using $x_1=1$, $y_1=\sqrt{3}$ and the relations \[ \left\{\begin{aligned} &x_{n+1}y_{n+1} - x_n = 0 \\ &x_{n+1}^2 + y_n = 2 \end{aligned}\right. \quad \forall n=1,2,\ldots. \] Let $L_x$ and $L_y$ be the limits of the sequences $\{x_n\}$ and $\{y_n\}$, respectiv... | [
"2"
] | [
"Two positive sequences $\\{x_n\\}$ and $\\{y_n\\}$ are defined using $x_1=1$, $y_1=\\sqrt{3}$ and the relations\n\\[\n\\left\\{\\begin{aligned}\n&x_{n+1}y_{n+1} - x_n = 0 \\\\\n&x_{n+1}^2 + y_n = 2\n\\end{aligned}\\right.\n\\quad \\forall n=1,2,\\ldots.\n\\]\nLet $L_x$ and $L_y$ be the limits of the sequences $\\{... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-70a633325483e2c3a0178c3769435040b72c8efb12ce0760f92465bba6f3594a | Two posts, one 12 ft high and the other 28 ft high, stand 30 ft apart. They are stayed by two wires attached to a single stake on the ground, running from the stake to the top of each post. Where should the stake be placed to minimize the total length of the wires? | [
"21 feet away from the taller post."
] | [
"Two posts, one 12 ft high and the other 28 ft high, stand 30 ft apart. They are stayed by two wires attached to a single stake on the ground, running from the stake to the top of each post. Where should the stake be placed to minimize the total length of the wires?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-890831297afcae8bc19f1c6aad85039e52fdddff773850d17e7001626266adbe | Two progressions are in arithmetic progression (AP) and geometric progression (GP) respectively. The first terms of both progressions are equal, and the third terms of both progressions are also equal. The common ratio of the GP is equal to the common difference of the AP. If the sum of the squares of all possible valu... | [
"8"
] | [
"Two progressions are in arithmetic progression (AP) and geometric progression (GP) respectively. The first terms of both progressions are equal, and the third terms of both progressions are also equal. The common ratio of the GP is equal to the common difference of the AP. If the sum of the squares of all possible... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 24,
"genselect": 6,
"tir": 2
} | 32 | 0 | 0 |
omr-ws-2d0cd093fc451d1c90745108da827bb0291af1b1d7341099811a7fca12db3dde | Two pulleys are connected by a belt that crosses over. The diameter of the larger pulley is 18 cm and the diameter of the smaller pulley is 12 cm. The centers of the two pulleys are 30 cm apart. How long is the belt? | [
"\\( 15\\pi-12\\arcsin\\left(\\frac{1}{10}\\right) \\)"
] | [
"Two pulleys are connected by a belt that crosses over. The diameter of the larger pulley is 18 cm and the diameter of the smaller pulley is 12 cm. The centers of the two pulleys are 30 cm apart. How long is the belt?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-502e645c00ccccc1de6dbd5a739629861afd6b27a52e020475723d72f419c939 | Two pulleys, one with radius 2 inches and the other with radius 8 inches, are connected by a belt. If the 2-inch pulley rotates at 3 revolutions per minute, determine the revolutions per minute of the 8-inch pulley. (Hint: The linear speeds of the pulleys are the same.) | [
"\\(\\frac{3}{4}\\)"
] | [
"Two pulleys, one with radius 2 inches and the other with radius 8 inches, are connected by a belt. If the 2-inch pulley rotates at 3 revolutions per minute, determine the revolutions per minute of the 8-inch pulley. (Hint: The linear speeds of the pulleys are the same.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-f67cbf5c4616477ae69b28faceda0ebb81dc5ff771428a537dbba63d2895491f | Two pumps transferred 64 m$^3$ of water. They began working at the same time and with the same capacity. After the first pump transferred 9 m$^3$ of water, it was stopped for 1 h 20 min. After that time, the capacity of the first pipe was increased by 1 m$^3$/h. Find the initial capacity of the pumps if the first pump ... | [
"3 m$^3$/h"
] | [
"Two pumps transferred 64 m$^3$ of water. They began working at the same time and with the same capacity. After the first pump transferred 9 m$^3$ of water, it was stopped for 1 h 20 min. After that time, the capacity of the first pipe was increased by 1 m$^3$/h. Find the initial capacity of the pumps if the first ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-590f4a654370c454044ecb72d662359d65fa2191db87e974a84b5308a8584266 | Two pupils alternately replace the dots in the expression \( x^3 + \ldots x^2 + \ldots x + \ldots = 0 \) with real numbers. The first pupil attempts to obtain an equation with exactly one real root. If the second pupil plays optimally to prevent this, what is the sum of all possible values of the coefficient of \( x^2 ... | [
""
] | [
"Two pupils alternately replace the dots in the expression \\( x^3 + \\ldots x^2 + \\ldots x + \\ldots = 0 \\) with real numbers. The first pupil attempts to obtain an equation with exactly one real root. If the second pupil plays optimally to prevent this, what is the sum of all possible values of the coefficient ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-057c50087229b4f6da772e49082236014d4ac68f4615f0818684f8cbe298bd56 | Two pupils of the 5th form and several pupils of the 6th form participated in a chess tournament. Each pupil played once with every other participant. The two pupils of the 5th form together had 8 points; each of the pupils of the 6th form had one and the same number of points (in the tournament a winner receives 1 poi... | [
"7 or 14"
] | [
"Two pupils of the 5th form and several pupils of the 6th form participated in a chess tournament. Each pupil played once with every other participant. The two pupils of the 5th form together had 8 points; each of the pupils of the 6th form had one and the same number of points (in the tournament a winner receives ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2863310caed8f0d8159af14d9dffc7a7c77a2f2f2e7c27094a90f2b9baf588e9 | Two pyramids with a common base $A_1 A_2 A_3 A_4 A_5 A_6 A_7$ and vertices $B$ and $C$ are given. The edges $BA_i$, $CA_i$ for $i = 1, 2, 3, \ldots, 7$, the diagonals of the common base, and $BC$ are colored either red or blue. Let $N$ be the number of monochromatic triangles formed. Calculate the minimum possible valu... | [
"9"
] | [
"Two pyramids with a common base $A_1 A_2 A_3 A_4 A_5 A_6 A_7$ and vertices $B$ and $C$ are given. The edges $BA_i$, $CA_i$ for $i = 1, 2, 3, \\ldots, 7$, the diagonals of the common base, and $BC$ are colored either red or blue. Let $N$ be the number of monochromatic triangles formed. Calculate the minimum possibl... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-603df9968b91d1393b3b2c90465282473745c46294694e69c71a8ae11f9ed9d1 | Two quadratic equations $1997x^2 + 1998x + 1 = 0$ and $x^2 + 1998x + 1997 = 0$ have a root in common. Find the product of the roots that are not in common. | [
"1"
] | [
"Two quadratic equations $1997x^2 + 1998x + 1 = 0$ and $x^2 + 1998x + 1997 = 0$ have a root in common. Find the product of the roots that are not in common."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-79421dca52fc4096562d408b9b74d90e077f990251673aa4464b28f03d3549ed | Two quadratic polynomials \( f_{1} \) and \( f_{2} \) satisfy \( f_{1}'(x) f_{2}'(x) \geq |f_{1}(x)| + |f_{2}(x)| \) for all \( x \in \mathbb{R} \). If \( f_{1}(x) = ax^2 + bx + c \) and \( f_{2}(x) = dx^2 + ex + f \), find the sum of all possible values of \( a + d \) given that \( f_{1} \cdot f_{2} = g^{2} \) for som... | [
"0"
] | [
"Two quadratic polynomials \\( f_{1} \\) and \\( f_{2} \\) satisfy \\( f_{1}'(x) f_{2}'(x) \\geq |f_{1}(x)| + |f_{2}(x)| \\) for all \\( x \\in \\mathbb{R} \\). If \\( f_{1}(x) = ax^2 + bx + c \\) and \\( f_{2}(x) = dx^2 + ex + f \\), find the sum of all possible values of \\( a + d \\) given that \\( f_{1} \\cdot ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d0bdbd41b095f9fc3320cb8c39b1bdafb2963b38b845dc4c0b4ce6215eadf665 | Two quadrilateral-based pyramids where the length of all its edges are the same, have their bases coincide, forming a new 3D figure called "8-plane" (octahedron). If the volume of such "8-plane" (octahedron) is $a^3\sqrt{2}$ cm$^3$, determine the volume of the largest sphere that can be fit inside such "8-plane" (octah... | [
"\\dfrac{\\sqrt{6}}{9} \\pi a^3"
] | [
"Two quadrilateral-based pyramids where the length of all its edges are the same, have their bases coincide, forming a new 3D figure called \"8-plane\" (octahedron). If the volume of such \"8-plane\" (octahedron) is $a^3\\sqrt{2}$ cm$^3$, determine the volume of the largest sphere that can be fit inside such \"8-pl... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-215e03a021e351be13bb8339f0177393e37be71633b9f0b4c59ed4107cd7a74e | Two quadrilaterals $ABCD$ and $A'B'C'D'$ are called orthologic if all corresponding four triangles formed by taking three of their vertices are orthologic. Let $H$ be the affine homography which maps $ABCD$ to $A'B'C'D'$. Given that $A(1,2)$, $B(3,4)$, $C(5,6)$, $D(7,8)$, $A'(9,10)$, $B'(11,12)$, $C'(13,14)$, and $D'(1... | [
"1"
] | [
"Two quadrilaterals $ABCD$ and $A'B'C'D'$ are called orthologic if all corresponding four triangles formed by taking three of their vertices are orthologic. Let $H$ be the affine homography which maps $ABCD$ to $A'B'C'D'$. Given that $A(1,2)$, $B(3,4)$, $C(5,6)$, $D(7,8)$, $A'(9,10)$, $B'(11,12)$, $C'(13,14)$, and ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d88ba01e976896a884e521d68927689ed8c35abea2cd4139872c7c31bfbc17cf | Two quadrilaterals $ABCD$ and $A_1B_1C_1D_1$ are mutually symmetric with respect to the point $P$. It is known that $A_1BCD$, $AB_1CD$, and $ABC_1D$ are cyclic quadrilaterals. If $AB = 10$, $BC = 15$, $CD = 20$, and $DA = 25$, find the length of $AD_1$. | [
"25"
] | [
"Two quadrilaterals $ABCD$ and $A_1B_1C_1D_1$ are mutually symmetric with respect to the point $P$. It is known that $A_1BCD$, $AB_1CD$, and $ABC_1D$ are cyclic quadrilaterals. If $AB = 10$, $BC = 15$, $CD = 20$, and $DA = 25$, find the length of $AD_1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-882b4189c795189da5759730a138f6a1340927a37eff095aad78f2fa8f5cc692 | Two quadrilaterals $ABCD$ and $EFGH$ have corresponding angles equal. If the lengths of the sides of $ABCD$ are $a, b, c,$ and $d$ in order, and the lengths of the sides of $EFGH$ are $e, f, g,$ and $h$ in order, find the value of $\frac{a}{e} + \frac{b}{f} + \frac{c}{g} + \frac{d}{h}$ given that $\frac{a}{e} = \frac{b... | [
"4k"
] | [
"Two quadrilaterals $ABCD$ and $EFGH$ have corresponding angles equal. If the lengths of the sides of $ABCD$ are $a, b, c,$ and $d$ in order, and the lengths of the sides of $EFGH$ are $e, f, g,$ and $h$ in order, find the value of $\\frac{a}{e} + \\frac{b}{f} + \\frac{c}{g} + \\frac{d}{h}$ given that $\\frac{a}{e}... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-637501c9fe7d53f51e8dfd1a358e01fa7c9f2878dcaffa488c797aea3be1a2e0 | Two quadrilaterals $ABCD$ with coordinates $A (5, 2)$, $B (1, -4)$, $C (-2, -3)$, and $D (-1, k)$ exist with numerical area $18.5$. One quadrilateral is convex and one is concave. Find the sum of the possible values of $k$. Express your answer as a common or improper fraction reduced to lowest terms. | [
"\\(-\\frac{60}{7}\\)"
] | [
"Two quadrilaterals $ABCD$ with coordinates $A (5, 2)$, $B (1, -4)$, $C (-2, -3)$, and $D (-1, k)$ exist with numerical area $18.5$. One quadrilateral is convex and one is concave. Find the sum of the possible values of $k$. Express your answer as a common or improper fraction reduced to lowest terms."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-81fe0d68fc51c9c1383172f1b13dad7594e62804852b90d15d51cca530284f53 | Two quadrilaterals have equal areas, perimeters, and corresponding angles. If one of the quadrilaterals has side lengths $a$, $b$, $c$, and $d$, and the other has side lengths $a$, $b$, $x$, and $y$, find the value of $x+y$ given that $a=5$, $b=12$, $c=13$, and $d=10$. | [
"23"
] | [
"Two quadrilaterals have equal areas, perimeters, and corresponding angles. If one of the quadrilaterals has side lengths $a$, $b$, $c$, and $d$, and the other has side lengths $a$, $b$, $x$, and $y$, find the value of $x+y$ given that $a=5$, $b=12$, $c=13$, and $d=10$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a1d724fcff4ab0f009f5ca78c48dcf1bce6a15dbacbdc8d5f06665d168dad165 | Two queens are randomly placed on an 8 by 8 chessboard on different squares. What is the probability that they do not attack each other? | [
"\\(\\frac{13}{36}\\)"
] | [
"Two queens are randomly placed on an 8 by 8 chessboard on different squares. What is the probability that they do not attack each other?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f2de9b96441df7bf16c7d21a3eabc3b7b2b750bb08009cf7d70789a5019ce47f | Two racing cars simultaneously start on a round track with velocities \( v_1 \) and \( v_2 \) respectively. After each circle, the first car doubles its velocity, and the second car triples its velocity. The car is said to be the favorite if there exists a positive integer \( N \) such that this car wins any race in \(... | [
"\\( a > \\frac{4}{3} \\)"
] | [
"Two racing cars simultaneously start on a round track with velocities \\( v_1 \\) and \\( v_2 \\) respectively. After each circle, the first car doubles its velocity, and the second car triples its velocity. The car is said to be the favorite if there exists a positive integer \\( N \\) such that this car wins any... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 5
} | 7 | 0 | 0 |
omr-ws-69b2305b9a7357f20c9f13aecc711eac07cc6d3a6f8f71d33b337ab08b6d41f0 | Two radii OA and OB of a circle c with midpoint O are perpendicular. Another circle touches circle c at point Q and the radii OA and OB at points C and D, respectively. Determine $ \angle{AQC}$. | [
"\\( 45^\\circ \\) or \\( 90^\\circ \\)"
] | [
"Two radii OA and OB of a circle c with midpoint O are perpendicular. Another circle touches circle c at point Q and the radii OA and OB at points C and D, respectively. Determine $ \\angle{AQC}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8624c3f3f1d1d3894510750d30bdc4b1f4acf9115294ce6ea489b216fec46710 | Two railroad workers are together in a 0.6 mile mountain tunnel. One walks east and the other west in order to be out of the tunnel before the train comes through at 30 mph. Each man reaches his end in 3 minutes. If the man walking east reaches the entrance just before the train enters, and the train passes the other m... | [
"5"
] | [
"Two railroad workers are together in a 0.6 mile mountain tunnel. One walks east and the other west in order to be out of the tunnel before the train comes through at 30 mph. Each man reaches his end in 3 minutes. If the man walking east reaches the entrance just before the train enters, and the train passes the ot... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8266ec0e2e15b7b0f0eac03bd4872a48c92a8dd91571319aab74bb3bb84d25af | Two railroad workers are together in a 0.8 mile tunnel. One jogs east and the other west in order to be out of the tunnel before the Express comes through at 40 mph. Each man reaches his respective end in three minutes. The man jogging east reaches the east entrance just as the train enters, and the front of the train ... | [
"The man jogging east jogged at 9 mph and the man jogging west jogged at 7 mph."
] | [
"Two railroad workers are together in a 0.8 mile tunnel. One jogs east and the other west in order to be out of the tunnel before the Express comes through at 40 mph. Each man reaches his respective end in three minutes. The man jogging east reaches the east entrance just as the train enters, and the front of the t... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-b13035e4fffea684375ded91f3a8e02b8cf0b2f41a8bd21e01ee4cc39e78bb72 | Two railway crews working together completed a repair job on a track section in 6 days. To do 40\% of the work the first crew alone would require two days more than the second crew alone would require tn complete $13\frac{1}{3}\%$ of the whole job. Determine how many days it would take each crew to repair the whole tra... | [
"The first crew can complete the whole repair job in 10 days, the second, in 15 days"
] | [
"Two railway crews working together completed a repair job on a track section in 6 days. To do 40\\% of the work the first crew alone would require two days more than the second crew alone would require tn complete $13\\frac{1}{3}\\%$ of the whole job. Determine how many days it would take each crew to repair the w... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f405221bd84d306b4143fa2bec0cb6911400c4e4db6f79b15968f30719b3deea | Two railway stations are at a distance of 96 km from each other. One train covers this distance 40 minutes faster than does the other. The speed of the first train is 12 km/h higher than that of the second. Determine the speed of both trains. | [
"Tthe speed of the first train is equal to 48 km/h, that of the second, to 36 km/h"
] | [
"Two railway stations are at a distance of 96 km from each other. One train covers this distance 40 minutes faster than does the other. The speed of the first train is 12 km/h higher than that of the second. Determine the speed of both trains."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e6850a3e083ee4d94aeda84fa79bacf3df665a82bdf8ac3eaa61e24b66621352 | Two random integers $1 \leq x, y \leq 10100$ are selected. The probability that both $x^{xy} - 1$ and $y^{xy} - 1$ are divisible by $101$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ share no common divisors. What is the remainder when $m+n$ is divided by $1000$? | [
"737"
] | [
"Two random integers $1 \\leq x, y \\leq 10100$ are selected. The probability that both $x^{xy} - 1$ and $y^{xy} - 1$ are divisible by $101$ can be expressed as $\\frac{m}{n}$ where $m$ and $n$ share no common divisors. What is the remainder when $m+n$ is divided by $1000$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5ac7f935b089eabb561518e514de95446594074081b1dd0bac3119e7e7e03881 | Two random integers, $a$ and $b,$ are independently chosen, with replacement, from $1$ to $1000,$ inclusive. What is the probability that both $2^a + 2^b$ and $3^a + 3^b$ are multiples of $5$? | [
"\\(\\frac{1}{4}\\)"
] | [
"Two random integers, $a$ and $b,$ are independently chosen, with replacement, from $1$ to $1000,$ inclusive. What is the probability that both $2^a + 2^b$ and $3^a + 3^b$ are multiples of $5$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-a446cfc1d2ef0ef7ffd7d90a8377087d13a30a85768eac650fa28ae2ae0c1d83 | Two random numbers $x$ and $y$ are drawn independently from the closed interval $[0, 2]$. What is the probability that $x + y > 1$? | [
"\\dfrac{7}{8}"
] | [
"Two random numbers $x$ and $y$ are drawn independently from the closed interval $[0, 2]$. What is the probability that $x + y > 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3d87b800d386722528f01d38ae86c1e2bc55a5594ea4c6880a2bc4e873a1f621 | Two random numbers between 0 and 1 are generated. Let \( p \) be the probability that these two numbers and 1 can form the sides of an obtuse triangle. Find \( 4p \). | [
"\\(\\pi - 2\\)."
] | [
"Two random numbers between 0 and 1 are generated. Let \\( p \\) be the probability that these two numbers and 1 can form the sides of an obtuse triangle. Find \\( 4p \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-fdd41423288d2f19afe647b70edeab9852dbd76cf60f644766f97db241889903 | Two random points are chosen, one from unit square $A$ with vertices $(0,0)$, $(0,1)$, $(1,0)$, and $(1,1)$, and another from unit square $B$ with vertices $(1,2)$, $(1,3)$, $(2,2)$, and $(2,3)$. What is the probability that the segment connecting these two points intersects the segment between $(1,2)$ and $(2,2)$? | [
"1 - \\dfrac{\\ln 2}{4}"
] | [
"Two random points are chosen, one from unit square $A$ with vertices $(0,0)$, $(0,1)$, $(1,0)$, and $(1,1)$, and another from unit square $B$ with vertices $(1,2)$, $(1,3)$, $(2,2)$, and $(2,3)$. What is the probability that the segment connecting these two points intersects the segment between $(1,2)$ and $(2,2)$... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-155b725bd32e75349590c75d60e2f65d7664646e22b3c15e5ee9845ec2b81fa5 | Two random points are independently thrown onto the interval $[0,1]$ according to a uniform distribution. These points divide the interval into three parts. What is the probability that these three parts can form a triangle? | [
"\\(\\frac{1}{4}\\)"
] | [
"Two random points are independently thrown onto the interval $[0,1]$ according to a uniform distribution. These points divide the interval into three parts. What is the probability that these three parts can form a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-448f9303d2ecf27d5878e7c528ae482ba44894354a4ccc90cc0d9972e9d910af | Two random points are selected uniformly on an equilateral triangle with area $1$. If the length of the line segment connecting the points is longer than half the side length of the triangle, the line segment is colored blue; otherwise, it is colored red. What is the area of the portion of the triangle which is only co... | [
"\\dfrac{1}{4}"
] | [
"Two random points are selected uniformly on an equilateral triangle with area $1$. If the length of the line segment connecting the points is longer than half the side length of the triangle, the line segment is colored blue; otherwise, it is colored red. What is the area of the portion of the triangle which is on... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-0a17407f1db39883c137085ecd116933f4c4f446ce2ab3445d9527e565db4c2b | Two random, not necessarily distinct, permutations of the digits $2017$ are selected and added together. What is the expected value of this sum? | [
"5555"
] | [
"Two random, not necessarily distinct, permutations of the digits $2017$ are selected and added together. What is the expected value of this sum?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 14
} | 31 | 0 | 0 |
omr-ws-8995c293e52041be76268f362120a0b5b725c8e9da56235542cee380f0fa1148 | Two rational numbers \( r \) and \( s \) are given. The numbers \( r + s \), \( r - s \), \( rs \), and \( \frac{s}{r} \) are computed and rearranged in increasing order to get the list: \( \frac{1}{12}, \frac{15}{16}, \frac{5}{3}, \frac{31}{12} \). Find \( r^2 + s^2 \). | [
"\\(\\frac{481}{144}\\)"
] | [
"Two rational numbers \\( r \\) and \\( s \\) are given. The numbers \\( r + s \\), \\( r - s \\), \\( rs \\), and \\( \\frac{s}{r} \\) are computed and rearranged in increasing order to get the list: \\( \\frac{1}{12}, \\frac{15}{16}, \\frac{5}{3}, \\frac{31}{12} \\). Find \\( r^2 + s^2 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-c48d87b8c31dceb57f21d409f903ebd0ac313d163bbf44c0d0ceb0e9ca9976ed | Two rays $r$ and $s$, which form an angle of $60^\circ$, emanate from the center $O$ of circle $\omega$, which has radius 2. Circle $\pi$ is tangent to both rays and internally tangent to $\omega$. Circle $\pi'$ is also tangent to both rays, but externally tangent to $\omega$. Let the tangent point of $\pi$ on $r$ be $... | [
"\\sqrt{3}"
] | [
"Two rays $r$ and $s$, which form an angle of $60^\\circ$, emanate from the center $O$ of circle $\\omega$, which has radius 2. Circle $\\pi$ is tangent to both rays and internally tangent to $\\omega$. Circle $\\pi'$ is also tangent to both rays, but externally tangent to $\\omega$. Let the tangent point of $\\pi$... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e9411bd4c4330ae3694f3befc5638ffdc51e786e76d6baea9f4bbdbdab7b170c | Two rays \( l_1 \) and \( l_2 \) are emitted from point \( O \). A line \( l \) intersects \( l_1 \) and \( l_2 \) at points \( X \) and \( Y \), respectively, forming a triangle \( OXY \) with a constant area \( c \). Let \( M \) be the midpoint of \( XY \). If \( l_1 \) and \( l_2 \) are given by the equations \( y =... | [
"x^2 - y^2 = \\pm 8"
] | [
"Two rays \\( l_1 \\) and \\( l_2 \\) are emitted from point \\( O \\). A line \\( l \\) intersects \\( l_1 \\) and \\( l_2 \\) at points \\( X \\) and \\( Y \\), respectively, forming a triangle \\( OXY \\) with a constant area \\( c \\). Let \\( M \\) be the midpoint of \\( XY \\). If \\( l_1 \\) and \\( l_2 \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-7d8ae33ec2a15b288fe139fe6a05f5efe8691e0ce537eb38dd37ad4838b4c8d1 | Two rays are drawn in square $ABCD$ from vertex $A$, dividing the angle by three equal parts. One ray intersects the diagonal $BD$ at point $N$, the second intersects the extension of $BC$ at point $M$. Through point $B$, draw a line perpendicular to $BD$ and intersecting line $MN$ at point $K$. Find $\angle KAC$. | [
"105^\\circ"
] | [
"Two rays are drawn in square $ABCD$ from vertex $A$, dividing the angle by three equal parts. One ray intersects the diagonal $BD$ at point $N$, the second intersects the extension of $BC$ at point $M$. Through point $B$, draw a line perpendicular to $BD$ and intersecting line $MN$ at point $K$. Find $\\angle KAC$... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 6,
"tir": 20
} | 70 | 0 | 0 |
omr-ws-1e0dbd2dc8f0cd4304db98510a4f620e4015fcfa7d62b8ccb5aa66fd160e1040 | Two rays begin at point A and form an angle of 43 degrees with one another. Lines l, m, and n (no two of which are parallel) each form an isosceles triangle with the original rays. Compute the largest angle of the triangle formed by lines l, m, and n. | [
"129^\\circ"
] | [
"Two rays begin at point A and form an angle of 43 degrees with one another. Lines l, m, and n (no two of which are parallel) each form an isosceles triangle with the original rays. Compute the largest angle of the triangle formed by lines l, m, and n."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-20127c95d5e7dafc8b9ae34c939c4e3d9422dbc24a27ae53b067405f1fce7fed | Two rays with a common endpoint O form a 30° angle. Point A lies on one ray, point B on the other ray, and AB = 2. The maximum possible length of OB is ... | [
"4"
] | [
"Two rays with a common endpoint O form a 30° angle. Point A lies on one ray, point B on the other ray, and AB = 2. The maximum possible length of OB is ..."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c69f69eab234076df3e423d92905c36c914e42232c389a26f86ff2a928db2c40 | Two rays with a common endpoint O form a 30° angle. Point A lies on one ray, point B on the other ray, and AB = 2. What is the maximum possible length of OB? | [
"4"
] | [
"Two rays with a common endpoint O form a 30° angle. Point A lies on one ray, point B on the other ray, and AB = 2. What is the maximum possible length of OB?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4cd325976dd5f352b92d8334110284f927295f6661ba34b007970cf9f8b7374a | Two rays with a common endpoint \(O\) form a 30-degree angle. Point \(A\) lies on one ray, point \(B\) on the other ray, and \(AB = 1\). What is the maximum possible length of \(OB\)? | [
"2"
] | [
"Two rays with a common endpoint \\(O\\) form a 30-degree angle. Point \\(A\\) lies on one ray, point \\(B\\) on the other ray, and \\(AB = 1\\). What is the maximum possible length of \\(OB\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dcc60f212ceb01fb7df8a7ab9262cdd8e6197571cc01c683650796c5a731f2aa | Two rays with common endpoint $O$ form a $30^\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB=1$. What is the maximum possible length of $\overline{OB}$? | [
"2"
] | [
"Two rays with common endpoint $O$ form a $30^\\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB=1$. What is the maximum possible length of $\\overline{OB}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-14b3e40de19ea260f931f3d59955cd4b8f3a019956e748dd0afa5dc9144caedf | Two real numbers $a$ and $b$ are chosen independently and uniformly at random from the interval $(0, 75)$. Let $O$ and $P$ be two points on the plane with $OP = 200$. Let $Q$ and $R$ be on the same side of line $OP$ such that the degree measures of $\angle POQ$ and $\angle POR$ are $a$ and $b$ respectively, and $\angle... | [
"041",
"41"
] | [
"Two real numbers $a$ and $b$ are chosen independently and uniformly at random from the interval $(0, 75)$. Let $O$ and $P$ be two points on the plane with $OP = 200$. Let $Q$ and $R$ be on the same side of line $OP$ such that the degree measures of $\\angle POQ$ and $\\angle POR$ are $a$ and $b$ respectively, and ... | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 91,
"genselect": 8
} | 99 | 0 | 0 |
omr-ws-65a04c92358851f79309455a5b2b1c8919799037f3e754b7637d8675df5c8494 | Two real numbers $a$ and $b$ are chosen such that the following system has a unique real solution: $$x + \frac{20}{y} = a$$ $$y + \frac{24}{x} = b$$ Find the product of all distinct possible values of $ab$. | [
"16"
] | [
"Two real numbers $a$ and $b$ are chosen such that the following system has a unique real solution: $$x + \\frac{20}{y} = a$$ $$y + \\frac{24}{x} = b$$ Find the product of all distinct possible values of $ab$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-45233956989b49e293d7ff8f8cba793f9a4aa2deb70f558d9e80d13deee82731 | Two real numbers $x$ and $y$ are chosen at random and are such that $|x|\le 1$, $0\le y\le 1$. What is the probability that $x^2<y$? | [
"$\\frac{2}{3}$"
] | [
"Two real numbers $x$ and $y$ are chosen at random and are such that $|x|\\le 1$, $0\\le y\\le 1$. What is the probability that $x^2<y$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-554623fa13e08b04a9bb8da776adb09672a45dbe56854a7e5418e6fd73a1accb | Two real numbers $x$ and $y$ satisfy $\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=1$. What is the value of $x + y$? | [
"0"
] | [
"Two real numbers $x$ and $y$ satisfy $\\left(x+\\sqrt{x^2+1}\\right)\\left(y+\\sqrt{y^2+1}\\right)=1$. What is the value of $x + y$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f11f008620edefcc6c059a418315ea67330af584d633302a6e6a237bd2627e5 | Two real numbers $x$ and $y$ satisfy $x-y=4$ and $x^3-y^3=28$. Compute $xy$. | [
"-3"
] | [
"Two real numbers $x$ and $y$ satisfy $x-y=4$ and $x^3-y^3=28$. Compute $xy$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d8cf291314326cce00b0d77d903cc6504f3275f9e0bff85e088169c04a9976ba | Two real numbers \( F \) and \( K \) are chosen randomly between 0 and 11 inclusive. What is the probability that \( |F - K| \ge 1 \)? | [
"\\(\\frac{100}{121}\\)"
] | [
"Two real numbers \\( F \\) and \\( K \\) are chosen randomly between 0 and 11 inclusive. What is the probability that \\( |F - K| \\ge 1 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-97790b1ad10087154b8e0c837f2f51a70175e907314f67bc9889ac45fd8c5b7f | Two real numbers \( s \) and \( t \) satisfy \( 19s^2 + 99s + 1 = 0 \) and \( t^2 + 99t + 19 = 0 \). Find the value of \( \frac{st + 4s + 1}{t} \). | [
"-5"
] | [
"Two real numbers \\( s \\) and \\( t \\) satisfy \\( 19s^2 + 99s + 1 = 0 \\) and \\( t^2 + 99t + 19 = 0 \\). Find the value of \\( \\frac{st + 4s + 1}{t} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 3
} | 21 | 0 | 0 |
omr-ws-aa70544c468de4d77d05b94bf6dda857fd53f0738e17e43f9cec3a44f4bc6045 | Two real numbers \( x \) and \( y \) are chosen at random in the interval \( (0,1) \) with respect to the uniform distribution. What is the probability that the closest integer to \( \frac{x}{y} \) is even? Express the answer in the form \( r + s\pi \), where \( r \) and \( s \) are rational numbers. | [
"\\( \\frac{5}{4} - \\frac{\\pi}{4} \\)"
] | [
"Two real numbers \\( x \\) and \\( y \\) are chosen at random in the interval \\( (0,1) \\) with respect to the uniform distribution. What is the probability that the closest integer to \\( \\frac{x}{y} \\) is even? Express the answer in the form \\( r + s\\pi \\), where \\( r \\) and \\( s \\) are rational number... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 16
} | 26 | 0 | 0 |
omr-ws-dce8eb8200d489097d49f74db5431fe88d3fe9d9c3bff1e3c38f2b2010cac19f | Two real numbers \( x \) and \( y \) are chosen at random in the interval \((0,1)\) with respect to the uniform distribution. What is the probability that the closest integer to \( \frac{x}{y} \) is even? Express the answer in the form \( r + s\pi \), where \( r \) and \( s \) are rational numbers. | [
"\\( \\frac{5}{4} - \\frac{1}{4}\\pi \\)"
] | [
"Two real numbers \\( x \\) and \\( y \\) are chosen at random in the interval \\((0,1)\\) with respect to the uniform distribution. What is the probability that the closest integer to \\( \\frac{x}{y} \\) is even? Express the answer in the form \\( r + s\\pi \\), where \\( r \\) and \\( s \\) are rational numbers.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-d939ba4c7eb128b2a91e044dcb6039f090e803cddc9958217dc031e782864e5d | Two real numbers \( x \) and \( y \) are chosen at random such that \( x, y \geq 0 \) and \( x + y = 2016.5 \). What is the probability that \( \lfloor x \rceil + \lfloor y \rceil = 2017 \)? Note: \( \lfloor x \rceil \) denotes the nearest integer to \( x \). | [
"\\dfrac{2016}{4033}"
] | [
"Two real numbers \\( x \\) and \\( y \\) are chosen at random such that \\( x, y \\geq 0 \\) and \\( x + y = 2016.5 \\). What is the probability that \\( \\lfloor x \\rceil + \\lfloor y \\rceil = 2017 \\)? Note: \\( \\lfloor x \\rceil \\) denotes the nearest integer to \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-782de2bc50f9871461f76c602214375ba417159bc270b844f215795a5e688839 | Two real numbers \( x \) and \( y \) are chosen at random. Find the probability that they satisfy \[ \lfloor x + y \rfloor \le x + y - \frac{1}{4}. \] | [
"\\(\\frac{3}{4}\\)"
] | [
"Two real numbers \\( x \\) and \\( y \\) are chosen at random. Find the probability that they satisfy\n\\[ \\lfloor x + y \\rfloor \\le x + y - \\frac{1}{4}. \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-15d17df27d6327074e8f2c6332cc91bc96e923bfb155e55ae2e2773af60c906f | Two real numbers \( x \) and \( y \) are chosen from the interval \([0,2]\). What is the probability that the sum of the floors of the squares of \( x \) and \( y \) is less than 3? | [
"\\dfrac{1 + \\sqrt{3} - \\sqrt{2}}{2}"
] | [
"Two real numbers \\( x \\) and \\( y \\) are chosen from the interval \\([0,2]\\). What is the probability that the sum of the floors of the squares of \\( x \\) and \\( y \\) is less than 3?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-426918f807733d85ad5c4df9d31b3c04aee0b043d09ebef09f0e0c34e56cd18b | Two real numbers \( x \) and \( y \) are chosen such that \( |x| < 2 \) and \( |y| < 3 \). Compute the probability that \( \left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 < 1 \). | [
"\\(\\frac{\\pi}{4}\\)"
] | [
"Two real numbers \\( x \\) and \\( y \\) are chosen such that \\( |x| < 2 \\) and \\( |y| < 3 \\). Compute the probability that \\( \\left(\\frac{x}{2}\\right)^2 + \\left(\\frac{y}{3}\\right)^2 < 1 \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-354091bdccbbce50ca4876845bf0edde1b333f8413de46de90a80dc26013435e | Two real numbers \( x \) and \( y \) are randomly selected on a number line between 0 and 12. Find the probability that their sum is greater than 15. | [
"9/32"
] | [
"Two real numbers \\( x \\) and \\( y \\) are randomly selected on a number line between 0 and 12. Find the probability that their sum is greater than 15."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-42df4918776e9cdd863e02d6a17e88dece88fe66ff10125c595020d38a20bd2c | Two real numbers \( x \) and \( y \) satisfy the equation \( x^2 + 3xy + y^2 = 2015 \). What is the maximum possible value of \( xy \)? | [
"403"
] | [
"Two real numbers \\( x \\) and \\( y \\) satisfy the equation \\( x^2 + 3xy + y^2 = 2015 \\). What is the maximum possible value of \\( xy \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-dd688e7b9a4ad590ade60a9de9534a712c966b8a784b5be459df3b1b981fb964 | Two real numbers \(a\) and \(b\) are chosen in the range \([0,10]\). What is the probability that \(|a-b| \ge 6\)? | [
"\\(\\frac{4}{25}\\)"
] | [
"Two real numbers \\(a\\) and \\(b\\) are chosen in the range \\([0,10]\\). What is the probability that \\(|a-b| \\ge 6\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-f4948314e7bbc269e530cd111f6a67149482c40032a41dfc051454b8b28e2882 | Two real numbers \(a\) and \(b\) are chosen independently and uniformly at random from the interval \((0, 75)\). Let \(O\) and \(P\) be two points on the plane with \(OP = 200\). Let \(Q\) and \(R\) be on the same side of line \(OP\) such that the degree measures of \(\angle POQ\) and \(\angle POR\) are \(a\) and \(b\)... | [
"41"
] | [
"Two real numbers \\(a\\) and \\(b\\) are chosen independently and uniformly at random from the interval \\((0, 75)\\). Let \\(O\\) and \\(P\\) be two points on the plane with \\(OP = 200\\). Let \\(Q\\) and \\(R\\) be on the same side of line \\(OP\\) such that the degree measures of \\(\\angle POQ\\) and \\(\\ang... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 5,
"tir": 2
} | 52 | 0 | 0 |
omr-ws-c70db71587c24a464ff5c9e4b21f9eaa33ab1186ff39946039cb3483c5024467 | Two real numbers \(a\) and \(b\) satisfy the inequality \(b + b^3 \leq a - a^3\). Find the maximum value of \(a + b\). | [
"\\(\\frac{\\sqrt{2}}{\\sqrt[4]{3}}\\)"
] | [
"Two real numbers \\(a\\) and \\(b\\) satisfy the inequality \\(b + b^3 \\leq a - a^3\\). Find the maximum value of \\(a + b\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 43,
"tir": 24
} | 67 | 0 | 0 |
omr-ws-042179d6a48e7c395d4e6516c33fe8300418e6c058d6864489fb6e9b6d06d019 | Two real numbers are chosen at random between 0 and 10, inclusive. What is the probability that they are within one unit of each other? (Round to the nearest hundredth) | [
"0.19"
] | [
"Two real numbers are chosen at random between 0 and 10, inclusive. What is the probability that they are within one unit of each other? (Round to the nearest hundredth)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-38442940542f6daadf6ca36fe816ab38e2def7151e64a98f49d1ea558ac1d17b | Two real sequences \(\{x_n\}\) and \(\{y_n\}\) satisfy the following recurrence relations: \[ x_0 = 1, \quad y_0 = 2007 \] \[ x_{n+1} = x_n - (x_n y_n + x_{n+1} y_{n+1} - 2)(y_n + y_{n+1}) \] \[ y_{n+1} = y_n - (x_n y_n + x_{n+1} y_{n+1} - 2)(x_n + x_{n+1}) \] Calculate the maximum possible value of \(x_{100}^2\). | [
"4028050"
] | [
"Two real sequences \\(\\{x_n\\}\\) and \\(\\{y_n\\}\\) satisfy the following recurrence relations:\n\\[ x_0 = 1, \\quad y_0 = 2007 \\]\n\\[ x_{n+1} = x_n - (x_n y_n + x_{n+1} y_{n+1} - 2)(y_n + y_{n+1}) \\]\n\\[ y_{n+1} = y_n - (x_n y_n + x_{n+1} y_{n+1} - 2)(x_n + x_{n+1}) \\]\n\nCalculate the maximum possible va... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 2
} | 2 | 0 | 0 |
omr-ws-4ced03fee13b30a2bd4e5a03feba724c7e9c0700613916adbce74f4f0bd17b0e | Two real sequences \(x_{1},x_{2},\cdots,\) and \(y_{1},y_{2},\cdots,\) are defined as follows: \[x_{1}=y_{1}=\sqrt{3},\] \[x_{n+1}=x_{n}+\sqrt{1+x_n^2},\] \[y_{n+1}=\frac{y_{n}}{1+\sqrt{1+y_n^2}}\] for all \(n \geq 1\). Calculate the value of \(x_{5}y_{5}\). | [
"2"
] | [
"Two real sequences \\(x_{1},x_{2},\\cdots,\\) and \\(y_{1},y_{2},\\cdots,\\) are defined as follows:\n\\[x_{1}=y_{1}=\\sqrt{3},\\]\n\\[x_{n+1}=x_{n}+\\sqrt{1+x_n^2},\\]\n\\[y_{n+1}=\\frac{y_{n}}{1+\\sqrt{1+y_n^2}}\\]\nfor all \\(n \\geq 1\\). Calculate the value of \\(x_{5}y_{5}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-a7218ab58b1f29a223e35cdf4e2387f5879fba7db7ccdfc3ad590401d54a5237 | Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | [
"12"
] | [
"Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-891601e06712eb4ffa72a4469ef8b10fa62c00965bb7bf17f544792b8f487b26 | Two rectangles $ABCD$ and $AEFG$ share a common vertex $A$. Additionally, point $D$ is on segment $FG$ and point $E$ is on segment $BC$. Given that $ABCD$ is a square of side length $17$, compute the area of rectangle $AEFG$ if $AF=15$. | [
"17"
] | [
"Two rectangles $ABCD$ and $AEFG$ share a common vertex $A$. Additionally, point $D$ is on segment $FG$ and point $E$ is on segment $BC$. Given that $ABCD$ is a square of side length $17$, compute the area of rectangle $AEFG$ if $AF=15$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2b7a77890eafffe2455b402c1e6d82181b497e7a7abca131873ac4708336fdbb | Two rectangles $ABCD$ and $APQR$ are shown such that $AB = AP = 2$ and $BC = PQ = 4$. Calculate angle $\angle DRQ$. | [
"\\(15^\\circ\\)"
] | [
"Two rectangles $ABCD$ and $APQR$ are shown such that $AB = AP = 2$ and $BC = PQ = 4$. Calculate angle $\\angle DRQ$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95f25be03cac7dcd9ce02cdda0b1c18afd75c095302dc9e518e3b33353b075b3 | Two rectangles have integer dimensions, and both have a perimeter of 144 cm. What is the greatest possible difference between the areas of two such rectangles? | [
"1225"
] | [
"Two rectangles have integer dimensions, and both have a perimeter of 144 cm. What is the greatest possible difference between the areas of two such rectangles?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-765fd25185a212db52ab105206b4d1adabab5daa89c4a27cf4d9fc3fa842670e | Two rectangles of different sizes have the same side ratio. They are positioned such that each vertex of the smaller rectangle lies on the sides of the larger rectangle. Determine the possible side ratios that allow this construction. | [
"1"
] | [
"Two rectangles of different sizes have the same side ratio. They are positioned such that each vertex of the smaller rectangle lies on the sides of the larger rectangle. Determine the possible side ratios that allow this construction."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-630ed8de66dc4162e627ba699de66264bf5654983347ce3b420e50d224ecaee2 | Two rectangles of unit area overlap to form a convex octagon. If the area of the octagon is represented by $A$, find the minimum possible value of $10A$. | [
"20\\sqrt{2} - 20"
] | [
"Two rectangles of unit area overlap to form a convex octagon. If the area of the octagon is represented by $A$, find the minimum possible value of $10A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-0fec143c1804db55fbe7217e32adad2c3fe9979a46eabebd1759325571e9f532 | Two rectangles, A, with vertices at (-2, 0), (0,0), (0,4), and (-2,4) and B, with vertices at (1,0), (5,0), (1,12), and (5, 12), are simultaneously bisected by a line in the plane. What is the slope of this line? | [
"1"
] | [
"Two rectangles, A, with vertices at (-2, 0), (0,0), (0,4), and (-2,4) and B, with vertices at (1,0), (5,0), (1,12), and (5, 12), are simultaneously bisected by a line in the plane. What is the slope of this line?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b843d47a1eb6763178126bd33044f88ee73f42a35e6de5587de6fd01d7ccacfb | Two rectangular cardboards are placed side by side on a flat surface such that their lower edges are aligned. The left cardboard has dimensions 4 units by 6 units, and the right cardboard has dimensions 5 units by 7 units. The left cardboard is rotated around its lower right corner until its upper right corner touches ... | [
"5"
] | [
"Two rectangular cardboards are placed side by side on a flat surface such that their lower edges are aligned. The left cardboard has dimensions 4 units by 6 units, and the right cardboard has dimensions 5 units by 7 units. The left cardboard is rotated around its lower right corner until its upper right corner tou... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-c47af44324b5baa249774019d052c0bf17fa13aeb9928a1b9f8c8ceff669f3cf | Two rectilinear railway lines $AA'$ and $BB'$ are mutually perpendicular and intersect at a point $C$, the distances $AC$ and $BC$ being equal to $a$ and $b$. Two trains whose speeds are, respectively, $v_1$ and $v_2$, start simultaneously from the points $A$ and $B$ toward $C$. In what time after the departure will th... | [
"t_0=\\frac{av_1+bv_2}{v_1^2+v_2^2}, r(t_0)=\\frac{|av_2-bv_1|}{\\sqrt{v_1^2+v_2^2}}"
] | [
"Two rectilinear railway lines $AA'$ and $BB'$ are mutually perpendicular and intersect at a point $C$, the distances $AC$ and $BC$ being equal to $a$ and $b$. Two trains whose speeds are, respectively, $v_1$ and $v_2$, start simultaneously from the points $A$ and $B$ toward $C$. In what time after the departure wi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1361c75191442ab3a06166b7f415a725b56e1dacfdc462300b9db0c88d77d024 | Two regular \( n \)-gons have their sides colored in two different colors (e.g., RED and BLUE). These two \( n \)-gons intersect to form a \( 2n \)-gon. If the total length of the RED sides in the \( 2n \)-gon is \( R \) and the total length of the BLUE sides is \( B \), find the value of \( R - B \). | [
"0"
] | [
"Two regular \\( n \\)-gons have their sides colored in two different colors (e.g., RED and BLUE). These two \\( n \\)-gons intersect to form a \\( 2n \\)-gon. If the total length of the RED sides in the \\( 2n \\)-gon is \\( R \\) and the total length of the BLUE sides is \\( B \\), find the value of \\( R - B \\)... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-307a0a088e1bae749b938acda71421d413ff759bcc0e5abc40d20bfd49899cb3 | Two regular convex polygons with \( n \) edges overlap each other such that the overlapping area forms a new convex polygon with \( 2n \) edges. The edges of this new polygon are labeled with the numbers \( 1, 2, \ldots, 2n \) in sequence. Calculate the difference between the sum of the edges labeled with even numbers ... | [
"n"
] | [
"Two regular convex polygons with \\( n \\) edges overlap each other such that the overlapping area forms a new convex polygon with \\( 2n \\) edges. The edges of this new polygon are labeled with the numbers \\( 1, 2, \\ldots, 2n \\) in sequence. Calculate the difference between the sum of the edges labeled with e... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-c8176e969a979ac7626a65b097be756f7856727af4bf1b26d567ef227160f82c | Two regular hexagons \( A_1A_2 \ldots A_6 \) and \( B_1B_2 \ldots B_6 \) are given, with \( A_1 \) coinciding with \( B_1 \) and \( A_2 \) with \( B_2 \). The hexagons can rotate in space with respect to the axis \( A_1A_2 \). The sides \( A_2A_3 \) and \( B_2B_3 \) are joined by a plane, and similarly for other sides.... | [
"\\( \\frac{s \\sqrt{3}}{2} \\cos \\left( \\frac{\\theta}{2} \\right) \\)"
] | [
"Two regular hexagons \\( A_1A_2 \\ldots A_6 \\) and \\( B_1B_2 \\ldots B_6 \\) are given, with \\( A_1 \\) coinciding with \\( B_1 \\) and \\( A_2 \\) with \\( B_2 \\). The hexagons can rotate in space with respect to the axis \\( A_1A_2 \\). The sides \\( A_2A_3 \\) and \\( B_2B_3 \\) are joined by a plane, and s... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-b859383cc60a3269c9a4be904ef5f0741dff540642434813c01f27f26e457d2c | Two regular hexagons are shown, one inside the other. If the points $A, B$ and $C$ belong on the same line and the perimeter of the larger hexagon is $120$ cm, calculate the perimeter of the minor hexagon (in cm). | [
"90"
] | [
"Two regular hexagons are shown, one inside the other. If the points $A, B$ and $C$ belong on the same line and the perimeter of the larger hexagon is $120$ cm, calculate the perimeter of the minor hexagon (in cm)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.