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-b5d0aa8b82fe2f8752154991f7b563a7eededee52663a58bef37b8fc3274dd5d | Two externally tangent circles with different radii are given. Their common tangents form a triangle. Find the area of this triangle in terms of the radii of the two circles. | [
"\\(\\frac{2Rr\\sqrt{Rr}}{R-r}\\)"
] | [
"Two externally tangent circles with different radii are given. Their common tangents form a triangle. Find the area of this triangle in terms of the radii of the two circles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 7,
"tir": 5
} | 48 | 0 | 0 |
omr-ws-eec7a76f6af20f68c91ca5edf5a97d0dbd48215e4922e3e0fd9fbea1d1e1262e | Two externally tangent congruent spheres are contained in a cube with edge length 1. Each sphere is tangent to exactly three faces of the cube. What is the radius of one of the spheres? | [
"\\(\\frac{3 - \\sqrt{3}}{4}\\)"
] | [
"Two externally tangent congruent spheres are contained in a cube with edge length 1. Each sphere is tangent to exactly three faces of the cube. What is the radius of one of the spheres?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-e92dcb49952e5f3a1a7d4bfd7cbdbfb3ea993e36a44a0c81f07bddb124bb38c0 | Two externally tangent unit circles are constructed inside square \(ABCD\), one tangent to \(AB\) and \(AD\), the other to \(BC\) and \(CD\). Compute the length of \(AB\). | [
"\\(2 + \\sqrt{2}\\)"
] | [
"Two externally tangent unit circles are constructed inside square \\(ABCD\\), one tangent to \\(AB\\) and \\(AD\\), the other to \\(BC\\) and \\(CD\\). Compute the length of \\(AB\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-c33a9ec476a717c7a3682bcc627c842ce417bb0123d8075706977ac39e50bb40 | Two factories have to process some raw material. If the output of the second factory were doubled, the time needed for the two factories to fulfil the assignment would be $\frac{2}{15}$ of the time needed for the first factory to complete the work alone. Which factory has more output and how many times more if each fac... | [
"The productivity of the second factory is twice the productivity of the first"
] | [
"Two factories have to process some raw material. If the output of the second factory were doubled, the time needed for the two factories to fulfil the assignment would be $\\frac{2}{15}$ of the time needed for the first factory to complete the work alone. Which factory has more output and how many times more if ea... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a84285b2407f1ccb91dd6d99dc16754cff5aeb3498cafd1098d4ad9a55eb7f3d | Two factories, $A$ and $B$, undertook to fulfil an order in 12 days. After two days factory $A$ was closed down for repairs, while factory $B$ continued fulfilment of the order. Knowing that $B$ has an efficiency of $66\frac{2}{3}\%$ of that of $A$, determine in how many days the order will be completed. | [
"The order will be completed in $27=(25+2)$ days"
] | [
"Two factories, $A$ and $B$, undertook to fulfil an order in 12 days. After two days factory $A$ was closed down for repairs, while factory $B$ continued fulfilment of the order. Knowing that $B$ has an efficiency of $66\\frac{2}{3}\\%$ of that of $A$, determine in how many days the order will be completed."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a5d5db69f3c24ef5ddb7e90ce53002ffee4535c3d74d81c07cf227e7eba4096f | Two fair 6-sided dice are rolled. What is the probability at least one of the dice shows a 1? | [
"\\dfrac{11}{36}"
] | [
"Two fair 6-sided dice are rolled. What is the probability at least one of the dice shows a 1?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a7ee327783fe824ed5d7989f0ffe55dd374ef5e3ef0d573783253bd340ca1b97 | Two fair dice are numbered 1, 2, 3, 5, 7, and 8. What is the probability that their sum will be an even number? | [
"\\dfrac{5}{9}"
] | [
"Two fair dice are numbered 1, 2, 3, 5, 7, and 8. What is the probability that their sum will be an even number?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-07cdc8cf0aa580080addd3e8ea50fbc672706fd03aed8083771dffd6f176d07d | Two fair dice each with faces numbered 1 through 6 are rolled at the same time. Each die has five exposed faces, which are summed. What is the probability that the least common multiple of the sums of the exposed faces is a multiple of 6? | [
"\\(\\frac{5}{12}\\)"
] | [
"Two fair dice each with faces numbered 1 through 6 are rolled at the same time. Each die has five exposed faces, which are summed. What is the probability that the least common multiple of the sums of the exposed faces is a multiple of 6?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-a5d73d933594a577f1ed09b42a4a8308a7ab120cf27b7cd2d18b222d296d48d2 | Two fair dice were rolled and the sum of the obtained numbers was even. What is the probability that both numbers were equal? | [
"\\(\\frac{1}{3}\\)"
] | [
"Two fair dice were rolled and the sum of the obtained numbers was even. What is the probability that both numbers were equal?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-04c7ebbb908d9d9bf66b79452b3c9f84345008bdbdd31f5a8e3ff851024eeeda | Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac{3}{4}$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\f... | [
"17"
] | [
"Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\\frac{3}{4}$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-69ce17dfba8f7a39d8b8e8bef16391701e7caee24c299cf82aea615a6e4c0e52 | Two fair dice, each with at least 6 faces, are rolled. On each face of each die is printed a distinct integer from 1 to the number of faces on that die, inclusive. The probability of rolling a sum of 7 is $\frac{3}{4}$ of the probability of rolling a sum of 10, and the probability of rolling a sum of 12 is $\frac{1}{12... | [
"17"
] | [
"Two fair dice, each with at least 6 faces, are rolled. On each face of each die is printed a distinct integer from 1 to the number of faces on that die, inclusive. The probability of rolling a sum of 7 is $\\frac{3}{4}$ of the probability of rolling a sum of 10, and the probability of rolling a sum of 12 is $\\fra... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 30
} | 62 | 0 | 0 |
omr-ws-b11ce4f79ad2d0b688a03621f1be4142efe4712a892e0502e3b6a1fbe963827d | Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 11? | [
"\\frac{11}{12}"
] | [
"Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 11?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-91fee1fd884631674879857684eca10da456d86e2db95c9ea3e87f503038ef10 | Two families of parallel lines are given in the plane, consisting of 15 and 11 lines, respectively. In each family, any two neighboring lines are at a unit distance from one another; the lines of the first family are perpendicular to the lines of the second family. Let \( V \) be the set of 165 intersection points of t... | [
"605"
] | [
"Two families of parallel lines are given in the plane, consisting of 15 and 11 lines, respectively. In each family, any two neighboring lines are at a unit distance from one another; the lines of the first family are perpendicular to the lines of the second family. Let \\( V \\) be the set of 165 intersection poin... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-0343f44f5b578d223f44673200d5cd6f2d6c42cacea3c2e1d1a33fe0aa181eaa | Two farmers agree that pigs are worth $\$300$ and that goats are worth $\$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $\$390$ debt could be paid with two pigs, with one goat received in change.) What i... | [
"\\$30"
] | [
"Two farmers agree that pigs are worth $\\$300$ and that goats are worth $\\$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $\\$390$ debt could be paid with two pigs, with one goat received in change.... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a68b338f557e63c0165da522a2bc81fa4e760d33e699ffc6590c6ef5f40400f2 | Two farmers, Johnny and Mac, are selling their apples at a market. Johnny sells apples at the price of three for a dollar, and Mac sells apples at the price of two for a dollar. Each has thirty apples left to sell when they ask Granny to sell the rest of their apples at a price of five for two dollars. Granny sells all... | [
"The extra dollar was lost because the combined selling price undervalued Mac's apples more than it overvalued Johnny's, creating a net deficit of \\$1."
] | [
"Two farmers, Johnny and Mac, are selling their apples at a market. Johnny sells apples at the price of three for a dollar, and Mac sells apples at the price of two for a dollar. Each has thirty apples left to sell when they ask Granny to sell the rest of their apples at a price of five for two dollars. Granny sell... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4a62ee94c46b9a9a7d55008c6814a47097610b66aeb0801075f4ececc01e8c8b | Two ferry boats start moving at the same time from opposite sides of the Hudson River, one boat going from NY to NJ and the other from NJ to NY. They meet at a point 720 yards from the nearest shore. After arriving at their destinations, each boat remains for 10 minutes to change passengers before starting on the retur... | [
"1760 yards"
] | [
"Two ferry boats start moving at the same time from opposite sides of the Hudson River, one boat going from NY to NJ and the other from NJ to NY. They meet at a point 720 yards from the nearest shore. After arriving at their destinations, each boat remains for 10 minutes to change passengers before starting on the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 5
} | 22 | 0 | 0 |
omr-ws-dc3554d466756930d4490c86d95085886c367e454657136e8f6fef3c6bd5b74a | Two ferryboats leave at the same time from opposite banks of a river. One is faster than the other and they meet at 720 meters from the nearest bank. Once they arrive at the opposite bank, each ferryboat stops for 10 minutes and then goes back. On the return trip, the ferryboats meet again but at 400 meters from the ne... | [
"1760 meters"
] | [
"Two ferryboats leave at the same time from opposite banks of a river. One is faster than the other and they meet at 720 meters from the nearest bank. Once they arrive at the opposite bank, each ferryboat stops for 10 minutes and then goes back. On the return trip, the ferryboats meet again but at 400 meters from t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 39,
"genselect": 5,
"tir": 5
} | 49 | 0 | 0 |
omr-ws-b0d0557377f4524d2c7c824e3c8b18ceb2202f48843f8fef979bebc51ef4573a | Two figures \( F \) and \( G \) have a center of half-turn at point \( O \). Given that the area of \( F \cap G \) is 24 square units, find the total area of the region that includes \( F \cup G \) but excludes \( F \cap G \) if the area of \( F \) is 40 square units and the area of \( G \) is 36 square units. | [
"28"
] | [
"Two figures \\( F \\) and \\( G \\) have a center of half-turn at point \\( O \\). Given that the area of \\( F \\cap G \\) is 24 square units, find the total area of the region that includes \\( F \\cup G \\) but excludes \\( F \\cap G \\) if the area of \\( F \\) is 40 square units and the area of \\( G \\) is 3... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-771aa99c6843c52c078b2b7cd3fd22c27353b6b4e4fa3421bb2b4318ef12e79b | Two finite systems of arcs are given on a circle of length $1$. The total length of the arcs of each system is $1/2$. Rotate one of the systems around the center of the circle to maximize the total length of its intersection with the arcs of the second system. If $L$ is the maximum total length of intersection achieved... | [
"1"
] | [
"Two finite systems of arcs are given on a circle of length $1$. The total length of the arcs of each system is $1/2$. Rotate one of the systems around the center of the circle to maximize the total length of its intersection with the arcs of the second system. If $L$ is the maximum total length of intersection ach... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a7525b9d65760e392be4545c0e85b3e9f34b3c17474f7197771987cebb84a8e6 | Two fishermen each had a certain number of trophies. The first fisherman said to the second, "If you give me 7 of your trophies, I will have twice as many trophies as you." The second fisherman said, "If you give me 7 of your trophies, I will have as many as you." How many trophies does each fisherman have? | [
"First fisherman = 49, Second fisherman = 35"
] | [
"Two fishermen each had a certain number of trophies. The first fisherman said to the second, \"If you give me 7 of your trophies, I will have twice as many trophies as you.\" The second fisherman said, \"If you give me 7 of your trophies, I will have as many as you.\" How many trophies does each fisherman have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 27
} | 28 | 0 | 0 |
omr-ws-b875aae02ea32ddf7c509bfeb23f7678f26e92be3fdb32d649caaf99d2d729c8 | Two fixed circles $\omega_1$ and $\omega_2$ pass through point $O$. A circle of radius $R$ centered at $O$ meets $\omega_1$ at points $A$ and $B$, and meets $\omega_2$ at points $C$ and $D$. Let $X$ be the common point of lines $AC$ and $BD$. If $\omega_1$ and $\omega_2$ have radii $r_1$ and $r_2$ respectively, find th... | [
"r_1 x - r_2 y = 0"
] | [
"Two fixed circles $\\omega_1$ and $\\omega_2$ pass through point $O$. A circle of radius $R$ centered at $O$ meets $\\omega_1$ at points $A$ and $B$, and meets $\\omega_2$ at points $C$ and $D$. Let $X$ be the common point of lines $AC$ and $BD$. If $\\omega_1$ and $\\omega_2$ have radii $r_1$ and $r_2$ respective... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eac2b4857fa44e5eaed67467604396f6c8992c19bf8222960e9be4b81180ecda | Two fixed circles are touched by a variable circle at points \( P \) and \( Q \). Let \( A \) and \( B \) be the centers of the two fixed circles. If the radii of the fixed circles are \( r_1 \) and \( r_2 \) respectively, and the distance between \( A \) and \( B \) is \( d \), find the product of the distances from \... | [
"r_1 r_2"
] | [
"Two fixed circles are touched by a variable circle at points \\( P \\) and \\( Q \\). Let \\( A \\) and \\( B \\) be the centers of the two fixed circles. If the radii of the fixed circles are \\( r_1 \\) and \\( r_2 \\) respectively, and the distance between \\( A \\) and \\( B \\) is \\( d \\), find the product ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-d797712e9e67f4b2103d43edfc76de140b53e984d94660f7708b427f0efe1a97 | Two fixed circles touch each other at point A. If a variable line through A intersects the circles at points P and Q, find the value of \(\frac{AP}{AQ}\) given that the radii of the two circles are 3 and 5, respectively. | [
"\\dfrac{3}{5}"
] | [
"Two fixed circles touch each other at point A. If a variable line through A intersects the circles at points P and Q, find the value of \\(\\frac{AP}{AQ}\\) given that the radii of the two circles are 3 and 5, respectively."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-713eec1f5d7693d917ee8215f249940e31d039fe9b6401ad23a4c63b193a1dac | Two fixed circles with centers $O_1$ and $O_2$ and radii $r_1$ and $r_2$ respectively are given. Let $S$ be the center of similitude of these two circles. If the distance from $S$ to $O_1$ is $d_1$ and the distance from $S$ to $O_2$ is $d_2$, find the value of $\frac{d_1}{d_2}$ in terms of $r_1$ and $r_2$. | [
"\\dfrac{r_1}{r_2}"
] | [
"Two fixed circles with centers $O_1$ and $O_2$ and radii $r_1$ and $r_2$ respectively are given. Let $S$ be the center of similitude of these two circles. If the distance from $S$ to $O_1$ is $d_1$ and the distance from $S$ to $O_2$ is $d_2$, find the value of $\\frac{d_1}{d_2}$ in terms of $r_1$ and $r_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-386c79148cdfa62dabf5544449dd4e18e4a71b210561d7314d8e432dff1b1943 | Two fixed lines \( l_1 \) and \( l_2 \) are perpendicular to each other at a point \( Y \). Points \( X \) and \( O \) are on \( l_2 \) and both are on one side of line \( l_1 \). We draw the circle \( \omega \) with center \( O \) and radius \( OY \). A variable point \( Z \) is on line \( l_1 \). Line \( OZ \) cuts c... | [
"The locus of the point \\( S \\) is a parabola with the focal axis \\( l_2 \\), vertex at the midpoint of \\( \\overline{XY} \\), and the length of the latus rectum \\( \\frac{2R^2}{XY} \\). \n\nSince the problem asks for the locus and not a numerical value or a specific mathematical expression, the answer is best... | [
"Two fixed lines \\( l_1 \\) and \\( l_2 \\) are perpendicular to each other at a point \\( Y \\). Points \\( X \\) and \\( O \\) are on \\( l_2 \\) and both are on one side of line \\( l_1 \\). We draw the circle \\( \\omega \\) with center \\( O \\) and radius \\( OY \\). A variable point \\( Z \\) is on line \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 2,
"tir": 2
} | 19 | 0 | 0 |
omr-ws-92ad709152a0cbb7ed30c7ac4d22fa0fdd54afcdb9c921ca81cc9d388c108113 | Two fixed points $A$ and $B$ and a moving point $M$ are taken on a circle. On the extension of the line segment $AM$ a segment $MN = MB$ is laid off outside the circle. Find the locus of points $N$. | [
"two circular arcs: the arc \\( BE \\) and the arc \\( BF \\)"
] | [
"Two fixed points $A$ and $B$ and a moving point $M$ are taken on a circle. On the extension of the line segment $AM$ a segment $MN = MB$ is laid off outside the circle. Find the locus of points $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9fac0a3f7b3ffa0238b706975cf764aff119c3b69a85fb3e1818963cb70553f6 | Two flagpoles are placed a distance \( x \) apart. The height of the first pole is \( a \) and the height of the second pole is \( b \). A string is tied from the top of the first pole to the bottom of the second pole, and another string is tied from the top of the second pole to the bottom of the first pole. Express t... | [
"\\dfrac{ab}{a + b}"
] | [
"Two flagpoles are placed a distance \\( x \\) apart. The height of the first pole is \\( a \\) and the height of the second pole is \\( b \\). A string is tied from the top of the first pole to the bottom of the second pole, and another string is tied from the top of the second pole to the bottom of the first pole... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d266be67d38431cd86526bd2a68241dec0f4647977343f19bc31bae7b93c2d6e | Two flashing lights at an intersection flash 40 times and 60 times every twenty seconds, respectively. If they start flashing together, how many times will they flash at the same time after 4 hours? | [
"14400"
] | [
"Two flashing lights at an intersection flash 40 times and 60 times every twenty seconds, respectively. If they start flashing together, how many times will they flash at the same time after 4 hours?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-ae7a4ade4fbf8b32c837ee10ab811a842ce85c5d1881367d8ab0e5d74c85787e | Two forces, one of 120 lbs and the other of 200 lbs, act on the same object at angles 30 and -30, respectively, with the positive x-axis. Find the direction of the resultant of these two forces. | [
"-8.21^\\circ"
] | [
"Two forces, one of 120 lbs and the other of 200 lbs, act on the same object at angles 30 and -30, respectively, with the positive x-axis. Find the direction of the resultant of these two forces."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a682a7a4a9e3a36bc29d2cac3fbb1bc416ccd0a1b613e70e2c61f214c4b6ca39 | Two four-digit numbers $ACCC$ and $CCCB$ satisfy $\frac{ACCC}{CCCB}=\frac{2}{5}$. Find the value of $A \times B \times C$. | [
"60"
] | [
"Two four-digit numbers $ACCC$ and $CCCB$ satisfy $\\frac{ACCC}{CCCB}=\\frac{2}{5}$. Find the value of $A \\times B \\times C$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-cf1b04156774674e96940d13d8e6d560591705befc447f2816ab5ee024bf1383 | Two friends $A$ and $B$ are initially at points $(0,0)$ and $(12,7)$ respectively on an infinite grid plane. $A$ takes steps of size $4$ units and $B$ takes steps of size $6$ units along the grid lines. Let $P(x, y)$ be a point on the grid. Determine the number of distinct points $P(x, y)$ that $A$ can reach such that ... | [
"0"
] | [
"Two friends $A$ and $B$ are initially at points $(0,0)$ and $(12,7)$ respectively on an infinite grid plane. $A$ takes steps of size $4$ units and $B$ takes steps of size $6$ units along the grid lines. Let $P(x, y)$ be a point on the grid. Determine the number of distinct points $P(x, y)$ that $A$ can reach such ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-026d3b43abb7ab86a616807b74f850a955e7902179f85a9c3bce6358e15d654b | Two friends $A$ and $B$ are removing stones from a stack. In each turn, they can remove either 2, 3, or 4 stones. The one who cannot make a move loses. $A$ plays first. For what values of the number of stones will $A$ win? | [
"\\(6n + k\\) where \\(k = 2, 3, 4, 5\\) and \\(n\\) is an integer."
] | [
"Two friends $A$ and $B$ are removing stones from a stack. In each turn, they can remove either 2, 3, or 4 stones. The one who cannot make a move loses. $A$ plays first. For what values of the number of stones will $A$ win?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e0dfea15ff088f86abd3cb279360ac842e472d774976227a24e14bbf76454985 | Two friends $A$ and $B$ each receive a number from the set $\{1, 2, \ldots, 250\}$. They take turns announcing positive integers such that the sum of all announced numbers is 20. Demonstrate that there exists a strategy that $A$ and $B$ can agree on beforehand to ensure that each can determine the other's number. | [
"20"
] | [
"Two friends $A$ and $B$ each receive a number from the set $\\{1, 2, \\ldots, 250\\}$. They take turns announcing positive integers such that the sum of all announced numbers is 20. Demonstrate that there exists a strategy that $A$ and $B$ can agree on beforehand to ensure that each can determine the other's numbe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-e8331a4338c94df98051fdc8a39fd5aec9f21c8599f7aa83e1d3f5cf94b1c89f | Two friends A and B each receive a number from the set $\{1, 2, ..., 250\}$ without knowing each other's number. They must guess the number of their friend by taking turns to announce positive integers, with the sum of all announced numbers being 20. If A and B follow an optimal strategy, what is the maximum number of ... | [
"62500"
] | [
"Two friends A and B each receive a number from the set $\\{1, 2, ..., 250\\}$ without knowing each other's number. They must guess the number of their friend by taking turns to announce positive integers, with the sum of all announced numbers being 20. If A and B follow an optimal strategy, what is the maximum num... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-bd0b60be231262e0dd7b30aa934dbdd23244b5c5671536c2f687938f3266b6ba | Two friends agree to meet between noon and 1:00 PM. Each arrives at a random time in the interval and will wait up to 15 minutes for the other before leaving. What is the probability that they meet? | [
"\\dfrac{7}{16}"
] | [
"Two friends agree to meet between noon and 1:00 PM. Each arrives at a random time in the interval and will wait up to 15 minutes for the other before leaving. What is the probability that they meet?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a0d4272b29668d85b97b69b7eea81cf15951999c460ea70bf504d2225f875249 | Two friends are playing a game on a board that initially contains the numbers 1, 2, and 3. On each turn, a player writes a number that does not exceed 2022, is not already on the board, and is either the sum or the product of two numbers already on the board. The player who writes 2022 on the board wins. Which player h... | [
"\\text{The second player has a winning strategy.}"
] | [
"Two friends are playing a game on a board that initially contains the numbers 1, 2, and 3. On each turn, a player writes a number that does not exceed 2022, is not already on the board, and is either the sum or the product of two numbers already on the board. The player who writes 2022 on the board wins. Which pla... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-4c1e12d12dd6b9d4f9dd8c61248de38f96fff4dea818430bc5ad860f21e91d7c | Two friends are playing tic-tac-toe. If Amy wins $\frac{3}{8}$ of the time, Lily wins $\frac{3}{10}$ of the time, and they tie the rest of the time, then what fraction of the time do they tie? | [
"\\frac{13}{40}"
] | [
"Two friends are playing tic-tac-toe. If Amy wins $\\frac{3}{8}$ of the time, Lily wins $\\frac{3}{10}$ of the time, and they tie the rest of the time, then what fraction of the time do they tie?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-52f1cffa80e958305df2a90520f37817afabfdc19028647d114bc6b1586d07c8 | Two friends get home at 2:30 P.M. and go to bed at 10 P.M. Each friend will go to a park twice for 15 minutes during this time. What is the probability that they will be able to see each other at any random time between these two time intervals? Assume that the travel to and from the park takes no time. | [
"\\dfrac{113}{225}"
] | [
"Two friends get home at 2:30 P.M. and go to bed at 10 P.M. Each friend will go to a park twice for 15 minutes during this time. What is the probability that they will be able to see each other at any random time between these two time intervals? Assume that the travel to and from the park takes no time."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 1
} | 1 | 0 | 0 |
omr-ws-de5190dfc3f6277ab2ea2be4a0d5e5afef2ee1cc6216e3aad3b7a3963820ea69 | Two friends get home at 2:30 P.M. and go to bed at 10 P.M. Each friend will visit the park twice for 15 minutes each visit. What is the probability that they will see each other at any random time during their park visits? (Assume travel time to and from the park is negligible.) | [
"\\dfrac{27}{196}"
] | [
"Two friends get home at 2:30 P.M. and go to bed at 10 P.M. Each friend will visit the park twice for 15 minutes each visit. What is the probability that they will see each other at any random time during their park visits? (Assume travel time to and from the park is negligible.)"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-0a18bec801c7399bc7782ea29fc46002d8608c4816ed733cc7cfeb62f0ea5ae9 | Two friends live 4 miles apart on the same "east-west" street, and you live halfway between them. You hear an explosion, your friend to the east hears it 6 seconds later, and your friend to the north hears it 8 seconds after you. Find the equations of two hyperbolas that would locate the explosion. Sound travels at a r... | [
"$\\sqrt{(x+21120)^{2}+y^{2}}-\\sqrt{x^{2}+y^{2}}= 6600$ and $\\sqrt{(x-21120)^{2}+y^{2}}-\\sqrt{x^{2}+y^{2}}=8800$"
] | [
"Two friends live 4 miles apart on the same \"east-west\" street, and you live halfway between them. You hear an explosion, your friend to the east hears it 6 seconds later, and your friend to the north hears it 8 seconds after you. Find the equations of two hyperbolas that would locate the explosion. Sound travels... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b57efaddbffd50f07e9b291ae7a3b5d1c0965185ce321aa909a678ff99192b51 | Two friends play a game starting with the number $2015$ in a table. In each turn, a player writes the difference between the current number and one of its factors. The game is lost by the player who reaches $0$. Which player can secure victory? | [
"The second player can secure victory."
] | [
"Two friends play a game starting with the number $2015$ in a table. In each turn, a player writes the difference between the current number and one of its factors. The game is lost by the player who reaches $0$. Which player can secure victory?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-58dbba94b8f12c71ec8482900079b002dfe9f4bfeb60564a29e7ee961eb0b8ed | Two friends start at the same point in NYC and bike apart. One friend bikes south at 10 mph for 30 minutes, while the other friend bikes west at 15 mph continuously. Write a function of time, in hours, for the distance between the two friends. | [
"\\[ f(t) = \\begin{cases} \n5t\\sqrt{13} & \\text{if } t \\leq \\frac{1}{2} \\\\\n5\\sqrt{9t^2 + 1} & \\text{if } t \\geq \\frac{1}{2} \n\\end{cases} \\]"
] | [
"Two friends start at the same point in NYC and bike apart. One friend bikes south at 10 mph for 30 minutes, while the other friend bikes west at 15 mph continuously. Write a function of time, in hours, for the distance between the two friends."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-737ae2625932e1d6458476cdf80d1034a9d76587b3aab1008867e0cf5456e537 | Two friends take turns picking chocolates from a pile of $n$ chocolates. Each turn, a player can take either 1, 3, or 8 chocolates. Determine for which values of $n$ the player who starts the game will be the winner. | [
"n \\not\\equiv 0, 2, 4, 6 \\pmod{11}"
] | [
"Two friends take turns picking chocolates from a pile of $n$ chocolates. Each turn, a player can take either 1, 3, or 8 chocolates. Determine for which values of $n$ the player who starts the game will be the winner."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-46403326ada1cd2f97fa214796e4df35dc8ed5641737d6a95a4368f157b9013b | Two friends walk towards each other on a straight road at constant but different speeds. Each friend releases a dog that runs at the same constant speed towards the other person and then returns to its owner. Which dog returns to its owner first: the dog from the faster walker or the dog from the slower walker? | [
"\\text{Both dogs return to their owners at the same time.}"
] | [
"Two friends walk towards each other on a straight road at constant but different speeds. Each friend releases a dog that runs at the same constant speed towards the other person and then returns to its owner. Which dog returns to its owner first: the dog from the faster walker or the dog from the slower walker?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-6d77c819150297c27d5bca0b9134fcbfec00008ecc4fecb79179350cb61cee8f | Two friends, Alexandra and Brianna, live at the corners marked by $A$ and $B$ on a grid map of Squareville, where each city block is of the same length. They start walking toward each other's house at the same time, with the same speed, and independently choose a path to the other's house with uniform distribution out ... | [
"\\(\\frac{517}{1152}\\)"
] | [
"Two friends, Alexandra and Brianna, live at the corners marked by $A$ and $B$ on a grid map of Squareville, where each city block is of the same length. They start walking toward each other's house at the same time, with the same speed, and independently choose a path to the other's house with uniform distribution... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-05db4315cefcb621cb131a3028dd4335a8c8b51cd4cad079eeff7dfc8866ea74 | Two friends, Marco and Ian, are talking about their ages. Ian says, "My age is a zero of a polynomial with integer coefficients." Having seen the polynomial \(p(x)\) Ian was talking about, Marco exclaims, "You mean, you are seven years old? Oops, sorry I miscalculated! \(p(7) = 77\) and not zero." "Yes, I am older than... | [
"14"
] | [
"Two friends, Marco and Ian, are talking about their ages. Ian says, \"My age is a zero of a polynomial with integer coefficients.\" Having seen the polynomial \\(p(x)\\) Ian was talking about, Marco exclaims, \"You mean, you are seven years old? Oops, sorry I miscalculated! \\(p(7) = 77\\) and not zero.\" \"Yes, I... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 28,
"genselect": 8,
"tir": 2
} | 38 | 0 | 0 |
omr-ws-45851be04951dc72f9f1efbc527688ab77151a1ad8f82b9fc7f2f15e7ee85f96 | Two friends, Mark and James, are playing a game where every time Mark wins a round, he wins the next round with probability \( p \), and if he loses, he wins the next round with probability \( q \). What is the expected score after playing 1,000,000 rounds? | [
"\\(\\frac{1000000q}{1+q-p}\\)"
] | [
"Two friends, Mark and James, are playing a game where every time Mark wins a round, he wins the next round with probability \\( p \\), and if he loses, he wins the next round with probability \\( q \\). What is the expected score after playing 1,000,000 rounds?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-694892d4484b1f7d04b2b5d370bd1e21b78b6351fb9925c0d05eda77e4d3b107 | Two friends, Tom and Peter, play a game where Peter reads out the numbers from 1 to 8 in any order, and Tom writes them down on a piece of paper, forming a 9-digit number by placing each subsequent number either to the left or right of the previously written numbers. Tom aims to maximize the number, while Peter aims to... | [
"912345678"
] | [
"Two friends, Tom and Peter, play a game where Peter reads out the numbers from 1 to 8 in any order, and Tom writes them down on a piece of paper, forming a 9-digit number by placing each subsequent number either to the left or right of the previously written numbers. Tom aims to maximize the number, while Peter ai... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-8aba805f0f76d4787cfce68f1e38e19b2ffbcfa63223529be1de6a8d60d42998 | Two functions \( f, g: \mathbb{N} \to \mathbb{N} \) satisfy the following conditions: 1. For every \( n \in \mathbb{N} \), \( \left| f(n) - n \right| < 2018 \sqrt{n} \). 2. For every \( n \in \mathbb{N} \), \( 2f(n)^2 = n^2 + g(n)^2 \). Let \( S \) be the set of all \( n \in \mathbb{N} \) such that \( f(n) = n \). Find... | [
"254620"
] | [
"Two functions \\( f, g: \\mathbb{N} \\to \\mathbb{N} \\) satisfy the following conditions:\n1. For every \\( n \\in \\mathbb{N} \\), \\( \\left| f(n) - n \\right| < 2018 \\sqrt{n} \\).\n2. For every \\( n \\in \\mathbb{N} \\), \\( 2f(n)^2 = n^2 + g(n)^2 \\).\n\nLet \\( S \\) be the set of all \\( n \\in \\mathbb{N... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-0955efa66ae8c82bf7fe4970ccc911c60071db6d59ad36d7c5984be46942d165 | Two gamblers take turns rolling a fair N-sided die, with N at least 5. The faces of the die are labeled with the numbers 1 to N. The first player wins if they roll an N or an N-1, and the second player wins if they roll a 1, 2, or 3. Play continues, with the players alternating rolls until one of them wins. What is the... | [
"\\(\\frac{2N}{5N-6}\\)"
] | [
"Two gamblers take turns rolling a fair N-sided die, with N at least 5. The faces of the die are labeled with the numbers 1 to N. The first player wins if they roll an N or an N-1, and the second player wins if they roll a 1, 2, or 3. Play continues, with the players alternating rolls until one of them wins. What i... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-795bdfb50b65ff3fa7344b4f3eef03276a47fa8907ee50a22860a2cc2e2bb0cb | Two girls start moving at the same time from the same point A and go in opposite directions on a track whose equation is \( \frac{(x-7)^2}{3} + \frac{(y+19)^2}{5} = 1 \). Their respective speeds are \( 7 \frac{1}{3} \) feet per second and \( 9 \frac{1}{5} \) feet per second. When the girls meet at point A again for the... | [
"123"
] | [
"Two girls start moving at the same time from the same point A and go in opposite directions on a track whose equation is \\( \\frac{(x-7)^2}{3} + \\frac{(y+19)^2}{5} = 1 \\). Their respective speeds are \\( 7 \\frac{1}{3} \\) feet per second and \\( 9 \\frac{1}{5} \\) feet per second. When the girls meet at point ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-0e6933ad0509dfd6d4e102a924c786d88d38afe988b8bbd479156e9745208ab6 | Two given circles \( C_1 \) and \( C_2 \) intersect at points \( A \) and \( B \). Let \( C \) be the second intersection of \( C_2 \) and the diameter of \( C_1 \) passing through \( B \). Let \( D \) be the second intersection of \( C_1 \) and the diameter of \( C_2 \) passing through \( B \). Let \( C_3 \) be the ci... | [
"no_valid_answer_found"
] | [
"Two given circles \\( C_1 \\) and \\( C_2 \\) intersect at points \\( A \\) and \\( B \\). Let \\( C \\) be the second intersection of \\( C_2 \\) and the diameter of \\( C_1 \\) passing through \\( B \\). Let \\( D \\) be the second intersection of \\( C_1 \\) and the diameter of \\( C_2 \\) passing through \\( B... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-79884761204ed2ca89386d67d302829af68eab4d541c04e9e0c0ffcf29cc1adb | Two given circles subtend equal angles at a point \( P \). Find the locus of \( P \). | [
"The radical axis of the two circles."
] | [
"Two given circles subtend equal angles at a point \\( P \\). Find the locus of \\( P \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a460fa3cc67d404d61d91a2ed22cb226d019bd98fb5ed82c95389cae70bebb06 | Two given lines $\ell_1$ and $\ell_2$ intersect at $X$. Two circles $C_1$ and $C_2$ intersect at two points $A$ and $B$ other than $\ell_2$. Line $\ell_1$ intersects $C_1$ at $C$ and $D$, and $C_2$ at $E$ and $F$, respectively. Take an arbitrary point $Y$ on $\ell_2$ and draw the circumcircles $C_3$ of $YCD$ and $C_4$ ... | [
"-3"
] | [
"Two given lines $\\ell_1$ and $\\ell_2$ intersect at $X$. Two circles $C_1$ and $C_2$ intersect at two points $A$ and $B$ other than $\\ell_2$. Line $\\ell_1$ intersects $C_1$ at $C$ and $D$, and $C_2$ at $E$ and $F$, respectively. Take an arbitrary point $Y$ on $\\ell_2$ and draw the circumcircles $C_3$ of $YCD$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-6d057f2a623f785f3050c534d63f62523b216d5dd7a26f30f0117a3cd85a422b | Two graphs $G_1$ and $G_2$ of quadratic polynomials intersect at points $A$ and $B$. Let $O$ be the vertex of $G_1$. Lines $OA$ and $OB$ intersect $G_2$ again at points $C$ and $D$. If the $y$-coordinates of $C$ and $D$ are $y_C$ and $y_D$ respectively, find the value of $y_C - y_D$. | [
"0"
] | [
"Two graphs $G_1$ and $G_2$ of quadratic polynomials intersect at points $A$ and $B$. Let $O$ be the vertex of $G_1$. Lines $OA$ and $OB$ intersect $G_2$ again at points $C$ and $D$. If the $y$-coordinates of $C$ and $D$ are $y_C$ and $y_D$ respectively, find the value of $y_C - y_D$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c7fc61d26ac32bc2c0f72ed2044609265f6ec84add627f880cc0b8953cc71628 | Two grasshoppers sit at opposite ends of the interval $[0, 1]$. A finite number of points (greater than zero) in the interval are marked. In a move, a grasshopper selects a marked point and jumps over it to the equidistant point on the other side, provided it lies within the interval. Alternatively, the grasshopper may... | [
"3"
] | [
"Two grasshoppers sit at opposite ends of the interval $[0, 1]$. A finite number of points (greater than zero) in the interval are marked. In a move, a grasshopper selects a marked point and jumps over it to the equidistant point on the other side, provided it lies within the interval. Alternatively, the grasshoppe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-f24e2f95bd1d2be78f383bfbab2affdced9d8adb9efdcb5ba815bb20cd0e7f65 | Two grasshoppers sit at opposite ends of the interval $[0, 1]$. A finite number of points (greater than zero) in the interval are marked. In a move, each grasshopper must jump over a marked point to the equidistant point on the other side, provided it lies within the interval. After two moves, calculate the maximum num... | [
"2"
] | [
"Two grasshoppers sit at opposite ends of the interval $[0, 1]$. A finite number of points (greater than zero) in the interval are marked. In a move, each grasshopper must jump over a marked point to the equidistant point on the other side, provided it lies within the interval. After two moves, calculate the maximu... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-4bb66f0ed72b67d79abb03443673e12c8db1075a6c43ae40783ed14123011e8c | Two grasshoppers start at points $A$ and $B$ on the number line, where $0 < A, B < 1$. Each minute, both grasshoppers jump to double their current positions (i.e., the first grasshopper jumps to $2A, 4A, 8A, \ldots$ and the second grasshopper jumps to $2B, 4B, 8B, \ldots$). Given that points are colored as follows: eve... | [
"9"
] | [
"Two grasshoppers start at points $A$ and $B$ on the number line, where $0 < A, B < 1$. Each minute, both grasshoppers jump to double their current positions (i.e., the first grasshopper jumps to $2A, 4A, 8A, \\ldots$ and the second grasshopper jumps to $2B, 4B, 8B, \\ldots$). Given that points are colored as follo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-7623afd42345d57c898d538db96d600665746ed930b87885176370c8282dd9b3 | Two groups of tourists start out from $A$ in the direction of $B$ at the same time. The first group goes by bus (at 20 km/h) and reaches $C$, midway between $A$ and $B$, and then continues on foot. The second group starts out on foot, but in one hour boards a car which proceeds at 30 km/h and reaches $B$. The first gro... | [
"30 km"
] | [
"Two groups of tourists start out from $A$ in the direction of $B$ at the same time. The first group goes by bus (at 20 km/h) and reaches $C$, midway between $A$ and $B$, and then continues on foot. The second group starts out on foot, but in one hour boards a car which proceeds at 30 km/h and reaches $B$. The firs... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b0c597e28db718317141139edb180a1b451d3a1316477e1939c259150344e603 | Two half-circles share a common base. Points \( A \) and \( B \) are on the larger half-circle such that the line segment \( AB \) is tangent to the smaller half-circle and \( |AB| = 24 \) cm. Determine the area of the red part, which is the area of the larger half-circle minus the area of the smaller half-circle. | [
"\\( 72 \\pi \\)"
] | [
"Two half-circles share a common base. Points \\( A \\) and \\( B \\) are on the larger half-circle such that the line segment \\( AB \\) is tangent to the smaller half-circle and \\( |AB| = 24 \\) cm. Determine the area of the red part, which is the area of the larger half-circle minus the area of the smaller half... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 14
} | 32 | 0 | 0 |
omr-ws-5a26d7e849dcde100657dad381f1c5ee5c63e25a1fd41835f8a9ed14efb9e80e | Two half-lines \(a\) and \(b\) with a common endpoint \(O\) form an acute angle \(\alpha\). Points \(A\) on \(a\) and \(B\) on \(b\) are such that \(OA = OB\). Let \(b'\) be the line through \(A\) parallel to \(b\). Let \(\beta\) be the circle with center \(B\) and radius \(BO\). A sequence of half-lines \(c_1, c_2, c_... | [
"\\dfrac{\\alpha}{3}"
] | [
"Two half-lines \\(a\\) and \\(b\\) with a common endpoint \\(O\\) form an acute angle \\(\\alpha\\). Points \\(A\\) on \\(a\\) and \\(B\\) on \\(b\\) are such that \\(OA = OB\\). Let \\(b'\\) be the line through \\(A\\) parallel to \\(b\\). Let \\(\\beta\\) be the circle with center \\(B\\) and radius \\(BO\\). A ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13465f12848eab7e540c1cb2c1f1e4a32c193773607e0b15efbcc05dac3031fc | Two half-silvered mirrors are spaced a little distance apart and are parallel. Each mirror passes exactly half the light hitting it and reflects the other half. A laser pointer is shone straight at the mirror pair. What fraction of the light leaving the laser gets through both mirrors, either on the first try or after ... | [
"\\(\\frac{1}{3}\\)"
] | [
"Two half-silvered mirrors are spaced a little distance apart and are parallel. Each mirror passes exactly half the light hitting it and reflects the other half. A laser pointer is shone straight at the mirror pair. What fraction of the light leaving the laser gets through both mirrors, either on the first try or a... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-fa03efbbd2126b03814c0b21981c7bcfce2f627face3912d065ea8ddbfd6ee83 | Two hallways of widths $a$ and $b$ meet at right angles. What is the greatest length of a ladder which can be carried horizontally around the corner? | [
"\\(\\left(a^\\frac{2}{3}+b^\\frac{2}{3}\\right)^\\frac{3}{2}\\)"
] | [
"Two hallways of widths $a$ and $b$ meet at right angles. What is the greatest length of a ladder which can be carried horizontally around the corner?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-e8c229fa4a35127496e11816e09887ab5e79246becc2feb80f37ead488c3a77e | Two hexagons with side length one share a common side. Points A and B are in different hexagons and are as far apart as possible. What is the direct distance from A to B? | [
"$\\sqrt{13}$\n...\n\nFrom the image, we see we can find the distance AB by using the Pythagorean Theorum. We know 1 side is 1 already. We need to find AC, the other side.\n\nIf we draw segment DE perpendicular to AC, since each angle of a regular hexagon is $120^\\circ$, we have two separate 30-60-90 triangles (pr... | [
"Two hexagons with side length one share a common side. Points A and B are in different hexagons and are as far apart as possible. What is the direct distance from A to B?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-9aa7de8898162d02ef4d4b2bb3dd1356aef3dac573706309ede0e57407e7ea91 | Two hikers leave $A$ and $B$ at the same time in the direction towards each other. When they meet, it turns ont that the first hiker has covered $a$ km more than the second. If they continue on their ways at the same rate as before, the first hiker will arrive at $B$ in $m$ hours and the second will arrive at $A$ in $n... | [
"The speed of the first hiker is $\\displaystyle\\frac{a\\sqrt n}{n\\sqrt m-m\\sqrt n}$ and that of the second, $\\displaystyle\\frac{a\\sqrt m}{n\\sqrt m-m\\sqrt n}$ km/h"
] | [
"Two hikers leave $A$ and $B$ at the same time in the direction towards each other. When they meet, it turns ont that the first hiker has covered $a$ km more than the second. If they continue on their ways at the same rate as before, the first hiker will arrive at $B$ in $m$ hours and the second will arrive at $A$ ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc9b3500fa765be81c007dcacf6ce438d7f04363f03fd2dd6c23f50e4dbf6445 | Two hikers start out at the same time and, walking towards each other, meet in 3 hours and 20 minutes. How long will it take for each hiker to cover the whole distance, if the first of them arrives at the starting point of the second 5 hours after the second arrives at the starting point of the first? | [
"The first hiker covers the whole distance in 10 hours and the second, in 5 hours"
] | [
"Two hikers start out at the same time and, walking towards each other, meet in 3 hours and 20 minutes. How long will it take for each hiker to cover the whole distance, if the first of them arrives at the starting point of the second 5 hours after the second arrives at the starting point of the first?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4bc11909396214839c6ad932f2cd29d387c4bf599c842dac505d63cf17bc0b2 | Two hikers start out towards each other, one from $A$ and the other, from $B$. The first hiker starts from $A$ six hours after the second leaves $B$ and when they meet it turns out, that he has covered 12 km less than the second hiker. After the encounter the hikers continue walking at the same rate as before and the f... | [
"$AB=84$ km. The rate of the first hiker is 6 km/h and that of the second one is 4 km/h"
] | [
"Two hikers start out towards each other, one from $A$ and the other, from $B$. The first hiker starts from $A$ six hours after the second leaves $B$ and when they meet it turns out, that he has covered 12 km less than the second hiker. After the encounter the hikers continue walking at the same rate as before and ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f380a5477f82c6aa4326712ff72dbf3c30bb3ca0077925049e8dc75c8c291a6 | Two hikers start walking at noon, the first from point A to point B, and the second from point B to point A. They each walk at a constant speed and meet at 3 pm. The first hiker arrives at point B 2.5 hours before the second hiker arrives at point A. When did the second hiker get to point A? | [
"7:30"
] | [
"Two hikers start walking at noon, the first from point A to point B, and the second from point B to point A. They each walk at a constant speed and meet at 3 pm. The first hiker arrives at point B 2.5 hours before the second hiker arrives at point A. When did the second hiker get to point A?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-5d6b95a75eaef125a7128c16ac229c3b203a4ae18a54d1906b2284bb07ea4767 | Two hikers, $A$ and $B$, start at points $(a,0)$ and $(b,0)$ respectively, and travel on a polygonal curve (mountain range) in the upper half-plane. The curve is such that their heights above the horizontal axis are always the same. If $a=1$ and $b=2023$, find the minimum number of segments the polygonal curve must hav... | [
"2"
] | [
"Two hikers, $A$ and $B$, start at points $(a,0)$ and $(b,0)$ respectively, and travel on a polygonal curve (mountain range) in the upper half-plane. The curve is such that their heights above the horizontal axis are always the same. If $a=1$ and $b=2023$, find the minimum number of segments the polygonal curve mus... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f278014864d8c1f37689a13d5609428c69ebc93ed1854649f6f35d59512d10fc | Two hikers, Alan and Ben, are 11.4 miles apart and walking towards each other. They meet in 3 hours. Find the rate of each hiker if Ben walks 0.6 mph faster than Alan. | [
"Alan's speed is 1.6 mph and Ben's speed is 2.2 mph."
] | [
"Two hikers, Alan and Ben, are 11.4 miles apart and walking towards each other. They meet in 3 hours. Find the rate of each hiker if Ben walks 0.6 mph faster than Alan."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 24
} | 33 | 0 | 0 |
omr-ws-2095f506b6a520a814d4f9a8a712447520865c3175e83bff9031d96285b638d4 | Two houses \( A \) and \( B \) have been selected from the sides of a \( 1993 \times 1993 \) grid such that their distance is even. Let \( f(A) \) be the number of ways to fill the grid without house \( A \) using dominoes, and \( f(B) \) be the number of ways to fill the grid without house \( B \) using dominoes. Calc... | [
"0"
] | [
"Two houses \\( A \\) and \\( B \\) have been selected from the sides of a \\( 1993 \\times 1993 \\) grid such that their distance is even. Let \\( f(A) \\) be the number of ways to fill the grid without house \\( A \\) using dominoes, and \\( f(B) \\) be the number of ways to fill the grid without house \\( B \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b20d61252f808e522e3a214bcf6947d8acd0547cf008abf3c866e4fe8e39d35 | Two humans boarded a Maglev train and expected to get to their final destination, 4000 miles away, in four hours. The driver drove the train at a constant speed of 1000 miles per hour throughout the first hour. After the first quarter of the journey, the driver slowed the train down by 100 miles per hour for the second... | [
"766"
] | [
"Two humans boarded a Maglev train and expected to get to their final destination, 4000 miles away, in four hours. The driver drove the train at a constant speed of 1000 miles per hour throughout the first hour. After the first quarter of the journey, the driver slowed the train down by 100 miles per hour for the s... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 16
} | 37 | 0 | 0 |
omr-ws-0cf2678ac7189315916f125dc1096d3b0a6cc24933b24f215687e0fee538a4c2 | Two hundred balls are distributed into one hundred boxes such that no box contains more than one hundred balls and each box contains at least one ball. Let $N$ be the number of ways to select a subset of these boxes such that the total number of balls in the selected boxes is exactly one hundred. Find the remainder whe... | [
"376"
] | [
"Two hundred balls are distributed into one hundred boxes such that no box contains more than one hundred balls and each box contains at least one ball. Let $N$ be the number of ways to select a subset of these boxes such that the total number of balls in the selected boxes is exactly one hundred. Find the remainde... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-2fac82072256b660ac12188f54c8ffb8fdc7caa471f526b408c8d239665943f0 | Two hundred students participated in a mathematical contest. They had six problems to solve. It is known that each problem was correctly solved by at least 120 participants. Determine the minimum number of pairs of students such that each problem was solved by at least one of the two students in each pair. | [
"100"
] | [
"Two hundred students participated in a mathematical contest. They had six problems to solve. It is known that each problem was correctly solved by at least 120 participants. Determine the minimum number of pairs of students such that each problem was solved by at least one of the two students in each pair."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-5c115cdea3dc2d326c0e4e70392f232b299e1fb20f6168d452470ae4f6fbe48b | Two hunters $P$ and $Q$ each have a 50% chance of hitting a duck when they shoot. Hunter $P$ shoots at 50 ducks and Hunter $Q$ shoots at 51 ducks. What is the probability that Hunter $Q$ bags more ducks than Hunter $P$? | [
"\\(\\frac{1}{2}\\)"
] | [
"Two hunters $P$ and $Q$ each have a 50% chance of hitting a duck when they shoot. Hunter $P$ shoots at 50 ducks and Hunter $Q$ shoots at 51 ducks. What is the probability that Hunter $Q$ bags more ducks than Hunter $P$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-97b8bf4ab1e1e45562f3799faf1056d654eeb1d82eeb228fca3705cba695525c | Two hunters A and B got 10 birds for shooting. The sum of the squares of their shots fired is 2880, and the product of the number of shots fired by each is 48 times the product of the number of birds killed by each. If A had fired as often as B and B had fired as often as A, then B would have shot five more birds than ... | [
"Hunter A killed 4 birds and Hunter B killed 6 birds."
] | [
"Two hunters A and B got 10 birds for shooting. The sum of the squares of their shots fired is 2880, and the product of the number of shots fired by each is 48 times the product of the number of birds killed by each. If A had fired as often as B and B had fired as often as A, then B would have shot five more birds ... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 15
} | 16 | 0 | 0 |
omr-ws-160f37aa07a4fea1dc1d7e7dddc4cfe8b745e9115e0c463946078553193cb535 | Two identical CDs regularly cost a total of $\$28.$ What is the cost in dollars of five of these CDs? | [
"70"
] | [
"Two identical CDs regularly cost a total of $\\$28.$ What is the cost in dollars of five of these CDs?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9553241f30a83736de73075947c13f5c79689201836f159b5939bf920b878c06 | Two identical balls of radius $\sqrt{15}$ and two identical balls of a smaller radius are located on a plane so that each ball touches the other three. Find the area of the surface of the ball with the smaller radius. | [
"\\(420\\pi - 240\\pi\\sqrt{3}\\)"
] | [
"Two identical balls of radius $\\sqrt{15}$ and two identical balls of a smaller radius are located on a plane so that each ball touches the other three. Find the area of the surface of the ball with the smaller radius."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 25
} | 42 | 0 | 0 |
omr-ws-88d96fd9751d4d73db26fa60c9a014e5bb8e4726597bc4dd0df19f8553e01d98 | Two identical circles $\omega_{a}$ and $\omega_{b}$ with radius $1$ have centers that are $\tfrac{4}{3}$ units apart. Two externally tangent circles $\omega_1$ and $\omega_2$ of radius $r_1$ and $r_2$ respectively are each internally tangent to both $\omega_a$ and $\omega_b$. If $r_1 + r_2 = \tfrac{1}{2}$, what is $r_1... | [
"\\(\\frac{1}{18}\\)"
] | [
"Two identical circles $\\omega_{a}$ and $\\omega_{b}$ with radius $1$ have centers that are $\\tfrac{4}{3}$ units apart. Two externally tangent circles $\\omega_1$ and $\\omega_2$ of radius $r_1$ and $r_2$ respectively are each internally tangent to both $\\omega_a$ and $\\omega_b$. If $r_1 + r_2 = \\tfrac{1}{2}$,... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 22
} | 44 | 0 | 0 |
omr-ws-849234df4b863e5ea60593830e9cf2f0d2bf2eba0ef219ce49a3c5bc7763d1fd | Two identical circular spinners each have 10 sectors numbered from 1 to 10 inclusive. Each turn, both spinners are spun, and each number that is selected is crossed out if it hasn't been crossed out before. What is the expected value of the number of turns it takes for all of the numbers on the paper to be crossed out? | [
"\\(\\frac{7027288137}{471793508}\\)"
] | [
"Two identical circular spinners each have 10 sectors numbered from 1 to 10 inclusive. Each turn, both spinners are spun, and each number that is selected is crossed out if it hasn't been crossed out before. What is the expected value of the number of turns it takes for all of the numbers on the paper to be crossed... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 14
} | 14 | 0 | 0 |
omr-ws-18562cd05d13b5faeb2e46a6fb07ee37e4879fffbc41c8db87b93fbf8c45b703 | Two identical components with lifetimes \( T_1 \) and \( T_2 \) are connected in parallel in a system. The distributions of \( T_1 \) and \( T_2 \) are independent and follow an exponential distribution with mean \( \frac{1}{a} \) where \( a > 0 \). When one component fails, the lifetime distribution of the other chang... | [
"\\[ P(T \\geq t) = \\begin{cases} \n\\frac{1}{c-2a}(ce^{-2at} - 2a e^{-ct}) & \\text{if } c \\neq 2a \\\\\ne^{-ct}(ct+1) & \\text{if } c = 2a \n\\end{cases} \\]"
] | [
"Two identical components with lifetimes \\( T_1 \\) and \\( T_2 \\) are connected in parallel in a system. The distributions of \\( T_1 \\) and \\( T_2 \\) are independent and follow an exponential distribution with mean \\( \\frac{1}{a} \\) where \\( a > 0 \\). When one component fails, the lifetime distribution ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 4,
"tir": 14
} | 19 | 0 | 0 |
omr-ws-29556d104f8b75fb38f514a2ca23600170b8a0fe177d179ec6214a503a6fcd89 | Two identical cylinders of infinite length, both with diameter $9 \sqrt{2}$ cm, lie on a plane and intersect at right angles. What is the maximum volume, in $cm^{3}$, of a rectangular prism that can be placed entirely inside the intersection region? | [
"729"
] | [
"Two identical cylinders of infinite length, both with diameter $9 \\sqrt{2}$ cm, lie on a plane and intersect at right angles. What is the maximum volume, in $cm^{3}$, of a rectangular prism that can be placed entirely inside the intersection region?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-83e0d0f399c549d6d25b40c3a98c55b6fa249ba7daf17429a96a1e2ce1d98544 | Two identical decks each have 36 cards. One deck is shuffled and placed on top of the second. For each card in the top deck, count the number of cards between it and the corresponding card in the bottom deck. What is the sum of these numbers? | [
"630"
] | [
"Two identical decks each have 36 cards. One deck is shuffled and placed on top of the second. For each card in the top deck, count the number of cards between it and the corresponding card in the bottom deck. What is the sum of these numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a8c9ce2907e7cd59f5728566d684a61fc38546a86e237efc03911d5e27cbec72 | Two identical jars are filled with a solution of alcohol and water. The ratio of alcohol to water in the first jar is \( p:1 \), and in the second jar, it is \( q:1 \). The contents of the two jars are combined. What is the ratio of alcohol to water in this mixture? | [
"\\(\\frac{2pq + p + q}{p + q + 2}\\)"
] | [
"Two identical jars are filled with a solution of alcohol and water. The ratio of alcohol to water in the first jar is \\( p:1 \\), and in the second jar, it is \\( q:1 \\). The contents of the two jars are combined. What is the ratio of alcohol to water in this mixture?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-b7ea18729682c2960718573920011cb9504f271880ce510484757c19dfed1f56 | Two identical jars are filled with alcohol solutions, the ratio of the volume of alcohol to the volume of water being \( p : 1 \) in one jar and \( q : 1 \) in the other jar. If the entire contents of the two jars are mixed together, what is the ratio of the volume of alcohol to the volume of water in the mixture? | [
"\\(\\frac{p+q+2pq}{p+q+2}\\)"
] | [
"Two identical jars are filled with alcohol solutions, the ratio of the volume of alcohol to the volume of water being \\( p : 1 \\) in one jar and \\( q : 1 \\) in the other jar. If the entire contents of the two jars are mixed together, what is the ratio of the volume of alcohol to the volume of water in the mixt... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-149a5806232ec894b227eaaddd18d97c11c1d034ca66e85afbd81be4d31931d7 | Two identical jars are filled with alcohol solutions, with the ratio of the volume of alcohol to the volume of water being \( p:1 \) in one jar and \( q:1 \) in the other jar. If the entire contents of the two jars are mixed together, what is the ratio of the volume of alcohol to the volume of water in the mixture? | [
"\\(\\frac{2pq + p + q}{p + q + 2}\\)"
] | [
"Two identical jars are filled with alcohol solutions, with the ratio of the volume of alcohol to the volume of water being \\( p:1 \\) in one jar and \\( q:1 \\) in the other jar. If the entire contents of the two jars are mixed together, what is the ratio of the volume of alcohol to the volume of water in the mix... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-dfc1db5fed573604dae1b60316e4ca24f1757b98378efc555a50321177bad8cc | Two identical jars are filled with alcohol solutions. The ratio of the volume of alcohol to the volume of water in one jar is \( p:1 \) and in the other jar is \( q:1 \). If the entire contents of the two jars are mixed together, find the ratio of the volume of alcohol to the volume of water in the mixture. | [
"\\( 2pq + p + q : p + q + 2 \\)"
] | [
"Two identical jars are filled with alcohol solutions. The ratio of the volume of alcohol to the volume of water in one jar is \\( p:1 \\) and in the other jar is \\( q:1 \\). If the entire contents of the two jars are mixed together, find the ratio of the volume of alcohol to the volume of water in the mixture."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-d603811bac14382d3886b38865f0601d973a15aebcc811ab3e87091d605e0e2d | Two identical rectangles with integer dimensions are placed on top of each other, with their centers coinciding and at right angles. The resulting figure is a fat cross with an area of $703687 \text{ cm}^2$. Find the perimeter of the cross. | [
"3376 cm"
] | [
"Two identical rectangles with integer dimensions are placed on top of each other, with their centers coinciding and at right angles. The resulting figure is a fat cross with an area of $703687 \\text{ cm}^2$. Find the perimeter of the cross."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 8,
"tir": 9
} | 21 | 0 | 0 |
omr-ws-61e6c0cdc1ca465ea6618f488c125ebb3fa7dc99aea38644658c6ed3ee386fa0 | Two identical regular hexagons are given. Divide each hexagon into three pieces using two straight cuts such that the six pieces can be rearranged to form an equilateral triangle without any gaps or overlaps. If the side length of each hexagon is $s$, find the side length of the equilateral triangle formed by rearrangi... | [
"2\\sqrt{3}s"
] | [
"Two identical regular hexagons are given. Divide each hexagon into three pieces using two straight cuts such that the six pieces can be rearranged to form an equilateral triangle without any gaps or overlaps. If the side length of each hexagon is $s$, find the side length of the equilateral triangle formed by rear... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-247d2715712cbeb17153c252248073b1018bd5b4b62a7add490b27cbf8ab0c94 | Two identical regular quadrilateral pyramids are placed so that their bases coincide but their vertices do not. Let's look at the eight planes in which the side faces of these pyramids are located. How many parts do these planes divide space into? | [
"59"
] | [
"Two identical regular quadrilateral pyramids are placed so that their bases coincide but their vertices do not. Let's look at the eight planes in which the side faces of these pyramids are located. How many parts do these planes divide space into?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b73e79ee0bb7985e6994010f2d7fb69ea5647127622c0b3510f6f66393d07e6 | Two identical regular triangles $ABC$ and $CDE$ with side 1 lie on the plane so that they have only one point $C$ in common and the angle $BCD < \pi /3$. Point $K$ is the midpoint of the side $AC$, point $L$ is the midpoint of the segment $CE$, point $M$ is the midpoint of the segment $BD$. The area of the triangle $KL... | [
"$\\frac{2-\\sqrt 3}{\\sqrt 5}$"
] | [
"Two identical regular triangles $ABC$ and $CDE$ with side 1 lie on the plane so that they have only one point $C$ in common and the angle $BCD < \\pi /3$. Point $K$ is the midpoint of the side $AC$, point $L$ is the midpoint of the segment $CE$, point $M$ is the midpoint of the segment $BD$. The area of the triang... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d97cc5a9836184eec964521caa67e16502a385b0334488b1f69f18ddb39c2dc | Two identical right-angled triangles \(ABC\) and \(NCM\) are located such that the right angle vertex \(C\) of one triangle falls on the hypotenuse of the other. If \(BC = 10\), find the length of \(MB\). | [
"5\\sqrt{2}"
] | [
"Two identical right-angled triangles \\(ABC\\) and \\(NCM\\) are located such that the right angle vertex \\(C\\) of one triangle falls on the hypotenuse of the other. If \\(BC = 10\\), find the length of \\(MB\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-bc2d27a24f49de70fdb31a83bd7d90ed2872909e016fb0c6e242c6a9654d8258 | Two identical ships start simultaneously from two piers: the first ship starts from pier $A$ down the river and the other from pier $B$ up the river. By the time they meet, the first ship covers thrice the distance covered by the second ship. Each ship arrives at the point of destination, stops there for some time, and... | [
"28 km, 20 km/h"
] | [
"Two identical ships start simultaneously from two piers: the first ship starts from pier $A$ down the river and the other from pier $B$ up the river. By the time they meet, the first ship covers thrice the distance covered by the second ship. Each ship arrives at the point of destination, stops there for some tim... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6da3e989279656f788ce9076392ae1f25007b149d01c17a3012bc95a066757b | Two identical squares, each with a side length of $5\text{cm}$, are each divided into $5$ regions through intersection with some lines. Color the regions of the first square with five colors and the regions of the second with the same five colors. After superimposing the two squares, calculate the minimum possible sum ... | [
"5"
] | [
"Two identical squares, each with a side length of $5\\text{cm}$, are each divided into $5$ regions through intersection with some lines. Color the regions of the first square with five colors and the regions of the second with the same five colors. After superimposing the two squares, calculate the minimum possibl... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-7227a1ee256ed831ee9fa64994bf1bfe6a13f5aa83b187b6121fece5ba6cf939 | Two identical uniform rods \( AB \) and \( CD \), each with mass \( m \) and length \( 2a \), are placed a distance \( b \) apart such that \( ABCD \) forms a rectangle. Calculate the gravitational attraction between the two rods. What is the limiting value of this attraction as \( a \) tends to zero? | [
"\\(\\frac{G m^2}{b}\\)"
] | [
"Two identical uniform rods \\( AB \\) and \\( CD \\), each with mass \\( m \\) and length \\( 2a \\), are placed a distance \\( b \\) apart such that \\( ABCD \\) forms a rectangle. Calculate the gravitational attraction between the two rods. What is the limiting value of this attraction as \\( a \\) tends to zero... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-38a6091febbf204d85b8a55e63dc9788c449fcb78bee1675eadcd6e883b18953 | Two identical vessels are filled with alcohol. We draw off a liters of alcohol from the first vessel and add that amount of water. Then we draw off $a$ liters of the resulting mixture of water and alcohol and add that amount of water. In the case of the second vessel, we draw off $2a$ liters of alcohol and add that amo... | [
"1/6"
] | [
"Two identical vessels are filled with alcohol. We draw off a liters of alcohol from the first vessel and add that amount of water. Then we draw off $a$ liters of the resulting mixture of water and alcohol and add that amount of water. In the case of the second vessel, we draw off $2a$ liters of alcohol and add tha... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-886f8b3f94b6fc4bacba89440cba9da18c7c3648e3146a8e3944734bfef5105d | Two identical vessels, each of 30 litres, contain a total of only 30 liters of alcohol. Water is added to the top of one vessel, the resulting mixture is added to the top of the other vessel and then 12 liters of the new mixture are poured from the second vessel into the first. How much alcohol did each vessel contain ... | [
"20 \\text{ liters and } 10 \\text{ liters}"
] | [
"Two identical vessels, each of 30 litres, contain a total of only 30 liters of alcohol. Water is added to the top of one vessel, the resulting mixture is added to the top of the other vessel and then 12 liters of the new mixture are poured from the second vessel into the first. How much alcohol did each vessel con... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19c5318951cd26874b419f4d6d51ab3fd3d6ff60b4b3e47739c83e2827faf408 | Two identical watches show different times. The angle between the hour hands is currently 42 degrees. What is the angle between the minute hands? | [
"} 144 \\]\n\nSo, the final answer is:"
] | [
"Two identical watches show different times. The angle between the hour hands is currently 42 degrees. What is the angle between the minute hands?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-333b5de0f77a2ac31fc9b0a932ea341f9fab3a8982124701b6092be7dd14f3df | Two independent customers are scheduled to arrive in the afternoon, with their arrival times uniformly distributed between 2 pm and 8 pm. What is the expected time of the first (earlier) arrival? | [
"4 pm"
] | [
"Two independent customers are scheduled to arrive in the afternoon, with their arrival times uniformly distributed between 2 pm and 8 pm. What is the expected time of the first (earlier) arrival?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-00debddf83dcb7119f68afa2497c97fac54ea64ee9169bd91869f07dfae2b700 | Two individuals A and B start at position 0 on the number line. Their positions are denoted by \(a\) and \(b\) respectively. They take a step according to the equation \(a^2 + b^4 = t\), where \(t\) is the second struck by the clock from the start of the game. If they take \(n\) consecutive steps (first step at time \(... | [
"3"
] | [
"Two individuals A and B start at position 0 on the number line. Their positions are denoted by \\(a\\) and \\(b\\) respectively. They take a step according to the equation \\(a^2 + b^4 = t\\), where \\(t\\) is the second struck by the clock from the start of the game. If they take \\(n\\) consecutive steps (first ... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
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