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-f831e56d2a51a2f002c401827ef5a715c4572b53b17acf6de366851d0f7a38d6 | A team of athletes, \( \frac{1}{3} \) of which are snowboarders, descended from the mountain. Some of the athletes boarded a cable car having no more than 10 people capacity. The remaining athletes went down the mountain by foot and their number was estimated as more than 45% but less than 50% of the total number of at... | [
"5"
] | [
"A team of athletes, \\( \\frac{1}{3} \\) of which are snowboarders, descended from the mountain. Some of the athletes boarded a cable car having no more than 10 people capacity. The remaining athletes went down the mountain by foot and their number was estimated as more than 45% but less than 50% of the total numb... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2e41d4cf94c047740e25106e06fa4304092e6fa13a385e66bfb620b8d1647967 | A team of metal workers can do a certain job of machining parts 15 h quicker than a team of apprentices. If a team of apprentices works for 18 h on the job and then a team of metal workers continues with the task for 6 h, only 0.6 of the total job will be done. How much time will it take a team of apprentices to comple... | [
"45 h"
] | [
"A team of metal workers can do a certain job of machining parts 15 h quicker than a team of apprentices. If a team of apprentices works for 18 h on the job and then a team of metal workers continues with the task for 6 h, only 0.6 of the total job will be done. How much time will it take a team of apprentices to c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f90fcc6f25c85fa9737563c49b9ab1c739342d233e05ee9bc784dc2db42ea51b | A team of three tractors (two tractors of make $A$ and one of make $B$) plough a field of 400 hectares in 10 days when the three tractors work together. The tractor of make $B$ can plough the field $8\frac{1}{3}$ days quicker than one tractor of make $A$ can do the job. How many hectares a day can a tractor of make $A$... | [
"Tractor of make $A$ does 12 hectares, tractor of make $B$, 16 hectares"
] | [
"A team of three tractors (two tractors of make $A$ and one of make $B$) plough a field of 400 hectares in 10 days when the three tractors work together. The tractor of make $B$ can plough the field $8\\frac{1}{3}$ days quicker than one tractor of make $A$ can do the job. How many hectares a day can a tractor of ma... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7ddfcaefc946adf49a3fa47ccd528d52eb61ac27a169e2633adfe25d98992226 | A tech company is hired to install cables between 25 computers such that for any set of 4 computers (A, B, C, D), there is a way to share information between any pair of them using only the cables between this set. Determine the minimum number of cables the company needs to install. | [
"150"
] | [
"A tech company is hired to install cables between 25 computers such that for any set of 4 computers (A, B, C, D), there is a way to share information between any pair of them using only the cables between this set. Determine the minimum number of cables the company needs to install."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-9401efb3abc189865524b7e0776ac4c7ab78ef9dd27e0635cfcc31976d6ffc85 | A teenager heard the hall clock striking the hour. At some moment between 15 and 20 minutes later, the minute hand hid the hour hand. To the nearest second, what time was it then? | [
"3:16:22 AM"
] | [
"A teenager heard the hall clock striking the hour. At some moment between 15 and 20 minutes later, the minute hand hid the hour hand. To the nearest second, what time was it then?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-462c7a3659ab1ad854f35e126907b507d9d2b6968d727158d8d3d41a765b60a7 | A teenager wrote his own age after his father's age to form a four-digit number. From this four-digit number, he subtracted the absolute value of the difference of their ages to get 4,289. What is the sum of their ages? | [
"59"
] | [
"A teenager wrote his own age after his father's age to form a four-digit number. From this four-digit number, he subtracted the absolute value of the difference of their ages to get 4,289. What is the sum of their ages?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-8fc34efd4ecbf280b74ed0fb8a60ae41ed34edd5fc84242850d40d310c096c80 | A teenager wrote his own age after that of his father, forming a four-digit number. From this number, he subtracted the absolute value of the difference of their ages to get 4289. Find the sum of the father's and son's ages. | [
"59"
] | [
"A teenager wrote his own age after that of his father, forming a four-digit number. From this number, he subtracted the absolute value of the difference of their ages to get 4289. Find the sum of the father's and son's ages."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-093f7e4c981557064678ebfd0a3911d4238b7d585c74cdd9e66404374014b70e | A telecom company has a faulty system that transposes adjacent digits in 3-digit extensions. To correct this, a computer detects and reverses the transposition. What is the greatest number of 3-digit extensions the company can assign under this plan? | [
"340"
] | [
"A telecom company has a faulty system that transposes adjacent digits in 3-digit extensions. To correct this, a computer detects and reverses the transposition. What is the greatest number of 3-digit extensions the company can assign under this plan?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-9cf61aa162d5b977db5b171d8e002cc4ceac9fb40bd6ec2587aebe2046009ff6 | A telephone number has the form \(ABC - DEF - GHIJ\), where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, \(A > B > C\), \(D > E > F\), and \(G > H > I > J\). Furthermore, \(D\), \(E\), and \(F\) are consecutive even digits; \(G\), \(H\), \(I\), and \(... | [
"8"
] | [
"A telephone number has the form \\(ABC - DEF - GHIJ\\), where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, \\(A > B > C\\), \\(D > E > F\\), and \\(G > H > I > J\\). Furthermore, \\(D\\), \\(E\\), and \\(F\\) are consecutive even digits; \\(G\\),... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-02eb711e051b0c3b841d0468b21442b14d80ed8b83b406cb7d3d11a6ccd9371d | A telephone number with 7 digits is called a Beautiful Number if the digits in the first three numbers (which must be different) repeat in the next three digits or the last three digits. For example, some beautiful numbers are $7133719$, $7131735$, $7130713$, $1739317$, $5433354$. If the digits are taken from $0, 1, 2,... | [
"73872"
] | [
"A telephone number with 7 digits is called a Beautiful Number if the digits in the first three numbers (which must be different) repeat in the next three digits or the last three digits. For example, some beautiful numbers are $7133719$, $7131735$, $7130713$, $1739317$, $5433354$. If the digits are taken from $0, ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-8c0fe348be247edfecbf58be8d338a39f7ba77047e5b4d0f9451cb86451cb511 | A telephone operator makes 755 connections in a day with a probability of 0.001 of connecting to the wrong number each time a connection is made. What is the probability that in one day at least one connection is wrong, to the nearest thousandth? | [
"0.530"
] | [
"A telephone operator makes 755 connections in a day with a probability of 0.001 of connecting to the wrong number each time a connection is made. What is the probability that in one day at least one connection is wrong, to the nearest thousandth?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-77cd3ef73ab76a153e462c009016b64172768d93142c54012cb0731b33bf8b59 | A telephone pole is supported by a steel cable which extends from the top of the pole to a point on the ground 3 meters from its base. When Leah walks 2.5 meters from the base of the pole toward the point where the cable is attached to the ground, her head just touches the cable. Leah is 1.5 meters tall. How many meter... | [
"9"
] | [
"A telephone pole is supported by a steel cable which extends from the top of the pole to a point on the ground 3 meters from its base. When Leah walks 2.5 meters from the base of the pole toward the point where the cable is attached to the ground, her head just touches the cable. Leah is 1.5 meters tall. How many ... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-789940455ae14b98f5c2e46dc3f4302121cbf4603b79bd191aa53ef80e318327 | A television's price was discounted $40\%$ and then slashed another $40\%$ during a liquidation sale. By what percent has the original price been reduced? | [
"64\\%"
] | [
"A television's price was discounted $40\\%$ and then slashed another $40\\%$ during a liquidation sale. By what percent has the original price been reduced?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-adf8ed468f607eadb00581464373f7faccbd320596cb1cc8176928e44bb1c05b | A ten-button lock can be opened by depressing any order of the correct five buttons. If the lock is redesigned so that combinations can consist of as few as one button or as many as nine buttons, how many additional combinations would this allow? | [
"770"
] | [
"A ten-button lock can be opened by depressing any order of the correct five buttons. If the lock is redesigned so that combinations can consist of as few as one button or as many as nine buttons, how many additional combinations would this allow?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 10
} | 31 | 0 | 0 |
omr-ws-ac69904ec13251555fc0f66a4f0c66ab75a5a80155a5ed4675cc5d035f974e05 | A tennis net is made of strings forming a grid of small congruent squares, with the size of the net being $100 \times 10$ small squares. What is the maximal number of edges of small squares that can be cut without breaking the net into two pieces? (If an edge is cut, the cut is made in the middle, not at the ends.) | [
"891"
] | [
"A tennis net is made of strings forming a grid of small congruent squares, with the size of the net being $100 \\times 10$ small squares. What is the maximal number of edges of small squares that can be cut without breaking the net into two pieces? (If an edge is cut, the cut is made in the middle, not at the ends... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c358c28a72233d3bdd77b47f44d69a1eb4d860d61cf168c3d17b14e2a7826997 | A tennis net is made of strings tied up together which make a grid consisting of small squares. The size of the net is $100 \times 10$ small squares. What is the maximal number of sides of small squares which can be cut without breaking the net into two separate pieces? (The side is cut only in the middle, not at the e... | [
"\\( 891 \\)"
] | [
"A tennis net is made of strings tied up together which make a grid consisting of small squares. The size of the net is $100 \\times 10$ small squares. What is the maximal number of sides of small squares which can be cut without breaking the net into two separate pieces? (The side is cut only in the middle, not at... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-bd456843d4f0cbd21a299b722afce99cc43b49927e4156724cbab4f17638d4b3 | A tennis player has a 75% probability to win a point on his own serve. What is the probability that he wins his serving game? | [
"0.9494"
] | [
"A tennis player has a 75% probability to win a point on his own serve. What is the probability that he wins his serving game?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 42,
"tir": 5
} | 47 | 0 | 0 |
omr-ws-aa14a2773ee99ccfdfb07847a9d8faee899933e05f3084852a94b359beefdcad | A tennis player's win ratio at the start of a weekend is exactly 0.500. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than 0.503. What is the largest number of matches she could have won before the weekend began? | [
"164"
] | [
"A tennis player's win ratio at the start of a weekend is exactly 0.500. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than 0.503. What is the largest number of matches she could have won before the weekend began?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 51
} | 83 | 0 | 0 |
omr-ws-46cf4189d308b36c883ef24e128eb63be3a2643692d162afcc6bd046a3c448f3 | A tennis player's win ratio before the weekend was exactly $0.500$. Over the weekend, she won three matches and lost one. If her win ratio after the weekend was greater than $0.503$, what is the largest number of games she could have won prior to the weekend? | [
"164"
] | [
"A tennis player's win ratio before the weekend was exactly $0.500$. Over the weekend, she won three matches and lost one. If her win ratio after the weekend was greater than $0.503$, what is the largest number of games she could have won prior to the weekend?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 24
} | 56 | 0 | 0 |
omr-ws-7ddbf6f82bef87ce28056a70adb85e2b8e8ee5ec821b17fe89ace424f1e8ee33 | A tennis star plays at least one match every day, but no more than 20 matches total, over a period of 2 weeks. Let $S$ be the number of matches played from the start of the period up to and including each day. Find the smallest positive integer $k$ such that there must exist two days $i$ and $j$ where $S_j - S_i = 7$. | [
"13"
] | [
"A tennis star plays at least one match every day, but no more than 20 matches total, over a period of 2 weeks. Let $S$ be the number of matches played from the start of the period up to and including each day. Find the smallest positive integer $k$ such that there must exist two days $i$ and $j$ where $S_j - S_i =... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-4db9d4e45233eb39542a893311ed4733e052ad6432a4ea8ca729ecbed85261f5 | A tent canopy has a cross-sectional area which can be defined by the area between the curves $y=2x^2$ and $y=x^3$. Find the total volume the tent canopy encompasses (in square units) by rotating it about the $x$-axis. | [
"\\dfrac{256}{35}\\pi"
] | [
"A tent canopy has a cross-sectional area which can be defined by the area between the curves $y=2x^2$ and $y=x^3$. Find the total volume the tent canopy encompasses (in square units) by rotating it about the $x$-axis."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cbbeedce8da14053ad1dbcbbed407926fa4c702f89aa9a1189004bf1541ef8c3 | A termite sits at the point $(0, 0, 0)$, at the center of the octahedron $|x| + |y| + |z| \le 5$. The termite can only move a unit distance in either direction parallel to one of the $x$, $y$, or $z$ axes: each step it takes moves it to an adjacent lattice point. How many distinct paths, consisting of $5$ steps, can th... | [
"1566"
] | [
"A termite sits at the point $(0, 0, 0)$, at the center of the octahedron $|x| + |y| + |z| \\le 5$. The termite can only move a unit distance in either direction parallel to one of the $x$, $y$, or $z$ axes: each step it takes moves it to an adjacent lattice point. How many distinct paths, consisting of $5$ steps, ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-68c713056d554aaa9bd10f1790271d379d520d0c8a9aa8c52394dda086feb34f | A terrain ( $ABCD$ ) has a rectangular trapezoidal shape with $\angle A = 90^\circ$. The lengths are $AB = 30$ m, $AD = 20$ m, and $DC = 45$ m. The land must be divided into two areas of the same area by drawing a line parallel to $AD$. At what distance from $D$ should this line be drawn? | [
"18.75"
] | [
"A terrain ( $ABCD$ ) has a rectangular trapezoidal shape with $\\angle A = 90^\\circ$. The lengths are $AB = 30$ m, $AD = 20$ m, and $DC = 45$ m. The land must be divided into two areas of the same area by drawing a line parallel to $AD$. At what distance from $D$ should this line be drawn?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-32faf074063e7b6240f5e33c5c859ac8b7de32e932a5b03ef73fb8b99a4f451a | A test consists of $30$ true or false questions. After the test (answering all $30$ questions), Victor gets his score: the number of correct answers. Victor is allowed to take the test (the same questions) several times. Can Victor work out a strategy that insures him to get a perfect score after: (a) $30$th attempt? (... | [
"No"
] | [
"A test consists of $30$ true or false questions. After the test (answering all $30$ questions), Victor gets his score: the number of correct answers. Victor is allowed to take the test (the same questions) several times. Can Victor work out a strategy that insures him to get a perfect score after:\n(a) $30$th atte... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ab1e970c4d00681c4c5634acf8efa4898f008714a10c9204a36db25f40fe907f | A test consists of 30 true or false questions. Victor can take the test multiple times and is informed of his score (number of correct answers) after each attempt. Can Victor devise a strategy to guarantee a perfect score by the 30th attempt? | [
"Yes"
] | [
"A test consists of 30 true or false questions. Victor can take the test multiple times and is informed of his score (number of correct answers) after each attempt. Can Victor devise a strategy to guarantee a perfect score by the 30th attempt?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-be276cddf2c47c4d2dd63284dfcf78cc1729e1fccbf13caac9da5c410015f611 | A test consists of 4 questions, each with 3 possible answers. A group of students takes the test, and it is observed that for every set of three students, there exists at least one question on which their answers are all different. What is the maximum number of students that could have taken the test? | [
"9"
] | [
"A test consists of 4 questions, each with 3 possible answers. A group of students takes the test, and it is observed that for every set of three students, there exists at least one question on which their answers are all different. What is the maximum number of students that could have taken the test?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-5306010287e5a6fbc30313dc929c46afa980521112f827ce4bbcd4eaed7be457 | A test contains 5 multiple choice questions, each with 4 options. Among 2000 sheets of answer papers, find the minimum value of \( n \) such that any \( n \) sheets guarantee the existence of 4 sheets where any two of them have at most 3 questions with the same answers. | [
"1025",
"769"
] | [
"A test contains 5 multiple choice questions, each with 4 options. Among 2000 sheets of answer papers, find the minimum value of \\( n \\) such that any \\( n \\) sheets guarantee the existence of 4 sheets where any two of them have at most 3 questions with the same answers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-6a5171bcedb1b0bfebf75cad5803e94993814a4b5ec801fc55201887bb7394ba | A test has twenty questions. Seven points are awarded for each correct answer, two points are deducted for each incorrect answer, and no points are awarded or deducted for each unanswered question. Joana obtained 87 points. How many questions did she not answer? | [
"5"
] | [
"A test has twenty questions. Seven points are awarded for each correct answer, two points are deducted for each incorrect answer, and no points are awarded or deducted for each unanswered question. Joana obtained 87 points. How many questions did she not answer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-0bb0c3df6ca419dd25c8f297f631b9146f318b984688e212bb389da9cd6a1ce9 | A test was given to a group of students. Among the grades the students received there are only 1, 2, 3 and 4. The same number of students received grades 1.0, 2.0, and 4.0, and there were more 3.0 grades than the other grades taken together. Less than 10 students received grades higher than 2.0. How many 2.0 grades and... | [
"Two 2.0 grades and seven 3.0 grades"
] | [
"A test was given to a group of students. Among the grades the students received there are only 1, 2, 3 and 4. The same number of students received grades 1.0, 2.0, and 4.0, and there were more 3.0 grades than the other grades taken together. Less than 10 students received grades higher than 2.0. How many 2.0 grade... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3fb61f9dc9f7eb08e8cd16e4d7e8dca765b28543d7b30c96db95f2b098b39b41 | A tetrahedron $ABCD$ has all edges of length less than $100$ and contains two nonintersecting spheres of diameter $1$. Let $r$ be the maximum possible radius of a sphere that can fit inside the tetrahedron without intersecting the given spheres. Calculate the greatest integer less than or equal to $100r$. | [
"2040"
] | [
"A tetrahedron $ABCD$ has all edges of length less than $100$ and contains two nonintersecting spheres of diameter $1$. Let $r$ be the maximum possible radius of a sphere that can fit inside the tetrahedron without intersecting the given spheres. Calculate the greatest integer less than or equal to $100r$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-a03be4e5dd6a527552a2e240b854edccc78c9b53bfdedba9e8a77933a0550efa | A tetrahedron $ABCD$ has edges $AB, AC,$ and $AD$ that are pairwise orthogonal. Given $AB=3$ and $CD=\sqrt{2}$, find the minimum possible value of $$BC^6 + BD^6 - AC^6 - AD^6.$$ | [
"1998"
] | [
"A tetrahedron $ABCD$ has edges $AB, AC,$ and $AD$ that are pairwise orthogonal. Given $AB=3$ and $CD=\\sqrt{2}$, find the minimum possible value of\n$$BC^6 + BD^6 - AC^6 - AD^6.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-fb2af5b3e7ce5631abdbea99de18269ff964c1a6ff00c542fbe305ed6dd19603 | A tetrahedron $ABCD$ has side lengths $AB=6$, $CD=8$, and $BC=DA=5$. Let $V$ be the maximum possible volume of the tetrahedron. If $V^4=2^n3^m$ for some integers $m$ and $n$, find the value of $mn$. | [
"-60"
] | [
"A tetrahedron $ABCD$ has side lengths $AB=6$, $CD=8$, and $BC=DA=5$. Let $V$ be the maximum possible volume of the tetrahedron. If $V^4=2^n3^m$ for some integers $m$ and $n$, find the value of $mn$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06b601a08045d27c3836a67bc4521e6dfa1bf48a9772dfce6835af8cea35cc9e | A tetrahedron $ABCD$ has six edges whose lengths, in some order, are $4$, $6$, $9$, $13$, $18$, and $21$. If edge $AB$ is $21$, how long is edge $CD$? | [
"6"
] | [
"A tetrahedron $ABCD$ has six edges whose lengths, in some order, are $4$, $6$, $9$, $13$, $18$, and $21$. If edge $AB$ is $21$, how long is edge $CD$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9dd5538609f5e5d46b5f5bfe9a89a00ddd40c3779450e9c1de6a9984041da7be | A tetrahedron $ABCD$ is circumscribed around a sphere $\omega$ of radius $r$, tangent to the faces $ABC, BCD, CDA, DAB$ at points $D_1, A_1, B_1, C_1$ respectively. The lines $AA_1, BB_1, CC_1, DD_1$ intersect the sphere $\omega$ for the second time at points $A_2, B_2, C_2, D_2$ respectively. If $r = 1$, calculate the... | [
"32"
] | [
"A tetrahedron $ABCD$ is circumscribed around a sphere $\\omega$ of radius $r$, tangent to the faces $ABC, BCD, CDA, DAB$ at points $D_1, A_1, B_1, C_1$ respectively. The lines $AA_1, BB_1, CC_1, DD_1$ intersect the sphere $\\omega$ for the second time at points $A_2, B_2, C_2, D_2$ respectively. If $r = 1$, calcul... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-9ab212a55f79f26f6f947f6cac46e19ead490c1e4f40fa5162fc1992b6c8c8a4 | A tetrahedron $ABCD$ is inscribed in a hemisphere with radius $r$ such that the vertices $A$, $B$, and $C$ lie on the periphery of the base of the hemisphere and $D$ lies at the vertex of the hemisphere. Let the volume of the tetrahedron be denoted by $V$ and the perimeter of the triangle $ABC$ by $u$. Find the maximum... | [
"324"
] | [
"A tetrahedron $ABCD$ is inscribed in a hemisphere with radius $r$ such that the vertices $A$, $B$, and $C$ lie on the periphery of the base of the hemisphere and $D$ lies at the vertex of the hemisphere. Let the volume of the tetrahedron be denoted by $V$ and the perimeter of the triangle $ABC$ by $u$. Find the ma... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-57eba64bcd0cde98243b9f104ee1c380eec878fde8b8a4297be5abf13e6234b7 | A tetrahedron $ABCD$ satisfies $\angle BAC = \angle CAD = \angle DAB = 90^\circ$. Let $a$, $b$, and $c$ be the lengths of the edges $AB$, $AC$, and $AD$, respectively. Calculate the value of $\frac{area(BCD)^2}{a^2 + b^2 + c^2}$. | [
"\\dfrac{a^2b^2 + a^2c^2 + b^2c^2}{4(a^2 + b^2 + c^2)}"
] | [
"A tetrahedron $ABCD$ satisfies $\\angle BAC = \\angle CAD = \\angle DAB = 90^\\circ$. Let $a$, $b$, and $c$ be the lengths of the edges $AB$, $AC$, and $AD$, respectively. Calculate the value of $\\frac{area(BCD)^2}{a^2 + b^2 + c^2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-460a41e0e955b733d70531107b10f2dc23bb8230503e42da0cdcf6fa95356f88 | A tetrahedron $ABCD$ with acute-angled edges is inscribed in a sphere centered at $O$. The straight line passing through point $O$ perpendicular to the plane $ABC$ intersects the sphere at point $E$ such that $D$ and $E$ lie on opposite sides relative to plane $ABC$. Line $DE$ intersects plane $ABC$ at point $F$, lying... | [
"\\(50^\\circ\\)"
] | [
"A tetrahedron $ABCD$ with acute-angled edges is inscribed in a sphere centered at $O$. The straight line passing through point $O$ perpendicular to the plane $ABC$ intersects the sphere at point $E$ such that $D$ and $E$ lie on opposite sides relative to plane $ABC$. Line $DE$ intersects plane $ABC$ at point $F$, ... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-e7bd0a6a4ebecceb6b126f51f5996c2315802b66ce594d2bd0f9737382ddfad0 | A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposite side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ at point $P$ that lies inside the triangle $ABC$. I... | [
"2"
] | [
"A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposite side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ at point $P$ that lies inside the triangle $AB... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-b6c2d71aca06c9c832fc91c976981e83bff58b09e80b6045dfb99e04ef5d379a | A tetrahedron $TABC$ with volume $1$ has a top vertex $T$ and a base that is an equilateral triangle $ABC$. The projection $T'$ of $T$ onto the base is the center of $ABC$. Point $I$ is the midpoint of $TT'$. A congruent tetrahedron $T'A'B'C'$ is generated by reflecting the original tetrahedron through $I$. Find the vo... | [
"\\(\\frac{1}{2}\\)"
] | [
"A tetrahedron $TABC$ with volume $1$ has a top vertex $T$ and a base that is an equilateral triangle $ABC$. The projection $T'$ of $T$ onto the base is the center of $ABC$. Point $I$ is the midpoint of $TT'$. A congruent tetrahedron $T'A'B'C'$ is generated by reflecting the original tetrahedron through $I$. Find t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 3
} | 8 | 0 | 0 |
omr-ws-216d664fd716fb961b4eec2bdafd030ce51cc0ab8028307f34200c8bb9cb7fd9 | A tetrahedron ABCD has a side length of 2. A nondegenerate cube of side length \( s \) has 4 vertices on face BCD and 4 vertices on the inscribed sphere of the tetrahedron. Find \( s \). | [
"\\(\\frac{2\\sqrt{6}}{9}\\)"
] | [
"A tetrahedron ABCD has a side length of 2. A nondegenerate cube of side length \\( s \\) has 4 vertices on face BCD and 4 vertices on the inscribed sphere of the tetrahedron. Find \\( s \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 8
} | 25 | 0 | 0 |
omr-ws-d627387bf5aae17aea019751041cc3b7f4a10841ee1f9c81b4fd75af96b1eca8 | A tetrahedron ABCD is formed from the following four triangles: - Face ABC has sides of lengths 5, 16, and 18. - Face ABD has sides of lengths 5, 12, and 13. - Face BCD has sides of lengths 12, 16, and 20. - Face ACD has sides of lengths 13, 18, and 20. What is the volume of the tetrahedron? | [
"154.11"
] | [
"A tetrahedron ABCD is formed from the following four triangles:\n- Face ABC has sides of lengths 5, 16, and 18.\n- Face ABD has sides of lengths 5, 12, and 13.\n- Face BCD has sides of lengths 12, 16, and 20.\n- Face ACD has sides of lengths 13, 18, and 20.\n\nWhat is the volume of the tetrahedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 19
} | 29 | 0 | 0 |
omr-ws-2aa3e0e267bf89e56974bb555df8b1a87182a9d8b300c7dbb128cff58159d3ba | A tetrahedron \(ABCD\) is inscribed in a sphere \(S\). Find the locus of points \(P\) on \(S\) such that \[ \frac{AP}{PA_{1}} + \frac{BP}{PB_{1}} + \frac{CP}{PC_{1}} + \frac{DP}{PD_{1}} = 4, \] where \(A_{1}, B_{1}, C_{1}, D_{1}\) are the other intersection points of \(AP\), \(BP\), \(CP\), \(DP\) with \(S\). | [
"S"
] | [
"A tetrahedron \\(ABCD\\) is inscribed in a sphere \\(S\\). Find the locus of points \\(P\\) on \\(S\\) such that\n\\[ \\frac{AP}{PA_{1}} + \\frac{BP}{PB_{1}} + \\frac{CP}{PC_{1}} + \\frac{DP}{PD_{1}} = 4, \\]\nwhere \\(A_{1}, B_{1}, C_{1}, D_{1}\\) are the other intersection points of \\(AP\\), \\(BP\\), \\(CP\\),... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0a16aa8a2669f77bd4e25cc8c643e5b24ca796bf64e740386867cc13d3b4b490 | A tetrahedron \(ABCD\) is such that \(\angle ACD + \angle BCD = \angle BAC + \angle CAD + \angle DAB = \angle ABC + \angle CBD + \angle DAB = 180^\circ\). Given that \(\angle ACB = \alpha\) and \(AC + CB = k\), compute the total surface area of the tetrahedron \(ABCD\). | [
"\\dfrac{k^2 \\sin \\alpha}{2}"
] | [
"A tetrahedron \\(ABCD\\) is such that \\(\\angle ACD + \\angle BCD = \\angle BAC + \\angle CAD + \\angle DAB = \\angle ABC + \\angle CBD + \\angle DAB = 180^\\circ\\). Given that \\(\\angle ACB = \\alpha\\) and \\(AC + CB = k\\), compute the total surface area of the tetrahedron \\(ABCD\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-cffa4799e174d0e5163bd5f4fb6fc336c7e18f7ff6611554fb444bebb533052c | A tetrahedron \(ABCS\) has three sides \(\overline{AS}, \overline{BS},\) and \(\overline{CS}\) which are all perpendicular to each other at vertex \(S\). The areas of triangles \(\triangle ABS\), \(\triangle BCS\), and \(\triangle ACS\) are \(10\), \(11\), and \(12\) respectively. Find the area of \(\triangle ABC\). | [
"\\(\\sqrt{365}\\)"
] | [
"A tetrahedron \\(ABCS\\) has three sides \\(\\overline{AS}, \\overline{BS},\\) and \\(\\overline{CS}\\) which are all perpendicular to each other at vertex \\(S\\). The areas of triangles \\(\\triangle ABS\\), \\(\\triangle BCS\\), and \\(\\triangle ACS\\) are \\(10\\), \\(11\\), and \\(12\\) respectively. Find th... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-c3571967537470ad4fda72ee84e212cee86c71d2b303f4ae4d57cf413f43aa2a | A tetrahedron has at most one edge longer than 1. What is the maximum total length of its edges? | [
"5 + \\sqrt{3}"
] | [
"A tetrahedron has at most one edge longer than 1. What is the maximum total length of its edges?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-dbbe92934290e3f26ace43826c70901f9d9e4d75bfabe6ebdd60f8db4b359c88 | A tetrahedron has the property that two opposite edges have equal length $a$ and are directionally perpendicular to each other. Additionally, when the midpoints of these two edges are connected, the resulting segment is perpendicular to both edges and has length $b$. Find the area of the tetrahedron in terms of $a$ and... | [
"a \\sqrt{a^2 + 4b^2}"
] | [
"A tetrahedron has the property that two opposite edges have equal length $a$ and are directionally perpendicular to each other. Additionally, when the midpoints of these two edges are connected, the resulting segment is perpendicular to both edges and has length $b$. Find the area of the tetrahedron in terms of $a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-7c37b53c6c77b1c3d2f87adf8c1d05daed7597093bfc54416d9f009afba46945 | A tetrahedron is called a MEMO-tetrahedron if all six side lengths are different positive integers where one of them is \( 2 \) and one of them is \( 3 \). Let \( l(T) \) be the sum of the side lengths of the tetrahedron \( T \). (a) Find all positive integers \( n \) so that there exists a MEMO-tetrahedron \( T \) wit... | [
"n \\geq 27"
] | [
"A tetrahedron is called a MEMO-tetrahedron if all six side lengths are different positive integers where one of them is \\( 2 \\) and one of them is \\( 3 \\). Let \\( l(T) \\) be the sum of the side lengths of the tetrahedron \\( T \\).\n(a) Find all positive integers \\( n \\) so that there exists a MEMO-tetrahe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-5114a43e9a2c68e7129aad849e45c83fbfefab2367216a660ca1580a567e9e2a | A tetrahedron is constructed by connecting the centers of the 4 sides of a regular tetrahedron. What is the ratio of the volume of the smaller tetrahedron to the volume of the original tetrahedron? | [
"\\(\\frac{1}{27}\\)"
] | [
"A tetrahedron is constructed by connecting the centers of the 4 sides of a regular tetrahedron. What is the ratio of the volume of the smaller tetrahedron to the volume of the original tetrahedron?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-f3afb98f10b0a0b0ce6ee5adb1e820fdf5578a2be9fdb422d54e7e7262996fc3 | A tetrahedron is formed by the planes \(x=0\), \(y=0\), \(z=0\), and a plane tangent to the surface \(f(x,y) = \frac{c}{xy}\). What is the volume of the tetrahedron in terms of \(c\)? | [
"\\(\\frac{9c}{2}\\)"
] | [
"A tetrahedron is formed by the planes \\(x=0\\), \\(y=0\\), \\(z=0\\), and a plane tangent to the surface \\(f(x,y) = \\frac{c}{xy}\\). What is the volume of the tetrahedron in terms of \\(c\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-ac7e13bfd63320cc86a31d8560d95681ce462b585293ca19e0d0473b23dd2993 | A tetrahedron is formed from an equilateral triangle of side length 4 cm by folding along the line segments that connect the midpoints of the sides. Compute the length of the altitude of the tetrahedron. | [
"\\(\\frac{2 \\sqrt 6}{3}\\)"
] | [
"A tetrahedron is formed from an equilateral triangle of side length 4 cm by folding along the line segments that connect the midpoints of the sides. Compute the length of the altitude of the tetrahedron."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-e66df211e861d975ea94e514119bb16f51a550e6b53f8a740d92de28dd36d10b | A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point P is selected at random inside the circumscribed sphere. What is ... | [
"0.2"
] | [
"A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point P is selected at random inside the circumscribed sphere. Wha... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-3746c5223dbe14217297ffad5f113b8eef07e3bf396f6e1156ed589d09aa8269 | A tetrahedron's four faces are right, similar triangles. The shortest segment on the tetrahedron has length 1. What is the volume? | [
"\\( \\frac{1}{6}\\sqrt{\\frac{\\sqrt{5}-1}{2}} \\)"
] | [
"A tetrahedron's four faces are right, similar triangles. The shortest segment on the tetrahedron has length 1. What is the volume?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8dadd3c3469b96a0d30324abf614b1d3d0f0619630ddcbfcbe08e4a9d46f3434 | A theater group gave 40 performances, with each performance involving 10 members of the group. For any two members of the group, either they never performed together or they performed together in exactly one performance. Determine the minimum number of members in the group. | [
"64"
] | [
"A theater group gave 40 performances, with each performance involving 10 members of the group. For any two members of the group, either they never performed together or they performed together in exactly one performance. Determine the minimum number of members in the group."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-1d6bdc51d95a9ff566516362996378e7d5bc7ffad2d521caf6890110bf015c5d | A theater is inviting $2670$ total students from various schools to see a movie. Each individual school invited is only allowed to bring at most $38$ students. If the theater has $193$ seats in each row, how many rows do they need to set aside for schools in order to guarantee that all students from the same school can... | [
"15"
] | [
"A theater is inviting $2670$ total students from various schools to see a movie. Each individual school invited is only allowed to bring at most $38$ students. If the theater has $193$ seats in each row, how many rows do they need to set aside for schools in order to guarantee that all students from the same schoo... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-d8bc03bea25c11c7223718dfad3bc281402de3f35517332cd66b8502f7b50f01 | A theater is inviting a maximum of 2470 students from various schools to show a movie. Each school has a quota that is the same across all schools and cannot exceed 38 students. Students from the same school must sit in the same row. If the theater has 193 seats in each row, what is the minimum number of rows needed to... | [
"13"
] | [
"A theater is inviting a maximum of 2470 students from various schools to show a movie. Each school has a quota that is the same across all schools and cannot exceed 38 students. Students from the same school must sit in the same row. If the theater has 193 seats in each row, what is the minimum number of rows need... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-f318610276428bf41e6f4c435bf71df36f0f1bdb727fcf443ef77f8621d16c7b | A theater that seats 150 people is hosting a drug abuse problem presentation. Admission is $2 for adults and $1 for students. Every two adults must bring at least one student. How many adults and students should attend to maximize the total proceeds? | [
"100 adults and 50 students"
] | [
"A theater that seats 150 people is hosting a drug abuse problem presentation. Admission is $2 for adults and $1 for students. Every two adults must bring at least one student. How many adults and students should attend to maximize the total proceeds?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-2bbdcc02237b5d3f218c1055d8c02d7e42f28f918886082ff977b6b8866956c7 | A thermometer initially reading 70°F is placed in an oven pre-heated to a constant temperature. The thermometer reads 110°F after 1/2 minute and 145°F after 1 minute. Using the formula \(\frac{dT}{dt} = k(T - T_m)\), where \(T_m\) is the oven temperature, determine the temperature of the oven. | [
"390"
] | [
"A thermometer initially reading 70°F is placed in an oven pre-heated to a constant temperature. The thermometer reads 110°F after 1/2 minute and 145°F after 1 minute. Using the formula \\(\\frac{dT}{dt} = k(T - T_m)\\), where \\(T_m\\) is the oven temperature, determine the temperature of the oven."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-389e4614eaa2dc7e09dc02305cab4a6f8b5a9fcdb1458bb90031d681cb5d4c52 | A thief enters a library with a $100 forged bill and buys a magazine and journal for $10. The library owner gives the thief $90 in change. Later, the library owner discovers the bill is forged and is reimbursed $100 by the restaurant owner. How much money did the library owner lose in total? | [
"0"
] | [
"A thief enters a library with a $100 forged bill and buys a magazine and journal for $10. The library owner gives the thief $90 in change. Later, the library owner discovers the bill is forged and is reimbursed $100 by the restaurant owner. How much money did the library owner lose in total?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-8851350fdce83d223410b0baa5b526970da4df6258067c872bd7b2616436e7d3 | A thief is given 100 white marbles and 100 black marbles, and 2 urns in which to place all the marbles however he likes. The king will choose an urn at random and then pull out a single marble at random. If the marble is white, the thief goes free; if it is black (or the urn is empty), the thief’s sentence is upheld. W... | [
"\\(\\frac{149}{199}\\)"
] | [
"A thief is given 100 white marbles and 100 black marbles, and 2 urns in which to place all the marbles however he likes. The king will choose an urn at random and then pull out a single marble at random. If the marble is white, the thief goes free; if it is black (or the urn is empty), the thief’s sentence is uphe... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-aa21d1d33ede17626dc73a45dc8c8f7930073d3ad03002d86ae9016bd5e8f077 | A thief is trying to unlock a security password of length $n$ using numbers from the set $\{1, 2, \ldots, n\}$. If a number in the sequence is correct, it lights up, but if a number is incorrect, all previously lit numbers turn off. Determine the minimum number of attempts the thief must make to guarantee unlocking the... | [
"2^n - 1"
] | [
"A thief is trying to unlock a security password of length $n$ using numbers from the set $\\{1, 2, \\ldots, n\\}$. If a number in the sequence is correct, it lights up, but if a number is incorrect, all previously lit numbers turn off. Determine the minimum number of attempts the thief must make to guarantee unloc... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-4a7215013fb4ea49b7fc81d51171d78be09308c2deaf6899c109d889c94703c6 | A thin circular path with diameter \(AB\) is laid on horizontal ground. A vertical flagpole is erected with its base at a point \(D\) on the diameter \(AB\). The angles of elevation of the top of the flagpole from \(A\) and \(B\) are \(\alpha\) and \(\beta\) respectively (both are acute). The point \(C\) lies on the ci... | [
"0"
] | [
"A thin circular path with diameter \\(AB\\) is laid on horizontal ground. A vertical flagpole is erected with its base at a point \\(D\\) on the diameter \\(AB\\). The angles of elevation of the top of the flagpole from \\(A\\) and \\(B\\) are \\(\\alpha\\) and \\(\\beta\\) respectively (both are acute). The point... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-71b77226b4aaa9d8aa86a9641bac763df04b1b9bee17cae90903856af9a80c71 | A thin cylinder of length $2a$ is randomly placed horizontally on a round table of radius $R > a$. Calculate the probability that exactly $k$ out of the two ends of the cylinder are off the table, where $k = 0, 1, 2$. | [
"1 - \\frac{(R - a)^2}{R^2} - \\frac{2a(2R - a)}{\\pi R^2}"
] | [
"A thin cylinder of length $2a$ is randomly placed horizontally on a round table of radius $R > a$. Calculate the probability that exactly $k$ out of the two ends of the cylinder are off the table, where $k = 0, 1, 2$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c576c85022463b66dd53d49806496753ed3b0c8751742a0196836c449ed13212 | A thin spherical shell $S$ of constant thickness and density has a total mass $M$ and center $O$. Let $P$ be a point outside $S$ at a distance $r$ from $O$. Calculate the gravitational force $F$ exerted on a mass $m$ located at $P$ due to the shell $S$ and express $F$ in terms of $M$, $m$, $r$, and the gravitational co... | [
"F = \\frac{G M m}{r^2}"
] | [
"A thin spherical shell $S$ of constant thickness and density has a total mass $M$ and center $O$. Let $P$ be a point outside $S$ at a distance $r$ from $O$. Calculate the gravitational force $F$ exerted on a mass $m$ located at $P$ due to the shell $S$ and express $F$ in terms of $M$, $m$, $r$, and the gravitation... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f5999279712e49a20806615c603b939ac344605cbcc0bc0c88cdc16f7a0fa195 | A three-digit integer $\overline{A2C}$, where $A$ and $C$ are digits, not necessarily distinct, and $A \ne 0$, is toxic if it is a multiple of $9$. For example, $126$ is toxic because $126 = 9 \cdot 14$. Find the sum of all toxic integers. | [
"5445"
] | [
"A three-digit integer $\\overline{A2C}$, where $A$ and $C$ are digits, not necessarily distinct, and $A \\ne 0$, is toxic if it is a multiple of $9$. For example, $126$ is toxic because $126 = 9 \\cdot 14$. Find the sum of all toxic integers."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d45d8526214a14e955811124ee8121e032934f904d8226bfbc3bbcc19f0a7a26 | A three-digit number $N$ has a total of 6 factors and 3 is one of its prime factors. What is the maximum possible value of $N$? | [
"981"
] | [
"A three-digit number $N$ has a total of 6 factors and 3 is one of its prime factors. What is the maximum possible value of $N$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-79bde5445599c010f4999c9b0d402d8585cd5d3c292970e88873dfe053a9ca6f | A three-digit number $\overline{abc}$ is called Ecuadorian if it meets the following conditions: - $\overline{abc}$ does not end in $0$. - $\overline{abc}$ is a multiple of $36$. - $\overline{abc} - \overline{cba}$ is positive and a multiple of $36$. Determine all the Ecuadorian numbers. | [
"864, 612"
] | [
"A three-digit number $\\overline{abc}$ is called Ecuadorian if it meets the following conditions:\n- $\\overline{abc}$ does not end in $0$.\n- $\\overline{abc}$ is a multiple of $36$.\n- $\\overline{abc} - \\overline{cba}$ is positive and a multiple of $36$.\nDetermine all the Ecuadorian numbers."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 30
} | 30 | 0 | 0 |
omr-ws-8ab9a767dfdc023773da12e4ee09faab8eb86db1c92d70b0e68ccc1809efdcf8 | A three-digit number $a$ is called mysterious if the decimal representations of $a$, $3a$, and $5a$ together contain all the digits from 1 to 9 exactly once, and do not contain the digit 0. Identify the conditions for a number to be mysterious. | [
"\\( a = 129 \\)"
] | [
"A three-digit number $a$ is called mysterious if the decimal representations of $a$, $3a$, and $5a$ together contain all the digits from 1 to 9 exactly once, and do not contain the digit 0. Identify the conditions for a number to be mysterious."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-16b052b3e723c4ef7f5092ddbe691c02e77228689096518ae28709dd79640b6d | A three-digit number \( m \) is chosen such that its hundreds digit is the sum of its tens and units digits. What is the smallest positive integer \( n \) such that \( n \) cannot divide \( m \)? | [
"23"
] | [
"A three-digit number \\( m \\) is chosen such that its hundreds digit is the sum of its tens and units digits. What is the smallest positive integer \\( n \\) such that \\( n \\) cannot divide \\( m \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-fdb700e1585cfa18756c544741ebf27229e223bf74e4815f65eeef5fa420f1a7 | A three-digit number \(ABC\) is 29 times the sum of its digits. When the first digit is made the last digit, the new number \(BCA\) is 68 times the sum of the digits. When the new first digit is made the last digit, the new number \(CAB\) is 14 times the sum of the digits. What is the sum of the squares of the digits? | [
"41"
] | [
"A three-digit number \\(ABC\\) is 29 times the sum of its digits. When the first digit is made the last digit, the new number \\(BCA\\) is 68 times the sum of the digits. When the new first digit is made the last digit, the new number \\(CAB\\) is 14 times the sum of the digits. What is the sum of the squares of t... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-ba0e610a2f089724dc68f8f1add119362517b2f40d9977a4d59a787247b7992c | A three-digit number does not contain the digit $9$. When each digit is increased by $1$, the product of the digits of the new number is three times the product of the digits of the original number. How many such numbers are there? | [
"15"
] | [
"A three-digit number does not contain the digit $9$. When each digit is increased by $1$, the product of the digits of the new number is three times the product of the digits of the original number. How many such numbers are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7f8f591531baaba0383a2376e058ad4d0ff580a7dc4d8437198bfecbcf689a38 | A three-digit number in base 7, when expressed in base 9, has its digits reversed in order. Find the number in base 7 and base 10. | [
"\\( 503_7 \\) and \\( 248 \\)"
] | [
"A three-digit number in base 7, when expressed in base 9, has its digits reversed in order. Find the number in base 7 and base 10."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 30
} | 37 | 0 | 0 |
omr-ws-03ca349b5e7aa4c5731576d87d6c1939f4b338d1960989103c8da014795e6b12 | A three-digit number is called balanced if any of its digits is the average of the other two. For example, $528$ is balanced since $5 = \frac{2+8}{2}$. How many balanced three-digit numbers are there? | [
"121"
] | [
"A three-digit number is called balanced if any of its digits is the average of the other two. For example, $528$ is balanced since $5 = \\frac{2+8}{2}$. How many balanced three-digit numbers are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-612deb7c5ea444d85eaf7f1194195fd969debb49540b7591384bb9d23e0e0a93 | A three-digit number is said to be "firm" when it is equal to the product of its unit digit by the number formed by the remaining digits. For example, $153$ is firm because $153 = 3 \times 51$. How many "firm" numbers are there? | [
"3"
] | [
"A three-digit number is said to be \"firm\" when it is equal to the product of its unit digit by the number formed by the remaining digits. For example, $153$ is firm because $153 = 3 \\times 51$. How many \"firm\" numbers are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-f9d11efedb97eec349120fa402796ce6370025edaf68ff49ba76845d3c760ebe | A three-digit number was written on a blackboard, followed by a multiplication sign, and then another three-digit number. If the multiplication sign is erased, the resulting six-digit number is exactly 3 times the product of the original two three-digit numbers. What is this six-digit number? | [
"167334"
] | [
"A three-digit number was written on a blackboard, followed by a multiplication sign, and then another three-digit number. If the multiplication sign is erased, the resulting six-digit number is exactly 3 times the product of the original two three-digit numbers. What is this six-digit number?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-eac184659c51d76d7d0a1e0bb086596b6dbea3ce6419533560088103246b2dcd | A three-digit prime number is such that the digit in the units place is equal to the sum of the other two digits, and if the other digits are interchanged, we still have a three-digit prime number. Find the total number of such primes. | [
"2"
] | [
"A three-digit prime number is such that the digit in the units place is equal to the sum of the other two digits, and if the other digits are interchanged, we still have a three-digit prime number. Find the total number of such primes."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b08fe1a95d99cece696f9420a82e1b373dff6629b47736bbdedb72a2f3b675c0 | A three-dimensional solid has six vertices and eight faces. Two of these faces are parallel equilateral triangles with side length 1, \(\vartriangle A_1A_2A_3\) and \(\vartriangle B_1B_2B_3\). The other six faces are isosceles right triangles — \(\vartriangle A_1B_2A_3\), \(\vartriangle A_2B_3A_1\), \(\vartriangle A_3B... | [
"\\(\\frac{\\sqrt{6}}{8}\\)"
] | [
"A three-dimensional solid has six vertices and eight faces. Two of these faces are parallel equilateral triangles with side length 1, \\(\\vartriangle A_1A_2A_3\\) and \\(\\vartriangle B_1B_2B_3\\). The other six faces are isosceles right triangles — \\(\\vartriangle A_1B_2A_3\\), \\(\\vartriangle A_2B_3A_1\\), \\... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f16c68a8fd8376013f7f1fbcfc570a2b73b2d9868ac30c7c2646f9a5cf67bd98 | A three-layer number is defined as a number \( n \) whose divisors can be partitioned into three disjoint sets \( A \), \( B \), and \( C \) such that the sum of the elements in each set is equal. Find the smallest three-layer number \( n \) that is greater than 1000. | [
"1080"
] | [
"A three-layer number is defined as a number \\( n \\) whose divisors can be partitioned into three disjoint sets \\( A \\), \\( B \\), and \\( C \\) such that the sum of the elements in each set is equal. Find the smallest three-layer number \\( n \\) that is greater than 1000."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-f2a691ff9abea598b6e0930212ba7bbb444acfb3653a6c90167c20ae294786b5 | A three-man jury has two members each of whom independently has probability $p$ of making the correct decision and a third member who flips a coin for each decision (majority rules). A one-man jury has probability $p$ of making the correct decision. Which jury has the better probability of making the correct decision? | [
"p"
] | [
"A three-man jury has two members each of whom independently has probability $p$ of making the correct decision and a third member who flips a coin for each decision (majority rules). A one-man jury has probability $p$ of making the correct decision. Which jury has the better probability of making the correct decis... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 6,
"tir": 18
} | 32 | 0 | 0 |
omr-ws-38ce30e50ff07e00d7ce3a8c0e775a3493d7b973dcfd21abdd2f407b3c5e94c5 | A three-month $960 note with interest at 10% dated July 12 is discounted at 12.5% on August 20. Find the Proceeds and the amount of discount. | [
"18.11"
] | [
"A three-month $960 note with interest at 10% dated July 12 is discounted at 12.5% on August 20. Find the Proceeds and the amount of discount."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-ab1f53b51b62692b74b7c6c66f8f2569be396c502b81a0e41721d531baa4972c | A three-quarter sector of a circle of radius $4$ inches along with its interior is the 2-D net that forms the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? | [
"3\\pi\\sqrt{7}"
] | [
"A three-quarter sector of a circle of radius $4$ inches along with its interior is the 2-D net that forms the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8c0673d59a155848c7a0a5abafce945026e7dbf0cf3937289d720aff8192484f | A throws three dice in a special game of ludo. If it is known that he needs a sum of 15 or more to win, what are his chances of winning? | [
"\\(\\frac{17}{216}\\)"
] | [
"A throws three dice in a special game of ludo. If it is known that he needs a sum of 15 or more to win, what are his chances of winning?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad1e433b4561f77c57232715583da38a8c409b385071d21a9b20c1abf13a42a4 | A tightrope is stretched 30 feet above the ground between the Jay and Tee buildings, which are 50 feet apart. A tightrope walker, walking at a constant rate of 2 feet per second from point A to point B, is illuminated by a spotlight 70 feet above point A. a. How fast is the shadow of the tightrope walker’s feet moving ... | [
"a) \\(\\frac{20}{7}\\) ft/sec, b) 35 feet, c) \\(\\frac{35}{8}\\) ft/sec"
] | [
"A tightrope is stretched 30 feet above the ground between the Jay and Tee buildings, which are 50 feet apart. A tightrope walker, walking at a constant rate of 2 feet per second from point A to point B, is illuminated by a spotlight 70 feet above point A.\n\na. How fast is the shadow of the tightrope walker’s feet... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48af9779dd209bf1691631563ec6794ff068a9993d1aba5069bbf44e98c8a180 | A tightrope walker is in the middle of a 16-inch rope (i.e., 8 inches in front and 8 inches behind). Every second, she moves 1 inch forward with a probability of $\frac{3}{4}$ and 1 inch backward with a probability of $\frac{1}{4}$. Given that she has an unlimited amount of time, what is the probability that she reache... | [
"\\(\\frac{1}{6561}\\)"
] | [
"A tightrope walker is in the middle of a 16-inch rope (i.e., 8 inches in front and 8 inches behind). Every second, she moves 1 inch forward with a probability of $\\frac{3}{4}$ and 1 inch backward with a probability of $\\frac{1}{4}$. Given that she has an unlimited amount of time, what is the probability that she... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-54f7536c2013e9677ca12c16e0e76b4294ecda56613fd8e053e220a18dca8202 | A tightrope walker is walking at a constant rate of 2 feet per second from point $A$ to point $B$ on a tightrope stretched 30 feet above the ground between two buildings 50 feet apart. A spotlight 70 feet above point $A$ illuminates the walker. How far from point $A$ is the tightrope walker when the shadow of her feet ... | [
"35"
] | [
"A tightrope walker is walking at a constant rate of 2 feet per second from point $A$ to point $B$ on a tightrope stretched 30 feet above the ground between two buildings 50 feet apart. A spotlight 70 feet above point $A$ illuminates the walker. How far from point $A$ is the tightrope walker when the shadow of her ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-5ed39735381661d3627aeb6c2037fcc41ee93ab7ba02c55bdabc24ae7d0d62d5 | A tightrope walker is walking at a constant rate of 2 feet per second from point $A$ to point $B$ on a tightrope stretched 30 feet above the ground between two buildings 50 feet apart. A spotlight 70 feet above point $A$ illuminates the walker. How fast is the shadow of the tightrope walker's feet moving along the grou... | [
"\\(\\frac{20}{7}\\) feet per second"
] | [
"A tightrope walker is walking at a constant rate of 2 feet per second from point $A$ to point $B$ on a tightrope stretched 30 feet above the ground between two buildings 50 feet apart. A spotlight 70 feet above point $A$ illuminates the walker. How fast is the shadow of the tightrope walker's feet moving along the... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 19
} | 31 | 0 | 0 |
omr-ws-1b63a9019764876337e6cfc0463a8549e7b8d00001e37ef13bafc449605801c5 | A tightrope walker is walking at a constant rate of 2 feet per second from point $A$ to point $B$ on a tightrope stretched 30 feet above the ground between two buildings 50 feet apart. A spotlight 70 feet above point $A$ illuminates the walker. How fast is the shadow of the tightrope walker's feet moving up the wall of... | [
"\\(\\frac{35}{8}\\) ft per second"
] | [
"A tightrope walker is walking at a constant rate of 2 feet per second from point $A$ to point $B$ on a tightrope stretched 30 feet above the ground between two buildings 50 feet apart. A spotlight 70 feet above point $A$ illuminates the walker. How fast is the shadow of the tightrope walker's feet moving up the wa... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 3,
"tir": 20
} | 33 | 0 | 0 |
omr-ws-5af1992ae6199d49ca63e9e2caba77fa3cd2575e9fa969829ca21fbbc6df3de9 | A tightrope walker stands in the center of a rope of length 16 meters. Every minute she walks forward one meter with probability 2/3 and backward with probability 1/3. What is the probability that she reaches the forward end before getting to the backward end? | [
"\\(\\frac{256}{257}\\)"
] | [
"A tightrope walker stands in the center of a rope of length 16 meters. Every minute she walks forward one meter with probability 2/3 and backward with probability 1/3. What is the probability that she reaches the forward end before getting to the backward end?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-ddd16c714067eb5ae3833102ac2160c78f1f401288667f5c2e79b236761da0f6 | A tile $\textbf{T}$ is defined as special if it appears identical after a $90^\circ$ rotation. Given a tile $\textbf{T}$ that consists of a $4 \times 4$ grid of squares, where each square can be either black or white, let $N$ be the number of distinct special tiles $\textbf{T}$ that can be created. Find the remainder w... | [
"16"
] | [
"A tile $\\textbf{T}$ is defined as special if it appears identical after a $90^\\circ$ rotation. Given a tile $\\textbf{T}$ that consists of a $4 \\times 4$ grid of squares, where each square can be either black or white, let $N$ be the number of distinct special tiles $\\textbf{T}$ that can be created. Find the r... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-514c4539d14793b7667beba09a49ec4109f7baa1c251b86ae1fbdf2f1000d7d7 | A tiler is tiling a bathroom shower floor which is \(5 \times 3\) tiles using \(1 \times 3\) tiles and \(1 \times 2\) tiles. How many ways are there to tile the floor? | [
"34"
] | [
"A tiler is tiling a bathroom shower floor which is \\(5 \\times 3\\) tiles using \\(1 \\times 3\\) tiles and \\(1 \\times 2\\) tiles. How many ways are there to tile the floor?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-d2bf341daa5d0b8f49e9d95e2736a301ef1223ea73f1b457dec80b63134d8180 | A tin sheet with area $2a^2$ is used to make a closed cuboid box. Determine the dimensions of the box that maximize its volume, given that the sheet can only be deformed with straight lines and cut, but not connected. | [
"\\dfrac{a}{\\sqrt{3}}"
] | [
"A tin sheet with area $2a^2$ is used to make a closed cuboid box. Determine the dimensions of the box that maximize its volume, given that the sheet can only be deformed with straight lines and cut, but not connected."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-1df8fe79dd81fd4589825c5fbc882aa501501b059e9dbe12efd1957c98289405 | A tin sheet with dimensions $b \cdot c$ is used to make an open cuboid box (without a lid). Determine the dimensions of the box that maximize its volume, given that the perimeter of the sheet is known. | [
"\\left( \\frac{P}{6}, \\frac{P}{6}, \\frac{P}{24} \\right)"
] | [
"A tin sheet with dimensions $b \\cdot c$ is used to make an open cuboid box (without a lid). Determine the dimensions of the box that maximize its volume, given that the perimeter of the sheet is known."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-f078c6cf8cad89756a9c8435d035335ecf485e108afea726c2d2a30d51cb459c | A tire company claims that the lifetimes of its tires average 50,000 miles with a standard deviation of 5,000 miles. You sample 100 tires and test the hypothesis that the mean tire lifetime is at least 50,000 miles against the alternative that it is less. If the true mean lifetime is 49,500 miles, at what significance ... | [
"0.4364"
] | [
"A tire company claims that the lifetimes of its tires average 50,000 miles with a standard deviation of 5,000 miles. You sample 100 tires and test the hypothesis that the mean tire lifetime is at least 50,000 miles against the alternative that it is less. If the true mean lifetime is 49,500 miles, at what signific... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 5
} | 10 | 0 | 0 |
omr-ws-fd9c21ce392eefe26179a9b4af77a957f5ef9f146670f105ba625e907a0d3819 | A toddler is dropping eggs from a window of a building which is 20 feet above the ground. A neighbor, whose window is 15 feet horizontally from the toddler's window and at the same height, shines a flashlight on one of the falling eggs. How fast is the shadow of this egg moving on the ground when the egg is halfway to ... | [
"76 feet / second"
] | [
"A toddler is dropping eggs from a window of a building which is 20 feet above the ground. A neighbor, whose window is 15 feet horizontally from the toddler's window and at the same height, shines a flashlight on one of the falling eggs. How fast is the shadow of this egg moving on the ground when the egg is halfwa... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-f6ea3e680287768d08645bdc2f99fb9fa21759ef6e0b1738cab808aac7e5659a | A token is initially placed in the top right corner of a $2019 \times 2019$ grid. The token can only move to an adjacent cell (sharing an edge) every minute. What is the expected number of minutes it takes for the token to reach the bottom left corner of the grid? | [
"2018 \\times 2019^2"
] | [
"A token is initially placed in the top right corner of a $2019 \\times 2019$ grid. The token can only move to an adjacent cell (sharing an edge) every minute. What is the expected number of minutes it takes for the token to reach the bottom left corner of the grid?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-8b0464b9565755c381892e25f67227b9414eaa25e3994d0f29993bb636f78fe5 | A token is placed at $(1,1)$. In each move, we can either double one of its coordinates, or replace the larger coordinate with the absolute difference between the coordinates. Find all possible positions that the token can move to. | [
"\\((m, n)\\) where \\(\\text{gcd}(m, n) = 2^k\\) for some nonnegative integer \\(k\\), and \\((2^k, 0)\\) and \\((0, 2^k)\\) for any nonnegative integer \\(k\\)."
] | [
"A token is placed at $(1,1)$. In each move, we can either double one of its coordinates, or replace the larger coordinate with the absolute difference between the coordinates. Find all possible positions that the token can move to."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 26,
"genselect": 4,
"tir": 9
} | 39 | 0 | 0 |
omr-ws-24e1962d74c49dac0fb75350cf9d08490e694a6a07e87d90976bb3c27653b908 | A token is placed at a vertex of a regular 20-gon. Each day, the token rolls to a vertex adjacent to the vertex it is on at random. Let \( E \) be the expected number of days that must pass before the token has visited each vertex at least once. Find the largest integer not exceeding \( E \). | [
"190"
] | [
"A token is placed at a vertex of a regular 20-gon. Each day, the token rolls to a vertex adjacent to the vertex it is on at random. Let \\( E \\) be the expected number of days that must pass before the token has visited each vertex at least once. Find the largest integer not exceeding \\( E \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 4
} | 25 | 0 | 0 |
omr-ws-827bc7c10ba934fb0de6b8b4112d82aa7053e7c5787b0d382fb2bf539ffa0c18 | A token is placed at each vertex of a regular $2n$-gon. A move consists of choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves, it turns out that every two tokens have been swapped exactly once. Determine the number of edges that have never b... | [
"n"
] | [
"A token is placed at each vertex of a regular $2n$-gon. A move consists of choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves, it turns out that every two tokens have been swapped exactly once. Determine the number of edges that have ne... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-2e742f119dd5cb3c345875d73b99037c6e78620853464da4e4fca34890048af4 | A token is placed at each vertex of a regular n-gon. A move consists of choosing an edge or a diagonal of the n-gon and swapping the two tokens placed at the endpoints of that segment. After a finite number of moves, every two tokens have been swapped exactly once. Find all possible values of n such that every edge and... | [
"\\( n = 2^k \\) for \\( k \\in \\mathbb{N} \\)"
] | [
"A token is placed at each vertex of a regular n-gon. A move consists of choosing an edge or a diagonal of the n-gon and swapping the two tokens placed at the endpoints of that segment. After a finite number of moves, every two tokens have been swapped exactly once. Find all possible values of n such that every edg... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48087b0659813b24f4932ef123e84dfae0aaa18441457bb27aca0637ffc89f26 | A token is placed in the corner of an $n \times n$ grid. Alice and Bob alternate moves, where each move cannot be followed by a move in the opposite direction. Bob's goal is to move the token to a previously visited position, which ends the game. What is the maximum number of moves Alice can make before Bob ends the ga... | [
"2n - 2"
] | [
"A token is placed in the corner of an $n \\times n$ grid. Alice and Bob alternate moves, where each move cannot be followed by a move in the opposite direction. Bob's goal is to move the token to a previously visited position, which ends the game. What is the maximum number of moves Alice can make before Bob ends ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-3b984b2f935534c12f46009971f12059dc911be38f6950906e05c08701ffe01a | A token is placed on a cell adjacent to a corner cell of an $n \times n$ chess-board. Each of two players moves it in turn to an orthogonally neighboring cell. A player loses if they move the token to a cell that was already visited (including the starting cell). Determine who wins if the token starts adjacent to a cor... | [
"\\text{First player wins if } n \\text{ is even; second player wins if } n \\text{ is odd}"
] | [
"A token is placed on a cell adjacent to a corner cell of an $n \\times n$ chess-board. Each of two players moves it in turn to an orthogonally neighboring cell. A player loses if they move the token to a cell that was already visited (including the starting cell). Determine who wins if the token starts adjacent to... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
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