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-6dbaeaa605f85b40a632762bebeb5f07a9e701edb651e2009d179f4a3f8e119f | Two men are betting money on a game of tic-tac-toe. A man pays $1 to randomly pick 4 spaces. If he gets tic-tac-toe, he wins $3. On average, how much can the player expect to win or lose in dollars? | [
"\\dfrac{1}{7}"
] | [
"Two men are betting money on a game of tic-tac-toe. A man pays $1 to randomly pick 4 spaces. If he gets tic-tac-toe, he wins $3. On average, how much can the player expect to win or lose in dollars?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9b34422d6db29ca673fc45e1d9c4ffbb91a2d5f2e5a90c5d46c23e58b480f25b | Two men are in rivers at coordinates $(0, 0)$ and $(0, 5)$, respectively, in Cartesian Lego City. The rivers flow at $10$ miles per hour in the positive $x$ direction. The ocean, located at $x > 14$, is fatal. A helicopter starting at $(0, 3)$ must rescue both people by reaching the same position as each person at some... | [
"**\n - The minimal speed \\( S \\) is \\( \\sqrt{109} \\) mph.\n - Therefore, \\( S^2 = 109 \\)."
] | [
"Two men are in rivers at coordinates $(0, 0)$ and $(0, 5)$, respectively, in Cartesian Lego City. The rivers flow at $10$ miles per hour in the positive $x$ direction. The ocean, located at $x > 14$, is fatal. A helicopter starting at $(0, 3)$ must rescue both people by reaching the same position as each person at... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b169e11cebbc0bfe8cdb3c9f4f886bdbada25c2f4b384fd4d3d471da49bcdd8 | Two men measure the angle of elevation of a helicopter at the same time. One angle is measured as 35 degrees, the other as 50 degrees. If the two men are 100 feet apart and the helicopter lies over the line joining them, how high is the helicopter? | [
"44.1 feet"
] | [
"Two men measure the angle of elevation of a helicopter at the same time. One angle is measured as 35 degrees, the other as 50 degrees. If the two men are 100 feet apart and the helicopter lies over the line joining them, how high is the helicopter?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-6bfe75775894729791a1c96f21ebdd2d4543cda79a9242c236c086f620a5ad21 | Two men set out at the same time to walk towards each other from points \( M \) and \( N \), which are 72 miles apart. The first man walks at a rate of 4 mph. The second man walks 2 miles the first hour, \( 2\frac{1}{2} \) miles the second hour, 3 miles the third hour, and so on in arithmetic progression. At what point... | [
"E"
] | [
"Two men set out at the same time to walk towards each other from points \\( M \\) and \\( N \\), which are 72 miles apart. The first man walks at a rate of 4 mph. The second man walks 2 miles the first hour, \\( 2\\frac{1}{2} \\) miles the second hour, 3 miles the third hour, and so on in arithmetic progression. A... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-41fd477e38c809e38df15c322d93fd4677551e8e087b0c5caf7609b54c7a9faa | Two men start out at the same time from $A$ in the direction of $B$, which is 100 km from $A$, one on a bicycle, the other on foot. Also at the same time a car starts out from $B$ and goes in the direction of $A$. An hour later the car meets the cyclist and, continuing another $14\frac{2}{17}$ km, it meets the man on f... | [
"20 km/h and 80 km/h"
] | [
"Two men start out at the same time from $A$ in the direction of $B$, which is 100 km from $A$, one on a bicycle, the other on foot. Also at the same time a car starts out from $B$ and goes in the direction of $A$. An hour later the car meets the cyclist and, continuing another $14\\frac{2}{17}$ km, it meets the ma... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-10995291222d8b0fda02da6de72d4a6a84f3990af467e464c52733fd5ef42071 | Two men starting at a point on a circular 1-mile race track walk in opposite directions with uniform speeds and meet in 6 minutes. If they walk in the same direction, it requires one hour for the faster walker to gain a lap. What is the rate of the slower walker? | [
"4.5 miles/hour",
"4.5 mph",
"\\(\\frac{9}{2}\\) miles per hour"
] | [
"Two men starting at a point on a circular 1-mile race track walk in opposite directions with uniform speeds and meet in 6 minutes. If they walk in the same direction, it requires one hour for the faster walker to gain a lap. What is the rate of the slower walker?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 41
} | 89 | 0 | 0 |
omr-ws-b8eb8766ad1778b86ef279a93f17b86e1602c01b14bed0f8a5aa306978465d12 | Two messengers left points $A$ and $B$ simultaneously to meet each other. After some time they met. Had the first messenger started an hour earlier, and the second messenger half an hour later, then they would have met 48 minutes earlier. Had the first messenger started half an hour later, and the second messenger an h... | [
"8 km/h and 7 km/h"
] | [
"Two messengers left points $A$ and $B$ simultaneously to meet each other. After some time they met. Had the first messenger started an hour earlier, and the second messenger half an hour later, then they would have met 48 minutes earlier. Had the first messenger started half an hour later, and the second messenger... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-939f55a8312d00e82f16333da9ba523f2d39735951a1e82cfd15ae1dcdcf94d5 | Two metal balls with diameters of 14 cm and 10 cm are placed in a cylinder with a diameter of 22 cm. After pouring 5 dm³ of water into the cylinder, will the water flood the balls? | [
"No"
] | [
"Two metal balls with diameters of 14 cm and 10 cm are placed in a cylinder with a diameter of 22 cm. After pouring 5 dm³ of water into the cylinder, will the water flood the balls?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-e21e83a137b335c23399b4b3c64e23bf3c073134e67c74c9ad3a74687977396f | Two microphones are placed 6000 feet apart in the ocean. One microphone picks up a whale's sound 0.5 seconds after the other microphone picks up the same sound. The speed of sound in water is about 5000 feet per second. Find the equation of the hyperbola that describes the possible locations of the whale. | [
"\\(\\frac{x^2}{1562500} - \\frac{y^2}{7437500} = 1\\)"
] | [
"Two microphones are placed 6000 feet apart in the ocean. One microphone picks up a whale's sound 0.5 seconds after the other microphone picks up the same sound. The speed of sound in water is about 5000 feet per second. Find the equation of the hyperbola that describes the possible locations of the whale."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-e488656794a114a837468dd234c5e1f9a36e6e12a125297a96a454941de7111b | Two microphones are placed 6000 feet apart in the ocean. One microphone picks up a whale's sound 0.5 seconds after the other microphone picks up the same sound. The speed of sound in water is about 5000 feet per second. What is the shortest distance that the whale could be to either microphone? | [
"1750 feet"
] | [
"Two microphones are placed 6000 feet apart in the ocean. One microphone picks up a whale's sound 0.5 seconds after the other microphone picks up the same sound. The speed of sound in water is about 5000 feet per second. What is the shortest distance that the whale could be to either microphone?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 25
} | 73 | 0 | 0 |
omr-ws-be3ddfccc4c7e796426c51fb441c014684ae0a11fd1e63a15ba0248e3c2f6bd4 | Two miles upstream from his starting point, a canoeist passed a log floating in the river's current. After paddling upstream for one more hour, he paddled back and reached his starting point just as the log arrived. Find the speed of the current. | [
"1 mph"
] | [
"Two miles upstream from his starting point, a canoeist passed a log floating in the river's current. After paddling upstream for one more hour, he paddled back and reached his starting point just as the log arrived. Find the speed of the current."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-7e0c51778a087d359d15de9700f5084ab60c34cfb698501658e8ac3b0e0657f7 | Two mirrors (= half planes with a common border line) in space are inclined at an angle $\alpha$ to each other. A candle and an observer stand between the two mirrors. How many images of the candle does the observer see? | [
"\\(\\frac{360^\\circ}{\\alpha} - 1\\)"
] | [
"Two mirrors (= half planes with a common border line) in space are inclined at an angle $\\alpha$ to each other. A candle and an observer stand between the two mirrors. How many images of the candle does the observer see?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 5,
"tir": 3
} | 38 | 0 | 0 |
omr-ws-7d0c3fc2b3016560fd0661307cd5de9ac87f5237bb3ec1f2126c097812bce903 | Two mirrors are joined at a $30^{\circ}$ angle, and a laser is aimed at one of the mirrors (in a plane perpendicular to the line of intersection of the mirrors). What is the maximum number of times the reflected beam of light will hit the two mirrors? | [
"6"
] | [
"Two mirrors are joined at a $30^{\\circ}$ angle, and a laser is aimed at one of the mirrors (in a plane perpendicular to the line of intersection of the mirrors). What is the maximum number of times the reflected beam of light will hit the two mirrors?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de91636f97476b9d94dcbf440640740a7f8bed289606537c8c216d5902e5094d | Two mirrors are joined at a hinge and opened to a 17-degree angle. A laser beam runs parallel to the bottom mirror and perpendicular to the hinge, bouncing off both mirrors repeatedly until it escapes. What angle does the beam make with the lower mirror when it exits? | [
"10^\\circ"
] | [
"Two mirrors are joined at a hinge and opened to a 17-degree angle. A laser beam runs parallel to the bottom mirror and perpendicular to the hinge, bouncing off both mirrors repeatedly until it escapes. What angle does the beam make with the lower mirror when it exits?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-08291d7de3488cb3cdc9d602d56bc21f99a62594d3b96afc6c7d29d433fb4e67 | Two mirrors are placed to create a 15 degree angle. A beam of light travels from point S to point R. Given the distances $OT=50$, $ST=5$, $RU=5$, and $OU=20$, find the value of $x+v+z$. | [
"60"
] | [
"Two mirrors are placed to create a 15 degree angle. A beam of light travels from point S to point R. Given the distances $OT=50$, $ST=5$, $RU=5$, and $OU=20$, find the value of $x+v+z$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-727e5ffc288995dcb2d01a6c8463cbc711b3517fbef2e04401f1a3324a04cd7b | Two mirrors form an angle $\angle ABC = 60^\circ$ between them. A light beam $PP_1$ forms an angle of $45^\circ$ with side $BA$. After reflecting three times from the sides, the ray $P_3P_4$ leaves the system of mirrors, forming an angle $\beta$ with the side $AB$. Find the angle $\beta$. | [
"15^\\circ"
] | [
"Two mirrors form an angle $\\angle ABC = 60^\\circ$ between them. A light beam $PP_1$ forms an angle of $45^\\circ$ with side $BA$. After reflecting three times from the sides, the ray $P_3P_4$ leaves the system of mirrors, forming an angle $\\beta$ with the side $AB$. Find the angle $\\beta$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 6
} | 11 | 0 | 0 |
omr-ws-c12e04a0d050492b9e7c9cc6dc723394ead41007b3a3abc606aef7ccc818d0a3 | Two mirrors form an angle of 30 degrees at point V. A ray of light, coming from a source S, is issued parallel to one of the mirrors and is reflected by the other mirror at point A. After a certain amount of reflection, the beam returns to S. If AS and AV are each 1 meter in length, what is the total distance traveled ... | [
"\\(2 + \\sqrt{3}\\)"
] | [
"Two mirrors form an angle of 30 degrees at point V. A ray of light, coming from a source S, is issued parallel to one of the mirrors and is reflected by the other mirror at point A. After a certain amount of reflection, the beam returns to S. If AS and AV are each 1 meter in length, what is the total distance trav... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-333c814537f68eed5cd8d218e82fcabd86c2945e78db7d496801f4c2735b2205 | Two missiles are initially 5000 km apart and travel towards each other along a straight line, with one traveling at 2000 km/hour and the other at 1000 km/hour. How many kilometers are they apart one minute before impact? | [
"50 km"
] | [
"Two missiles are initially 5000 km apart and travel towards each other along a straight line, with one traveling at 2000 km/hour and the other at 1000 km/hour. How many kilometers are they apart one minute before impact?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a3267ac1a61f35f24416838c1a47e571b79aef638e204d56e6f8e0a55db08c8e | Two mixtures of 28% acid and 33% acid are combined along with 2 gallons of water to form a mixture of 30% acid. Ten more gallons of the 33% mixture than the 28% mixture were used. How many gallons of acid should be added to the 30% mixture to obtain a 40% acid mixture? | [
"12"
] | [
"Two mixtures of 28% acid and 33% acid are combined along with 2 gallons of water to form a mixture of 30% acid. Ten more gallons of the 33% mixture than the 28% mixture were used. How many gallons of acid should be added to the 30% mixture to obtain a 40% acid mixture?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ca927603fc01ebe3eff7b8f4fea857ebd86a3999fba2b948dd18864d0a7d6311 | Two mobiles start simultaneously from two distinct points A and B, which are 165 km apart. The first mobile travels from A to B at a constant speed of 30 km/h, and the second mobile travels from B to A at a constant speed of 25 km/h. Find the distance between the two mobiles 12 minutes before they meet. | [
"11"
] | [
"Two mobiles start simultaneously from two distinct points A and B, which are 165 km apart. The first mobile travels from A to B at a constant speed of 30 km/h, and the second mobile travels from B to A at a constant speed of 25 km/h. Find the distance between the two mobiles 12 minutes before they meet."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-047d35ebabf9a68d59a7c541b20f9c379dafa6d394d4e892b48f04397c2e74a7 | Two more than three times $B$ is equal to 20. What is the value of $B$? | [
"6"
] | [
"Two more than three times $B$ is equal to 20. What is the value of $B$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-465fbbf224a5a54ab5dd4560bb8990eb7c2a6cb749e4c5be4fa3c3b9e045d811 | Two motor-cyclists began the race simultaneously from the same start and in the same direction, one at 80 km/h and the other at 60 km/h. A third motor-cyclist started half an hour later from the same point and in the same direction. Find the speed of the third motor-cyclist if it is known that he overtook the first mot... | [
"100 km/h"
] | [
"Two motor-cyclists began the race simultaneously from the same start and in the same direction, one at 80 km/h and the other at 60 km/h. A third motor-cyclist started half an hour later from the same point and in the same direction. Find the speed of the third motor-cyclist if it is known that he overtook the firs... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-824ce529f2e828b93b73cf7fbd43c991e3a6e3490fcb9a08adbaff18094bdd91 | Two motorists met at 10 a.m. at point X. After this meeting, one proceeded east and the other north. Exactly at noon, they were 60 km apart. Find their speeds if the difference of their speeds is 6 km/h. | [
"18 km/h and 24 km/h"
] | [
"Two motorists met at 10 a.m. at point X. After this meeting, one proceeded east and the other north. Exactly at noon, they were 60 km apart. Find their speeds if the difference of their speeds is 6 km/h."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 28
} | 34 | 0 | 0 |
omr-ws-0ce06a15497a0c9d7e554dd403a0bc68b4f49892546f857ac7b5498e64c4571b | Two moving bodies \( M_1 \) and \( M_2 \) are displaced uniformly on two coplanar straight lines. Describe the union of all straight lines \( M_1M_2 \). | [
"The union of all straight lines \\( M_1M_2 \\) is either a point or a parabola."
] | [
"Two moving bodies \\( M_1 \\) and \\( M_2 \\) are displaced uniformly on two coplanar straight lines. Describe the union of all straight lines \\( M_1M_2 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-e098926c7d6e58ae92918c6fd6eef8c46c9ee03d01407d4c1893ce645a58d6fb | Two moving points \( B \) and \( C \) are on the circle \( x^2 + y^2 = 25 \). If \( A(2, 0) \) and \( BAC \) is a straight line, find the value of \( k \) such that the equation of the locus of \( M \) is \( X^2 + Y^2 - 2X = k \), where \( M(X, Y) \) is the midpoint of \( BC \). | [
"0"
] | [
"Two moving points \\( B \\) and \\( C \\) are on the circle \\( x^2 + y^2 = 25 \\). If \\( A(2, 0) \\) and \\( BAC \\) is a straight line, find the value of \\( k \\) such that the equation of the locus of \\( M \\) is \\( X^2 + Y^2 - 2X = k \\), where \\( M(X, Y) \\) is the midpoint of \\( BC \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-54bcbb1ed2a43447ddbf37f80700728cb1fbb9dd13cfe09ddd4369ed680b0d50 | Two mutually perpendicular generatrices of a right circular cone are known to divide the circle of the base into two arcs, one of which is half the other. Find the volume of the cone if its height is $h$. | [
"$2/3\\pi h^3$"
] | [
"Two mutually perpendicular generatrices of a right circular cone are known to divide the circle of the base into two arcs, one of which is half the other. Find the volume of the cone if its height is $h$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e4ac71ca11efd4cb4b3b012e161f93fb3613e4888dd1da2259da527d84cbfc19 | Two mutually perpendicular intersecting chords $AB$ and $CD$ are drawn in a circle. It is known that $| AB | = | BC | = | CD |$. Find what is larger, the area of the circle or that of the square with side $AB$. | [
"The area of the square is larger than that of the circle"
] | [
"Two mutually perpendicular intersecting chords $AB$ and $CD$ are drawn in a circle. It is known that $| AB | = | BC | = | CD |$. Find what is larger, the area of the circle or that of the square with side $AB$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-287d2ccb4337dbcca3cd8f9b2a397d708f458b2c2d1c6982a38502cdd07e9566 | Two mutually perpendicular lines $\ell$ and $m$ intersect each other at a point on the circumference of a circle, dividing it into three arcs. A point $M_i$ ($i = 1, 2, 3$) is taken on each arc such that the tangent line to the circumference at $M_i$ intersects $\ell$ and $m$ at two points equidistant from $M_i$. If th... | [
"\\dfrac{3\\sqrt{3}}{4} r^2"
] | [
"Two mutually perpendicular lines $\\ell$ and $m$ intersect each other at a point on the circumference of a circle, dividing it into three arcs. A point $M_i$ ($i = 1, 2, 3$) is taken on each arc such that the tangent line to the circumference at $M_i$ intersects $\\ell$ and $m$ at two points equidistant from $M_i$... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-3a4ec30fb99154d674a15b7891af032b05b5d80db60100955ef42bfdafede027 | Two natural numbers \(a\) and \(b\) are such that \(1 < a < 100\) and \(1 < b < 100\). A is told the product \(ab\) and B is told their sum \(a + b\). A says, "I don't know the numbers but I know that you don't." B then says, "Now I know them." A then says, "Now I do too." What are \(a\) and \(b\)? | [
"13"
] | [
"Two natural numbers \\(a\\) and \\(b\\) are such that \\(1 < a < 100\\) and \\(1 < b < 100\\). A is told the product \\(ab\\) and B is told their sum \\(a + b\\). A says, \"I don't know the numbers but I know that you don't.\" B then says, \"Now I know them.\" A then says, \"Now I do too.\" What are \\(a\\) and \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-d4208fc17918fc0003b69463844c801926fa699b4d06cdf7d8c5781870fc06c4 | Two naval fleets face each other off at the sea: one composed of 7 submarines and the other composed of 3 destroyers and a battleship. It takes 2 hits by a destroyer/battleship to sink a submarine, 3 submarine hits to sink a destroyer, and 4 hits to sink a battleship. The battle rages in a sequence of "moves," wherein ... | [
"50"
] | [
"Two naval fleets face each other off at the sea: one composed of 7 submarines and the other composed of 3 destroyers and a battleship. It takes 2 hits by a destroyer/battleship to sink a submarine, 3 submarine hits to sink a destroyer, and 4 hits to sink a battleship. The battle rages in a sequence of \"moves,\" w... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-556c9c99659824123f76378ab1ba60c85c11c2dfb1e88c62d683d27c5933cb9a | Two non-congruent isosceles triangles with integer side lengths have the same perimeter and the same area. The ratio of the lengths of the bases of these two triangles is $3:2$. Find the smallest possible value of their common perimeter. | [
"76"
] | [
"Two non-congruent isosceles triangles with integer side lengths have the same perimeter and the same area. The ratio of the lengths of the bases of these two triangles is $3:2$. Find the smallest possible value of their common perimeter."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-c9e383cfb9d67e425789fdba70b6123f677c5ee05077ec914bfa35d55aaa7026 | Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is \( N \). What is the smallest possible value of \( N \)? | [
"104"
] | [
"Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is \\( N \\). What is the smallest possible value of \\( N \\)?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 3
} | 3 | 0 | 0 |
omr-ws-5304c899124a5a14765a263eab1a24baa6739cfd60c1fc2fed28bbf74083c968 | Two non-equal circles \(C_1\) and \(C_2\) with centers \(K\) and \(L\) intersect at points \(A\) and \(B\). A line through \(B\) meets the first circle at point \(C\) and the second at point \(D\). If \(M\) and \(N\) are the midpoints of \(KL\) and \(CD\) respectively, calculate the ratio \(\frac{MA}{MN}\). | [
"1"
] | [
"Two non-equal circles \\(C_1\\) and \\(C_2\\) with centers \\(K\\) and \\(L\\) intersect at points \\(A\\) and \\(B\\). A line through \\(B\\) meets the first circle at point \\(C\\) and the second at point \\(D\\). If \\(M\\) and \\(N\\) are the midpoints of \\(KL\\) and \\(CD\\) respectively, calculate the ratio... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93ec07af2e649b7bc3c236271fc9cbb25513e6b012348faadc5877289e1b7a37 | Two non-intersecting circles $C_1$ and $C_2$ have radii in the ratio $1:2$. A third circle intersects $C_1$ at points $A$ and $B$ and $C_2$ at points $C$ and $D$. The lines $AB$ and $CD$ intersect at point $P$. Determine the ratio of the lengths of the tangents from $P$ to the circles $C_1$ and $C_2$. | [
"\\( \\frac{1}{2} \\)"
] | [
"Two non-intersecting circles $C_1$ and $C_2$ have radii in the ratio $1:2$. A third circle intersects $C_1$ at points $A$ and $B$ and $C_2$ at points $C$ and $D$. The lines $AB$ and $CD$ intersect at point $P$. Determine the ratio of the lengths of the tangents from $P$ to the circles $C_1$ and $C_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-4a4136064a8e689c4817716b71e931be46587fa2553153067a3a90c35cb442a8 | Two non-intersecting circles $C_1$ and $C_2$ have radii in the ratio $1:2$. A third circle intersects $C_1$ at points $A$ and $B$ and $C_2$ at points $C$ and $D$. The lines $AB$ and $CD$ intersect at point $P$. What is the ratio of the lengths of the tangents from $P$ to the circles $C_1$ and $C_2$? | [
"1"
] | [
"Two non-intersecting circles $C_1$ and $C_2$ have radii in the ratio $1:2$. A third circle intersects $C_1$ at points $A$ and $B$ and $C_2$ at points $C$ and $D$. The lines $AB$ and $CD$ intersect at point $P$. What is the ratio of the lengths of the tangents from $P$ to the circles $C_1$ and $C_2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2ba78ff1b03bbee0beb65cb80ff84115ad3eed27cd6e550bfb0746347b3bdbec | Two non-intersecting circles $\omega_a$ and $\omega_b$ have centers $A$ and $B$ respectively. Let $P$ be a variable point in the plane. Let the polars $\ell_a$ and $\ell_b$ of $P$ with respect to $\omega_a$ and $\omega_b$ intersect at $Q$. If the circle on diameter $\overline{PQ}$ passes through two fixed points $K$ an... | [
"AB"
] | [
"Two non-intersecting circles $\\omega_a$ and $\\omega_b$ have centers $A$ and $B$ respectively. Let $P$ be a variable point in the plane. Let the polars $\\ell_a$ and $\\ell_b$ of $P$ with respect to $\\omega_a$ and $\\omega_b$ intersect at $Q$. If the circle on diameter $\\overline{PQ}$ passes through two fixed p... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2d8c39fdacee35ebc73ac4dd418356f1733094830f9785b17bb3c9ce9ea804fc | Two non-intersecting circles of radius $1$ and $3$ are inscribed in the angle $\angle POQ$. The point $P$ is the point of contact of the side of the corner with the first circle, and $Q$ is the point of contact of the other side of the angle with the second circle. The common internal tangent of the circles intersects ... | [
"6"
] | [
"Two non-intersecting circles of radius $1$ and $3$ are inscribed in the angle $\\angle POQ$. The point $P$ is the point of contact of the side of the corner with the first circle, and $Q$ is the point of contact of the other side of the angle with the second circle. The common internal tangent of the circles inter... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 2
} | 15 | 0 | 0 |
omr-ws-711bff6bdfca27ec28b5ccdd285d08e889eabff8f2a6f3242aa4b179039c3bde | Two non-intersecting circles of the same radius \( r \) are given on the plane. A line intersects the first circle at points \( A \) and \( B \), and the second circle at points \( C \) and \( D \) such that \( AB = BC = CD = 14 \). Another line intersects these circles at points \( E \), \( F \), \( G \), and \( H \) ... | [
"13"
] | [
"Two non-intersecting circles of the same radius \\( r \\) are given on the plane. A line intersects the first circle at points \\( A \\) and \\( B \\), and the second circle at points \\( C \\) and \\( D \\) such that \\( AB = BC = CD = 14 \\). Another line intersects these circles at points \\( E \\), \\( F \\), ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 35,
"genselect": 7,
"tir": 1
} | 43 | 0 | 0 |
omr-ws-7e3958ef79cc55be383528dd067c99ac3397a930f8ab081b85a39eafa0464bdb | Two non-intersecting circles, $\Gamma$ and $\Omega$, are given. Let the point $x$ vary on $\Gamma$. We draw the unique circle tangent to $\Gamma$ at $x$, which is also tangent to $\Omega$ at $x'$. Let $P(x)$ be the transformation that maps $x$ to $x'$. If $A$, $B$, and $C$ are three distinct points on $\Gamma$ such tha... | [
"\\dfrac{3}{2}"
] | [
"Two non-intersecting circles, $\\Gamma$ and $\\Omega$, are given. Let the point $x$ vary on $\\Gamma$. We draw the unique circle tangent to $\\Gamma$ at $x$, which is also tangent to $\\Omega$ at $x'$. Let $P(x)$ be the transformation that maps $x$ to $x'$. If $A$, $B$, and $C$ are three distinct points on $\\Gamm... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bb09681d02aced15531f21ee1b39677948aa5005f08bda0cdde84127aa86b9c1 | Two non-intersecting circles, $\omega$ and $\Omega$, have centers $C_\omega$ and $C_\Omega$ respectively, with the radius of $\Omega$ being strictly larger than the radius of $\omega$. The two common external tangents of $\omega$ and $\Omega$ intersect at a point $P$, and an internal tangent of the two circles intersec... | [
"12"
] | [
"Two non-intersecting circles, $\\omega$ and $\\Omega$, have centers $C_\\omega$ and $C_\\Omega$ respectively, with the radius of $\\Omega$ being strictly larger than the radius of $\\omega$. The two common external tangents of $\\omega$ and $\\Omega$ intersect at a point $P$, and an internal tangent of the two cir... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-b33c4f031a6ae6ee31806203b78b70bdfa8e517a47648e41f24e9498497ff1ca | Two non-intersecting circles, not lying inside each other, are drawn in the plane. Two lines pass through a point P which lies outside each circle. The first line intersects the first circle at A and A′ and the second circle at B and B′; here A and B are closer to P than A′ and B′, respectively, and P lies on segment A... | [
"10"
] | [
"Two non-intersecting circles, not lying inside each other, are drawn in the plane. Two lines pass through a point P which lies outside each circle. The first line intersects the first circle at A and A′ and the second circle at B and B′; here A and B are closer to P than A′ and B′, respectively, and P lies on segm... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c463ae144127cd30fda02e4b4f1ccb628a8796e66d8e71f814aad6e1faa790f9 | Two non-overlapping squares with side lengths $a$ and $b$ are placed inside a unit square. If $a + b$ is maximized under these conditions, find the value of $a + b$. | [
"1"
] | [
"Two non-overlapping squares with side lengths $a$ and $b$ are placed inside a unit square. If $a + b$ is maximized under these conditions, find the value of $a + b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-27ec13103558111fe944153d5b06baa6a881e6eda708b71fb668725b71c13703 | Two non-overlapping squares with side lengths \(a\) and \(b\) are placed inside a larger square with side length \(x\). If \(a + b = kx\), find the maximum possible value of \(k\). | [
"1"
] | [
"Two non-overlapping squares with side lengths \\(a\\) and \\(b\\) are placed inside a larger square with side length \\(x\\). If \\(a + b = kx\\), find the maximum possible value of \\(k\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-056fe3c7b71bef706ba867d3653e8a2a234aad161be1df43652063a014a9ade8 | Two non-overlapping squares with sides \(a\) and \(b\) are placed inside a square of side 1. Find the maximum possible value of \(a + b\). | [
"1"
] | [
"Two non-overlapping squares with sides \\(a\\) and \\(b\\) are placed inside a square of side 1. Find the maximum possible value of \\(a + b\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-8630ddc41c8ac96e02f6af7d8f00bc900ee629cf6e7ece8eeed742b3bac59899 | Two noncongruent circles \( k_1 \) and \( k_2 \) are exterior to each other. Their common tangents intersect the line through their centers at points \( A \) and \( B \). Let \( P \) be any point on \( k_1 \). If \( D \) is the diameter of \( k_2 \) with one endpoint on line \( PA \) and the other on \( PB \), calculat... | [
"10"
] | [
"Two noncongruent circles \\( k_1 \\) and \\( k_2 \\) are exterior to each other. Their common tangents intersect the line through their centers at points \\( A \\) and \\( B \\). Let \\( P \\) be any point on \\( k_1 \\). If \\( D \\) is the diameter of \\( k_2 \\) with one endpoint on line \\( PA \\) and the othe... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-e03aca82502220ba020f2e0f9dba9abbdcf0c9212da54fa56889a6d3f1ba685e | Two noncongruent circles \( k_1 \) and \( k_2 \) are exterior to each other. Their common tangents intersect the line through their centers at points \( A \) and \( B \). Let \( P \) be any point on \( k_1 \). If \( D \) is the diameter of \( k_2 \) with one endpoint on line \( PA \) and the other on line \( PB \), fin... | [
"2r_2"
] | [
"Two noncongruent circles \\( k_1 \\) and \\( k_2 \\) are exterior to each other. Their common tangents intersect the line through their centers at points \\( A \\) and \\( B \\). Let \\( P \\) be any point on \\( k_1 \\). If \\( D \\) is the diameter of \\( k_2 \\) with one endpoint on line \\( PA \\) and the othe... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a3c7fef7580fe8ca7727efc9406b8072fac30c851b1da9fdc1f41fe6d4cd679a | Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $8: 7$. Find the minimum possible value of their common perimeter. | [
"676"
] | [
"Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $8: 7$. Find the minimum possible value of their common perimeter.",
"Two noncongruent integer-sided isosceles triangles have the same perimeter and the same... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 17
} | 49 | 0 | 0 |
omr-ws-1444b5ee8cfb47b32689be84be4bcc39008ff8072a5edb6aaef0f66e24a566a6 | Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $8:7$. Find the minimum possible value of their common perimeter. | [
"676"
] | [
"Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $8:7$. Find the minimum possible value of their common perimeter."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-a26d63f3f52ac29477a92f81f783b475693bd0219806b2db9fd70cca815be7be | Two noncongruent isosceles triangles with integer sides have equal perimeters and equal areas, and the ratio of their bases is 9:8. What is the common perimeter of the smallest pair of such triangles? | [
"868"
] | [
"Two noncongruent isosceles triangles with integer sides have equal perimeters and equal areas, and the ratio of their bases is 9:8. What is the common perimeter of the smallest pair of such triangles?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-f916a6fd5dc8d1e2e6930e82ef93d82858a00d24d3393dc01a1890108d9bdf0a | Two noncongruent triangles have the same area but different perimeters. Triangle A has side lengths of 8 cm, 8 cm, and 6 cm. Triangle B has two sides with lengths of 8 cm each. Find the length of the third side of Triangle B. Round your answer to the nearest hundredth of a cm. | [
"14.83"
] | [
"Two noncongruent triangles have the same area but different perimeters. Triangle A has side lengths of 8 cm, 8 cm, and 6 cm. Triangle B has two sides with lengths of 8 cm each. Find the length of the third side of Triangle B. Round your answer to the nearest hundredth of a cm."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-4cc72dc9d3c066a16cb47dca1d2b0c9b89ca89cb4a50a79c5b5a01f213710dd4 | Two nonhorizontal, nonvertical lines in the $xy$-coordinate plane intersect to form a $45^\circ$ angle. One line has a slope equal to 6 times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines? | [
"\\dfrac{3}{2}"
] | [
"Two nonhorizontal, nonvertical lines in the $xy$-coordinate plane intersect to form a $45^\\circ$ angle. One line has a slope equal to 6 times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1eed1bd3e2a3b3ed54518e25b1c80d321d001a7b023c8024ffb326b3da6b1c34 | Two nonintersecting circles with radii 3 and 4 have an internal tangent of length \( x \) and an external tangent of length \( y \), where \( x \) and \( y \) are positive integers. Find the sum of all possible values of \( x + y \). | [
"44"
] | [
"Two nonintersecting circles with radii 3 and 4 have an internal tangent of length \\( x \\) and an external tangent of length \\( y \\), where \\( x \\) and \\( y \\) are positive integers. Find the sum of all possible values of \\( x + y \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ac68cdafaaa61601f55a2b73cdfa327258d526d1f832a9936ad0883f8c1ebdfb | Two nonintersecting circles with radii \( R \) and \( r \) are each tangent to both sides of the same angle. Construct an isosceles triangle such that its base lies on one side of the angle, the vertex is on the other side, and each leg touches one of the circles. Express the length of the altitude to the base of this ... | [
"\\( R + r \\)"
] | [
"Two nonintersecting circles with radii \\( R \\) and \\( r \\) are each tangent to both sides of the same angle. Construct an isosceles triangle such that its base lies on one side of the angle, the vertex is on the other side, and each leg touches one of the circles. Express the length of the altitude to the base... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-254a008a4c7b2d580323856d5b0e1cb6343daf048a3322285ebfb473502b8fe3 | Two nonnegative real numbers \( x \) and \( y \) have a constant sum \( a \). Find the minimum value of \( x^m + y^m \), where \( m \) is a given positive integer. | [
"\\(\\frac{a^m}{2^{m-1}}\\)"
] | [
"Two nonnegative real numbers \\( x \\) and \\( y \\) have a constant sum \\( a \\). Find the minimum value of \\( x^m + y^m \\), where \\( m \\) is a given positive integer."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-f05f22c7a13139512a9cb906d2ed89555d66d7b7695b126efe29d7701fdfc706 | Two nonoverlapping squares with sides \(a\) and \(b\) are placed inside a square of side 1. Find the maximum possible value of \(a + b\). | [
"1"
] | [
"Two nonoverlapping squares with sides \\(a\\) and \\(b\\) are placed inside a square of side 1. Find the maximum possible value of \\(a + b\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-b50c94131475a40c837196e9f54a1e3af6221ed093232541ead94a5b73546f56 | Two normal dice have their faces randomly reassigned to their sides such that no two faces occupy the same side of a die. The dice are then rolled. Determine the probability that the sum of the top faces is 7. | [
"\\(\\frac{2}{11}\\)"
] | [
"Two normal dice have their faces randomly reassigned to their sides such that no two faces occupy the same side of a die. The dice are then rolled. Determine the probability that the sum of the top faces is 7."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-958f2fcde8f084427590186c969c8be2050d65662e0493fb3c7a9958e093b980 | Two not necessarily distinct real numbers \(a\) and \(b\) are chosen uniformly at random from the range \((0, 1)\). The probability that \(\{a\} + \{b\} = \{a^2\} + \{b^2\}\) can be expressed in the form \(\frac{a - b\sqrt{c}}{d}\), where \(a\) and \(d\) are relatively prime positive integers and \(c\) is not divisible... | [
"23"
] | [
"Two not necessarily distinct real numbers \\(a\\) and \\(b\\) are chosen uniformly at random from the range \\((0, 1)\\). The probability that \\(\\{a\\} + \\{b\\} = \\{a^2\\} + \\{b^2\\}\\) can be expressed in the form \\(\\frac{a - b\\sqrt{c}}{d}\\), where \\(a\\) and \\(d\\) are relatively prime positive intege... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d7f971ff90af92153df65063fcf17b00ee628c1e81e776726cb3aa1f692b8f0c | Two not necessarily equal non-intersecting wooden disks, one gray and one black, are glued to a plane. An infinite angle with one gray side and one black side can be moved along the plane so that the disks remain outside the angle, while the colored sides of the angle are tangent to the disks of the same color (the tan... | [
"\\left( \\dfrac{4}{3}, 0 \\right)"
] | [
"Two not necessarily equal non-intersecting wooden disks, one gray and one black, are glued to a plane. An infinite angle with one gray side and one black side can be moved along the plane so that the disks remain outside the angle, while the colored sides of the angle are tangent to the disks of the same color (th... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-06952d95dfea573c371016ecbe601bb3de7ccf1641ae5120a83ad5c93b492913 | Two numbers $(z_1, z_2) \in \mathbb{C}^* \times \mathbb{C}^*$ have the property $(P)$ if there exists a real number $a \in [-2,2]$ such that $z_1^2 - az_1z_2 + z_2^2 = 0$. Let $z_1 = 1 + i$ and $z_2 = 1 - i$. Determine the smallest positive integer $n$ such that $(z_1^n, z_2^n)$ satisfy the property $(P)$ with $a = 0$. | [
"1"
] | [
"Two numbers $(z_1, z_2) \\in \\mathbb{C}^* \\times \\mathbb{C}^*$ have the property $(P)$ if there exists a real number $a \\in [-2,2]$ such that $z_1^2 - az_1z_2 + z_2^2 = 0$. Let $z_1 = 1 + i$ and $z_2 = 1 - i$. Determine the smallest positive integer $n$ such that $(z_1^n, z_2^n)$ satisfy the property $(P)$ wit... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-94ccf8c9371f62fc4478998ebe2d3fd87827fd87f4b6f20911ee58e44283825e | Two numbers $90$ and $m$ share exactly three positive divisors. What is the greatest of these three common divisors? | [
"9"
] | [
"Two numbers $90$ and $m$ share exactly three positive divisors. What is the greatest of these three common divisors?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cf0f6f66a9e60d36c3aaf0b5d5d9c1e01fe30d6fb3340826f7bb470630326f8b | Two numbers $x$ and $y$ are chosen at random from the segment $[0,1]$. Find the probability that $y \leq 4x$. | [
"\\(\\frac{7}{8}\\)"
] | [
"Two numbers $x$ and $y$ are chosen at random from the segment $[0,1]$. Find the probability that $y \\leq 4x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 29
} | 77 | 0 | 0 |
omr-ws-091ecc8ad9af97dce14b3f55724e7d2cf225ef17cffae38018a390b8a485fe59 | Two numbers \( (z_1, z_2) \in \mathbb{C}^* \times \mathbb{C}^* \) have the property \( (P) \) if there is a real number \( a \in [-2, 2] \) such that \( z_1^2 - az_1z_2 + z_2^2 = 0 \). Let \( z_1 = 1 + i \) and \( z_2 = 1 - i \). Find the sum of all possible values of \( a \) for which \( (z_1^n, z_2^n) \) satisfy the ... | [
"0"
] | [
"Two numbers \\( (z_1, z_2) \\in \\mathbb{C}^* \\times \\mathbb{C}^* \\) have the property \\( (P) \\) if there is a real number \\( a \\in [-2, 2] \\) such that \\( z_1^2 - az_1z_2 + z_2^2 = 0 \\). Let \\( z_1 = 1 + i \\) and \\( z_2 = 1 - i \\). Find the sum of all possible values of \\( a \\) for which \\( (z_1^... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bb764845a340a6d5b369104eb06a1a3e628a4d67672a2031f0902cbf07658f4a | Two numbers \( X \) and \( Y \) are chosen randomly with replacement from the set \( \{1, 2, \ldots, n\} \) where \( n \geq 4 \). What is the probability \( p \) that \( (X^2 - Y^2) \mod 3 = 0 \)? | [
"\\( \\frac{r^2 + (n - r)^2}{n^2} \\) where \\( r = \\lfloor \\frac{n}{3} \\rfloor \\)"
] | [
"Two numbers \\( X \\) and \\( Y \\) are chosen randomly with replacement from the set \\( \\{1, 2, \\ldots, n\\} \\) where \\( n \\geq 4 \\). What is the probability \\( p \\) that \\( (X^2 - Y^2) \\mod 3 = 0 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 7,
"tir": 19
} | 62 | 0 | 0 |
omr-ws-ab3be5700ae70fe84b19d7f23fe396286f5e4ca3a5cad7156dfa7aed4f69d563 | Two numbers \( a \) and \( b \) are selected from the set \(\{1, 2, \ldots, 3m\}\). What is the probability that \( 3 \) divides \( a^3 + b^3 \)? | [
"\\(\\frac{1}{3}\\)"
] | [
"Two numbers \\( a \\) and \\( b \\) are selected from the set \\(\\{1, 2, \\ldots, 3m\\}\\). What is the probability that \\( 3 \\) divides \\( a^3 + b^3 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-fe7c6bf54f484d43b3005ae76f730977a10e878eeae20ba4087c4cb2be382c32 | Two numbers \( x \) and \( y \) are chosen at random (without replacement) from the numbers 1, 2, 3, ..., 2004. What is the probability that \( x^3 + y^3 \) is divisible by 3? | [
"\\(\\frac{667}{2003}\\)"
] | [
"Two numbers \\( x \\) and \\( y \\) are chosen at random (without replacement) from the numbers 1, 2, 3, ..., 2004. What is the probability that \\( x^3 + y^3 \\) is divisible by 3?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3d6ba0672dc1b56c4540fb0fe6f306af39fa514787b336ea91982b9f830d2443 | Two numbers \( x \) and \( y \) are chosen at random (without replacement) from the set \(\{1, 2, 3, \ldots, 2004\}\). Find the probability that \( x^3 + y^3 \) is divisible by 3. | [
"\\(\\frac{1}{3}\\)"
] | [
"Two numbers \\( x \\) and \\( y \\) are chosen at random (without replacement) from the set \\(\\{1, 2, 3, \\ldots, 2004\\}\\). Find the probability that \\( x^3 + y^3 \\) is divisible by 3."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-c519cefe67f11b0234a033ab9f3359dd5d8d88c3d4b8a7b6fcc3f74874196d47 | Two numbers \( x \) and \( y \) are randomly selected from the set \(\{1, 2, \ldots, 20\}\). Find the probability that \( |x - y| \geq 10 \). | [
"\\(\\frac{11}{38}\\)"
] | [
"Two numbers \\( x \\) and \\( y \\) are randomly selected from the set \\(\\{1, 2, \\ldots, 20\\}\\). Find the probability that \\( |x - y| \\geq 10 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 6
} | 28 | 0 | 0 |
omr-ws-8369fa69e4fa7fdd4c1a432a7c4d3f0430e9e23679393836819eb48fc508fc6e | Two numbers \(a\) and \(b\) are chosen randomly from the set of whole numbers. What is the probability that \(a^2 + b^2\) is divisible by 5? | [
"\\(\\frac{1}{5}\\)"
] | [
"Two numbers \\(a\\) and \\(b\\) are chosen randomly from the set of whole numbers. What is the probability that \\(a^2 + b^2\\) is divisible by 5?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-db81a889d3310433f95dcb0b89f429f3dc9c05d6b7775152bf8c134d13f76b16 | Two numbers \(a\) and \(b\) satisfy the condition \(\text{lcm} (\gcd(24, a), \gcd(a, b)) = \text{lcm} (\gcd(24, b), \gcd(a, b))\). If \(a = 2b\), what is the smallest possible value of \(a + b\)? | [
"24"
] | [
"Two numbers \\(a\\) and \\(b\\) satisfy the condition \\(\\text{lcm} (\\gcd(24, a), \\gcd(a, b)) = \\text{lcm} (\\gcd(24, b), \\gcd(a, b))\\). If \\(a = 2b\\), what is the smallest possible value of \\(a + b\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5c56037d2e85d7de806bdaed4ff84a491c08e8e19670e82647eb460a5125b434 | Two numbers are chosen at random from the set \(\{1, 2, 3, \ldots, 188, 189\}\). Find the probability that the sum of the squares of these two chosen numbers is also a perfect square. | [
"\\(\\frac{71}{8883}\\)"
] | [
"Two numbers are chosen at random from the set \\(\\{1, 2, 3, \\ldots, 188, 189\\}\\). Find the probability that the sum of the squares of these two chosen numbers is also a perfect square."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 8,
"tir": 26
} | 37 | 0 | 0 |
omr-ws-76c52728c484e2b5a4f628b095d2917be015a11bbf4a822af84721ccfdedf590 | Two numbers are chosen between 0 and 2. What is the probability that the sum of the numbers is between 2 and 3? | [
"\\dfrac{3}{8}"
] | [
"Two numbers are chosen between 0 and 2. What is the probability that the sum of the numbers is between 2 and 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b39d07202166ecde858610ab0cbaf18519b24074d84d1079a3fd1977ead81ace | Two numbers are chosen from $1, 3, 5, 7, \ldots, 147, 149, 151$ and multiplied together. Find the number of ways in which the product is a multiple of $5$. | [
"1020"
] | [
"Two numbers are chosen from $1, 3, 5, 7, \\ldots, 147, 149, 151$ and multiplied together. Find the number of ways in which the product is a multiple of $5$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-8eea5afe643fa2fc0f3264508c74d1f5fd6a130cff7c7669f882a10316eba399 | Two numbers are drawn at random from the set \(\{1, 2, 3, \ldots, 100\}\). Find the probability that one of the drawn numbers is half of the other. | [
"\\(\\frac{1}{99}\\)"
] | [
"Two numbers are drawn at random from the set \\(\\{1, 2, 3, \\ldots, 100\\}\\). Find the probability that one of the drawn numbers is half of the other."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-32c183c1e4fa39ed0c926cfeceecb27062d352d6232e267743314aa99b04f858 | Two numbers are drawn from the set \(\{1, 2, 3, \ldots, n\}\) with replacement. Then, three numbers are drawn from the same set with replacement. What is the probability that the sum of the two numbers is greater than the sum of the three numbers? | [
"\\dfrac{(n - 1)(n)(n + 1)(3n + 1)}{40n^5}"
] | [
"Two numbers are drawn from the set \\(\\{1, 2, 3, \\ldots, n\\}\\) with replacement. Then, three numbers are drawn from the same set with replacement. What is the probability that the sum of the two numbers is greater than the sum of the three numbers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d2e248e31826de5b230095c7c1d11719b7e0dec1f4c90867003ac247a01da0d | Two numbers are drawn from two bags. The first bag contains numbers 1 to 6, and the second bag contains numbers 1 to 3. The numbers drawn cannot be the same. What is the formula to calculate the number of possible combinations? | [
"6 \\times 3 - 3"
] | [
"Two numbers are drawn from two bags. The first bag contains numbers 1 to 6, and the second bag contains numbers 1 to 3. The numbers drawn cannot be the same. What is the formula to calculate the number of possible combinations?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e4e96faa19b7a6adbaa31afb25b6d5db13ff9cdebd530bc5667c87f864753d49 | Two numbers are independently selected from the set of positive integers less than or equal to 5. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | [
"\\frac{3}{5}"
] | [
"Two numbers are independently selected from the set of positive integers less than or equal to 5. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e4d8718144ba274c40f8bf9c8626490973690ce82884a9bb0e42b2e1f6bcd8f6 | Two numbers are randomly selected from the interval $[0, 1]$. Given $\alpha \in [0, 1]$, what is the probability that the smaller of the two numbers does not exceed $\alpha$? | [
"\\(2\\alpha - \\alpha^2\\)"
] | [
"Two numbers are randomly selected from the interval $[0, 1]$. Given $\\alpha \\in [0, 1]$, what is the probability that the smaller of the two numbers does not exceed $\\alpha$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-d32e6ebf57f8dca4afc353dcbb21611a7bcb484ced22db71a9daf75a04b1f191 | Two numbers are randomly selected from the interval \((0, \frac{3}{2})\). Find the probability that their product is less than a constant real number \(k\) (where \(0 < k < \frac{3}{2}\)). | [
"\\(\\frac{4k}{9}\\left(1 - \\ln\\frac{4k}{9}\\right)\\)"
] | [
"Two numbers are randomly selected from the interval \\((0, \\frac{3}{2})\\). Find the probability that their product is less than a constant real number \\(k\\) (where \\(0 < k < \\frac{3}{2}\\))."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 22
} | 36 | 0 | 0 |
omr-ws-a02e8016518ac9e15a9302451006dbef9ec1a5ec6c91947de74a2d3cc76e0ad8 | Two numbers are selected at random from the numbers 1, 2, 3, ..., 10 without replacement. What is the probability that the minimum of the two numbers is less than 5? Consider whether the sample space should be $\binom{10}{2}$ or $10 \times 9$. | [
"\\(\\frac{2}{3}\\)"
] | [
"Two numbers are selected at random from the numbers 1, 2, 3, ..., 10 without replacement. What is the probability that the minimum of the two numbers is less than 5? Consider whether the sample space should be $\\binom{10}{2}$ or $10 \\times 9$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-c0cfb3e1bfefeced9e108bebd6bc9005527236190d2c1b9b51621219f8018f0a | Two numbers are selected from the first 50 natural numbers. In how many ways can this be done such that the first number is less than the second number and the sum of the numbers is divisible by 5? | [
"245"
] | [
"Two numbers are selected from the first 50 natural numbers. In how many ways can this be done such that the first number is less than the second number and the sum of the numbers is divisible by 5?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 12
} | 29 | 0 | 0 |
omr-ws-581b93c48d16721eefbb5dc49a1a4db106d583253e1cbf229e0d0bb49467d5a7 | Two numbers are such that the sum of their cubes is 5 and the sum of their squares is 3. Find the sum of the numbers. | [
"\\( 2 \\) or \\( -1 + \\sqrt{6} \\)",
"\\( 2, -1 + \\sqrt{6}, -1 - \\sqrt{6} \\)"
] | [
"Two numbers are such that the sum of their cubes is 5 and the sum of their squares is 3. Find the sum of the numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 4,
"tir": 4
} | 35 | 0 | 0 |
omr-ws-f320f9dfa04142dbf317b3d5371ea5a5d71620b6b1d7636864c460fcccf91f21 | Two numbers are such that their difference, their sum, and their product are in the ratio $1: 7: 24$. Determine the product of these two numbers. | [
"48"
] | [
"Two numbers are such that their difference, their sum, and their product are in the ratio $1: 7: 24$. Determine the product of these two numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-fe9dbacfd264d177bf3e1e3b9305c26db0425b50087c3f5d5b4546b6d9e52491 | Two numbers have a difference of 3 and a sum of 31. What is the larger of the two numbers? | [
"17"
] | [
"Two numbers have a difference of 3 and a sum of 31. What is the larger of the two numbers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ac45210050f47430ad08f4753b9c7730265cb8a633816f31b2852cfbe437917a | Two numbers sum to 36. The difference between the numbers is 71. What are both numbers? | [
"\\( \\frac{107}{2} \\) and \\( -\\frac{35}{2} \\)"
] | [
"Two numbers sum to 36. The difference between the numbers is 71. What are both numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 25
} | 31 | 0 | 0 |
omr-ws-2a8a4c27f740488240443029a19503aa93a407dda99e73da7af761f98154c3c5 | Two numbers, \( x \) and \( y \), are chosen at random such that each is between 0 and 1. What is the probability that the chosen numbers are the side lengths of a triangle with positive area? | [
"\\dfrac{1}{2}"
] | [
"Two numbers, \\( x \\) and \\( y \\), are chosen at random such that each is between 0 and 1. What is the probability that the chosen numbers are the side lengths of a triangle with positive area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-5bd96c384eaa0cf2f42040d2cc627f08c22b3e102afacf8cb5a0a274eed671cf | Two numbers, \(a\) and \(b\), form a compatible pair if \(s(a) - 2d' = b\) and \(s(b) - 2d'' = a\), where \(s(n)\) is the sum of the proper divisors of \(n\), \(d'\) is one of the proper divisors of \(a\), and \(d''\) is one of the proper divisors of \(b\). Find other compatible pairs besides \(30\) and \(40\). | [
"(42, 52), (24, 28), (80, 104), (48, 60)"
] | [
"Two numbers, \\(a\\) and \\(b\\), form a compatible pair if \\(s(a) - 2d' = b\\) and \\(s(b) - 2d'' = a\\), where \\(s(n)\\) is the sum of the proper divisors of \\(n\\), \\(d'\\) is one of the proper divisors of \\(a\\), and \\(d''\\) is one of the proper divisors of \\(b\\). Find other compatible pairs besides \... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d291ca1349fc15f28375f0701144749da0618e085080f3c6c3a8cc19b17ed4a2 | Two objects start at the origin in the plane and move simultaneously and independently. With each step, they move either up, down, left, or right. Find the probability that the objects are in the same position after $n$ steps. | [
"\\dfrac{\\dbinom{2n}{n}^2}{16^n}"
] | [
"Two objects start at the origin in the plane and move simultaneously and independently. With each step, they move either up, down, left, or right. Find the probability that the objects are in the same position after $n$ steps."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-70c8ad52f69662d44de748f63f090a478fdcc6ebb407f3ea8bef61b286e2cc39 | Two oblique cylinders of the same height \( h \) have a circle of radius \( a \) as the lower base and their upper bases are tangent. Calculate the shared volume. | [
"\\dfrac{4}{3} a^2 h"
] | [
"Two oblique cylinders of the same height \\( h \\) have a circle of radius \\( a \\) as the lower base and their upper bases are tangent. Calculate the shared volume."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0344011a85bb684eacd018b70da7ac5d606b2311d15c9c976c9c6fcb7a6f41d7 | Two odd numbers \(a\) and \(b\) are called Eulerian brothers if \(a - \phi(a) = b - \phi(b)\). Given that all prime numbers greater than 2 are Eulerian brothers, find a closed formula or method to determine all composite Eulerian brothers for a given composite number \(n\). | [
"m \\text{ is composite and } m - \\phi(m) = n - \\phi(n)"
] | [
"Two odd numbers \\(a\\) and \\(b\\) are called Eulerian brothers if \\(a - \\phi(a) = b - \\phi(b)\\). Given that all prime numbers greater than 2 are Eulerian brothers, find a closed formula or method to determine all composite Eulerian brothers for a given composite number \\(n\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-5d4c468d16b19345751d8028b1e3ef90d61e369902e608e26faa65afc5c532c6 | Two of six persons (A, B, C, D, E, F) stole apples. The statements made by the individuals are as follows: - A said "B and C." - D said "E and F." - E said "A and B." - B said "F and D." - C said "B and E." Four of the five persons interrogated had named one apple thief correctly and one incorrectly. The fifth had name... | [
"D: the thieves are b and d"
] | [
"Two of six persons (A, B, C, D, E, F) stole apples. The statements made by the individuals are as follows:\n- A said \"B and C.\"\n- D said \"E and F.\"\n- E said \"A and B.\"\n- B said \"F and D.\"\n- C said \"B and E.\"\nFour of the five persons interrogated had named one apple thief correctly and one incorrectl... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 27
} | 40 | 0 | 0 |
omr-ws-1e850aee52c43de1e90b8f0a61b508a814c93463502fab0f928d32924b569d83 | Two of the altitudes of a right triangle have lengths 4 and 5. What is the largest possible length of the third altitude? | [
"\\(\\frac{20}{3}\\)"
] | [
"Two of the altitudes of a right triangle have lengths 4 and 5. What is the largest possible length of the third altitude?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-3f7a6193866f5d35b5de040005b60c5583e2f64b416d0597b4b87e6cfb214965 | Two of the altitudes of a scalene triangle \(ABC\) have lengths 4 and 12. If the length of the third altitude is also an integer, what is the biggest it can be? | [
"5"
] | [
"Two of the altitudes of a scalene triangle \\(ABC\\) have lengths 4 and 12. If the length of the third altitude is also an integer, what is the biggest it can be?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-4898337d49ab7182a0e09ba3a11fd2c5c8dd16bd77263f254a6d353947f13bfc | Two of the altitudes of the scalene triangle ABC have lengths 4 and 12. If the length of the third altitude is also an integer, what is the biggest it can be? | [
"5"
] | [
"Two of the altitudes of the scalene triangle ABC have lengths 4 and 12. If the length of the third altitude is also an integer, what is the biggest it can be?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-20fb8c0ac4f8c517e1dd17ce20cb29ae51a8c56e0c1fa5ef295f9f1b3d87cfae | Two of the altitudes of the scalene triangle \(ABC\) have lengths 4 and 12. If the length of the third altitude is also an integer, what is the biggest it can be? | [
"5"
] | [
"Two of the altitudes of the scalene triangle \\(ABC\\) have lengths 4 and 12. If the length of the third altitude is also an integer, what is the biggest it can be?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-ba2b9d19776329b30f5858494161d5546596bf80c95e956ab76d5b99b11b4659 | Two of the altitudes of triangle $ABC$ are acute and equal to $4$ and $12$, respectively. If the length of the third altitude of the triangle is an integer, what is the maximum length of the third altitude? | [
"4"
] | [
"Two of the altitudes of triangle $ABC$ are acute and equal to $4$ and $12$, respectively. If the length of the third altitude of the triangle is an integer, what is the maximum length of the third altitude?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-87f6f34c64b313132cef6da401f7f2f48bb2c3ae588acadb995e3f46b00c0c6c | Two of the angles of a triangle are 15 degrees and 60 degrees. The triangle is inscribed in a circle with a radius of 6. What is the area of this triangle? | [
"\\( 9 \\sqrt{3} \\)"
] | [
"Two of the angles of a triangle are 15 degrees and 60 degrees. The triangle is inscribed in a circle with a radius of 6. What is the area of this triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-fd9af59ece102384177554fe266930a52ebf2203451f26eda6bc24cffc4116ad | Two of the edge lengths of a tetrahedron are 1 and $\sqrt{2}$. If its vertices are also vertices of some cube, how many such tetrahedra exist? | [
"3"
] | [
"Two of the edge lengths of a tetrahedron are 1 and $\\sqrt{2}$. If its vertices are also vertices of some cube, how many such tetrahedra exist?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-198020d341cb153fa8123d0242ebdb990279c3f048bb6fba721d16d7fd51fb9c | Two of the edge lengths of a tetrahedron are 1 and \(\sqrt{2}\). If its vertices are also vertices of some cube, how many such tetrahedra exist? | [
"8"
] | [
"Two of the edge lengths of a tetrahedron are 1 and \\(\\sqrt{2}\\). If its vertices are also vertices of some cube, how many such tetrahedra exist?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-a72425aa439d9a5d5511a02c42a4089e0e83b4938e23a19a34c10c504d74dfb6 | Two of the numbers in a three-term combination are 17 and 24. There are 40 possibilities for the third number. If the order of the numbers is unknown, how long will it take to try all possible combinations at a rate of ten seconds per try? | [
"40 minutes"
] | [
"Two of the numbers in a three-term combination are 17 and 24. There are 40 possibilities for the third number. If the order of the numbers is unknown, how long will it take to try all possible combinations at a rate of ten seconds per try?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 1,
"tir": 11
} | 43 | 0 | 0 |
omr-ws-59cce14720d27800c74782976e1ffcb4e6ecd546e9957dc8fec22985d491bb67 | Two of the numbers in a three-term combination for Rick's gym locker are 17 and 24. The third number is unknown but is between 1 and 40. How long will it take Rick to try all possible combinations if each attempt takes 10 seconds? | [
"2340 seconds"
] | [
"Two of the numbers in a three-term combination for Rick's gym locker are 17 and 24. The third number is unknown but is between 1 and 40. How long will it take Rick to try all possible combinations if each attempt takes 10 seconds?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 24,
"genselect": 6,
"tir": 18
} | 48 | 0 | 0 |
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