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-172a9441acdb9dd2a6310e501b7d1cf698d350c9e9796ccab9e773e372726cca | Two disjoint circles are inscribed in an angle. They touch one side of the corner at points $K$ and $L$, the other at points $M$ and $N$. $C$ is the midpoint of the segment $KL$, and $A$ and $B$ are the points of intersection of the segments $CM$ and $CN$ with the circles. If the radius of the circle touching at $K$ an... | [
"1"
] | [
"Two disjoint circles are inscribed in an angle. They touch one side of the corner at points $K$ and $L$, the other at points $M$ and $N$. $C$ is the midpoint of the segment $KL$, and $A$ and $B$ are the points of intersection of the segments $CM$ and $CN$ with the circles. If the radius of the circle touching at $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b597159223cb46b80fe79e8fb3f2abba01c089eb14ed492010014e0cd175ca7 | Two disjoint circles with radii $r_1$ and $r_2$ are arranged such that one of their common internal tangents is perpendicular to one of their common external tangents. Find the area of the triangle formed by these tangents and the third common tangent of the given circles. | [
"\\(\\frac{r_{1}r_{2}(r_{1}+r_{2})}{r_{1}-r_{2}}\\) or \\(\\frac{r_{1}r_{2}(r_{1}-r_{2})}{r_{1}+r_{2}}\\)"
] | [
"Two disjoint circles with radii $r_1$ and $r_2$ are arranged such that one of their common internal tangents is perpendicular to one of their common external tangents. Find the area of the triangle formed by these tangents and the third common tangent of the given circles."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-29e9dfba177431dd21542511e16229d19dc1551a0a7c09e3b2ad88be79d0d1a2 | Two disjoint squares are located inside a square of side length 1. If the lengths of the sides of the two squares are \(a\) and \(b\), find the maximum possible value of \(a + b\). | [
"1"
] | [
"Two disjoint squares are located inside a square of side length 1. If the lengths of the sides of the two squares are \\(a\\) and \\(b\\), find the maximum possible value of \\(a + b\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-5140a2f63ed5cee1592408fca57807839b34761fdd1b66f5b867a517155bba27 | Two disks $A$ and $B$ are divided into $2n$ equal sectors. Disk $A$ has $n$ sectors painted blue and $n$ sectors painted red. Disk $B$ has its sectors painted blue or red arbitrarily. Find the maximum number of sectors that can always be guaranteed to have the same color when the two disks are optimally overlapped. | [
"n"
] | [
"Two disks $A$ and $B$ are divided into $2n$ equal sectors. Disk $A$ has $n$ sectors painted blue and $n$ sectors painted red. Disk $B$ has its sectors painted blue or red arbitrarily. Find the maximum number of sectors that can always be guaranteed to have the same color when the two disks are optimally overlapped... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-37d8726d1536a8e5e226aea6f23212ee8e91f85438e9a8b8622a696e445e8f63 | Two disks of radius 1 are drawn so that each disk's circumference passes through the center of the other disk. What is the circumference of the region in which they overlap? | [
"\\(\\frac{4\\pi}{3}\\)"
] | [
"Two disks of radius 1 are drawn so that each disk's circumference passes through the center of the other disk. What is the circumference of the region in which they overlap?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-aae772171ae21162cb87ffde528766c7a7c1a934a915d600463f55c09c6d4b64 | Two distinct 2-digit prime numbers \( p \) and \( q \) can be written one after the other in two different ways to form two 4-digit numbers. If the two 4-digit numbers formed are both divisible by the average value of \( p \) and \( q \), find all possible pairs \(\{p, q\}\). | [
"\\{13, 53\\}, \\{19, 47\\}, \\{23, 43\\}, \\{29, 37\\}"
] | [
"Two distinct 2-digit prime numbers \\( p \\) and \\( q \\) can be written one after the other in two different ways to form two 4-digit numbers. If the two 4-digit numbers formed are both divisible by the average value of \\( p \\) and \\( q \\), find all possible pairs \\(\\{p, q\\}\\)."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 28
} | 30 | 0 | 0 |
omr-ws-9466093cb92378427e2ccc03dfe01b707e389db250fdf2545382532a8e673b09 | Two distinct circles $K_1$ and $K_2$ are drawn in the plane. They intersect at points $A$ and $B$, where $AB$ is the diameter of $K_1$. A point $P$ on $K_2$ and inside $K_1$ is also given. Using only a "T-square" (i.e. an instrument which can produce a straight line joining two points and the perpendicular to a line th... | [
"CD"
] | [
"Two distinct circles $K_1$ and $K_2$ are drawn in the plane. They intersect at points $A$ and $B$, where $AB$ is the diameter of $K_1$. A point $P$ on $K_2$ and inside $K_1$ is also given. Using only a \"T-square\" (i.e. an instrument which can produce a straight line joining two points and the perpendicular to a ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-f1fef89b4fc1eb50dee14f4ed7ccbe0d3dd13695a3afd76dcd470b386e67ea59 | Two distinct circles $\Gamma_1$ and $\Gamma_2$ intersect at points $M$ and $N$. A line $\ell$ intersects $\Gamma_1$ at points $A$ and $C$ and intersects $\Gamma_2$ at points $B$ and $D$ such that $A, B, C, D$ are distinct and lie on $\ell$ in this order. Let $X$ be a point on the line $MN$ such that $M$ lies between $X... | [
"1"
] | [
"Two distinct circles $\\Gamma_1$ and $\\Gamma_2$ intersect at points $M$ and $N$. A line $\\ell$ intersects $\\Gamma_1$ at points $A$ and $C$ and intersects $\\Gamma_2$ at points $B$ and $D$ such that $A, B, C, D$ are distinct and lie on $\\ell$ in this order. Let $X$ be a point on the line $MN$ such that $M$ lies... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-f8c90a301d0f1301c71798cdd3e68a202bbddb536d74a47bb901b4b33ffcf817 | Two distinct circles \(\omega_1\) and \(\omega_2\) meet at two distinct points \(P\) and \(Q\). A line \(\ell\) intersects \(\omega_1\) at points \(A\) and \(C\) and intersects \(\omega_2\) at points \(B\) and \(D\) such that the points \(A, B, C,\) and \(D\) are all distinct and lie on \(\ell\) in this order. Let \(T\... | [
"\\dfrac{5}{3}"
] | [
"Two distinct circles \\(\\omega_1\\) and \\(\\omega_2\\) meet at two distinct points \\(P\\) and \\(Q\\). A line \\(\\ell\\) intersects \\(\\omega_1\\) at points \\(A\\) and \\(C\\) and intersects \\(\\omega_2\\) at points \\(B\\) and \\(D\\) such that the points \\(A, B, C,\\) and \\(D\\) are all distinct and lie... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-c35853937cc9a8eb753ac9b6222594e72322bd24c84a4a123a8a89aed5cf5fef | Two distinct circles with radius 1 are tangent to each other. How many circles with radius 2 can be drawn that are tangent to both of these circles? | [
"3"
] | [
"Two distinct circles with radius 1 are tangent to each other. How many circles with radius 2 can be drawn that are tangent to both of these circles?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8283d0673cbd0ac7f664038f112c49f6acb9861c204441e609e893b3e12551f9 | Two distinct integers, $x$ and $y$, are randomly chosen from the set $\{1,2,3,4,5,6,7,8,9,10\}$. What is the probability that $xy-x-y$ is even? | [
"\\frac{2}{9}"
] | [
"Two distinct integers, $x$ and $y$, are randomly chosen from the set $\\{1,2,3,4,5,6,7,8,9,10\\}$. What is the probability that $xy-x-y$ is even?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2384768dbce8e0e51fe170e5a2c5ab3ee75c227e4a3836812b2b463c45effca7 | Two distinct lines pass through the center of three concentric circles with radii 3, 2, and 1. The area of the shaded region is \( \frac{8}{13} \) of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? | [
"\\dfrac{8\\pi}{21}"
] | [
"Two distinct lines pass through the center of three concentric circles with radii 3, 2, and 1. The area of the shaded region is \\( \\frac{8}{13} \\) of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-d61341fc7da010e454b651a0b95993ffe6624799a77e2455978e7f4d72d91e69 | Two distinct lines pass through the center of three concentric circles with radii 3, 2, and 1. The area of the shaded region is \( \frac{8}{13} \) of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: \( \pi \) radians is \( 180 \) degrees.) | [
"\\dfrac{8\\pi}{21}"
] | [
"Two distinct lines pass through the center of three concentric circles with radii 3, 2, and 1. The area of the shaded region is \\( \\frac{8}{13} \\) of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: \\( \\pi \\) radians is \\( 180 \\) degrees.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-42e37b26d5877ec56033e0ac6c673ee4a864bcfcbdfb5e3ff9ed42657acd6e90 | Two distinct lines pass through the center of three concentric circles with radii 3, 2, and 1. The area of the shaded region is \(\frac{8}{13}\) of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? | [
"\\dfrac{8\\pi}{21}"
] | [
"Two distinct lines pass through the center of three concentric circles with radii 3, 2, and 1. The area of the shaded region is \\(\\frac{8}{13}\\) of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-188c9cf71ae88735e1466d72c2ab8ed956a6355126220437d9a66c425b494778 | Two distinct non-tangent circles are drawn so that they intersect each other. A third circle, distinct from the previous two, is drawn. Let \( P \) be the number of points of intersection between any two circles. How many possible values of \( P \) are there? | [
"5"
] | [
"Two distinct non-tangent circles are drawn so that they intersect each other. A third circle, distinct from the previous two, is drawn. Let \\( P \\) be the number of points of intersection between any two circles. How many possible values of \\( P \\) are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-69a80c198d9d90eefb1f019113b6bfeca7f8094299f16d8ff5643b620e678a79 | Two distinct notes are called an interval and if they are $0$, $5$, or $7$ semitones apart modulo $12$, they are a perfect interval. If a piano has $88$ notes, and consecutive notes are $1$ semitone apart, how many perfect intervals can be played? | [
"924"
] | [
"Two distinct notes are called an interval and if they are $0$, $5$, or $7$ semitones apart modulo $12$, they are a perfect interval. If a piano has $88$ notes, and consecutive notes are $1$ semitone apart, how many perfect intervals can be played?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0be11e1a371549039c9fe54f5c409b254793b22cacb41515831b950fe709268a | Two distinct numbers $a$ and $b$ are randomly and uniformly chosen from the set $\{3, 8, 16, 18, 24\}$. What is the probability that there exist integers $c$ and $d$ such that $ac + bd = 6$? | [
"\\(\\frac{7}{10}\\)"
] | [
"Two distinct numbers $a$ and $b$ are randomly and uniformly chosen from the set $\\{3, 8, 16, 18, 24\\}$. What is the probability that there exist integers $c$ and $d$ such that $ac + bd = 6$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-43613e17195b956403a1a0f159ca3a9db2f03d89b2a3d8ebc757042e01fc66be | Two distinct numbers \( a \) and \( b \) are chosen randomly from the set \( \{ 2, 2^2, 2^3, \ldots, 2^{25} \} \). What is the probability that \( \log_{a} b \) is an integer? | [
"\\(\\frac{31}{300}\\)"
] | [
"Two distinct numbers \\( a \\) and \\( b \\) are chosen randomly from the set \\( \\{ 2, 2^2, 2^3, \\ldots, 2^{25} \\} \\). What is the probability that \\( \\log_{a} b \\) is an integer?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-5e11bc3e6a310bb2cd154e0e66835ffeeca090ec1c0cc19a2f740a9ca6abcf20 | Two distinct numbers \( a \) and \( b \) are chosen randomly from the set \( \{2, 2^2, 2^3, \ldots, 2^{25}\} \). What is the probability that \( \log_a b \) is an integer? | [
"\\dfrac{31}{300}"
] | [
"Two distinct numbers \\( a \\) and \\( b \\) are chosen randomly from the set \\( \\{2, 2^2, 2^3, \\ldots, 2^{25}\\} \\). What is the probability that \\( \\log_a b \\) is an integer?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6d213166b6896fa05dc32c442f9890f6a9be65dde154109af8857750f9691a3a | Two distinct numbers are selected from the set $\{1, 2, 3, \dots, 36, 37\}$ so that the sum of the remaining 35 numbers is the product of these two numbers. What is the difference of these two numbers? | [
"10"
] | [
"Two distinct numbers are selected from the set $\\{1, 2, 3, \\dots, 36, 37\\}$ so that the sum of the remaining 35 numbers is the product of these two numbers. What is the difference of these two numbers?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-7e09e7bfbffb90215c91c4dc331847ae779bdf76dfb25e0fdf0799e17255f7be | Two distinct numbers are selected from the set $\{1,2,3,4,\dots,36,37\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers? | [
"10"
] | [
"Two distinct numbers are selected from the set $\\{1,2,3,4,\\dots,36,37\\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b5401c83d03f1c1d2d2ac997c8b5c840009930845f5fcd2d99a28b06d9dc157b | Two distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5\}$. What is the probability that the smaller one divides the larger one? Express your answer as a common fraction. | [
"\\frac{1}{2}"
] | [
"Two distinct numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 5\\}$. What is the probability that the smaller one divides the larger one? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-12f56e3b16e21fc4f5e15b35c2f86f054b9f56832ec96fa497f79ecbf077950a | Two distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5\}$. What is the probability that their product is an even number? Express your answer as a common fraction. | [
"\\frac{7}{10}"
] | [
"Two distinct numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 5\\}$. What is the probability that their product is an even number? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-50e1837acd7cac8adceca921a3e30f28942b7c7b7090c2872f67bd28a62384bf | Two distinct numbers are written on each vertex of a convex 100-gon. Let $S$ be the set of all possible values of the sum of the numbers on any two adjacent vertices before any numbers are removed. After removing one number from each vertex so that the remaining numbers on any two adjacent vertices differ, let $T$ be t... | [
"0"
] | [
"Two distinct numbers are written on each vertex of a convex 100-gon. Let $S$ be the set of all possible values of the sum of the numbers on any two adjacent vertices before any numbers are removed. After removing one number from each vertex so that the remaining numbers on any two adjacent vertices differ, let $T$... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-41c7a358e5c85245cdecd1e0e799269fc5f792b957e297a5736d142477f95a2d | Two distinct points D and E are chosen randomly from 15 points equally spaced around a circle centered at O such that each pair of points D and E has the same probability of being chosen. The probability that the perpendicular bisectors of OD and OE intersect strictly inside the circle can be expressed as a/b, where a ... | [
"11"
] | [
"Two distinct points D and E are chosen randomly from 15 points equally spaced around a circle centered at O such that each pair of points D and E has the same probability of being chosen. The probability that the perpendicular bisectors of OD and OE intersect strictly inside the circle can be expressed as a/b, whe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 2
} | 48 | 0 | 0 |
omr-ws-3e8d67db4f53c3b62f89e0b18d79cfe397284223775848bb10ef7f9afe626d1b | Two distinct points \( A \) and \( B \) are chosen at random from 15 points equally spaced around a circle centered at \( O \) such that each pair of points \( A \) and \( B \) has the same probability of being chosen. The probability that the perpendicular bisectors of \( OA \) and \( OB \) intersect strictly inside t... | [
"11"
] | [
"Two distinct points \\( A \\) and \\( B \\) are chosen at random from 15 points equally spaced around a circle centered at \\( O \\) such that each pair of points \\( A \\) and \\( B \\) has the same probability of being chosen. The probability that the perpendicular bisectors of \\( OA \\) and \\( OB \\) intersec... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9f18d87816ecfbbf73ef6bbadc910ac8499b2f1616ca380526a702e54ad7afc | Two distinct points \( A \) and \( B \) are marked on the left half of the parabola \( y = x^2 \). Consider any pair of parallel lines which pass through \( A \) and \( B \) and intersect the right half of the parabola at points \( C \) and \( D \). Let \( K \) be the intersection point of the diagonals \( AC \) and \(... | [
"0"
] | [
"Two distinct points \\( A \\) and \\( B \\) are marked on the left half of the parabola \\( y = x^2 \\). Consider any pair of parallel lines which pass through \\( A \\) and \\( B \\) and intersect the right half of the parabola at points \\( C \\) and \\( D \\). Let \\( K \\) be the intersection point of the diag... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-7b86e1b5c5040adf846f8a84ef72cab397f5d72101680b89a27c9edbda9791be | Two distinct points are selected at random from the nine-point set \( S = \{(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)\} \). What is the probability that their midpoint belongs to \( S \)? | [
"\\(\\frac{2}{9}\\)"
] | [
"Two distinct points are selected at random from the nine-point set \\( S = \\{(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)\\} \\). What is the probability that their midpoint belongs to \\( S \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-91be23ca00d23635200923027e996b359516ff97958e678ffc05a102845570aa | Two distinct positive integers $a$ and $b$ are written on the board. The smaller of them is erased and replaced with the number $\frac{ab}{|a-b|}$. This process is repeated as long as the two numbers are not equal. After performing this operation $k$ times, the two numbers on the board become equal. Find the sum of all... | [
"2"
] | [
"Two distinct positive integers $a$ and $b$ are written on the board. The smaller of them is erased and replaced with the number $\\frac{ab}{|a-b|}$. This process is repeated as long as the two numbers are not equal. After performing this operation $k$ times, the two numbers on the board become equal. Find the sum ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-43c223f1aa53c75cc8af79bfa7696b4b684deb98b4eb014782d80101d40043fc | Two distinct positive integers $x$ and $y$ are factors of 36. If $x\cdot y$ is not a factor of 36, what is the smallest possible value of $x\cdot y$? | [
"8"
] | [
"Two distinct positive integers $x$ and $y$ are factors of 36. If $x\\cdot y$ is not a factor of 36, what is the smallest possible value of $x\\cdot y$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c992c4a458fa775d0af058741cca1a192ddfb404b42015335407354b09908060 | Two distinct positive integers between 1 and 9 are chosen. The sum is revealed to logician A, and the product is revealed to logician B. Both A and B are given this information and the information contained in the conversation below. The conversation goes as follows: B: "I can't tell what they are." A: "I can't tell wh... | [
"18"
] | [
"Two distinct positive integers between 1 and 9 are chosen. The sum is revealed to logician A, and the product is revealed to logician B. Both A and B are given this information and the information contained in the conversation below. The conversation goes as follows:\n\nB: \"I can't tell what they are.\"\nA: \"I c... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 44,
"tir": 27
} | 71 | 0 | 0 |
omr-ws-856f6db1bc72010310ab1fda05489adb5cddcb8ea1010e712686e3430604521b | Two distinct positive integers from 1 to 50 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 5? | [
"\\frac{89}{245}"
] | [
"Two distinct positive integers from 1 to 50 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 5?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b4fda6624d64031d90eac907faf1e4c1dec32c04d70d2da8fb3ceea059de646f | Two distinct positive real numbers \( A \) and \( B \) are chosen at random with \( B > A \). What is the expected value of \( \frac{B}{A} \)? | [
"\\infty"
] | [
"Two distinct positive real numbers \\( A \\) and \\( B \\) are chosen at random with \\( B > A \\). What is the expected value of \\( \\frac{B}{A} \\)?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-1c37f09b5bb0ed660bce99a01e0839d4ef236f7f52c3f0b9e3f2a587b7ee5c95 | Two distinct primes, each greater than 20, are multiplied. What is the least possible product of these two primes? | [
"667"
] | [
"Two distinct primes, each greater than 20, are multiplied. What is the least possible product of these two primes?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-77e8bb825c742e2d15c86d0b0b962ec812c224582e08413d9e2943ef7b24909d | Two distinct regular hexagons \( ABCDEF \) and \( AGHIJK \) both have side length 1. If quadrilateral \( CIDJ \) is a parallelogram, compute the area of the intersection of the two hexagons. | [
"\\dfrac{2\\sqrt{3}}{3}"
] | [
"Two distinct regular hexagons \\( ABCDEF \\) and \\( AGHIJK \\) both have side length 1. If quadrilateral \\( CIDJ \\) is a parallelogram, compute the area of the intersection of the two hexagons."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-ba92a549c9184360f7793ad0e48ceab6c0f1176ff51cad086ffd10b3b53f4bc0 | Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra? | [
"\\( \\frac{1}{6} \\)",
"\\(\\frac{1}{6}\\)",
"\\dfrac{1}{6}"
] | [
"Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 37,
"tir": 37
} | 74 | 0 | 0 |
omr-ws-fc9cc7b366459db157303e12fa5d040b170022a0736d476e159bb19ef07333b9 | Two distinct roots of the equation $f(x) = 2x^2 - 8nx + 10x - n^2 + 35n - 76$, where the constant $n$ is an integer, are positive primes. Find the sum of the roots of $f(x)$. | [
"7"
] | [
"Two distinct roots of the equation $f(x) = 2x^2 - 8nx + 10x - n^2 + 35n - 76$, where the constant $n$ is an integer, are positive primes. Find the sum of the roots of $f(x)$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-67b7e305498c2ebfed2a53deb6e70da552439afb421fcd6d97bebd933f954dbc | Two distinct two-digit prime numbers less than 60 are chosen. When their sum is subtracted from their product, which of the following values is impossible to attain? (A) 1137 (B) 1287 (C) 1439 (D) 1559 (E) 1623 | [
"1137"
] | [
"Two distinct two-digit prime numbers less than 60 are chosen. When their sum is subtracted from their product, which of the following values is impossible to attain?\n(A) 1137 (B) 1287 (C) 1439 (D) 1559 (E) 1623"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-63ff2382ea9e920fa0ac9890359657a3b505cc120bacc3b2b153e3898d951533 | Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8$, and the second term of both series can be written in the form $\frac{\sqrt{m}-n}p$, where $m$, $n$, and $p$ are positive integers and $m$ is not divisible by the square of an... | [
"518"
] | [
"Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8$, and the second term of both series can be written in the form $\\frac{\\sqrt{m}-n}p$, where $m$, $n$, and $p$ are positive integers and $m$ is not divisible by the squar... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-370914500909edf8d868c324ab0379df69ea14c717c31e2b10e1e46435c0a682 | Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $\frac{1}{8}$, and the second term of both series can be written in the form $\frac{\sqrt{m}-n}{p}$, where $m$, $n$, and $p$ are positive integers and $m$ is not divisible by the sq... | [
"518"
] | [
"Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $\\frac{1}{8}$, and the second term of both series can be written in the form $\\frac{\\sqrt{m}-n}{p}$, where $m$, $n$, and $p$ are positive integers and $m$ is not divisible b... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-f2dd68fbb5fa3104cdaf464490a16150a5e33ad99c55748cd6f4fd11e5a121bb | Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $\frac{1}{8}$. If the second term of both the series can be written in the form $\frac{\sqrt{m} - n}{p}$, where $m$, $n$, and $p$ are positive integers and $m$ is not divisible by t... | [
"518"
] | [
"Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $\\frac{1}{8}$. If the second term of both the series can be written in the form $\\frac{\\sqrt{m} - n}{p}$, where $m$, $n$, and $p$ are positive integers and $m$ is not divisi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-c0e20f0738dc741dcbe2d3073f5ff492fb2865dd5458a5f27d9f178182378754 | Two drivers leave towns A and B, which are 100km apart, at the same time and meet exactly 30 minutes later. If the speed limit is 90km/h, calculate the minimum speed (in km/h) that at least one of the drivers must have exceeded the speed limit by. | [
"10"
] | [
"Two drivers leave towns A and B, which are 100km apart, at the same time and meet exactly 30 minutes later. If the speed limit is 90km/h, calculate the minimum speed (in km/h) that at least one of the drivers must have exceeded the speed limit by."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c58cf9633dcb6e27c74ece01dc2a2ac90d2f904e91861aa9b91a30312459f2cb | Two drivers start at the same location and drive in perpendicular directions. Driver A travels due north at 30 miles per hour, and Driver B travels due west at 50 miles per hour. At what rate, in miles per hour, is the distance between the two drivers changing 4 hours into their journey? | [
"58.3 mph"
] | [
"Two drivers start at the same location and drive in perpendicular directions. Driver A travels due north at 30 miles per hour, and Driver B travels due west at 50 miles per hour. At what rate, in miles per hour, is the distance between the two drivers changing 4 hours into their journey?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-d4087803de7094ec788693f067857152819e5fe9aea35bc6ee31db2ee4286d59 | Two drivers start driving towards each other from points A and B at sunrise, each at a constant speed. They meet at noon and continue driving. The first driver arrives at point B at 4:00 p.m., and the second driver arrives at point A at 9:00 p.m. At what time did the sun rise that day? | [
"6:00 a.m."
] | [
"Two drivers start driving towards each other from points A and B at sunrise, each at a constant speed. They meet at noon and continue driving. The first driver arrives at point B at 4:00 p.m., and the second driver arrives at point A at 9:00 p.m. At what time did the sun rise that day?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-5f6f6c871b3ad8322714aea0252575857b56337babb38310cc08b8253a4a3fe0 | Two drivers travel towards each other from two towns 240 km apart. They meet after 6 hours. The first driver takes 5 hours to travel the distance that the second driver takes 3 hours to travel. What are the speeds of each driver? | [
"25"
] | [
"Two drivers travel towards each other from two towns 240 km apart. They meet after 6 hours. The first driver takes 5 hours to travel the distance that the second driver takes 3 hours to travel. What are the speeds of each driver?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-9ab69b92df675ad71e5e1d6c5befdf50ed2cf9cf3ab585fb74f8fd1171b344d1 | Two drugs are being tested. Of 200 patients given drug A, 60 are cured; and of 1100 given drug B, 170 are cured. If we assume a homogeneous group of patients, find the probabilities of successful treatment with A or B. Now closer investigation reveals that the 200 patients given drug A were in fact 100 men, of whom 50 ... | [
"Men with drug A: 0.5, Women with drug A: 0.1, Men with drug B: 0.6, Women with drug B: 0.11"
] | [
"Two drugs are being tested. Of 200 patients given drug A, 60 are cured; and of 1100 given drug B, 170 are cured. If we assume a homogeneous group of patients, find the probabilities of successful treatment with A or B. Now closer investigation reveals that the 200 patients given drug A were in fact 100 men, of who... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 26
} | 28 | 0 | 0 |
omr-ws-85a9287c8609ad2ef5b6f666c8c4d15ea22af2559394429295d0846adb1f1c85 | Two ducks race in the first quadrant of the xy-plane starting from the origin. The race ends when one duck reaches the line \( y = \frac{1}{2} \). The first duck follows the path \( y = \frac{x}{3} \) and the second duck follows the path \( y = \frac{x}{5} \). If their x-coordinates are the same at any time during the ... | [
"\\dfrac{5\\sqrt{65}}{39}"
] | [
"Two ducks race in the first quadrant of the xy-plane starting from the origin. The race ends when one duck reaches the line \\( y = \\frac{1}{2} \\). The first duck follows the path \\( y = \\frac{x}{3} \\) and the second duck follows the path \\( y = \\frac{x}{5} \\). If their x-coordinates are the same at any ti... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-99ba50e0b4acbb5f800dbaa4abfdaa18bb540433abd69fb3cae2818b11990fc3 | Two ducks race in the first quadrant of the xy-plane starting from the origin. The race ends when one of the ducks reaches the line \( y = \frac{1}{2} \). The first duck follows the path \( y = \frac{x}{3} \) and the second duck follows the path \( y = \frac{x}{5} \). If their x-coordinates are the same at any time dur... | [
"\\(\\frac{5\\sqrt{5}}{3\\sqrt{13}}\\)"
] | [
"Two ducks race in the first quadrant of the xy-plane starting from the origin. The race ends when one of the ducks reaches the line \\( y = \\frac{1}{2} \\). The first duck follows the path \\( y = \\frac{x}{3} \\) and the second duck follows the path \\( y = \\frac{x}{5} \\). If their x-coordinates are the same a... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-ac3a0b20eeccf8342030c9d135e6d6d0e24b4177c550043fc9608325c6876516 | Two dueling wizards are at an altitude of 100 above the sea. They cast spells in turn, and each spell is of the form "decrease the altitude by \(a\) for me and by \(b\) for my rival" where \(0 < a < b\). The set of spells is the same for both wizards, the spells may be cast in any order, and the same spell may be cast ... | [
"50"
] | [
"Two dueling wizards are at an altitude of 100 above the sea. They cast spells in turn, and each spell is of the form \"decrease the altitude by \\(a\\) for me and by \\(b\\) for my rival\" where \\(0 < a < b\\). The set of spells is the same for both wizards, the spells may be cast in any order, and the same spell... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-75eda4cb655d265e726accd9b6dbfdb6710a76a0d9c6f47aa83df3ba6e8410bc | Two edges of a regular tetrahedron are opposite if they are skew. Find the distance between the midpoints of 2 opposite edges of a regular tetrahedron with edge length 1. | [
"\\dfrac{\\sqrt{2}}{2}"
] | [
"Two edges of a regular tetrahedron are opposite if they are skew. Find the distance between the midpoints of 2 opposite edges of a regular tetrahedron with edge length 1."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ec910ffb32d48ba983154e84a9fb90a42caf81c3be527058f43b3bf24bde8167 | Two egg cartons, $A$ and $B$, are given. Carton $A$ has 6 spaces and 3 eggs, while carton $B$ has 12 spaces and 3 eggs. Two cells are selected at random from the total of 18 cells, and the contents of the selected cells are interchanged. (a) Find the number of selections/interchanges that produce a decrease in the numb... | [
"4"
] | [
"Two egg cartons, $A$ and $B$, are given. Carton $A$ has 6 spaces and 3 eggs, while carton $B$ has 12 spaces and 3 eggs. Two cells are selected at random from the total of 18 cells, and the contents of the selected cells are interchanged.\n(a) Find the number of selections/interchanges that produce a decrease in th... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-7f6cc6c349e39f4f81dbff4a9ee942da07ad89ab03ecad4d4ec80d419ce6af73 | Two ellipses $\Omega_1$ and $\Omega_2$ are in double contact with a third ellipse $\Gamma$. Suppose $\Omega_1$ and $\Omega_2$ intersect at four points forming a quadrilateral $ABCD$. Let $P$ be the intersection point of the diagonals $AC$ and $BD$, and let $Q$ and $R$ be the points of double contact between $\Omega_1$ ... | [
"\\left( \\dfrac{a^2 (r_y - q_y)}{q_x r_y - q_y r_x}, \\dfrac{b^2 (q_x - r_x)}{q_x r_y - q_y r_x} \\right)"
] | [
"Two ellipses $\\Omega_1$ and $\\Omega_2$ are in double contact with a third ellipse $\\Gamma$. Suppose $\\Omega_1$ and $\\Omega_2$ intersect at four points forming a quadrilateral $ABCD$. Let $P$ be the intersection point of the diagonals $AC$ and $BD$, and let $Q$ and $R$ be the points of double contact between $... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-ee9e56ec55900cd66b3b54978b40000ccc1f67d4cedd7b1fd89fd78d131912cd | Two ellipses intersect at points \(A, B, C, D\) in that order. Another ellipse is tangent to these ellipses at points \(M, N\) and \(P, Q\). Let the intersection point of lines \(AC\) and \(BD\) be \(X\). If the coordinates of \(M\) and \(N\) are \((1, 2)\) and \((3, 4)\) respectively, and the coordinates of \(P\) and ... | [
"9"
] | [
"Two ellipses intersect at points \\(A, B, C, D\\) in that order. Another ellipse is tangent to these ellipses at points \\(M, N\\) and \\(P, Q\\). Let the intersection point of lines \\(AC\\) and \\(BD\\) be \\(X\\). If the coordinates of \\(M\\) and \\(N\\) are \\((1, 2)\\) and \\((3, 4)\\) respectively, and the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-780076cfa7ff8a65af6308eeeababf97bf5e1f23b4168f29aa36664fb255d9e4 | Two ellipses with centers A and B are tangent at point X. Their external common tangent lines PQ and RS meet at point C. If the coordinates of A are (0,0) and the coordinates of B are (10,0), find the x-coordinate of point C. | [
"20"
] | [
"Two ellipses with centers A and B are tangent at point X. Their external common tangent lines PQ and RS meet at point C. If the coordinates of A are (0,0) and the coordinates of B are (10,0), find the x-coordinate of point C."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-1e42d708a6c7f0406fa0365da4f884078aa045089080ca58a19eacadd0f67e59 | Two ellipses, $(2x-5)^2 + 2y^2 = 10$ and $2x^2 + (y+3)^2 = 9$, intersect at two points. Find the equation of the line passing through the two points of intersection. | [
"\\( y = -\\frac{5}{3}x + \\frac{5}{4} \\)"
] | [
"Two ellipses, $(2x-5)^2 + 2y^2 = 10$ and $2x^2 + (y+3)^2 = 9$, intersect at two points. Find the equation of the line passing through the two points of intersection."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-c3e663a4c5cc25aef7684cc1baf3fb1a530d439025f041f465a34eca8aa7feee | Two engines are running in opposite directions on a circular track $C_1$ of radius $3$ km. When the faster engine reaches point $B$, it sees the slower engine at point $A$ and moves to a semicircular track $C_2$ of radius $1.5$ km to avoid a collision. If the faster engine and the slower engine reach points $A$ and $B$... | [
"\\(\\frac{2}{3}\\)"
] | [
"Two engines are running in opposite directions on a circular track $C_1$ of radius $3$ km. When the faster engine reaches point $B$, it sees the slower engine at point $A$ and moves to a semicircular track $C_2$ of radius $1.5$ km to avoid a collision. If the faster engine and the slower engine reach points $A$ an... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-26b8ff35f44d04252dddc4ef1c95e2994fc4ebbe95c45a191f71c88bbcd9b41c | Two equal circles $\Gamma_1$ and $\Gamma_2$ are tangent at point $A$. Quadrilateral $ABCD$ is inscribed in circle $\Gamma_1$, and quadrilateral $AEFG$ is inscribed in circle $\Gamma_2$. The sides satisfy $AB \parallel FG$, $BC \parallel GA$, $CD \parallel AE$, and $DA \parallel EF$. Let $BC$ and $AD$ intersect at point... | [
"1"
] | [
"Two equal circles $\\Gamma_1$ and $\\Gamma_2$ are tangent at point $A$. Quadrilateral $ABCD$ is inscribed in circle $\\Gamma_1$, and quadrilateral $AEFG$ is inscribed in circle $\\Gamma_2$. The sides satisfy $AB \\parallel FG$, $BC \\parallel GA$, $CD \\parallel AE$, and $DA \\parallel EF$. Let $BC$ and $AD$ inter... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-709de5345557fc2cf7d4301e35a1b122cbb0f5ffe62d7e497b92f1237444f2c0 | Two equal circles $\Omega_1$ and $\Omega_2$ intersect at points $A$ and $B$, and $M$ is the midpoint of $AB$. Two rays are drawn from $M$, lying in the same half-plane with respect to $AB$. The first ray intersects the circles $\Omega_1$ and $\Omega_2$ at points $X_1$ and $X_2$, and the second ray intersects them at po... | [
"2"
] | [
"Two equal circles $\\Omega_1$ and $\\Omega_2$ intersect at points $A$ and $B$, and $M$ is the midpoint of $AB$. Two rays are drawn from $M$, lying in the same half-plane with respect to $AB$. The first ray intersects the circles $\\Omega_1$ and $\\Omega_2$ at points $X_1$ and $X_2$, and the second ray intersects t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-91e4b75806a5663ecc1b60a54147a734f68d39ee36117e5e39fd64c7d3bbae19 | Two equal circles $\odot J$ and $\odot K$ intersect at points $A$ and $B$. Circles $\odot P$ and $\odot Q$ are inside $\odot J$ and are tangent to both $\odot J$ and $\odot K$. Additionally, $\odot P$ is tangent to $\odot Q$ at point $T$. If the radius of $\odot J$ is $r$, find the length of $AT$ in terms of $r$. | [
"\\dfrac{r\\sqrt{6}}{2}"
] | [
"Two equal circles $\\odot J$ and $\\odot K$ intersect at points $A$ and $B$. Circles $\\odot P$ and $\\odot Q$ are inside $\\odot J$ and are tangent to both $\\odot J$ and $\\odot K$. Additionally, $\\odot P$ is tangent to $\\odot Q$ at point $T$. If the radius of $\\odot J$ is $r$, find the length of $AT$ in term... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-169529420e091c88e9f300732bd0e1b4e813bf0d3c43736fa099ad09bdb10578 | Two equal circles $\omega_1$ and $\omega_2$ are given. Line $\ell$ intersects $\omega_1$ at points $A$ and $C$, while $\omega_2$ intersects at $B$ and $D$ (order of points on the line: $A$, $B$, $C$, $D$). Circles $\Gamma_1$ and $\Gamma_2$ touch $\omega_1$ internally, $\omega_2$ externally, and the straight line $\ell$... | [
"2d"
] | [
"Two equal circles $\\omega_1$ and $\\omega_2$ are given. Line $\\ell$ intersects $\\omega_1$ at points $A$ and $C$, while $\\omega_2$ intersects at $B$ and $D$ (order of points on the line: $A$, $B$, $C$, $D$). Circles $\\Gamma_1$ and $\\Gamma_2$ touch $\\omega_1$ internally, $\\omega_2$ externally, and the straig... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-6fd076ba83541c22e9d2f4d2028fb236409509d58b32f0d5cc6b91a4c8b2f5d5 | Two equal circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A point $P$ is chosen on the smaller arc $AB$ of the circle $\omega_2$. Straight $AP$ intersects the circle $\omega_1$ at point $C$. The line $CB$ intersects the large arc $AB$ of the circle $\omega_2$ at the point $D$. Angle bisector $CAD$ in... | [
"90^\\circ"
] | [
"Two equal circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. A point $P$ is chosen on the smaller arc $AB$ of the circle $\\omega_2$. Straight $AP$ intersects the circle $\\omega_1$ at point $C$. The line $CB$ intersects the large arc $AB$ of the circle $\\omega_2$ at the point $D$. Angle bisecto... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-9bb31393610acbf80696223a3417942cad635f6eb2b7a3108f6c17b7b7fffb2f | Two equal circles \( S_1 \) and \( S_2 \) are internally tangent to a circle \( S \) at points \( A_1 \) and \( A_2 \) respectively. Let \( C \) be any point on the circle \( S \), and let \( CA_1 \) intersect \( S_1 \) at \( B_1 \), and \( CA_2 \) intersect \( S_2 \) at \( B_2 \). If the radius of circle \( S \) is 10... | [
"20"
] | [
"Two equal circles \\( S_1 \\) and \\( S_2 \\) are internally tangent to a circle \\( S \\) at points \\( A_1 \\) and \\( A_2 \\) respectively. Let \\( C \\) be any point on the circle \\( S \\), and let \\( CA_1 \\) intersect \\( S_1 \\) at \\( B_1 \\), and \\( CA_2 \\) intersect \\( S_2 \\) at \\( B_2 \\). If the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-7bf26189a739c537d90cd595825abb430f1638d3735bae25c86f72749af7bdf7 | Two equal circles \( S_1 \) and \( S_2 \) intersect at two different points. The line \( \ell \) intersects \( S_1 \) at points \( A \) and \( C \), and \( S_2 \) at points \( B \) and \( D \) (in the order \( A, B, C, D \) on \( \ell \)). Define circles \( \Gamma_1 \) and \( \Gamma_2 \) such that both touch \( S_1 \) ... | [
"32"
] | [
"Two equal circles \\( S_1 \\) and \\( S_2 \\) intersect at two different points. The line \\( \\ell \\) intersects \\( S_1 \\) at points \\( A \\) and \\( C \\), and \\( S_2 \\) at points \\( B \\) and \\( D \\) (in the order \\( A, B, C, D \\) on \\( \\ell \\)). Define circles \\( \\Gamma_1 \\) and \\( \\Gamma_2 ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-495fef4b9cee55a250e245f1a87b3bf4090ca547dc86a5c6030c7ab3a5530109 | Two equal circles \( w_1 \) and \( w_2 \) are given, with the center of \( w_2 \), \( O \), lying on \( w_1 \). Triangle \( ABC \) is inscribed in \( w_1 \) such that \( AC \) and \( BC \) are tangent to \( w_2 \). If \( \angle A = \alpha \) and \( \angle B = \beta \), calculate the value of \( \cos \alpha + \cos \beta... | [
"1"
] | [
"Two equal circles \\( w_1 \\) and \\( w_2 \\) are given, with the center of \\( w_2 \\), \\( O \\), lying on \\( w_1 \\). Triangle \\( ABC \\) is inscribed in \\( w_1 \\) such that \\( AC \\) and \\( BC \\) are tangent to \\( w_2 \\). If \\( \\angle A = \\alpha \\) and \\( \\angle B = \\beta \\), calculate the val... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-d91611ee15bfe3c25689d124173501f39314ed2a41f3238d00c826b45a1444c5 | Two equal circles are tangent externally at point T. Chord $\overline{TM}$ in circle O is perpendicular to chord $\overline{TN}$ in circle Q. If the radius of each circle is 5 units, calculate the length of segment $\overline{MN}$. | [
"10"
] | [
"Two equal circles are tangent externally at point T. Chord $\\overline{TM}$ in circle O is perpendicular to chord $\\overline{TN}$ in circle Q. If the radius of each circle is 5 units, calculate the length of segment $\\overline{MN}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d61fcb81fed9d6281b506d3b0701311b893d7cfb3791a80cdbdb1647b3ce0009 | Two equal circles intersect at points $A$ and $B$. Let $P$ be a point on one of the circles different from $A$ and $B$. The lines $PA$ and $PB$ intersect the other circle again at points $X$ and $Y$, respectively. Let $M$ be the midpoint of the arc $XY$ not containing $A$ and $B$. If the line passing through $P$ and pe... | [
"\\dfrac{7}{2}"
] | [
"Two equal circles intersect at points $A$ and $B$. Let $P$ be a point on one of the circles different from $A$ and $B$. The lines $PA$ and $PB$ intersect the other circle again at points $X$ and $Y$, respectively. Let $M$ be the midpoint of the arc $XY$ not containing $A$ and $B$. If the line passing through $P$ a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-935c571d78cf920ee1cf8bc17c34f6d4e3da4bf69f18a337b79e5acad6c64349 | Two equal cones have a common vertex and are tangent along a common generatrix. The angle of the axial section of the cone is equal to $2\alpha $. Find the dihedral angle between two planes, each of which touches both cones but does not pass through their common generatrix. | [
"$\\arctan \\left[\\sin \\alpha /\\sqrt {\\cos 2\\alpha }\\right]$. If $2\\alpha >\\pi /2$, then the cones do not have common tangent planes (not passing through a common generatrix)"
] | [
"Two equal cones have a common vertex and are tangent along a common generatrix. The angle of the axial section of the cone is equal to $2\\alpha $. Find the dihedral angle between two planes, each of which touches both cones but does not pass through their common generatrix."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-695c45e4248a30fddec103a35006f5998b953ad5b377301e17853977f754248c | Two equal disjoint circles intersect two lines. Each straight line crosses the circle at four points, and the three formed segments (with the ends at neighboring intersection points) on each of the lines are equal. The segments do not intersect. The lengths of segments on one of the lines are $4$, and on the other - $2... | [
"4"
] | [
"Two equal disjoint circles intersect two lines. Each straight line crosses the circle at four points, and the three formed segments (with the ends at neighboring intersection points) on each of the lines are equal. The segments do not intersect. The lengths of segments on one of the lines are $4$, and on the other... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-088c46ccfdf5f738c9802be3a9fd9fea9db898a4e03675c813ce6b1b65677d18 | Two equal non-intersecting wooden disks, one gray and one black, are glued to a plane. A triangle with one gray side and one black side can be moved along the plane so that the disks remain outside the triangle, while the colored sides of the triangle are tangent to the disks of the same color (the tangency points are ... | [
"(2, 0)"
] | [
"Two equal non-intersecting wooden disks, one gray and one black, are glued to a plane. A triangle with one gray side and one black side can be moved along the plane so that the disks remain outside the triangle, while the colored sides of the triangle are tangent to the disks of the same color (the tangency points... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-65b0dd38917d20fe96a1baf44f6ffcd915d1db1f064a5682aca0787d71d95d04 | Two equal parallel chords are drawn 8 inches apart in a circle of radius 8 inches. What is the area of that part of the circle that lies between the chords? | [
"\\( 32\\sqrt{3} + 21\\frac{1}{3}\\pi \\)"
] | [
"Two equal parallel chords are drawn 8 inches apart in a circle of radius 8 inches. What is the area of that part of the circle that lies between the chords?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-fd11acb64f4ed18ca8d7b579d8dbfb62f9b60051331667c72b52638f04554675 | Two equal rectangles intersect in 8 points. Let the area of one rectangle be $A$. If the area of the common part is represented as a fraction of $A$, find the minimum possible value of this fraction. Express your answer as a common fraction. | [
"\\dfrac{1}{2}"
] | [
"Two equal rectangles intersect in 8 points. Let the area of one rectangle be $A$. If the area of the common part is represented as a fraction of $A$, find the minimum possible value of this fraction. Express your answer as a common fraction."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-9f9e0fd6030305486b072ee9639569327788e2d2e21259eed609a227ddd78dbb | Two equal segments $AB$ and $CD$ are perpendicular, and the point $C$ lies inside the segment $AB$. Point $X$ is such that the triangles $XAD$ and $XBC$ are isosceles with a vertex in $X$. If the length of $AB$ is 10 units, find the area of triangle $XAD$. | [
"\\dfrac{125}{4}"
] | [
"Two equal segments $AB$ and $CD$ are perpendicular, and the point $C$ lies inside the segment $AB$. Point $X$ is such that the triangles $XAD$ and $XBC$ are isosceles with a vertex in $X$. If the length of $AB$ is 10 units, find the area of triangle $XAD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-856a46af444356c4c68f58db078b2a4c19dbb0b1bb52c81d71350c1bff1a47d5 | Two equal segments $CM$ and $CN$ are marked on the sides $CA$ and $CB$ of an isosceles triangle $ABC$. Determine the length of the segments, knowing the perimeter $2P$ of the triangle $ABC$, its base $AB = 2a$ and the perimeter $2p$ of the rectangle $AMNB$ cut off by the straight line $MN$. | [
"$CM=CN=\\frac{(P-a)(P-p)}{P-2a}$"
] | [
"Two equal segments $CM$ and $CN$ are marked on the sides $CA$ and $CB$ of an isosceles triangle $ABC$. Determine the length of the segments, knowing the perimeter $2P$ of the triangle $ABC$, its base $AB = 2a$ and the perimeter $2p$ of the rectangle $AMNB$ cut off by the straight line $MN$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7a95a227c6e2c452f4893e4b93ca5fce7b941aaa63d74158e16e439ee8e7ab9e | Two equal spheres are tangent to each other and to the faces of a dihedral angle. A third sphere of smaller radius is also tangent to the faces of the dihedral angle and to both given spheres. Given the ratio $m$ of the radius of the smaller sphere to the radius of one of the equal spheres. Find the dihedral angle. Ove... | [
"$2\\arcsin\\left[(1-m)/\\sqrt {m^2+2m}\\right]$ for $1/4<m<l$"
] | [
"Two equal spheres are tangent to each other and to the faces of a dihedral angle. A third sphere of smaller radius is also tangent to the faces of the dihedral angle and to both given spheres. Given the ratio $m$ of the radius of the smaller sphere to the radius of one of the equal spheres. Find the dihedral angle... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22b5e040e4355d0d31201852078dcddae3737232232e0d333f17df4effcb077b | Two equal spheres of radius $r$ are tangent to each other and to the faces of a dihedral angle equal to $\alpha $. Find the radius of the sphere that is tangent to the faces of the dihedral angle and to both given spheres. | [
"The condition of the problem is satisfied by two spheres with radii $r\\left[1+2\\tan^2(\\alpha /2)\\pm \\tan(\\alpha /2)\\sqrt {3+4\\tan^2 (\\alpha /2)}\\right]$"
] | [
"Two equal spheres of radius $r$ are tangent to each other and to the faces of a dihedral angle equal to $\\alpha $. Find the radius of the sphere that is tangent to the faces of the dihedral angle and to both given spheres."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b36209b0b48ba1f819859d421f2ec0431197fe1af931bc98f9736687a59242fc | Two equal spheres of radius $r$ touch each other and the faces of the dihedral angle which is equal to $\alpha $. Find the radius of the sphere which touches the faces of the dihedral angle and the two given spheres. | [
"$r\\left(1+2\\tan^2 \\frac{\\alpha }{2}\\pm \\tan\\frac{\\alpha }{2} \\sqrt {3+4\\tan^2 \\frac{\\alpha }{2}}\\right)$"
] | [
"Two equal spheres of radius $r$ touch each other and the faces of the dihedral angle which is equal to $\\alpha $. Find the radius of the sphere which touches the faces of the dihedral angle and the two given spheres."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a93f7e132a921cb0f7c8e37bac95bbc1a4b53028e6396abea80aa01229557317 | Two equal squares ABCD and DEFG have vertex D in common. The angle CDG is 60 degrees. Find the measure of angle CFE. | [
"75 degrees"
] | [
"Two equal squares ABCD and DEFG have vertex D in common. The angle CDG is 60 degrees. Find the measure of angle CFE."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 5,
"tir": 9
} | 22 | 0 | 0 |
omr-ws-5fbcbf590012be448d836e4b29be3be196c2d1fe09535c24bd48cf9d6aec6b1a | Two equal squares form an octagon at their intersection. Two diagonals of this octagon divide it into 4 quadrilaterals. Calculate the area of the octagon formed by the intersection of the two squares if the side length of each square is 4 units. | [
"32(\\sqrt{2} - 1)"
] | [
"Two equal squares form an octagon at their intersection. Two diagonals of this octagon divide it into 4 quadrilaterals. Calculate the area of the octagon formed by the intersection of the two squares if the side length of each square is 4 units."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-55b9b1157424c03e7b01730139fbdd83291435ae505694796ef6fb40a3b498c4 | Two equal-radii circles with centers \( O_1 \) and \( O_2 \) intersect each other at points \( P \) and \( Q \). \( O \) is the midpoint of the common chord \( PQ \). Two lines \( AB \) and \( CD \) are drawn through \( P \) (where \( AB \) and \( CD \) are not coincident with \( PQ \)) such that \( A \) and \( C \) li... | [
"no_valid_answer_found"
] | [
"Two equal-radii circles with centers \\( O_1 \\) and \\( O_2 \\) intersect each other at points \\( P \\) and \\( Q \\). \\( O \\) is the midpoint of the common chord \\( PQ \\). Two lines \\( AB \\) and \\( CD \\) are drawn through \\( P \\) (where \\( AB \\) and \\( CD \\) are not coincident with \\( PQ \\)) suc... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0cda8e1bc03fd452058d59e524f5623e7a58bf10e2d606c9a58826e7fb0a44d6 | Two equal-radius circles \( T1 \) and \( T2 \) are centered at points \( O1 \) and \( O2 \). A point \( X \) is reflected through \( O1 \) and \( O2 \) to get points \( A1 \) and \( A2 \). The tangents from \( A1 \) to \( T1 \) touch \( T1 \) at points \( P1 \) and \( Q1 \), and the tangents from \( A2 \) to \( T2 \) t... | [
"1"
] | [
"Two equal-radius circles \\( T1 \\) and \\( T2 \\) are centered at points \\( O1 \\) and \\( O2 \\). A point \\( X \\) is reflected through \\( O1 \\) and \\( O2 \\) to get points \\( A1 \\) and \\( A2 \\). The tangents from \\( A1 \\) to \\( T1 \\) touch \\( T1 \\) at points \\( P1 \\) and \\( Q1 \\), and the tan... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-34afdb823920b436082a7bd9ed2d3ae98270d6e8d4950b9a2833469a81706e5c | Two equal-sized regular $n$-gons intersect to form a $2n$-gon $C$. If the side length of each $n$-gon is $s$, calculate the sum of the lengths of the sides of $C$ which form part of one $n$-gon. Express your answer in terms of $s$ and $n$. | [
"n s \\tan\\left(\\frac{\\pi}{2n}\\right)"
] | [
"Two equal-sized regular $n$-gons intersect to form a $2n$-gon $C$. If the side length of each $n$-gon is $s$, calculate the sum of the lengths of the sides of $C$ which form part of one $n$-gon. Express your answer in terms of $s$ and $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-075f07fdaf39db80bcaa03a29c333ff3e60d1d1205c7fe8d9c0e2299ac96ec08 | Two equally big squares with sides of 12 cm partly cover each other. One square's corner is placed at the center of the other square. Determine the area of the shaded part. | [
"36 cm^2"
] | [
"Two equally big squares with sides of 12 cm partly cover each other. One square's corner is placed at the center of the other square. Determine the area of the shaded part."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-08a0b98c347c08a7cbce22c294d47cb6ecaa4a3673255704ae7aaff14499bbc1 | Two equally oriented regular $2n$-gons $A_1A_2\ldots A_{2n}$ and $B_1B_2\ldots B_{2n}$ are given. The perpendicular bisectors $\ell_i$ of the segments $A_iB_i$ are drawn. Let the lines $\ell_i$ and $\ell_{i+1}$ intersect at the point $K_i$ (indices are reduced modulo $2n$). Denote by $m_i$ the line $K_iK_{i+n}$. Let $P... | [
"180"
] | [
"Two equally oriented regular $2n$-gons $A_1A_2\\ldots A_{2n}$ and $B_1B_2\\ldots B_{2n}$ are given. The perpendicular bisectors $\\ell_i$ of the segments $A_iB_i$ are drawn. Let the lines $\\ell_i$ and $\\ell_{i+1}$ intersect at the point $K_i$ (indices are reduced modulo $2n$). Denote by $m_i$ the line $K_iK_{i+n... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-129db7daef345ee6d60d58afc532aac83f8ebce6fae701bc73120f53750c92b7 | Two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player has a probability of $\frac{1}{2}$ of winning each game. What is the expected value of the number of games they will play? | [
"14"
] | [
"Two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player has a probability of $\\frac{1}{2}$ of winning each game. What is the expected value of the number of games they will play?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aacc1bb2deb8a8fd3c89e10cfaf501a8c7124b9525c9b083bc91d5f9ecc5b8c7 | Two equally strong tennis players, A and B, bet $50$ euros and play one set, where the player that wins six games first wins and receives the pot of $100$ euros. After the seventh game, the score is $4–3$ and player A has the advantage. They agree to play some (possibly random) number $M$ of extra 'independent' games i... | [
"11"
] | [
"Two equally strong tennis players, A and B, bet $50$ euros and play one set, where the player that wins six games first wins and receives the pot of $100$ euros. After the seventh game, the score is $4–3$ and player A has the advantage. They agree to play some (possibly random) number $M$ of extra 'independent' ga... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a7c82a3ec9d36cf0becafbca24bc0c46d4145f15bfa1154c3ccc1207ede141e8 | Two equilateral triangles \(ABC\) and \(A'B'C'\) lie on the same circle with circumradius \(R\). The lines \(AA'\), \(BB'\), and \(CC'\) intersect at points \(X\), \(Y\), and \(Z\). If the circumradius of triangle \(XYZ\) is \(r\), find the value of \(\frac{r}{R}\). | [
"\\sqrt{3}"
] | [
"Two equilateral triangles \\(ABC\\) and \\(A'B'C'\\) lie on the same circle with circumradius \\(R\\). The lines \\(AA'\\), \\(BB'\\), and \\(CC'\\) intersect at points \\(X\\), \\(Y\\), and \\(Z\\). If the circumradius of triangle \\(XYZ\\) is \\(r\\), find the value of \\(\\frac{r}{R}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-871489ebdc546ddcfefdca36dd6e8a7434ca5f6f7041425be1d3b8bca3dbbd08 | Two equilateral triangles \(ABC\) and \(CDE\) share a common vertex \(C\). Find the angle between the straight lines \(AD\) and \(BE\). | [
"\\(\\pi/3\\)"
] | [
"Two equilateral triangles \\(ABC\\) and \\(CDE\\) share a common vertex \\(C\\). Find the angle between the straight lines \\(AD\\) and \\(BE\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-3698822e784d76087fd8520111244ab88bf2a64ce2f2ba058f3df673daff0510 | Two equilateral triangles \(ABC\) and \(DEF\), each with side length 1, are drawn in two parallel planes such that when one plane is projected onto the other, the vertices of the triangles form a regular hexagon \(AFBDCE\). Line segments \(AE\), \(AF\), \(BF\), \(BD\), \(CD\), and \(CE\) are drawn, and each of these se... | [
"\\(\\frac{\\sqrt{2}}{3}\\)"
] | [
"Two equilateral triangles \\(ABC\\) and \\(DEF\\), each with side length 1, are drawn in two parallel planes such that when one plane is projected onto the other, the vertices of the triangles form a regular hexagon \\(AFBDCE\\). Line segments \\(AE\\), \\(AF\\), \\(BF\\), \\(BD\\), \\(CD\\), and \\(CE\\) are draw... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 8
} | 17 | 0 | 0 |
omr-ws-716e3d0d24a420716ed302dc1875b89c3d599c5ffa572af10455f60ea4b06aad | Two equilateral triangles are contained in a square whose side length is $2\sqrt{3}$. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus? | [
"\\(8\\sqrt{3} - 12\\)"
] | [
"Two equilateral triangles are contained in a square whose side length is $2\\sqrt{3}$. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 20,
"tir": 14
} | 34 | 0 | 0 |
omr-ws-dc510e5c9f0334300eb3bbef91d3260332746c6d2a7de58cc3cf74d51f8717ec | Two equilateral triangles are contained within a circle of radius 2. The midpoint of a side of the larger triangle is connected to the farthest vertex of the smaller triangle (with side length $1$) to create length $x$. What is $x$? | [
"\\dfrac{3 + \\sqrt{3}}{3}"
] | [
"Two equilateral triangles are contained within a circle of radius 2. The midpoint of a side of the larger triangle is connected to the farthest vertex of the smaller triangle (with side length $1$) to create length $x$. What is $x$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-64b7a730b930e56d80e7542a7114d0d97cc9a29e9e4572eab6fde2f9f3a372db | Two equilateral triangles are inscribed in a circle with radius $r$. Let $A$ be the area of the set consisting of all points interior to both triangles. If $r = 2$, find the minimum possible value of $2A$. | [
"18 - 6\\sqrt{3}"
] | [
"Two equilateral triangles are inscribed in a circle with radius $r$. Let $A$ be the area of the set consisting of all points interior to both triangles. If $r = 2$, find the minimum possible value of $2A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-b6b98327dfe2be0772acfce3e942edb2dcb7d19dbbe44fb9361fe7bff4274cc8 | Two equivalent decks of N different cards each are put into random order and matched against each other. If a card occupies the same position in both decks, we say that a matching has occurred. What is the probability that there is at least one match? | [
"\\( 1 - \\frac{1}{e} \\)"
] | [
"Two equivalent decks of N different cards each are put into random order and matched against each other. If a card occupies the same position in both decks, we say that a matching has occurred. What is the probability that there is at least one match?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 3,
"tir": 3
} | 7 | 0 | 0 |
omr-ws-bc6414d0783deca04bcb6fcac9c07bf93d176fa136506e77d6475953385df22f | Two evenly matched teams play a series of four games with each other. Each game may end in a win, a loss, or a tie, and each outcome is equally likely. What is the probability that each team wins as many games as it loses? | [
"\\(\\frac{19}{81}\\)"
] | [
"Two evenly matched teams play a series of four games with each other. Each game may end in a win, a loss, or a tie, and each outcome is equally likely. What is the probability that each team wins as many games as it loses?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 29
} | 76 | 0 | 0 |
omr-ws-1ef6937d42115681c642526a8a01b4a7870a05754e7c5919be70fbf238def30f | Two excavators were used to dig a trench. The first excavator needs three hours less to dig the whole trench than the second excavator needs. How many hours does it take each excavator working separately to dig the trench if the sum of the two times is $\frac{144}{35}$ of the time (in hours) it would take the two excav... | [
"7.5 hours and 10.5 hours"
] | [
"Two excavators were used to dig a trench. The first excavator needs three hours less to dig the whole trench than the second excavator needs. How many hours does it take each excavator working separately to dig the trench if the sum of the two times is $\\frac{144}{35}$ of the time (in hours) it would take the two... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7ee67db214c1396f4d78dbab8a9109175bf4067bc2df338ffcb05c58c8a0fd3c | Two excircles of triangle \(ABC\) are tangent to \(AB\) and \(AC\) at points \(P\) and \(Q\), respectively. Let \(L\) and \(M\) denote the midpoints of \(PQ\) and \(BC\), respectively. Let \(L_1\) and \(L_2\) be the reflections of point \(L\) about the midpoints of \(BM\) and \(CM\), respectively. Calculate the value o... | [
"14"
] | [
"Two excircles of triangle \\(ABC\\) are tangent to \\(AB\\) and \\(AC\\) at points \\(P\\) and \\(Q\\), respectively. Let \\(L\\) and \\(M\\) denote the midpoints of \\(PQ\\) and \\(BC\\), respectively. Let \\(L_1\\) and \\(L_2\\) be the reflections of point \\(L\\) about the midpoints of \\(BM\\) and \\(CM\\), re... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f5d33a0e98dfdc42a2a91de68c835dddfd3377bd89a9117598d34854cd4f440 | Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon... | [
"140"
] | [
"Two externally tangent circles $\\omega_1$ and $\\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\\Omega$ passing through $O_1$ and $O_2$ intersects $\\omega_1$ at $B$ and $C$ and $\\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a conv... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2451bbbe5e0f49a55210d01a5c0c6fc04478f4cff4d279dc97bd68ed53a7ca6e | Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$. Given that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon, find the a... | [
"120"
] | [
"Two externally tangent circles $\\omega_1$ and $\\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\\Omega$ passing through $O_1$ and $O_2$ intersects $\\omega_1$ at $B$ and $C$ and $\\omega_2$ at $A$ and $D$. Given that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon, ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-3a5cdec49849aa128bebbc1160fcc6062fe56725247a968a478b05db7662d571 | Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$. Given that $AB = 2$, $O_1O_2 = 15$, and $CD = 16$, find the area of the convex hexagon $ABO_1CD... | [
"120"
] | [
"Two externally tangent circles $\\omega_1$ and $\\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\\Omega$ passing through $O_1$ and $O_2$ intersects $\\omega_1$ at $B$ and $C$ and $\\omega_2$ at $A$ and $D$. Given that $AB = 2$, $O_1O_2 = 15$, and $CD = 16$, find the area of the convex hexago... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-bbb8cc18da16243a4746bd93ed2f21b21bbe22c0407afee0c7c9746a570adc84 | Two externally tangent circles \(\omega_1\) and \(\omega_2\) have centers \(O_1\) and \(O_2\), respectively. A third circle \(\Omega\) passing through \(O_1\) and \(O_2\) intersects \(\omega_1\) at \(B\) and \(C\) and \(\omega_2\) at \(A\) and \(D\). Given that \(AB = 2\), \(O_1O_2 = 15\), \(CD = 16\), and \(ABO_1CDO_2... | [
"120"
] | [
"Two externally tangent circles \\(\\omega_1\\) and \\(\\omega_2\\) have centers \\(O_1\\) and \\(O_2\\), respectively. A third circle \\(\\Omega\\) passing through \\(O_1\\) and \\(O_2\\) intersects \\(\\omega_1\\) at \\(B\\) and \\(C\\) and \\(\\omega_2\\) at \\(A\\) and \\(D\\). Given that \\(AB = 2\\), \\(O_1O_... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-7f249bcfff3ff40e1bafb77c848d861de711e28295c4cec53a388b04983f002a | Two externally tangent circles have radii 2 and 3. Two lines are drawn, each tangent to both circles but not at the point where the circles are tangent to each other. What is the area of the quadrilateral whose vertices are the four points of tangency between the circles and the lines? | [
"\\(\\frac{48\\sqrt{6}}{5}\\)",
"\\dfrac{48\\sqrt{6}}{5}"
] | [
"Two externally tangent circles have radii 2 and 3. Two lines are drawn, each tangent to both circles but not at the point where the circles are tangent to each other. What is the area of the quadrilateral whose vertices are the four points of tangency between the circles and the lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 14,
"tir": 13
} | 52 | 0 | 0 |
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