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-41e2a1317f6ccb4cfaedf7d4082a1b24d0d52464f9d4318acf031f0cac80d898 | A ray of light passing through the point $(1,2)$ reflects on the $x$-axis at point $A$ and the reflected ray passes through the point $(5,3)$. Find the coordinates of $A$. | [
"\\(\\left(\\frac{13}{5}, 0\\right)\\)"
] | [
"A ray of light passing through the point $(1,2)$ reflects on the $x$-axis at point $A$ and the reflected ray passes through the point $(5,3)$. Find the coordinates of $A$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-d34b2cc3cbf5d05de66aec024acb66a444fc867ecae6a172f723cba3ab8e3b61 | A ray of light starts from the center of a square and reflects off its sides, with the angle of reflection equal to the angle of incidence. The ray never reaches a vertex and returns to the center of the square after reflecting off the sides. Let $n$ be the number of times the ray reflects off the sides of the square b... | [
"4"
] | [
"A ray of light starts from the center of a square and reflects off its sides, with the angle of reflection equal to the angle of incidence. The ray never reaches a vertex and returns to the center of the square after reflecting off the sides. Let $n$ be the number of times the ray reflects off the sides of the squ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-c649322db9b55f6f85298ba370f5bbbcde510fec7d07039ee7dcf4cfe9219b5f | A ray of light, directed from the top of a square with side length $1$, reflects $2005$ times from its walls according to the law "the angle of incidence is equal to the angle of reflection", and ends its path at another vertex of the square. What is the smallest distance the ray could travel? | [
"\\sqrt{1002^2 + 1003^2}"
] | [
"A ray of light, directed from the top of a square with side length $1$, reflects $2005$ times from its walls according to the law \"the angle of incidence is equal to the angle of reflection\", and ends its path at another vertex of the square. What is the smallest distance the ray could travel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-ac0ccf0703b2e5350df4ae74463bed38d126de93535f1e8ae768700137059683 | A real $2024 \times 2024$ matrix $A$ is called nice if $(Av, v) = 1$ for every vector $v \in \mathbb{R}^{2024}$ with unit norm. Find an example of a nice non-identity matrix. | [
"\\( A = B + I \\) where \\( B \\) is a non-zero skew-symmetric matrix."
] | [
"A real $2024 \\times 2024$ matrix $A$ is called nice if $(Av, v) = 1$ for every vector $v \\in \\mathbb{R}^{2024}$ with unit norm. Find an example of a nice non-identity matrix."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 22
} | 41 | 0 | 0 |
omr-ws-740294774f015e3ce77ed7532e82b3d5eaa740aed1e16ddd6fcf8d12c7911d98 | A real estate agent has eight master keys to open several new homes, with only one master key opening any given house. If 40% of these homes are usually left unlocked, what is the probability that the real estate agent can get into a specific house if he selects three master keys at random before leaving the office? | [
"\\dfrac{5}{8}"
] | [
"A real estate agent has eight master keys to open several new homes, with only one master key opening any given house. If 40% of these homes are usually left unlocked, what is the probability that the real estate agent can get into a specific house if he selects three master keys at random before leaving the offic... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-09d29ce48e49a7666df2b99be12ee48fe278092f2786e5d70602da06e974207a | A real estate developer placed a penalty clause in the contract for a shopping mall to be built at a cost of $15 million. The clause specified that if the mall was finished late, the builder would be fined $10,000 for the first month late, and for each additional month late, $2,000 more than the fine from the previous ... | [
"\\$14600000"
] | [
"A real estate developer placed a penalty clause in the contract for a shopping mall to be built at a cost of $15 million. The clause specified that if the mall was finished late, the builder would be fined $10,000 for the first month late, and for each additional month late, $2,000 more than the fine from the prev... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 16
} | 27 | 0 | 0 |
omr-ws-215254a7008220a9a2998281fdbb82001f0d9096c51e0d867aad78a480569518 | A real estate investor is examining a triangular plot of land. She measures each angle of the field. The sum of the first and second angles is 144° more than the measure of the third angle. If the measure of the third angle is subtracted from the measure of the second angle, the result is thrice the measure of the firs... | [
"144"
] | [
"A real estate investor is examining a triangular plot of land. She measures each angle of the field. The sum of the first and second angles is 144° more than the measure of the third angle. If the measure of the third angle is subtracted from the measure of the second angle, the result is thrice the measure of the... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-35ca489972ed3900f307f3d7d51eb44ff26d8fdbf66548bed49f829065831a12 | A real estate office handles 50 apartment units. When the rent is $540 per month, all units are occupied. However, for each $30 increase in rent, one unit becomes vacant. Each occupied unit requires an average of $18 per month for service and repairs. What rent should be charged to obtain the most profit? | [
"$1020"
] | [
"A real estate office handles 50 apartment units. When the rent is $540 per month, all units are occupied. However, for each $30 increase in rent, one unit becomes vacant. Each occupied unit requires an average of $18 per month for service and repairs. What rent should be charged to obtain the most profit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 21
} | 42 | 0 | 0 |
omr-ws-40badb4fa0cc8856c06ac938cecd4d8c770f16120fd61bfc38877b41813fe5c5 | A real function \( f \) defined on pairs of nonnegative integers satisfies the following conditions: - \( f(0,0) = 0 \) - \( f(2x,2y) = f(2x+1,2y+1) = f(x,y) \) - \( f(2x+1,2y) = f(2x,2y+1) = f(x,y) + 1 \) Given that \( f(a,b) = n \), find the number of nonnegative integers \( x \) such that \( f(a,x) + f(b,x) = n \). | [
"\\( 2^n \\)"
] | [
"A real function \\( f \\) defined on pairs of nonnegative integers satisfies the following conditions:\n- \\( f(0,0) = 0 \\)\n- \\( f(2x,2y) = f(2x+1,2y+1) = f(x,y) \\)\n- \\( f(2x+1,2y) = f(2x,2y+1) = f(x,y) + 1 \\)\n\nGiven that \\( f(a,b) = n \\), find the number of nonnegative integers \\( x \\) such that \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 5
} | 22 | 0 | 0 |
omr-ws-24b021a0c538a03c82a516500844182100075f0e81c7fc1afcc87251f617239d | A real function \( f \) is always positive and satisfies \( \frac{f(x)}{f(y)} \leq 2^{(x-y)^2} \) for all \( x, y \in \mathbb{D} \), where \( \mathbb{D} \) is the domain of the function. Determine the nature of \( f(x) \). | [
"\\( f(x) = c \\) where \\( c > 0 \\)"
] | [
"A real function \\( f \\) is always positive and satisfies \\( \\frac{f(x)}{f(y)} \\leq 2^{(x-y)^2} \\) for all \\( x, y \\in \\mathbb{D} \\), where \\( \\mathbb{D} \\) is the domain of the function. Determine the nature of \\( f(x) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-c0e7432a9d788fdcf6b5e9e1a211080a537c4410c3796b1a33fa4e18c5511a2f | A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. Find the probability that the roots of the polynomial \[x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2\]are all real. | [
"\\frac{18}{19}"
] | [
"A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. Find the probability that the roots of the polynomial\n\\[x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2\\]are all real."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-af067028993d6f9ddcf2ca97d3e1b816b421b2a59ef6ac102d5c49de185237c7 | A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial $x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2$ are all real can be written in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | [
"037"
] | [
"A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial \n$x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2$ \nare all real can be written in the form $\\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-7ffe344700398425aa2631c9bd028e4bc6caee0a1f8a88f33a1f9313bf63fa95 | A real number $a$ is randomly chosen in the interval $-20 \leq a \leq 18$. What is the probability that all roots of the equation $x^3+(2a+1)x^2+(4a-1)x+2$ are real? | [
"\\dfrac{18}{19}"
] | [
"A real number $a$ is randomly chosen in the interval $-20 \\leq a \\leq 18$. What is the probability that all roots of the equation $x^3+(2a+1)x^2+(4a-1)x+2$ are real?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-801252f8b581a2ea1fe763a53c21ea517f14e7edef54131780ef03951303ccd5 | A real number $k$ is chosen at random from the open interval $(-10,10)$. What is the probability that the equation $x^2 - 5\left\lvert x \right\rvert = k$ has four real roots? | [
"\\dfrac{5}{16}"
] | [
"A real number $k$ is chosen at random from the open interval $(-10,10)$. What is the probability that the equation $x^2 - 5\\left\\lvert x \\right\\rvert = k$ has four real roots?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c007bf67310b744a338d0e2bf2892cf3477d4f92dc902bb322243b0cd5463963 | A real number $x$ is chosen randomly from the interval $(0,1)$. What is the probability that $\lfloor \log_5(3x) \rfloor = \lfloor \log_5(x) \rfloor$? | [
"\\(\\frac{1}{6}\\)"
] | [
"A real number $x$ is chosen randomly from the interval $(0,1)$. What is the probability that $\\lfloor \\log_5(3x) \\rfloor = \\lfloor \\log_5(x) \\rfloor$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-cb318206c491f69b31fcd199d04352ff50d5a94d8735aa72979a2cac40f978dc | A real number $x$ is chosen randomly from the interval $(0,1].$ Define $y=\lceil{ \log_4{x}}\rceil.$ Compute the reciprocal of the sum of the lengths of all subintervals of $(0,1]$ for which $y$ is odd. | [
"5"
] | [
"A real number $x$ is chosen randomly from the interval $(0,1].$ Define $y=\\lceil{ \\log_4{x}}\\rceil.$ Compute the reciprocal of the sum of the lengths of all subintervals of $(0,1]$ for which $y$ is odd."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ab50b0440b7f469fc45ef00a1106b382602472cc7535ca0dde9b66c9ff76c655 | A real number $x$ is randomly selected from $(0, 500).$ If we define \[ S = \frac{1}{x} \sum_{j=1}^{1000} \sum_{k=j}^{1000} \left\lfloor \frac{j+x}{k} \right\rfloor,\] there is a single nonzero digit in the decimal representation of the probability that $S \geq 1200.$ Determine the value of this digit. | [
"5"
] | [
"A real number $x$ is randomly selected from $(0, 500).$ If we define \\[ S = \\frac{1}{x} \\sum_{j=1}^{1000} \\sum_{k=j}^{1000} \\left\\lfloor \\frac{j+x}{k} \\right\\rfloor,\\] there is a single nonzero digit in the decimal representation of the probability that $S \\geq 1200.$ Determine the value of this digit."... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-17093b9cc4cae6cb28a7c71c0ac5c14ac8686aa1c8ec08300b2929c97e01c952 | A real number $y$ is chosen at random such that $0 \leq y \leq 100$. What is the probability that $y - \lfloor y \rfloor \geq \frac{1}{3}$? | [
"\\(\\frac{2}{3}\\)"
] | [
"A real number $y$ is chosen at random such that $0 \\leq y \\leq 100$. What is the probability that $y - \\lfloor y \\rfloor \\geq \\frac{1}{3}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-4e656a8fd4882104cec1e54d0d605e143c66160ba1ca21b52c180eae59a42abd | A real number \( a \) is called beautiful if there exist positive integers \( m \) and \( n \) and a fair assignment of \( 0 \)s and \( 1 \)s to each unit square of an \( m \times n \) chessboard such that for each row and column, the percentage of \( 1 \)s is not less than \( a \) and not greater than \( 100 - a \). F... | [
"75"
] | [
"A real number \\( a \\) is called beautiful if there exist positive integers \\( m \\) and \\( n \\) and a fair assignment of \\( 0 \\)s and \\( 1 \\)s to each unit square of an \\( m \\times n \\) chessboard such that for each row and column, the percentage of \\( 1 \\)s is not less than \\( a \\) and not greater... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-992041e9855e18bbdd6b26a016c6597e115fb9e31c4c310e60a5fc5a20c2175f | A real number \( a \) is chosen randomly and uniformly from the interval \([-20, 18]\). The probability that the roots of the polynomial \[ x^4 + 2ax^3 + (2a-2)x^2 + (-4a+3)x - 2 \] are all real can be written in the form \(\frac{m}{n}\), where \( m \) and \( n \) are relatively prime positive integers. Find \( m+n \). | [
"037"
] | [
"A real number \\( a \\) is chosen randomly and uniformly from the interval \\([-20, 18]\\). The probability that the roots of the polynomial \n\\[ x^4 + 2ax^3 + (2a-2)x^2 + (-4a+3)x - 2 \\]\nare all real can be written in the form \\(\\frac{m}{n}\\), where \\( m \\) and \\( n \\) are relatively prime positive inte... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-9865c3155dde4e0e306616d4f62609707fdf2ea6a4c0e7ac41377674883d91bc | A real number \( a \) is randomly and uniformly chosen from the interval \([-3, 4]\). What is the probability that all roots of the polynomial \( x^3 + ax^2 + ax + 1 \) are real? | [
"\\(\\frac{4}{7}\\)"
] | [
"A real number \\( a \\) is randomly and uniformly chosen from the interval \\([-3, 4]\\). What is the probability that all roots of the polynomial \\( x^3 + ax^2 + ax + 1 \\) are real?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bdc949345ca18905d375c55f93f552039b5631a05c3cfeda9488e9c1cbad8d8b | A real number \( a \geq -1 \) is called philosophical if there exists a sequence \( \epsilon_1, \epsilon_2, \dots \) with \( \epsilon_i \in \{-1, 1\} \) for all \( i \geq 1 \), such that the sequence \( a_1, a_2, a_3, \dots \), with \( a_1 = a \), satisfies \[ a_{n+1} = \epsilon_n \sqrt{a_n + 1} \quad \forall n \geq 1 ... | [
"\\(0, -1, \\frac{1 + \\sqrt{5}}{2}, \\frac{1 - \\sqrt{5}}{2}\\)"
] | [
"A real number \\( a \\geq -1 \\) is called philosophical if there exists a sequence \\( \\epsilon_1, \\epsilon_2, \\dots \\) with \\( \\epsilon_i \\in \\{-1, 1\\} \\) for all \\( i \\geq 1 \\), such that the sequence \\( a_1, a_2, a_3, \\dots \\), with \\( a_1 = a \\), satisfies\n\\[ a_{n+1} = \\epsilon_n \\sqrt{a... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 24,
"tir": 20
} | 44 | 0 | 0 |
omr-ws-931754caf7f66c5c6b0f86912fe4943bffb4d150e47deceb91087d16b363e8c7 | A real number \( c > 0 \) is given. Ana and Bob play a game on the \( x,y \)-coordinate plane. Bob chooses a function \( f: \mathbb{R} \rightarrow [0,2020] \), and the set of nice points is defined as \( \{(x,y) \in \mathbb{R}^2 \mid f(x + f(y)) = y\} \). Ana and Bob each choose a nice point to stand on, and they take ... | [
"4040"
] | [
"A real number \\( c > 0 \\) is given. Ana and Bob play a game on the \\( x,y \\)-coordinate plane. Bob chooses a function \\( f: \\mathbb{R} \\rightarrow [0,2020] \\), and the set of nice points is defined as \\( \\{(x,y) \\in \\mathbb{R}^2 \\mid f(x + f(y)) = y\\} \\). Ana and Bob each choose a nice point to stan... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-303e468848e134780b2e671e46c0882794f743abbad6685be687b7fa20577ada | A real number \( f(X) \neq 0 \) is assigned to each point \( X \) in space. For any tetrahedron \( ABCD \) with \( O \) as the center of the inscribed sphere, it is known that: \[ f(O) = f(A)f(B)f(C)f(D). \] Find the value of \( f(P) \) for any point \( P \) in space, given that \( f(A) = 2 \) for a specific point \( A... | [
"1"
] | [
"A real number \\( f(X) \\neq 0 \\) is assigned to each point \\( X \\) in space. For any tetrahedron \\( ABCD \\) with \\( O \\) as the center of the inscribed sphere, it is known that:\n\\[ f(O) = f(A)f(B)f(C)f(D). \\]\nFind the value of \\( f(P) \\) for any point \\( P \\) in space, given that \\( f(A) = 2 \\) f... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c7ce1ec5112fb53fb2550914ebbfadd76ac432b1c9e3ce6b7e69838cc5212ba6 | A real number \( x \) between 0 and 1 is chosen at random, and \( x \) has a continued fraction expansion \([a_0; a_1, a_2, \dots]\). Find the expected value of \(\frac{1}{a_1}\). | [
"\\(\\frac{\\pi^2}{6} - 1\\)"
] | [
"A real number \\( x \\) between 0 and 1 is chosen at random, and \\( x \\) has a continued fraction expansion \\([a_0; a_1, a_2, \\dots]\\). Find the expected value of \\(\\frac{1}{a_1}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-6c65eb6b7f749ad5bfc720c3452572015380a6e2bd26d00c93e5f37cba4c79e3 | A real number \( x \) is chosen uniformly at random from the interval \([0, 1]\). A real number \( y \) is chosen uniformly at random from the interval \([0, x]\). A real number \( z \) is chosen uniformly at random from the interval \([0, y]\). What is the probability that \( z > 0.5 \)? | [
"\\frac{1}{2} - \\frac{\\ln 2}{2} - \\frac{(\\ln 2)^2}{4}"
] | [
"A real number \\( x \\) is chosen uniformly at random from the interval \\([0, 1]\\). A real number \\( y \\) is chosen uniformly at random from the interval \\([0, x]\\). A real number \\( z \\) is chosen uniformly at random from the interval \\([0, y]\\). What is the probability that \\( z > 0.5 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-8da9b3522a711cc76d5192b566047fa0ab543965bf475dab07b05d31252c0181 | A real number greater than or equal to $0$ is written in each square of a $3 \times 3$ grid. For any three squares X, Y, and Z that satisfy the following positional relationship, the sum of the three numbers written in them is $1$ or more: - X and Y are in the same row and Y is to the right of X. - X and Z are in the s... | [
"2"
] | [
"A real number greater than or equal to $0$ is written in each square of a $3 \\times 3$ grid. For any three squares X, Y, and Z that satisfy the following positional relationship, the sum of the three numbers written in them is $1$ or more:\n- X and Y are in the same row and Y is to the right of X.\n- X and Z are ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-321f48b8cd9f23590c4e6e6b4637699d45dbf0448b8ba34deca362745eaa57dc | A real number is assigned to each vertex of a triangular prism such that the number on any vertex is the arithmetic mean of the numbers on the three adjacent vertices. If the sum of the numbers on the vertices of one triangular face is 18, find the sum of the numbers on all six vertices. | [
"36"
] | [
"A real number is assigned to each vertex of a triangular prism such that the number on any vertex is the arithmetic mean of the numbers on the three adjacent vertices. If the sum of the numbers on the vertices of one triangular face is 18, find the sum of the numbers on all six vertices."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-26fa1d0cc8270ccef6956650e8e25667698fc040e18150c45eb558d3e0e52a66 | A real number is chosen in the interval [0.75, 0.85]. The number is first rounded to the nearest hundredths, and then rounded again to the nearest tenths. What is the probability that the final result is 0.8? | [
"\\(\\frac{19}{20}\\)"
] | [
"A real number is chosen in the interval [0.75, 0.85]. The number is first rounded to the nearest hundredths, and then rounded again to the nearest tenths. What is the probability that the final result is 0.8?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-d83b6cd2f67c8a138683cff2d2c8569eae1b81522cf18a84bf1ec1fe5a611353 | A real number is placed in each cell of a grid with 3 rows and 4 columns. The average of the numbers in each column is 2016, and the average of the numbers in each row is a constant x. Compute x. | [
"2016"
] | [
"A real number is placed in each cell of a grid with 3 rows and 4 columns. The average of the numbers in each column is 2016, and the average of the numbers in each row is a constant x. Compute x."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-72dcb53a5ea046dcce0f77c64fd9eedabc6cad6528ae4149291bbd858b388fa1 | A real number is placed in each square of an infinite chessboard. Numbers in the same row or column at a distance of 1982 apart are equal. Each number is the average of its two horizontal neighbors or its two vertical neighbors. Let $S$ be the set of all distinct numbers appearing on the chessboard. If the smallest ele... | [
"2975"
] | [
"A real number is placed in each square of an infinite chessboard. Numbers in the same row or column at a distance of 1982 apart are equal. Each number is the average of its two horizontal neighbors or its two vertical neighbors. Let $S$ be the set of all distinct numbers appearing on the chessboard. If the smalles... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-9e1fbbc8b381c8267ab35a2f8f34bcf9935b4b147fbf5a9fa9ea0651fb481ffc | A real number is written next to each vertex of a regular pentagon. All five numbers are different. A triple of vertices is called successful if they form an isosceles triangle for which the number written on the top vertex is either larger than both numbers written on the base vertices, or smaller than both. Find the ... | [
"8"
] | [
"A real number is written next to each vertex of a regular pentagon. All five numbers are different. A triple of vertices is called successful if they form an isosceles triangle for which the number written on the top vertex is either larger than both numbers written on the base vertices, or smaller than both. Find... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 2
} | 24 | 0 | 0 |
omr-ws-87b17596365244c9bcbe7af120d414982087ba3d28eaeb11a544e73e9ff549fa | A real number is written on each square of a $2024 \times 2024$ chessboard, with the sum of all numbers being $2024$. The board is covered by $1 \times 2$ or $2 \times 1$ dominos such that no square is covered by two different dominos. For each domino, Aslı deletes the two numbers covered by it and writes $0$ on one of... | [
"\\(\\frac{3}{2}\\)"
] | [
"A real number is written on each square of a $2024 \\times 2024$ chessboard, with the sum of all numbers being $2024$. The board is covered by $1 \\times 2$ or $2 \\times 1$ dominos such that no square is covered by two different dominos. For each domino, Aslı deletes the two numbers covered by it and writes $0$ o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-383e8b1bc26d665a9883b19233a1e59800f0633132dcd4fc8eb517b06fd553ff | A real number sequence \(a_1, \cdots, a_{2021}\) satisfies the conditions: \[ a_1 = 1, \quad a_2 = 2, \quad a_{n+2} = \frac{2a_{n+1}^2}{a_n + a_{n+1}} \quad (1 \leq n \leq 2019). \] Let the minimum of \(a_1, \cdots, a_{2021}\) be \(m\), and the maximum of \(a_1, \cdots, a_{2021}\) be \(M\). Consider the 2021 degree pol... | [
"no_valid_answer_found"
] | [
"A real number sequence \\(a_1, \\cdots, a_{2021}\\) satisfies the conditions:\n\\[ a_1 = 1, \\quad a_2 = 2, \\quad a_{n+2} = \\frac{2a_{n+1}^2}{a_n + a_{n+1}} \\quad (1 \\leq n \\leq 2019). \\]\nLet the minimum of \\(a_1, \\cdots, a_{2021}\\) be \\(m\\), and the maximum of \\(a_1, \\cdots, a_{2021}\\) be \\(M\\). ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2cd7383406ebb2a1560eaa106514c38a632d62dbc523a8128cd98f02c73b7849 | A real sequence \((a_n)_{n=0}^\infty\) is defined recursively by \(a_0 = 2\) and the recursion formula \[ a_{n} = \begin{dcases} a_{n-1}^2 & \text{if } a_{n-1} < \sqrt{3} \\ \frac{a_{n-1}^2}{3} & \text{if } a_{n-1} \geq \sqrt{3}. \end{dcases} \] Another real sequence \((b_n)_{n=1}^\infty\) is defined in terms of the fi... | [
"\\dfrac{161}{256}"
] | [
"A real sequence \\((a_n)_{n=0}^\\infty\\) is defined recursively by \\(a_0 = 2\\) and the recursion formula\n\\[\na_{n} = \n\\begin{dcases}\na_{n-1}^2 & \\text{if } a_{n-1} < \\sqrt{3} \\\\\n\\frac{a_{n-1}^2}{3} & \\text{if } a_{n-1} \\geq \\sqrt{3}.\n\\end{dcases}\n\\]\nAnother real sequence \\((b_n)_{n=1}^\\inft... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-7a1868a6b16895f69f5b49f9cf6116e47b5215e2376e36ab9e8cce5d9442741e | A real symmetric $2018 \times 2018$ matrix $A = (a_{ij})$ satisfies $|a_{ij} - 2018| \leq 1$ for every $1 \leq i, j \leq 2018$. Denote the largest eigenvalue of $A$ by $\lambda(A)$. Find the maximum and minimum value of $\lambda(A)$. | [
"Minimum value is \\(2018 \\cdot 2017\\), Maximum value is \\(2018 \\cdot 2019\\)."
] | [
"A real symmetric $2018 \\times 2018$ matrix $A = (a_{ij})$ satisfies $|a_{ij} - 2018| \\leq 1$ for every $1 \\leq i, j \\leq 2018$. Denote the largest eigenvalue of $A$ by $\\lambda(A)$. Find the maximum and minimum value of $\\lambda(A)$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-e04233afd6a997107b729d8775c52b9c012484374dd7a7d4e4aac119fdcb7cc1 | A real-valued function \( f \) defined on a closed interval \([a, b]\) satisfies \( f(a) = f(b) = 0 \) and \( f(x) = f'(x) + f''(x) \) for all \( x \) in \([a, b]\). If \( f \) is non-zero at some point in \([a, b]\), find the value of \( f(c) \) where \( c \) is the point in \([a, b]\) where \( f \) attains its maximu... | [
"0"
] | [
"A real-valued function \\( f \\) defined on a closed interval \\([a, b]\\) satisfies \\( f(a) = f(b) = 0 \\) and \\( f(x) = f'(x) + f''(x) \\) for all \\( x \\) in \\([a, b]\\). If \\( f \\) is non-zero at some point in \\([a, b]\\), find the value of \\( f(c) \\) where \\( c \\) is the point in \\([a, b]\\) where... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-378fa01e5191ac52a7ec9be8942e6fb93b5dc18297d38c7e10ae4eb7523aa1dc | A real-valued function \( f \) defined on the closed interval \([a, b]\) satisfies \( f(a) = f(b) = 0 \) and \( f(x) = f'(x) + f''(x) \) for all \( x \) in \([a, b]\). If \( f(x) \) is not identically zero, find the value of \( f\left(\frac{a+b}{2}\right) \). | [
"0"
] | [
"A real-valued function \\( f \\) defined on the closed interval \\([a, b]\\) satisfies \\( f(a) = f(b) = 0 \\) and \\( f(x) = f'(x) + f''(x) \\) for all \\( x \\) in \\([a, b]\\). If \\( f(x) \\) is not identically zero, find the value of \\( f\\left(\\frac{a+b}{2}\\right) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-09320a00056854fbe29b98a1b6d95cfd4d4eacd2b7dd67175da7ee6f382357f0 | A real-valued function \( f \) is defined on the interval \((-1, 2)\) and is differentiable. Given that \( f(0) > 0 \), \( f(1) = 1 \), and \( f'(1) > 1 \), find the number of fixed points of \( f \) in the interval \((0, 1)\). | [
"1"
] | [
"A real-valued function \\( f \\) is defined on the interval \\((-1, 2)\\) and is differentiable. Given that \\( f(0) > 0 \\), \\( f(1) = 1 \\), and \\( f'(1) > 1 \\), find the number of fixed points of \\( f \\) in the interval \\((0, 1)\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cac9ba4418fc8f9d50a42089559c7ba245c9db75512b20378be477bea7b5baac | A real-valued function \( f \) is defined on the interval \((-1, 2)\). Suppose \( f \) is differentiable, \( f(0) > 0 \), and \( f(1) = 1 \). Given that \( f'(1) > 1 \), find the value of \( c \) in the interval \((0, 1)\) such that \( f(c) = c \). If there are multiple such values, provide the smallest one. | [
"1",
"\\dfrac{1}{2}"
] | [
"A real-valued function \\( f \\) is defined on the interval \\((-1, 2)\\). Suppose \\( f \\) is differentiable, \\( f(0) > 0 \\), and \\( f(1) = 1 \\). Given that \\( f'(1) > 1 \\), find the value of \\( c \\) in the interval \\((0, 1)\\) such that \\( f(c) = c \\). If there are multiple such values, provide the s... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-d82a993d476dc07f5c10b2404f7035b5ddc1dadda38b2752800f5ca2f8a7ef69 | A real-valued function \( f \) on \( \mathbb{Q} \) satisfies the following conditions for arbitrary \( \alpha, \beta \in \mathbb{Q} \): (i) \( f(0) = 0 \), (ii) \( f(\alpha) > 0 \) if \( \alpha \neq 0 \), (iii) \( f(\alpha \cdot \beta) = f(\alpha)f(\beta) \), (iv) \( f(\alpha + \beta) \leq f(\alpha) + f(\beta) \), (v) ... | [
"1"
] | [
"A real-valued function \\( f \\) on \\( \\mathbb{Q} \\) satisfies the following conditions for arbitrary \\( \\alpha, \\beta \\in \\mathbb{Q} \\):\n\n(i) \\( f(0) = 0 \\),\n(ii) \\( f(\\alpha) > 0 \\) if \\( \\alpha \\neq 0 \\),\n(iii) \\( f(\\alpha \\cdot \\beta) = f(\\alpha)f(\\beta) \\),\n(iv) \\( f(\\alpha + \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-2044c335a8dd48126f1fe611b727147f5cb7a99bea18f78b41b0863c9a3889c9 | A real-valued function \( f \) satisfies for all reals \( x \) and \( y \) the equality \[ f(xy) = f(x)y + x f(y). \] Given \( f(1) = 2 \), find the value of \( f\left(\frac{3}{4}\right) \). | [
"\\dfrac{3}{2}"
] | [
"A real-valued function \\( f \\) satisfies for all reals \\( x \\) and \\( y \\) the equality\n\\[ f(xy) = f(x)y + x f(y). \\]\nGiven \\( f(1) = 2 \\), find the value of \\( f\\left(\\frac{3}{4}\\right) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-fc4a9a9807e91375f69467ebc31f97b08e81ce7290f46dbf318782637c04c567 | A real-valued function \( f \) satisfies the relation \( f(x^2 + x) + 2f(x^2 - 3x + 2) = 9x^2 - 15x \) for all real values of \( x \). Find \( f(2013) \). | [
"6035"
] | [
"A real-valued function \\( f \\) satisfies the relation \\( f(x^2 + x) + 2f(x^2 - 3x + 2) = 9x^2 - 15x \\) for all real values of \\( x \\). Find \\( f(2013) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-7082e166b19b22dfc1c1c58c5c705108c40408d044969d4aba759426b29da0ad | A recent study found that 60$\%$ of men and 80$\%$ of women surveyed support increased funding for particular medical research. The study surveyed 100 men and 900 women. What was the overall percent of the people surveyed who supported increased funding? | [
"78\\%"
] | [
"A recent study found that 60$\\%$ of men and 80$\\%$ of women surveyed support increased funding for particular medical research. The study surveyed 100 men and 900 women. What was the overall percent of the people surveyed who supported increased funding?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b1371e0761a320a485681f2c95149a58f7d907901f90e2250501130f52435b09 | A recipe calls for $4 \frac{1}{2}$ cups of flour. If you only make half of the recipe, then how many cups of flour do you need? Express your answer as a mixed number. | [
"2\\frac{1}{4}"
] | [
"A recipe calls for $4 \\frac{1}{2}$ cups of flour. If you only make half of the recipe, then how many cups of flour do you need? Express your answer as a mixed number."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ad1f0156b1b567f6f5752b919d9d6afb4fc9e0a463b6de7b3447f81769ef9ab | A recipe for 30 cookies requires two cups of flour among its ingredients. Eduardo wants to bake five dozen cookies. How many cups of flour will he need to use in his adjusted recipe? | [
"4"
] | [
"A recipe for 30 cookies requires two cups of flour among its ingredients. Eduardo wants to bake five dozen cookies. How many cups of flour will he need to use in his adjusted recipe?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aedb2da255ab4642204a5213c920a70ed7f7a81a96d9110530c57e940fd29e5b | A record makes 1.2 revolutions per second. Find the angular speed of the record in radians per second. | [
"\\( 2.4\\pi \\) radians/second"
] | [
"A record makes 1.2 revolutions per second. Find the angular speed of the record in radians per second."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-c55489547f1bd47c1a7cc3cf5e80ab181dd2def048ea765418ec7b83d7892e1a | A rectangle "R" has a width of 800 m and a height of 600 m. A smaller rectangle "r" is to be created within "R" with the maximum possible area, such that the ratio of its width \( w \) to its height \( h \) is 1.43. Find the dimensions \( w \) and \( h \) of "r". | [
"\\( w \\approx 748.712 \\) meters, \\( h \\approx 523.575 \\) meters"
] | [
"A rectangle \"R\" has a width of 800 m and a height of 600 m. A smaller rectangle \"r\" is to be created within \"R\" with the maximum possible area, such that the ratio of its width \\( w \\) to its height \\( h \\) is 1.43. Find the dimensions \\( w \\) and \\( h \\) of \"r\"."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-f486aae646a5b8ef154f59e448dd4c4d977a05e8d24b91a02ddcdc624cc79452 | A rectangle $ ABCD $, with sides $ AB = 4 $ cm and $ AD = 6 $ cm, is given. How to use a compass and a ruler to draw a straight line through the center of the rectangle $ O $, which will cut the rectangle into two parts, from which you can make a rhombus? | [
"\\text{Draw a line through the center } O \\text{ perpendicular to one of the rectangle's diagonals.}"
] | [
"A rectangle $ ABCD $, with sides $ AB = 4 $ cm and $ AD = 6 $ cm, is given. How to use a compass and a ruler to draw a straight line through the center of the rectangle $ O $, which will cut the rectangle into two parts, from which you can make a rhombus?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-5f1a22bf2e87288e7d13fc626c799bd915cfa43c5ee11047716e9b9dc8b27c7f | A rectangle $A BCD$ lies in a semicircle so that its side $AB$ lies on the diameter which bound the semicircle, and the vertices $C$ and $D$ lie on the arc which bounds the semicircle. The radius of the semicircle is 5 cm long. Find the lengths of the sides of the rectangle $A BCD$ if its area is 24 cm$^2$ and the leng... | [
"3 cm, 8 cm"
] | [
"A rectangle $A BCD$ lies in a semicircle so that its side $AB$ lies on the diameter which bound the semicircle, and the vertices $C$ and $D$ lie on the arc which bounds the semicircle. The radius of the semicircle is 5 cm long. Find the lengths of the sides of the rectangle $A BCD$ if its area is 24 cm$^2$ and the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-92a10a28d2361bd9ff77da094ffb05e16580b5fe100733cd7a16f7a029edc9f3 | A rectangle $ABCD$ has center $X$, $AB = 20$ and $BC = 19$. Point $E$ is on ray $BC$ such that $\angle DAE = 30^\circ$. Let $O$ be the center of the circle passing through $D$, $C$, and $E$. Find $OX$. | [
"\\(3\\sqrt{29}\\)"
] | [
"A rectangle $ABCD$ has center $X$, $AB = 20$ and $BC = 19$. Point $E$ is on ray $BC$ such that $\\angle DAE = 30^\\circ$. Let $O$ be the center of the circle passing through $D$, $C$, and $E$. Find $OX$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3730494633a37a57d665abd72bb0f786328f1dab36fb48033e1bd6724fba21b2 | A rectangle $ABCD$ has dimensions $AB = 6$ and $BC = 8$. A circle centered at the same point as the rectangle has a radius of $1$. A particle starts on the circle, travels to $AB$, then $BC$, then $CD$, then $DA$, and finally returns to the circle. Find the shortest possible path the particle can take. | [
"\\(20\\)"
] | [
"A rectangle $ABCD$ has dimensions $AB = 6$ and $BC = 8$. A circle centered at the same point as the rectangle has a radius of $1$. A particle starts on the circle, travels to $AB$, then $BC$, then $CD$, then $DA$, and finally returns to the circle. Find the shortest possible path the particle can take."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-e9aae3ae30d989b64b51f64915d43c7a952446544bbdde4cc228c0ee52e41bc5 | A rectangle $ABCD$ has point $E$ on $AB$ and point $F$ on $BC$. If $\triangle ACE$, $\triangle EBF$, and $\triangle FCD$ each have an area of $1$, find the area of $\triangle CEF$. | [
"\\(\\frac{1}{2}(\\sqrt{5} - 1)\\)"
] | [
"A rectangle $ABCD$ has point $E$ on $AB$ and point $F$ on $BC$. If $\\triangle ACE$, $\\triangle EBF$, and $\\triangle FCD$ each have an area of $1$, find the area of $\\triangle CEF$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-342b8bd7e246d499c9c01ca6df8b10fb98fe2a067312d5c919e237558eacddf2 | A rectangle $ABCD$ has side lengths $AB = m$ and $AD = n$, where $m$ and $n$ are relatively prime and both odd. The rectangle is divided into unit squares, and the diagonal $AC$ intersects the sides of the unit squares at points $A_1 = A, A_2, A_3, \ldots, A_k = C$. Calculate the value of $A_1A_2 - A_2A_3 + A_3A_4 - \l... | [
"\\dfrac{\\sqrt{74}}{35}"
] | [
"A rectangle $ABCD$ has side lengths $AB = m$ and $AD = n$, where $m$ and $n$ are relatively prime and both odd. The rectangle is divided into unit squares, and the diagonal $AC$ intersects the sides of the unit squares at points $A_1 = A, A_2, A_3, \\ldots, A_k = C$. Calculate the value of $A_1A_2 - A_2A_3 + A_3A_... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-30994c01507e8f90320da7e876b3bbc463e9fcc93d51ad050752cf9bd267ad31 | A rectangle $ABCD$ is cut from a piece of paper and folded along a straight line so that the diagonally opposite vertices $A$ and $C$ coincide. Find the length of the resulting crease in terms of the length ($\ell$) and width ($w$) of the rectangle. | [
"\\(\\frac{w}{\\ell} \\sqrt{\\ell^2 + w^2}\\)"
] | [
"A rectangle $ABCD$ is cut from a piece of paper and folded along a straight line so that the diagonally opposite vertices $A$ and $C$ coincide. Find the length of the resulting crease in terms of the length ($\\ell$) and width ($w$) of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 1
} | 1 | 0 | 0 |
omr-ws-e1f7b84c25dd1d981d93c09be8f98573c558699468fd1444082edca47be06ea3 | A rectangle $ABCD$ is divided into two rectangles by a line parallel to side $AB$, resulting in areas in the ratio $1:2$ and perimeters in the ratio $3:5$. If the rectangle is instead divided by a line parallel to side $BC$ into two parts with areas in the ratio $1:2$, what is the ratio of their perimeters? | [
"\\(\\frac{19}{20}\\)"
] | [
"A rectangle $ABCD$ is divided into two rectangles by a line parallel to side $AB$, resulting in areas in the ratio $1:2$ and perimeters in the ratio $3:5$. If the rectangle is instead divided by a line parallel to side $BC$ into two parts with areas in the ratio $1:2$, what is the ratio of their perimeters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-ae6ec8c16ded7abd2e6904ae46a84da2318a6ee5da335eeea1f68c297491d9a7 | A rectangle $ABCD$ is given. On the side $AB$, $n$ different points are chosen strictly between $A$ and $B$. Similarly, $m$ different points are chosen on the side $AD$ between $A$ and $D$. Lines are drawn from these points parallel to the sides. How many rectangles are formed in this way? | [
"\\(\\frac{(n+2)(n+1)(m+2)(m+1)}{4}\\)"
] | [
"A rectangle $ABCD$ is given. On the side $AB$, $n$ different points are chosen strictly between $A$ and $B$. Similarly, $m$ different points are chosen on the side $AD$ between $A$ and $D$. Lines are drawn from these points parallel to the sides. How many rectangles are formed in this way?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 11
} | 24 | 0 | 0 |
omr-ws-3fd6f45b61ada5586f4b1cdb05f0ee402a55e33b7995eecced9c3cfe205ef03c | A rectangle $ABCD$ is inscribed in a circle with centre $O$. The exterior bisectors of $\angle ABD$ and $\angle ADB$ intersect at $P$; those of $\angle DAB$ and $\angle DBA$ intersect at $Q$; those of $\angle ACD$ and $\angle ADC$ intersect at $R$; and those of $\angle DAC$ and $\angle DCA$ intersect at $S$. If the rad... | [
"r(\\sqrt{2} - 1)"
] | [
"A rectangle $ABCD$ is inscribed in a circle with centre $O$. The exterior bisectors of $\\angle ABD$ and $\\angle ADB$ intersect at $P$; those of $\\angle DAB$ and $\\angle DBA$ intersect at $Q$; those of $\\angle ACD$ and $\\angle ADC$ intersect at $R$; and those of $\\angle DAC$ and $\\angle DCA$ intersect at $S... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-96fce4e2fbc1516b65bfe8455d6560f2a91a3771c2ccb49b10c151318669fb20 | A rectangle $ABCD$ with $AB$ and $DC$ horizontal is folded in half along $EF$, where $E$ and $F$ are the midpoints of $DC$ and $AB$, respectively. It is also folded along $DB$ and $CF$. If $DB$ and $CF$ meet at right angles, what does this imply about the rectangle? | [
"\\( a = \\sqrt{2} \\)"
] | [
"A rectangle $ABCD$ with $AB$ and $DC$ horizontal is folded in half along $EF$, where $E$ and $F$ are the midpoints of $DC$ and $AB$, respectively. It is also folded along $DB$ and $CF$. If $DB$ and $CF$ meet at right angles, what does this imply about the rectangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 27
} | 74 | 0 | 0 |
omr-ws-07029ad62376c23f24b30e963cc3928f527c83593de199aa6c8cfb0e1467d58d | A rectangle $ABCD$ with sides $AB = 7$ and $BC = 2$ is folded along the diagonal $BD$ so that point $A$ is transformed to another point $A'$ on the plane. Let $E$ be the intersection of $A'B$ and $CD$. What is the length of $A'E$? | [
"\\(\\frac{45}{14}\\)"
] | [
"A rectangle $ABCD$ with sides $AB = 7$ and $BC = 2$ is folded along the diagonal $BD$ so that point $A$ is transformed to another point $A'$ on the plane. Let $E$ be the intersection of $A'B$ and $CD$. What is the length of $A'E$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-88c62e26340ebc1e372819977109ab61cba37144a44620bccb4df301802035a0 | A rectangle $HOMF$ has sides $HO=11$ and $OM=5$. A triangle $\Delta ABC$ has $H$ as the orthocenter, $O$ as the circumcenter, $M$ as the midpoint of $BC$, and $F$ as the foot of the altitude from $A$. What is the length of $BC$? | [
"28"
] | [
"A rectangle $HOMF$ has sides $HO=11$ and $OM=5$. A triangle $\\Delta ABC$ has $H$ as the orthocenter, $O$ as the circumcenter, $M$ as the midpoint of $BC$, and $F$ as the foot of the altitude from $A$. What is the length of $BC$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-e11916613fab02d88b2d032aadccc111457fa7b002d2d2c931425515b6f8b2e5 | A rectangle $KMPT$ lies in a circular sector $OAB$ whose central angle is equal to $\pi /4$. The side $KM$ of the rectangle lies on the radius $OA$, the vertex $P$ lies on the arc $AB$, and the vertex $T$ lies on the radius $OB$. The side $KT$ is 3 cm longer than the side $KM$. The area of the rectangle $KMPT$ is 18 cm... | [
"$3\\sqrt {13}$"
] | [
"A rectangle $KMPT$ lies in a circular sector $OAB$ whose central angle is equal to $\\pi /4$. The side $KM$ of the rectangle lies on the radius $OA$, the vertex $P$ lies on the arc $AB$, and the vertex $T$ lies on the radius $OB$. The side $KT$ is 3 cm longer than the side $KM$. The area of the rectangle $KMPT$ is... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-476fa5f2c0639d440491c80445c3ec399dfada96a3217a9d398a653297b54ac4 | A rectangle $MNKB$ is inscribed in an isosceles right triangle $ABC$ with the right vertex angle $B$ so that two sides $MB$ and $KB$ of the rectangle lie on the legs and the vertex $N$ lies on the hypotenuse $AC$. In what ratio must the point $N$ divide the hypotenuse for the area of the rectangle to constitute 18\% of... | [
"$\\frac{|AN|}{|NC|}=\\frac{1}{9}$ or $=9$"
] | [
"A rectangle $MNKB$ is inscribed in an isosceles right triangle $ABC$ with the right vertex angle $B$ so that two sides $MB$ and $KB$ of the rectangle lie on the legs and the vertex $N$ lies on the hypotenuse $AC$. In what ratio must the point $N$ divide the hypotenuse for the area of the rectangle to constitute 18... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-52422a8bcaf49dbe9ba95f6b69245ff2d7eb7532c59290a04fe359f1f0bdfec6 | A rectangle $OPQR$ is given. Its sides $\overline{PQ}$ and $\overline{RO}$ are each divided by $n$ points into $n+1$ equal lengths, and the side $QR$ is divided by $m$ points into $m+1$ equal lengths. The points on $\overline{PQ}$ are denoted by $A_1, A_2, \ldots, A_n$, those on $\overline{QR}$ by $B_1, B_2, \ldots, B_... | [
"n^2 + mn + \\frac{m(m-1)}{2} + 2n + 1"
] | [
"A rectangle $OPQR$ is given. Its sides $\\overline{PQ}$ and $\\overline{RO}$ are each divided by $n$ points into $n+1$ equal lengths, and the side $QR$ is divided by $m$ points into $m+1$ equal lengths. The points on $\\overline{PQ}$ are denoted by $A_1, A_2, \\ldots, A_n$, those on $\\overline{QR}$ by $B_1, B_2, ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-56636ea4a6071d2fe3b6a2654c74b68882f2fcb2cf7ca3642392c5a107400072 | A rectangle $PQRS$ has dimensions $PQ = 49$ and $PS = 100$. The rectangle is divided into $4900$ squares of side length $1$. Point $T$ is on $QR$ such that $QT = 60$. How many of these $4900$ squares are cut by the lines $PT$ and $TS$? | [
"196"
] | [
"A rectangle $PQRS$ has dimensions $PQ = 49$ and $PS = 100$. The rectangle is divided into $4900$ squares of side length $1$. Point $T$ is on $QR$ such that $QT = 60$. How many of these $4900$ squares are cut by the lines $PT$ and $TS$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 9
} | 27 | 0 | 0 |
omr-ws-53f5cdb4cfaf3a1e81bac8fe6d987dc2c4a3d05612dee3b24d83f898c586b7c1 | A rectangle $R$ is partitioned into smaller rectangles with sides parallel to $R$, where each smaller rectangle has at least one side of integer length. If the dimensions of $R$ are $a$ and $b$, find the minimum possible value of $a+b$ such that $R$ does not have any side of integer length. | [
"3"
] | [
"A rectangle $R$ is partitioned into smaller rectangles with sides parallel to $R$, where each smaller rectangle has at least one side of integer length. If the dimensions of $R$ are $a$ and $b$, find the minimum possible value of $a+b$ such that $R$ does not have any side of integer length."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-a01b00aabf1779005ed21318d8def5b6187a7d471beeb96271573739e4502d1c | A rectangle $R$ with integer side lengths has a perimeter $P$ units and an area of $2P$ square units. Find the sum of all possible values of $P$. | [
"17"
] | [
"A rectangle $R$ with integer side lengths has a perimeter $P$ units and an area of $2P$ square units. Find the sum of all possible values of $P$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-90177af982ed39200c423ffdea45905824ae441350df24298e86bd0a07881bc1 | A rectangle $\mathcal{R}$ with odd integer side lengths is divided into smaller rectangles with integer side lengths. Let $N$ be the number of smaller rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. If $\mathcal{R}$ has dimensions $15 \times 21$, find the minimum possible... | [
"1"
] | [
"A rectangle $\\mathcal{R}$ with odd integer side lengths is divided into smaller rectangles with integer side lengths. Let $N$ be the number of smaller rectangles whose distances from the four sides of $\\mathcal{R}$ are either all odd or all even. If $\\mathcal{R}$ has dimensions $15 \\times 21$, find the minimum... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-1c66f069e9022b4b4c3bd6906fccb767d88d378089a8f66027e1145b5b43b12e | A rectangle $m \times n$ is painted with two colors (black and white). Let $N$ be the number of colorations such that there is a path of black squares joining the right side to the left side of the rectangle (a path is a sequence of adjacent squares). Let $M$ be the number of colorations with at least two disjoint path... | [
"\\dfrac{1}{2}"
] | [
"A rectangle $m \\times n$ is painted with two colors (black and white). Let $N$ be the number of colorations such that there is a path of black squares joining the right side to the left side of the rectangle (a path is a sequence of adjacent squares). Let $M$ be the number of colorations with at least two disjoin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-97ae7964272703517fc76de3c097166ce60f21594335324895dfe5b93b462bae | A rectangle 19 cm × 65 cm is divided by straight lines parallel to its sides into squares with side 1 cm. How many parts will this rectangle be divided into if you also draw a diagonal in it? | [
"84"
] | [
"A rectangle 19 cm × 65 cm is divided by straight lines parallel to its sides into squares with side 1 cm. How many parts will this rectangle be divided into if you also draw a diagonal in it?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9187d42cdad7b11c8252791fd6a133956f3a083277aaaf1d2775711e9b7cc8be | A rectangle ABCD has the equation of line AB as $x-2y-1=0$ and the equation of line BD as $x-7y+14=0$. Line AC passes through the point $M(2,1)$. Find the coordinates of points A, B, C, and D. | [
"\\( A(1,0), B(7,3), C(6,5), D(0,2) \\)"
] | [
"A rectangle ABCD has the equation of line AB as $x-2y-1=0$ and the equation of line BD as $x-7y+14=0$. Line AC passes through the point $M(2,1)$. Find the coordinates of points A, B, C, and D."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 9
} | 21 | 0 | 0 |
omr-ws-b63c85ec8b16d9fa23a582f927bc3d9c5578a37f196496be757da9eef4cdd55d | A rectangle MNOQ is drawn, with NO extended to R, forming triangle PQR. Given that QP = \(\frac{2}{3} QM\), \(\angle ORP = 45^\circ\), QR = \(4\sqrt{7}\), and S and T are the midpoints of QR and PR respectively with ST = 6, find the area of rectangle MNOQ. | [
"108 + 36\\sqrt{5}"
] | [
"A rectangle MNOQ is drawn, with NO extended to R, forming triangle PQR. Given that QP = \\(\\frac{2}{3} QM\\), \\(\\angle ORP = 45^\\circ\\), QR = \\(4\\sqrt{7}\\), and S and T are the midpoints of QR and PR respectively with ST = 6, find the area of rectangle MNOQ."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-966b5ee5bbdfcedba713e25949bcd349014d7779c0ffe1107ad3ce941d3d2599 | A rectangle OACB is drawn with the two axes as two sides and the origin O as a vertex. The length OA is four times the width OB. A circle is drawn passing through points B and C and touching OA at its midpoint, dividing the rectangle into three parts. Find the ratio of the areas of these three parts. | [
"6:7:7"
] | [
"A rectangle OACB is drawn with the two axes as two sides and the origin O as a vertex. The length OA is four times the width OB. A circle is drawn passing through points B and C and touching OA at its midpoint, dividing the rectangle into three parts. Find the ratio of the areas of these three parts."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-0af38f3cb657c37272b8fa1ab3e0a100c1f0339fb8f4d34927612b19c87b4fe6 | A rectangle \( D \) is partitioned into several (\(\ge 2\)) smaller rectangles with sides parallel to those of \( D \). Given that any line parallel to one of the sides of \( D \) and having common points with the interior of \( D \) also has common interior points with the interior of at least one of the smaller recta... | [
"2"
] | [
"A rectangle \\( D \\) is partitioned into several (\\(\\ge 2\\)) smaller rectangles with sides parallel to those of \\( D \\). Given that any line parallel to one of the sides of \\( D \\) and having common points with the interior of \\( D \\) also has common interior points with the interior of at least one of t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-04f1eca9327069a5e6ebbc024b3d90b8d1fe33146a00239fa36fc2f059ecc433 | A rectangle \( R \) is partitioned into smaller rectangles whose sides are parallel with the sides of \( R \). Let \( B \) be the set of all boundary points of all the rectangles in the partition, including the boundary of \( R \). Let \( S \) be the set of all (closed) segments whose points belong to \( B \). Let a ma... | [
"11"
] | [
"A rectangle \\( R \\) is partitioned into smaller rectangles whose sides are parallel with the sides of \\( R \\). Let \\( B \\) be the set of all boundary points of all the rectangles in the partition, including the boundary of \\( R \\). Let \\( S \\) be the set of all (closed) segments whose points belong to \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-2a59c527c5d9a78968b7a5eebc10ba98c078281ea228cc4df26004c3cfdf237d | A rectangle \(ABCD\) has dimensions \(4\sqrt{3} \times 4\) meters. Two quarter circles with radius 4 meters are drawn, centered at \(C\) and \(D\) respectively. Find the area of the overlapping region of the two quarter circles. | [
"\\(\\frac{16\\pi}{3} - 4\\sqrt{3}\\)"
] | [
"A rectangle \\(ABCD\\) has dimensions \\(4\\sqrt{3} \\times 4\\) meters. Two quarter circles with radius 4 meters are drawn, centered at \\(C\\) and \\(D\\) respectively. Find the area of the overlapping region of the two quarter circles."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-5825800a4c430ac208b8a986dbffbb5150ef50361422cc1ec652664287462632 | A rectangle \(ABCD\) has side lengths \(AB = m\) and \(AD = n\), where \(m\) and \(n\) are relatively prime and both odd. The rectangle is divided into unit squares, and the diagonal \(AC\) intersects the sides of the unit squares at points \(A_1 = A, A_2, A_3, \ldots, A_k = C\). Calculate the value of \[ A_1A_2 - A_2A... | [
"\\dfrac{\\sqrt{74}}{35}"
] | [
"A rectangle \\(ABCD\\) has side lengths \\(AB = m\\) and \\(AD = n\\), where \\(m\\) and \\(n\\) are relatively prime and both odd. The rectangle is divided into unit squares, and the diagonal \\(AC\\) intersects the sides of the unit squares at points \\(A_1 = A, A_2, A_3, \\ldots, A_k = C\\). Calculate the value... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-a1ebd0fdcc8ca30c0cad8aac1b54c5d7c7cfbd7762f771ea15e8cf3c86379ea1 | A rectangle \(ABCD\) is folded along \(MN\) so that points \(B\) and \(D\) coincide. It turned out that \(AD = AC'\). Find the ratio of the rectangle’s sides. | [
"\\(\\sqrt{3}:1\\)"
] | [
"A rectangle \\(ABCD\\) is folded along \\(MN\\) so that points \\(B\\) and \\(D\\) coincide. It turned out that \\(AD = AC'\\). Find the ratio of the rectangle’s sides."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 18
} | 36 | 0 | 0 |
omr-ws-7a4196804e01ebdf3976e50916b6150981735aad60b01f7cac80770e6c37ef46 | A rectangle \(ABCD\) is twice as wide as it is high. The rectangle \(EFGH\) is inscribed in \(ABCD\) such that \(E\) is the midpoint of \(AD\) and \(G\) is the midpoint of \(BC\). What is the size of angle \(DFE\)? | [
"45^\\circ"
] | [
"A rectangle \\(ABCD\\) is twice as wide as it is high. The rectangle \\(EFGH\\) is inscribed in \\(ABCD\\) such that \\(E\\) is the midpoint of \\(AD\\) and \\(G\\) is the midpoint of \\(BC\\). What is the size of angle \\(DFE\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-f3b28ccf2faa39755edc96537e4b2dec31adeef1768912b3e9c9aea1dba3ba7d | A rectangle \(ABCD\) lies on a semicircle \(E\) of radius \(r\) such that \(AB\) is on the diameter of the semicircle and the points \(C\) and \(D\) lie on the circumference. Determine the maximum perimeter of \(ABCD\). | [
"\\( 2r\\sqrt{5} \\)"
] | [
"A rectangle \\(ABCD\\) lies on a semicircle \\(E\\) of radius \\(r\\) such that \\(AB\\) is on the diameter of the semicircle and the points \\(C\\) and \\(D\\) lie on the circumference. Determine the maximum perimeter of \\(ABCD\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-b9e5e85857414eb9f448ccfce473803a38fdf463e25d4a9a752972d862be0ff0 | A rectangle \(ABCD\) with \(AB = 12\) and \(BC = 16\) is the base of a pyramid \(P\) with height 24. A plane parallel to \(ABCD\) divides \(P\) into a frustum \(F\) and a smaller pyramid \(P'\). If the volume of \(P\) is eight times that of \(P'\), find the value of \(XT\), where \(X\) is the center of the circum-spher... | [
"177"
] | [
"A rectangle \\(ABCD\\) with \\(AB = 12\\) and \\(BC = 16\\) is the base of a pyramid \\(P\\) with height 24. A plane parallel to \\(ABCD\\) divides \\(P\\) into a frustum \\(F\\) and a smaller pyramid \\(P'\\). If the volume of \\(P\\) is eight times that of \\(P'\\), find the value of \\(XT\\), where \\(X\\) is t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 7,
"tir": 8
} | 61 | 0 | 0 |
omr-ws-287099c166e631b47236a0426d73ca22cab4e85b64229cac491a71f01714dbb8 | A rectangle \(OACB\) is drawn with the two axes as two sides and the origin \(O\) as a vertex. The length \(OA\) is four times the width \(OB\). A circle is drawn passing through points \(B\) and \(C\) and touching \(OA\) at its midpoint. This circle divides the rectangle into three parts. Find the ratio of the areas o... | [
"7:6:7"
] | [
"A rectangle \\(OACB\\) is drawn with the two axes as two sides and the origin \\(O\\) as a vertex. The length \\(OA\\) is four times the width \\(OB\\). A circle is drawn passing through points \\(B\\) and \\(C\\) and touching \\(OA\\) at its midpoint. This circle divides the rectangle into three parts. Find the r... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-5f93f97235a65a65a4f78bf3fd066c6ff5a1a356ed964ac5b4d3053139e50f52 | A rectangle \(\mathcal{R}\) with odd integer side lengths is divided into small rectangles with integer side lengths. Let \(N\) be the number of small rectangles whose distances from the four sides of \(\mathcal{R}\) are either all odd or all even. Calculate the minimum possible value of \(N\). | [
"1"
] | [
"A rectangle \\(\\mathcal{R}\\) with odd integer side lengths is divided into small rectangles with integer side lengths. Let \\(N\\) be the number of small rectangles whose distances from the four sides of \\(\\mathcal{R}\\) are either all odd or all even. Calculate the minimum possible value of \\(N\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b034eb26bed3be2a215434e16512d7fdd77cd29c999f29fb9d7c8e217979f46 | A rectangle and an equilateral triangle with equal area are inscribed in a circle with radius $2$. Find the dimensions of the rectangle. | [
"\\( l = \\frac{3\\sqrt{3}}{\\sqrt{8 \\pm \\sqrt{37}}}, w = \\sqrt{8 \\pm \\sqrt{37}} \\)"
] | [
"A rectangle and an equilateral triangle with equal area are inscribed in a circle with radius $2$. Find the dimensions of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e34d8adf8a9b42ee4580b536d03eee82aa862b15bd703961ee9d7ff95cf8677 | A rectangle can be divided by parallel lines to its sides into 200 congruent squares and also into 288 congruent squares. Find the smallest possible number of congruent squares into which the rectangle can be divided, given that this number is greater than 288. | [
"338"
] | [
"A rectangle can be divided by parallel lines to its sides into 200 congruent squares and also into 288 congruent squares. Find the smallest possible number of congruent squares into which the rectangle can be divided, given that this number is greater than 288."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-8cf139f2e783e83e0cf58c4445b973386bb4051c94d4bdadbc29bcb3f06dd742 | A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$. | [
"324"
] | [
"A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-ce2e90b354e87b19bef448beccab8d55b62a6e8454c087f00a1cd6a56047eeff | A rectangle can be divided into \( n \) equal squares and into \( n + 98 \) equal squares. The area of the rectangle is \( n \). Find all possible values for the sides of the rectangle. | [
"\\( (48, 24) \\) and \\( (42, 3) \\)"
] | [
"A rectangle can be divided into \\( n \\) equal squares and into \\( n + 98 \\) equal squares. The area of the rectangle is \\( n \\). Find all possible values for the sides of the rectangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e6e5bb6f40c5d0e8e41c41edbd238962a85a85f031ddafd3f3139d5a003e0b5f | A rectangle contains 4 unit circles, each tangent to the adjacent sides of a different vertex. A larger circle with radius 4 is also inside the rectangle and is tangent to all 4 unit circles. How many different rectangles with unique areas can be formed under these conditions? | [
"1"
] | [
"A rectangle contains 4 unit circles, each tangent to the adjacent sides of a different vertex. A larger circle with radius 4 is also inside the rectangle and is tangent to all 4 unit circles. How many different rectangles with unique areas can be formed under these conditions?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-9954605d5bfb03c0f1ccfe97af21461b4cdac8a45f9c14ad47ced3aff7768077 | A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles with a base of two units, squares with a unit side, and parallelograms formed by two sides and two diagonals of unit squares (figures may be oriented in any way). If the rectangle has dimensions $10 \tim... | [
"2"
] | [
"A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles with a base of two units, squares with a unit side, and parallelograms formed by two sides and two diagonals of unit squares (figures may be oriented in any way). If the rectangle has dimensions $10... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-78da560e1991505b33b5544168e23bad222d15b1739d7aaea1cd140d92185c06 | A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles with a base of two units, squares with a unit side, and parallelograms formed by two sides and two diagonals of unit squares (figures may be oriented in any way). If the total area of the rectangle is 10... | [
"1"
] | [
"A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles with a base of two units, squares with a unit side, and parallelograms formed by two sides and two diagonals of unit squares (figures may be oriented in any way). If the total area of the rectangle ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-8df952b40bd746fca0506be415cda173b8e1c8201baae4a136f4d7c9b5ccefcd | A rectangle grid with integer side lengths greater than 1 is given. Smaller rectangles with odd integer areas and integer side lengths greater than 1 are cut out one by one. Finally, one single unit square is left. Find the least possible area of the initial grid before the cuttings. | [
"25"
] | [
"A rectangle grid with integer side lengths greater than 1 is given. Smaller rectangles with odd integer areas and integer side lengths greater than 1 are cut out one by one. Finally, one single unit square is left. Find the least possible area of the initial grid before the cuttings."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-448ba1ad4f4d28d0cd2473321675b9bd4ae31acb4141d528fa5f7939d6937362 | A rectangle has a length and width in the ratio 3:2. It is cut into 24 congruent squares. The sum of the perimeters of these squares is 240 cm. What is the perimeter of the original rectangle? | [
"50"
] | [
"A rectangle has a length and width in the ratio 3:2. It is cut into 24 congruent squares. The sum of the perimeters of these squares is 240 cm. What is the perimeter of the original rectangle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-1c51ef31dd5a7b36834e6a352fceb52267dcb712a7a48ea217fa6d51bccd5d4f | A rectangle has a length of 4 and a width of 2. Semicircles are drawn inward on each length of the rectangle. Find the area of the region inside both semicircles. | [
"\\(\\frac{8\\pi}{3} - 2\\sqrt{3}\\)"
] | [
"A rectangle has a length of 4 and a width of 2. Semicircles are drawn inward on each length of the rectangle. Find the area of the region inside both semicircles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-cf780ec5240d90a3d8c851bc095ffbec7bc3eb9b156735c533f644c1e4f6ff86 | A rectangle has a length that is 7.5 times its width. The rectangle is lined with 2D mirrors. A laser beam is released from one corner at an angle of 45 degrees with each side of the rectangle. Let \( r \) be the ratio of the length of the path traced by the laser beam (until it hits a corner) to the perimeter of the r... | [
"739"
] | [
"A rectangle has a length that is 7.5 times its width. The rectangle is lined with 2D mirrors. A laser beam is released from one corner at an angle of 45 degrees with each side of the rectangle. Let \\( r \\) be the ratio of the length of the path traced by the laser beam (until it hits a corner) to the perimeter o... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-85b6a50102882de8d74062b78d3767fa015bbcef33797e0ecd823b9e94d47e61 | A rectangle has a perimeter of 15 cm. What is its maximum volume if the rectangle starts spinning around one of its sides to form a cylinder? Note: The problem involves finding the maximum volume of a cylinder formed by rotating a rectangle with a given perimeter around one of its sides. | [
"196"
] | [
"A rectangle has a perimeter of 15 cm. What is its maximum volume if the rectangle starts spinning around one of its sides to form a cylinder?\n\nNote: The problem involves finding the maximum volume of a cylinder formed by rotating a rectangle with a given perimeter around one of its sides."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-be5f8f887a2e7b910fe01fe94b3c6bd4dca0a29025a5995b82e3f7df21ca7590 | A rectangle has a perimeter of 30 units and its dimensions are whole numbers. What is the maximum possible area of the rectangle in square units? | [
"56"
] | [
"A rectangle has a perimeter of 30 units and its dimensions are whole numbers. What is the maximum possible area of the rectangle in square units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-792051afd4ba8ef1c97fd4200ee5ec62545c011abe862eca003dd28909c88232 | A rectangle has a perimeter of 46 cm and a diagonal of 17 cm. Find the lengths of the sides of the rectangle. | [
"8 cm and 15 cm"
] | [
"A rectangle has a perimeter of 46 cm and a diagonal of 17 cm. Find the lengths of the sides of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 32
} | 33 | 0 | 0 |
omr-ws-27f920713386cf4cf06ac337a85118064d2448988ef076ab0399fc272e28c217 | A rectangle has a perimeter of 62 cm and an area of 240 cm². Find the lengths of its sides. | [
"15 cm and 16 cm"
] | [
"A rectangle has a perimeter of 62 cm and an area of 240 cm². Find the lengths of its sides."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 22
} | 27 | 0 | 0 |
omr-ws-742c0e7896378cf25b853447534ab2ffc4c80e0ffed1a6f81fe08d034f252f56 | A rectangle has a perimeter of 64 inches and each side has an integer length. How many non-congruent rectangles meet these criteria? | [
"16"
] | [
"A rectangle has a perimeter of 64 inches and each side has an integer length. How many non-congruent rectangles meet these criteria?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ac3029ed85054d3cfdd5a57ec510730cab16d8bf4307f1c86acd0bdc48aa1a9e | A rectangle has a width \( a \). One vertex is folded over to touch the opposite edge of the rectangle. Find the minimum length of the crease. | [
"\\(\\frac{3\\sqrt{3}a}{4}\\)"
] | [
"A rectangle has a width \\( a \\). One vertex is folded over to touch the opposite edge of the rectangle. Find the minimum length of the crease."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 1
} | 10 | 0 | 0 |
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