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-3363b32c5fe3385286d31188e3558dd8f995cbed93865bd31d7a8f80615ff4fb | A rope of length 10 meters is tied tautly from the top of a flagpole to the ground 6 meters away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 cm/min. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of Cancer... | [
"2"
] | [
"A rope of length 10 meters is tied tautly from the top of a flagpole to the ground 6 meters away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 cm/min. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of C... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-47f83bdd91facc71bda0af79dc6e7f80db6259062c819c9034c43b54b8c5ca45 | A rope was bent five times and then folded in half. A transverse cut was made that did not coincide with the fold lines, resulting in pieces of various lengths. Two of the pieces had lengths of 4 cm and 7 cm. Determine a possible original length of the rope. | [
"176, 240, 288"
] | [
"A rope was bent five times and then folded in half. A transverse cut was made that did not coincide with the fold lines, resulting in pieces of various lengths. Two of the pieces had lengths of 4 cm and 7 cm. Determine a possible original length of the rope."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04e1f8e7f52f0defbd3aab0bbed3e769114118215aab3408eb8a58128f5de8ba | A rope was folded in half three times and then cut at some point. Two of the resulting pieces are 9 cm and 4 cm long. What is the length of the original rope? List all possible lengths. | [
"36 cm"
] | [
"A rope was folded in half three times and then cut at some point. Two of the resulting pieces are 9 cm and 4 cm long. What is the length of the original rope? List all possible lengths."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a967dc5e0d5993ec5966b4428bf210c039d4d7d2a2f022b2c75a155bf7afc16d | A rotating beacon is located 1 kilometer off a straight shoreline. If the beacon rotates at a rate of 3 revolutions per minute, how fast (in kilometers per hour) does the beam of light appear to be moving to a viewer who is 1/2 kilometer down the shoreline? | [
"\\( 180\\pi \\sqrt{5} \\)"
] | [
"A rotating beacon is located 1 kilometer off a straight shoreline. If the beacon rotates at a rate of 3 revolutions per minute, how fast (in kilometers per hour) does the beam of light appear to be moving to a viewer who is 1/2 kilometer down the shoreline?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9bcebf3f90a3fb89ad46a05a66d75356ecffc9af470c859c16ddedf2f9fa8fb6 | A rotation of $m$ degrees clockwise around the point $(x,y)$, where $0<m<180$, transforms $\triangle ABC$ with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$ to $\triangle A'B'C'$ with vertices $A'(24,18)$, $B'(36,18)$, and $C'(24,2)$. Find $m+x+y$. | [
"108"
] | [
"A rotation of $m$ degrees clockwise around the point $(x,y)$, where $0<m<180$, transforms $\\triangle ABC$ with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$ to $\\triangle A'B'C'$ with vertices $A'(24,18)$, $B'(36,18)$, and $C'(24,2)$. Find $m+x+y$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 4
} | 36 | 0 | 0 |
omr-ws-c02357aa5f931dfa276b4d487ca4046b2eceb8e77ec479ec44b8be133d3d4bb9 | A rotation with center $(h,k)$ maps $A$ onto $A'$ and $B$ onto $B'$, where $A(2,3)$, $B(9,4)$, $A'(4,7)$, and $B'(9,12)$. What is the ordered pair $(h,k)$? | [
"(-3, 8)"
] | [
"A rotation with center $(h,k)$ maps $A$ onto $A'$ and $B$ onto $B'$, where $A(2,3)$, $B(9,4)$, $A'(4,7)$, and $B'(9,12)$. What is the ordered pair $(h,k)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-6f6b684d2e10a4edb0924ab0daf73e897ef3d04b7183ac6ee761a6b2d20473c1 | A round balloon of radius \( R \) subtends an angle \( A \) at the eye of the observer, and the angle of elevation of its center is \( B \). Find the height of the center of the balloon. | [
"\\( R \\csc\\left(\\frac{A}{2}\\right) \\sin(B) \\)"
] | [
"A round balloon of radius \\( R \\) subtends an angle \\( A \\) at the eye of the observer, and the angle of elevation of its center is \\( B \\). Find the height of the center of the balloon."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-86b7a4429a20ffdb84d4ee2eab2a82b3a0eab20bd0281287b880b4b946ed86a8 | A round fence with a circumference of 1 km is guarded by 100 guards. Each guard can see 10 meters to the left and 10 meters to the right of their position. Robbers attempt to climb over the fence one at a time. If a guard sees a robber, they catch the robber and remain at that position. Determine if it is possible to o... | [
"Yes"
] | [
"A round fence with a circumference of 1 km is guarded by 100 guards. Each guard can see 10 meters to the left and 10 meters to the right of their position. Robbers attempt to climb over the fence one at a time. If a guard sees a robber, they catch the robber and remain at that position. Determine if it is possible... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-6642dfecb0b22bfe50ff6fb8cd443fbf1055819c45d3c2e28504c44c1ed26b21 | A round hole of radius \(\sqrt{3}\) feet is bored through the center of a sphere of radius 2 feet. Find the volume of the piece cut out. | [
"\\(\\frac{4\\pi}{3}\\)"
] | [
"A round hole of radius \\(\\sqrt{3}\\) feet is bored through the center of a sphere of radius 2 feet. Find the volume of the piece cut out."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-249d6f98d3523da1902350d0142004bd85642d3740e4edaf315e23b11483ca21 | A round pizza is $\frac13$ of an inch thick and has a diameter of 12 inches. It is cut into 12 congruent pieces. What is the number of cubic inches in the volume of one piece? Express your answer in terms of $\pi$. | [
"\\pi"
] | [
"A round pizza is $\\frac13$ of an inch thick and has a diameter of 12 inches. It is cut into 12 congruent pieces. What is the number of cubic inches in the volume of one piece? Express your answer in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8adf42f86eb229633652a0989a615ad93b4b2ba3ec977506230b945c22c752c3 | A round table has $2N$ chairs around it. Due to social distancing guidelines, no two people are allowed to sit next to each other. How many different ways are there to choose seats around the table on which $N-1$ guests can be seated? | [
"\\(N^2\\)"
] | [
"A round table has $2N$ chairs around it. Due to social distancing guidelines, no two people are allowed to sit next to each other. How many different ways are there to choose seats around the table on which $N-1$ guests can be seated?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 9
} | 56 | 0 | 0 |
omr-ws-3fa6df5731af8d5c0eac02e8d407de03236bb1569607bb487ed0ec3d178f162d | A round table has a radius of 4. Six rectangular place mats are placed on the table such that each mat has a width of 1 and a length of \( x \). Each mat has two corners on the edge of the table, and these two corners are endpoints of the same side of length \( x \). The inner corners of each mat touch the inner corner... | [
"\\(\\frac{3 \\sqrt{7} - \\sqrt{3}}{2}\\)"
] | [
"A round table has a radius of 4. Six rectangular place mats are placed on the table such that each mat has a width of 1 and a length of \\( x \\). Each mat has two corners on the edge of the table, and these two corners are endpoints of the same side of length \\( x \\). The inner corners of each mat touch the inn... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 9
} | 30 | 0 | 0 |
omr-ws-011aadc2c5c4c9cd007e0da29db103392a57ab44805d1c82b8c956d7ee0d6f10 | A round table has room for \( n \) diners (\( n \ge 2 \)). There are napkins in three different colors. In how many ways can the napkins be placed, one for each seat, so that no two neighbors get napkins of the same color? | [
"\\( 2^n + 2(-1)^n \\)"
] | [
"A round table has room for \\( n \\) diners (\\( n \\ge 2 \\)). There are napkins in three different colors. In how many ways can the napkins be placed, one for each seat, so that no two neighbors get napkins of the same color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 15
} | 32 | 0 | 0 |
omr-ws-9b1809176586dbcf2e772986398386089738ab14fa7020e111cfcd603d05ffa1 | A round-robin tennis tournament consists of each player playing every other player exactly once. How many matches will be held during an 8-person round-robin tennis tournament? | [
"28"
] | [
"A round-robin tennis tournament consists of each player playing every other player exactly once. How many matches will be held during an 8-person round-robin tennis tournament?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5c44389000ad7b54e91c3a0ddc6419bc47d5fb8331257338033c0551123b5ef9 | A router distributes 60 distinguishable parcels to four different channels in one second. If the order in which the parcels pass through each channel does matter, determine the number of ways to distribute the parcels using a generating function and find the term of the coefficient that gives the correct answer. | [
"60! \\cdot \\dbinom{63}{3}"
] | [
"A router distributes 60 distinguishable parcels to four different channels in one second. If the order in which the parcels pass through each channel does matter, determine the number of ways to distribute the parcels using a generating function and find the term of the coefficient that gives the correct answer."
... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bc2d6cbfb970b111c1d3a94644bb75052d29462f39ce307d68d1270a1e0369f3 | A router distributes 60 distinguishable parcels to four different channels in one second. If the order in which the parcels pass through each channel does not matter, determine the number of ways to distribute the parcels using a generating function and find the term of the coefficient that gives the correct answer. | [
"4^{60}"
] | [
"A router distributes 60 distinguishable parcels to four different channels in one second. If the order in which the parcels pass through each channel does not matter, determine the number of ways to distribute the parcels using a generating function and find the term of the coefficient that gives the correct answe... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b8883ab06483d02ccbcb6bd86fc70dfd5040b1219f800e813c3f343f62807f4e | A row of $n$ integers from $1$ to $n$ (where $n$ is a positive even number) is written on the blackboard. Then $\frac{n}{2}$ consecutive numbers were wiped out from this row. The sum of the remaining numbers on the blackboard is $2021$. What is the sum of all possible values of $n$? | [
"180"
] | [
"A row of $n$ integers from $1$ to $n$ (where $n$ is a positive even number) is written on the blackboard. Then $\\frac{n}{2}$ consecutive numbers were wiped out from this row. The sum of the remaining numbers on the blackboard is $2021$. What is the sum of all possible values of $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-b00c1e289fa13800822a4e57e2faa75392db0c1ffa367b43e8dde707eecbd6ed | A row of 1000 numbers is written on a blackboard. We write a new row below the first according to the rule: We write under every number \( a \) a natural number that indicates how many times the number \( a \) is encountered in the first line. Then we write down the third line: under every number \( b \), we write a na... | [
"1000"
] | [
"A row of 1000 numbers is written on a blackboard. We write a new row below the first according to the rule: We write under every number \\( a \\) a natural number that indicates how many times the number \\( a \\) is encountered in the first line. Then we write down the third line: under every number \\( b \\), we... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-65ce16766b9467363be6d21cb075388472255ed3a6dc2fafc3093cd1d26aaf76 | A row of 1000 numbers is written on a blackboard. We write a new row below the first according to the rule: We write under every number \( a \) a natural number that indicates how many times the number \( a \) is encountered in the first line. Then we write down the third line: under every number \( b \), we write a na... | [
"1, 2, 3, \\ldots, 1000"
] | [
"A row of 1000 numbers is written on a blackboard. We write a new row below the first according to the rule: We write under every number \\( a \\) a natural number that indicates how many times the number \\( a \\) is encountered in the first line. Then we write down the third line: under every number \\( b \\), we... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-5e77bafbb6939c3937bacf25fb78fd620e3707ce0d4ba57d676c5d37073c2326 | A row of handcrafters consists of four carrying a folding rule and six carrying a hammer. If the chief randomly picks five handcrafters with replacement, what is the probability that exactly two of them are carrying a folding rule? | [
"\\dfrac{216}{625}"
] | [
"A row of handcrafters consists of four carrying a folding rule and six carrying a hammer. If the chief randomly picks five handcrafters with replacement, what is the probability that exactly two of them are carrying a folding rule?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e08ea304b395a0ded18ff768c0ee7a2f2e427fdf8ea5ae6c3f999dec628a022b | A row of holes labeled with positive integers starts with hole $1$ and continues indefinitely to the right. Some finite number of these holes have pegs in them. A series of operations is performed as follows: if holes $n$ and $n+1$ have pegs and hole $n+2$ is empty, then we can move the peg in hole $n$ to hole $n+2$ an... | [
"144"
] | [
"A row of holes labeled with positive integers starts with hole $1$ and continues indefinitely to the right. Some finite number of these holes have pegs in them. A series of operations is performed as follows: if holes $n$ and $n+1$ have pegs and hole $n+2$ is empty, then we can move the peg in hole $n$ to hole $n+... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-56ea66022521c25eaa204b81ac22632091e0d2114ce53bfd28975b1c77bcbba6 | A rug is made with three different colors. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is 1 foot wide on all four sides. What is the length in feet of the inner rectangle? | [
"2"
] | [
"A rug is made with three different colors. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is 1 foot wide on all four sides. What is the length in feet of the inner rectangle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6158137d3135cdcdff8a80657095c555b8636233d03b6be8989c94b4ee4d4fa6 | A rug is made with three differently colored regions that form an arithmetic progression in area. The inner rectangle is 1 foot wide, and each of the two shaded regions is 1 foot wide on all four sides. What is the length in feet of the inner rectangle? | [
"2"
] | [
"A rug is made with three differently colored regions that form an arithmetic progression in area. The inner rectangle is 1 foot wide, and each of the two shaded regions is 1 foot wide on all four sides. What is the length in feet of the inner rectangle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9747e0351931e90a2f56e047a244c0c55866c67b36288a27c2d4bab5270e34b1 | A rugby goal has posts at points $A$ and $B$. A point $C$ lies on the line $AB$ such that $AC = a$ and $BC = b$. A player wishes to place the ball at point $P$ on the line through $C$ perpendicular to $AB$. If $CP = y$ and the angle $APB = \theta$, find the value of $y^2$ that maximizes $\theta$. | [
"0"
] | [
"A rugby goal has posts at points $A$ and $B$. A point $C$ lies on the line $AB$ such that $AC = a$ and $BC = b$. A player wishes to place the ball at point $P$ on the line through $C$ perpendicular to $AB$. If $CP = y$ and the angle $APB = \\theta$, find the value of $y^2$ that maximizes $\\theta$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-0fc843a415888f8b454564e2e91c65c5746d50cd17dbc1c08a3cbc1c007c4c0a | A runner hopes to complete the 10,000 meter run in less than 30.0 minutes. After exactly 27 minutes, there are still 1100 meters to go. The runner must then accelerate at 0.20 $ m/s^2$ for how many seconds in order to achieve the desired time? | [
"3.11 seconds"
] | [
"A runner hopes to complete the 10,000 meter run in less than 30.0 minutes. After exactly 27 minutes, there are still 1100 meters to go. The runner must then accelerate at 0.20 $ m/s^2$ for how many seconds in order to achieve the desired time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-d24386d78c88075fca5931dd7a42a371ced86550a666040591bd13cbb253abaf | A runner running at 10 miles per hour and a bicyclist riding 20 miles per hour are headed toward each other. At the instant they are 15 miles apart, a hummingbird passes the runner flying in the direction of the bicyclist at 30 miles per hour. The hummingbird reverses direction upon reaching the bicyclist and continues... | [
"18"
] | [
"A runner running at 10 miles per hour and a bicyclist riding 20 miles per hour are headed toward each other. At the instant they are 15 miles apart, a hummingbird passes the runner flying in the direction of the bicyclist at 30 miles per hour. The hummingbird reverses direction upon reaching the bicyclist and cont... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-15cdfa5edacdc1e98a68dbce0f61dc6512e3a00b783cfcd5af1f95ddf6848920 | A runner sprints around a circular track of radius 100m at a constant speed of 7 m/s. The runner's friend is standing at a distance of 200m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200m? | [
"\\(\\frac{-7\\sqrt{15}}{4}\\)"
] | [
"A runner sprints around a circular track of radius 100m at a constant speed of 7 m/s. The runner's friend is standing at a distance of 200m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200m?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-5028ddf96d10ea916c2eac162a72d99a30588949905cfbe7aacb43621b2f220e | A running club buys a case of sweatshirts for \$660 to sell at a fundraiser. The club needs to sell all but 25 of the sweatshirts at a profit of \$10 per sweatshirt to regain the \$660 investment. How many sweatshirts must be sold, and at what price, to do this? | [
"30 sweatshirts at $22 each"
] | [
"A running club buys a case of sweatshirts for \\$660 to sell at a fundraiser. The club needs to sell all but 25 of the sweatshirts at a profit of \\$10 per sweatshirt to regain the \\$660 investment. How many sweatshirts must be sold, and at what price, to do this?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 17
} | 18 | 0 | 0 |
omr-ws-76ffbc732a84505e95521853f659b4a043c59e4b14d78bbd9a818312ca743ade | A running track is the ring formed by two concentric circles. It is 10 feet wide. What is the difference in the circumferences of the two circles? | [
"60 feet"
] | [
"A running track is the ring formed by two concentric circles. It is 10 feet wide. What is the difference in the circumferences of the two circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 2,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-f3617d8de1cf03b6bae510a04e7d10cb1135abd609b6a0e1b0139f34f02a8bdf | A safety box has 10 buttons labeled 0 through 9. The special code to open the box is a 4-digit number (digits can be repeated). What is the minimum number of button presses required to ensure that the safe will open, regardless of the secret code? | [
"10003"
] | [
"A safety box has 10 buttons labeled 0 through 9. The special code to open the box is a 4-digit number (digits can be repeated). What is the minimum number of button presses required to ensure that the safe will open, regardless of the secret code?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 18
} | 36 | 0 | 0 |
omr-ws-166acc4704d42a778d290de0010dd051a19d6480f109711d49d8d5ec266c3af6 | A sailing boat has three identical black flags and three identical red flags. Using at least one flag, how many different arrangements of these flags in a row are possible? | [
"68"
] | [
"A sailing boat has three identical black flags and three identical red flags. Using at least one flag, how many different arrangements of these flags in a row are possible?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-c043463ac85a453b1ff3bbb34f7bc0634941ce78bf56ed62cfa6c2b1bd82c1e5 | A sailor goes 8 km downstream in 40 minutes and returns in 1 hour. Determine the speed of the sailor in still water and the speed of the current. | [
"\\( s = 10 \\text{ km/h}, c = 2 \\text{ km/h} \\)"
] | [
"A sailor goes 8 km downstream in 40 minutes and returns in 1 hour. Determine the speed of the sailor in still water and the speed of the current."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-3fb4a0b6fad41589ceda05d133ebe4d2d78741b4a4bf3b762c92fe5af5047d4f | A sailor graphs the curve $y=x^2$, picks points $(0, 0)$ and $(5, 25)$ on the curve, and connects those points. What is the radius of the largest circular ball that can fit in the boat? | [
"$\\frac{25\\sqrt{26}}{208}$"
] | [
"A sailor graphs the curve $y=x^2$, picks points $(0, 0)$ and $(5, 25)$ on the curve, and connects those points. What is the radius of the largest circular ball that can fit in the boat?"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-072ff3659b6213e13ad61cafaaba50fd70d20201fa957f2b7c4ef7dd5e2ae1fc | A salesman and a customer together have 1999 rubles in coins and bills of denominations 1, 5, 10, 50, 100, 500, and 1000 rubles. The customer has enough money to buy a Cat in the Bag, which costs an integer number of rubles. If the Cat in the Bag costs exactly 1234 rubles, calculate the minimum number of transactions n... | [
"7"
] | [
"A salesman and a customer together have 1999 rubles in coins and bills of denominations 1, 5, 10, 50, 100, 500, and 1000 rubles. The customer has enough money to buy a Cat in the Bag, which costs an integer number of rubles. If the Cat in the Bag costs exactly 1234 rubles, calculate the minimum number of transacti... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-05ce1501a6ca0f3c5526aa7ca230c147859d88c9f903a473eaa853671541d396 | A sample consisting of five observations has an arithmetic mean of $10$ and a median of $12$. What is the smallest value that the range (largest observation minus smallest) can assume for such a sample? | [
"5"
] | [
"A sample consisting of five observations has an arithmetic mean of $10$ and a median of $12$. What is the smallest value that the range (largest observation minus smallest) can assume for such a sample?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2d5fcaeabe92211f08e0dc943e271d996b7208a8d0b848138489c569a5b5141c | A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\lfloor D\rfloor$? (For real $x$, $\lfloor x\rfloor... | [
"947"
] | [
"A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\\lfloor D\\rfloor$? (For real $x$, $\\lflo... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 8,
"tir": 11
} | 44 | 0 | 0 |
omr-ws-e53cb052c9775f87e6704eb74aff5470b786e43f269b1786b28ee5eef4b74ef1 | A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let \( D \) be the difference between the mode and the arithmetic mean of the sample. If \( D \) is as large as possible, what is \( \lfloor D \rfloor \)? (\( \lfloor x \rf... | [
"495"
] | [
"A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let \\( D \\) be the difference between the mode and the arithmetic mean of the sample. If \\( D \\) is as large as possible, what is \\( \\lfloor D \\rfloor \\)? (\\(... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-1d84dbc9a001d2c4358e3ac85c45b6b822b8a4f986c9b9758295b9f2aa299312 | A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let \( D \) be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of \( \lfloor D \rfloor \)? (For real \( x \), \( \... | [
"947"
] | [
"A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let \\( D \\) be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of \\( \\lfloor D \\rfloor \\)? (For real \\... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 7,
"tir": 9
} | 39 | 0 | 0 |
omr-ws-3fe21330432c656232aa7fdc077c37b4824ed044820815e111ad4a36bfa4d23c | A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode. Let \( D \) be the difference between the mode and the arithmetic mean of the sample. If \( D \) is as large as possible, what is the greatest integer less than or equal to \( D \)? | [
"947"
] | [
"A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode. Let \\( D \\) be the difference between the mode and the arithmetic mean of the sample. If \\( D \\) is as large as possible, what is the greatest integer less than or equal to \\( D \\)?"
... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 8,
"tir": 6
} | 45 | 0 | 0 |
omr-ws-0309801f0ed3f8324b5137bc918404abd2b9ca8d2af3a453fbc368e154ea4c86 | A sample of 270 adult Australians revealed that 43 were smokers. 1. What is the sample proportion of smokers? (3 dp) 2. What is the estimated standard error of the sample proportion? (4 dp) 3. Given that studies conducted in Australia in 2003 indicate that 20% of the adult population are smokers, what is the probabilit... | [
"0.9599"
] | [
"A sample of 270 adult Australians revealed that 43 were smokers.\n1. What is the sample proportion of smokers? (3 dp)\n2. What is the estimated standard error of the sample proportion? (4 dp)\n3. Given that studies conducted in Australia in 2003 indicate that 20% of the adult population are smokers, what is the pr... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-0977b7e77287017f1937cc3d049fcce1d48b12fee18e930c98a6eebb09f53959 | A sample of size \( r \) (with replacement) is taken from a population of \( n \) people. Find the probability that a given \( m \) people will be included in the sample. | [
"\\[\n\\begin{cases} \n0 & \\text{if } m > r \\\\\n\\frac{m!}{n^m} & \\text{if } m = r \\\\\n\\frac{C_{r}^m \\cdot m! \\cdot n^{r - m}}{n^r} & \\text{if } m < r \n\\end{cases}\n\\]"
] | [
"A sample of size \\( r \\) (with replacement) is taken from a population of \\( n \\) people. Find the probability that a given \\( m \\) people will be included in the sample."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-03414aed0b8cbe9730da09c526a8424cdbd27a653c58f0690dcb23beaa46a9df | A sample size of 10 is drawn from each of two normal populations having the same unknown variance. The mean and variance of the sample from the first population are 7 and 26, respectively, and those of the sample from the second population are 4 and 10, respectively. Test at a 5 percent significance level if the two po... | [
"Fail to reject the null hypothesis (the two populations have the same mean)."
] | [
"A sample size of 10 is drawn from each of two normal populations having the same unknown variance. The mean and variance of the sample from the first population are 7 and 26, respectively, and those of the sample from the second population are 4 and 10, respectively. Test at a 5 percent significance level if the t... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-13477393c06303fea33d43ab2172555bc84c6aaabba9f0680354e08fd96e6e1b | A satellite in a 100 mile high circular orbit around Earth has a velocity of approximately 17,500 mi/h. If this velocity is multiplied by \(\sqrt{2}\), the satellite will have the minimum velocity necessary to escape Earth's gravity and will follow a parabolic path with the center of the Earth at the focus. Given the r... | [
"394.4"
] | [
"A satellite in a 100 mile high circular orbit around Earth has a velocity of approximately 17,500 mi/h. If this velocity is multiplied by \\(\\sqrt{2}\\), the satellite will have the minimum velocity necessary to escape Earth's gravity and will follow a parabolic path with the center of the Earth at the focus. Giv... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-2d210af342fd1301ccbc6318ab76214212c5d290fcbe75ce6596b028408ac63f | A savings bank added 6 roubles to the account of a customer over a year. Having added another 44 roubles, the customer left his deposit for one more year. At the end of that year, the savings bank added a new number of per cent to the customer's account which constituted now, with the extra per cent, 257 roubles and 50... | [
"200 rubles, 3-per-cent"
] | [
"A savings bank added 6 roubles to the account of a customer over a year. Having added another 44 roubles, the customer left his deposit for one more year. At the end of that year, the savings bank added a new number of per cent to the customer's account which constituted now, with the extra per cent, 257 roubles a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-890ce19fdfc64b524d1b6e85ca08acedbcf7d340c60c47f7fb3000ea3655f9e1 | A scale drawing of a park shows that one inch represents 800 feet. A line segment in the drawing that is 4.75 inches long represents how many feet? | [
"3800"
] | [
"A scale drawing of a park shows that one inch represents 800 feet. A line segment in the drawing that is 4.75 inches long represents how many feet?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3fdc752096001ad31acf3c8366939371b689ef887068554fc071012a66eb4dfd | A scalene triangle \(ABC\) is inscribed in a circle \(\omega\). A line tangent to \(\omega\) at \(C\) meets the line \(AN\) at \(D\). Let \(I\) be the incenter of \(\triangle ABC\). The lines \(AI\) and \(BI\) meet the angle bisector of \(\angle CDB\) at \(Q\) and \(P\), respectively. Let \(M\) be the midpoint of the s... | [
"1"
] | [
"A scalene triangle \\(ABC\\) is inscribed in a circle \\(\\omega\\). A line tangent to \\(\\omega\\) at \\(C\\) meets the line \\(AN\\) at \\(D\\). Let \\(I\\) be the incenter of \\(\\triangle ABC\\). The lines \\(AI\\) and \\(BI\\) meet the angle bisector of \\(\\angle CDB\\) at \\(Q\\) and \\(P\\), respectively.... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-3d25e0170d1499321c57204ad87d927ff68412e4798460f7f303e01c8761d214 | A scalene triangle \(ABC\) is inscribed in a circle \(w\). The bisector of angle \(A\) meets \(BC\) at \(E\). Let \(M\) be the midpoint of the arc \(BAC\). The line \(ME\) intersects \(w\) again at \(D\). If \(O\) is the circumcenter of triangle \(AED\) and \(T\) is the intersection point of the tangent to \(w\) at \(D... | [
"2"
] | [
"A scalene triangle \\(ABC\\) is inscribed in a circle \\(w\\). The bisector of angle \\(A\\) meets \\(BC\\) at \\(E\\). Let \\(M\\) be the midpoint of the arc \\(BAC\\). The line \\(ME\\) intersects \\(w\\) again at \\(D\\). If \\(O\\) is the circumcenter of triangle \\(AED\\) and \\(T\\) is the intersection point... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f34cb9c62010df317c9cf77b3d658a7de18383d9c089749eebe5afc14435d298 | A scalene triangle \(ABC\) is inscribed within circle \(\omega\). The tangent to the circle at point \(C\) intersects line \(AB\) at point \(D\). Let \(I\) be the incenter of \(\triangle ABC\). Lines \(AI\) and \(BI\) intersect the bisector of \(\angle CDB\) at points \(Q\) and \(P\), respectively. Let \(M\) be the mid... | [
"1"
] | [
"A scalene triangle \\(ABC\\) is inscribed within circle \\(\\omega\\). The tangent to the circle at point \\(C\\) intersects line \\(AB\\) at point \\(D\\). Let \\(I\\) be the incenter of \\(\\triangle ABC\\). Lines \\(AI\\) and \\(BI\\) intersect the bisector of \\(\\angle CDB\\) at points \\(Q\\) and \\(P\\), re... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-3bebba8f6991f584142591ad80ce13f79ac70d5f8dbd89da6ae666f0c9a8de9b | A scalene triangle has side lengths which are prime numbers and the length of its perimeter is also prime. What is its smallest possible perimeter? | [
"23"
] | [
"A scalene triangle has side lengths which are prime numbers and the length of its perimeter is also prime. What is its smallest possible perimeter?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4da9a6ba99df1e5390fcb70cc05de5a5fee68d14deae69661bb4cf6c396fd05e | A scalene triangle with a 35-degree angle and side lengths 33 and 17 has triangles congruent to it pasted on its sides so that touching sides are congruent. More triangles are pasted on in the same fashion. After one billion triangles are pasted, the smallest circle that contains all of them is drawn. Compute the small... | [
"\\dfrac{1089\\pi}{4\\sin^2 35^\\circ}"
] | [
"A scalene triangle with a 35-degree angle and side lengths 33 and 17 has triangles congruent to it pasted on its sides so that touching sides are congruent. More triangles are pasted on in the same fashion. After one billion triangles are pasted, the smallest circle that contains all of them is drawn. Compute the ... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-fdd96dc665b1a209f2767925daa31a9e6eb37a10ea13ee576e439ecf987669a2 | A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called $\text{symmetric}$ if its look does not change when the entire square is rotated by a mul... | [
"1022"
] | [
"A scanning code consists of a $7 \\times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called $\\text{symmetric}$ if its look does not change when the entire square is rotated b... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-30040c6b00f32f7679a1d872e236a7c663f9e5ae9268b105623781d3ae6f2f42 | A scanning code consists of a $7 \times 7$ grid of squares, with some squares colored black and the rest colored white. The code is called symmetric if its look does not change when the entire square is rotated by a multiple of $90^\circ$ counterclockwise around its center, nor when it is reflected across a line joinin... | [
"1022"
] | [
"A scanning code consists of a $7 \\times 7$ grid of squares, with some squares colored black and the rest colored white. The code is called symmetric if its look does not change when the entire square is rotated by a multiple of $90^\\circ$ counterclockwise around its center, nor when it is reflected across a line... | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ab12ebd499c9f0fd4022e9cf2f787a66256ebe179952b64a7bb153d3bb5790b1 | A scanning code consists of a \(7 \times 7\) grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of 49 squares. A scanning code is called symmetric if its look does not change when the entire square is rotated by a multiple of ... | [
"1022"
] | [
"A scanning code consists of a \\(7 \\times 7\\) grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of 49 squares. A scanning code is called symmetric if its look does not change when the entire square is rotated by a mult... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-45a8b0792276f1a218bed33e264c407cdfeb581f8db84fa2859a49c11cbe221f | A scarlet bag contains the numbers 1, 5, and 7. A gray bag contains the numbers 2, 4, and 6. Two distinct numbers are drawn at random from each bag. Find the probability that the sum of the two numbers from the scarlet bag is greater than the sum of the two numbers from the gray bag. | [
"\\(\\frac{4}{9}\\)"
] | [
"A scarlet bag contains the numbers 1, 5, and 7. A gray bag contains the numbers 2, 4, and 6. Two distinct numbers are drawn at random from each bag. Find the probability that the sum of the two numbers from the scarlet bag is greater than the sum of the two numbers from the gray bag."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-a01b2df891a906f7b07fddebfa8d64ba3447a92a829168b078f8de34ecdf7916 | A school band found they could arrange themselves in rows of 6, 7, or 8 with no one left over. What is the minimum number of students in the band? | [
"168"
] | [
"A school band found they could arrange themselves in rows of 6, 7, or 8 with no one left over. What is the minimum number of students in the band?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a5607ddbbe9c80f0a4873af5dfd007a6e923b60f7429848b3f54092b1232f901 | A school board issues bonds in the amount of \$15,000,000 to be retired in 20 years. How much must be paid into a sinking fund each year at 4% compounded annually to pay off the total amount due? | [
"503726.26"
] | [
"A school board issues bonds in the amount of \\$15,000,000 to be retired in 20 years. How much must be paid into a sinking fund each year at 4% compounded annually to pay off the total amount due?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-32d86ffeffdc609bcd61281595fc72fa0c1b48c75516ff35b10ffa100bde832a | A school day consists of eight 67-minute classes and 3-minute breaks in between classes. How long is a school day, in minutes? | [
"557"
] | [
"A school day consists of eight 67-minute classes and 3-minute breaks in between classes. How long is a school day, in minutes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-9c410fbc2645c14eacfc4de335f04d6343b7771841b0e9039d9c84a7cded6782 | A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50$, $20$, $20$, $5$, and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class... | [
"-\\dfrac{27}{2}"
] | [
"A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50$, $20$, $20$, $5$, and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1d6d90295e369fb899213c7d0d60e9f84fccb7b23f5bc00c43ac6a1850689074 | A school has $2007$ male students and $2007$ female students. Each student cannot participate in more than $100$ clubs. Any two people of the opposite sex (one male and one female) join at least one club. Let $C$ be the number of clubs that have at least $11$ men and $11$ women. Find the minimum possible value of $C$. | [
"201"
] | [
"A school has $2007$ male students and $2007$ female students. Each student cannot participate in more than $100$ clubs. Any two people of the opposite sex (one male and one female) join at least one club. Let $C$ be the number of clubs that have at least $11$ men and $11$ women. Find the minimum possible value of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-92558d9d18017352bcef6ba2a3824030c2be538fa4c3297a188470069e8620bd | A school has $n$ students and $k$ classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Determine the minimum number of parts into which the students can be divided such that no two students in the same part are friends. ... | [
"1"
] | [
"A school has $n$ students and $k$ classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Determine the minimum number of parts into which the students can be divided such that no two students in the same part are frie... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-086ff9d2e2a06b04054c668cf5dc8f07482366d6fd0b8ff56a52cab2efda427e | A school has $n$ students and $k$ disjoint classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Determine the maximum number of groups, $G$, into which the students can be divided such that no two students in the same gr... | [
"n - k + 1"
] | [
"A school has $n$ students and $k$ disjoint classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Determine the maximum number of groups, $G$, into which the students can be divided such that no two students in the sa... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-d02fe63788df5e2bdb0f54a1f157e70b672be73a39e5708c10dfc38309132a5a | A school has $n$ students and some number of classes, with at least one student in each class, and each student can be in at most one class. Every day, you can choose up to $k$ students ($k>1$) from the list and the principal will tell you how many distinct classes the students are in. What is the minimum number of day... | [
"\\left\\lceil \\frac{n - 1}{k - 1} \\right\\rceil"
] | [
"A school has $n$ students and some number of classes, with at least one student in each class, and each student can be in at most one class. Every day, you can choose up to $k$ students ($k>1$) from the list and the principal will tell you how many distinct classes the students are in. What is the minimum number o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-024165f5debf37ed44b2ebe3d3828d1503389ed1f1b668846d2e3b98022c4d15 | A school has $n$ students and some super classes are provided for them. Each student can participate in any number of classes. Every class has at least two students participating in it. If two different classes have at least two common students, then the number of students in the first class is different from the numbe... | [
"79"
] | [
"A school has $n$ students and some super classes are provided for them. Each student can participate in any number of classes. Every class has at least two students participating in it. If two different classes have at least two common students, then the number of students in the first class is different from the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-e0791013ecd058b5d257878d9210f767cc67aa758d56212ebd0bc4894383a8cd | A school has $n$ students and some super classes are provided for them. Each student can participate in any number of classes. Every class has at least two students. If two different classes have at least two common students, then the number of students in the first class is different from the number of students in the... | [
"79"
] | [
"A school has $n$ students and some super classes are provided for them. Each student can participate in any number of classes. Every class has at least two students. If two different classes have at least two common students, then the number of students in the first class is different from the number of students i... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-b9dd88d872f3d1dbe916aa99b3711e809e8f85eab7798124cd0f7ebd9f3a0236 | A school has 100 students and 5 teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are 50, 20, 20, 5, and 5. Let \( t \) be the average value obtained if a teacher is picked at random and the number of students in their class is noted.... | [
"-13.5"
] | [
"A school has 100 students and 5 teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are 50, 20, 20, 5, and 5. Let \\( t \\) be the average value obtained if a teacher is picked at random and the number of students in their class is... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-c1ffdf5b288a79a35108bd6656ffa08dee21d9b0a3b43df7421779f65686d94b | A school has 1000 students. Each student joins at least one club, and there are \( m \) clubs. Each club must have at least 2 members. It is observed that if any two clubs have at least 2 common members, these two clubs have different numbers of members. Let \( a_i \) be the number of clubs with \( i \) members. Calcul... | [
"499000500"
] | [
"A school has 1000 students. Each student joins at least one club, and there are \\( m \\) clubs. Each club must have at least 2 members. It is observed that if any two clubs have at least 2 common members, these two clubs have different numbers of members. Let \\( a_i \\) be the number of clubs with \\( i \\) memb... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-6f62c97d8e526b054fcaacb07320116f5e817459919f77990795a52bb6f7f9a0 | A school has 12 volleyball teams and 5 volleyball fields. Each match takes 50 minutes, and each team takes at least a 10-minute break between matches. Assuming each team has different strength and the stronger team always beats the weaker one, at least how many hours will it take to rank all 12 teams' strengths? | [
"9.17"
] | [
"A school has 12 volleyball teams and 5 volleyball fields. Each match takes 50 minutes, and each team takes at least a 10-minute break between matches. Assuming each team has different strength and the stronger team always beats the weaker one, at least how many hours will it take to rank all 12 teams' strengths?"
... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e4b471210693ec8405c9d3629060c36ca1e44e163f4a75a3d79496c0820c4788 | A school has 2001 students. Between 80% and 85% study Spanish, between 30% and 40% study French, and no one studies neither. Find the smallest number \( m \) who could study both languages and the largest number \( M \) who could study both languages. | [
"\\( m = 201 \\), \\( M = 499 \\)"
] | [
"A school has 2001 students. Between 80% and 85% study Spanish, between 30% and 40% study French, and no one studies neither. Find the smallest number \\( m \\) who could study both languages and the largest number \\( M \\) who could study both languages."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-f332aba8a89f233fe1ac6035b6c7dfee4cb3512e31d55b8242f75cacdf4cd684 | A school has 3 calculators to loan to students during an examination. The number of students who need to borrow calculators for each day of the examination follows independent Poisson distributions with mean 2. Calculate the probability that the demand for calculators is not satisfied on the first two days of the exami... | [
"0.0204"
] | [
"A school has 3 calculators to loan to students during an examination. The number of students who need to borrow calculators for each day of the examination follows independent Poisson distributions with mean 2. Calculate the probability that the demand for calculators is not satisfied on the first two days of the ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0b9dd3f7ba25b6c7a341afebe7a281e794b17abc7193db6b64251847f2572579 | A school has 450 students. Each student has at least 100 friends among the others, and among any 200 students, there are always two that are friends. Determine the maximum number of two-seater kayaks that can be filled such that each kayak contains two friends. | [
"225"
] | [
"A school has 450 students. Each student has at least 100 friends among the others, and among any 200 students, there are always two that are friends. Determine the maximum number of two-seater kayaks that can be filled such that each kayak contains two friends."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-2ce0759667db4e0be1648809f87cb9b6dfeca9ca0615a1ff68fc0cf96fc184dc | A school has 50 students and four teachers. Each student has exactly one teacher, such that two teachers have 10 students each and the other two teachers have 15 students each. You survey each student in the school, asking the number of classmates they have (not including themself or the teacher). What is the average o... | [
"12"
] | [
"A school has 50 students and four teachers. Each student has exactly one teacher, such that two teachers have 10 students each and the other two teachers have 15 students each. You survey each student in the school, asking the number of classmates they have (not including themself or the teacher). What is the aver... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-af74b101574bc1209e2db08551b71823a1743dabca8368c1ec929eec1e076f54 | A school has 8 clubs, each with 30 students. In any two clubs, there are no more than 5 students who are members of both clubs. Determine the minimum possible number of distinct students participating in these clubs. | [
"114"
] | [
"A school has 8 clubs, each with 30 students. In any two clubs, there are no more than 5 students who are members of both clubs. Determine the minimum possible number of distinct students participating in these clubs."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-3951fef4804dbe9b2ac58a152f04ed819f69e557381ba6a5c6f8d00b08981aab | A school has \( n \) students and some classes for these students, with no limit on the number of classes a student can enroll in. Each class has at least 2 students. If two classes have 2 common students, the number of students in these classes must be different. Calculate the maximum possible number of classes, given... | [
"79"
] | [
"A school has \\( n \\) students and some classes for these students, with no limit on the number of classes a student can enroll in. Each class has at least 2 students. If two classes have 2 common students, the number of students in these classes must be different. Calculate the maximum possible number of classes... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-3c5d171678672ea7058b51a714d62bbaf787f8ab97c62a2f55bd5388a7c9ec29 | A school has a committee with 4 members, each with 2 desired duties. There are 4 distinct duties to be assigned, and there are exactly 2 ways to assign the duties such that each member gets one of their desired duties. How many possible combinations of desired duties are there for the 4 members? | [
"936"
] | [
"A school has a committee with 4 members, each with 2 desired duties. There are 4 distinct duties to be assigned, and there are exactly 2 ways to assign the duties such that each member gets one of their desired duties. How many possible combinations of desired duties are there for the 4 members?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 14
} | 14 | 0 | 0 |
omr-ws-340d0683999503bc7784f7abfc3d652ebb868245bbd9c7d247c88a8afa5d7731 | A school has a row of \( n \) open lockers, numbered 1 through \( n \). Starting at the beginning of the row, you walk past and close every second locker until reaching the end of the row. Then you turn around, walk back, and close every second locker that is still open. You continue in this manner back and forth along... | [
"649"
] | [
"A school has a row of \\( n \\) open lockers, numbered 1 through \\( n \\). Starting at the beginning of the row, you walk past and close every second locker until reaching the end of the row. Then you turn around, walk back, and close every second locker that is still open. You continue in this manner back and fo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 4,
"tir": 1
} | 6 | 0 | 0 |
omr-ws-78f3ae02ab04dd94b6ec372f5bb4f3fdb418e25e5acef59f03ba30888cb4466f | A school has a row of \( n \) open lockers, numbered 1 through \( n \). Starting at the beginning of the row, you walk past and close every second locker until reaching the end of the row. Then you turn around, walk back, and close every second locker that is still open. You continue in this manner back and forth along... | [
"2017"
] | [
"A school has a row of \\( n \\) open lockers, numbered 1 through \\( n \\). Starting at the beginning of the row, you walk past and close every second locker until reaching the end of the row. Then you turn around, walk back, and close every second locker that is still open. You continue in this manner back and fo... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-c62c7f816674cad9e5c796ada920622e65439230ebff15369edd3780eab1edb4 | A school has between 150 and 200 students enrolled. Every afternoon, all the students come together to participate in gym class. The students are separated into six distinct sections of students. If one student is absent from school, the sections can all have the same number of students. What is the sum of all possible... | [
"1575"
] | [
"A school has between 150 and 200 students enrolled. Every afternoon, all the students come together to participate in gym class. The students are separated into six distinct sections of students. If one student is absent from school, the sections can all have the same number of students. What is the sum of all pos... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e321fbf04c79489d3a56d300b88e99ad5a3d2f32e980ff86c0a881c023f68eb4 | A school has four class $11$ study groups. Each study group sends two students to a meeting. They will sit in a circle with no two students from one study group sitting close together. The number of ways is ... | [
"1488"
] | [
"A school has four class $11$ study groups. Each study group sends two students to a meeting. They will sit in a circle with no two students from one study group sitting close together. The number of ways is ..."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-06fa3dae0febcbd5a8b801a273136f65255fc04e687a21681a9b7d866e3fc275 | A school has three classes. Some pairs of children from different classes are enemies (there are no enemies in a class). It is known that every child from the first class has as many enemies in the second class as in the third; the same is true for other classes. Let $E$ be the number of pairs of children being enemies... | [
"1"
] | [
"A school has three classes. Some pairs of children from different classes are enemies (there are no enemies in a class). It is known that every child from the first class has as many enemies in the second class as in the third; the same is true for other classes. Let $E$ be the number of pairs of children being en... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f4a59447f7b5eb0da6bc4fc2c3d652dd30928562633c5ec5e267baf653d90c6 | A school has three senior classes of $M$ students each. Every student knows at least $\frac{3}{4}M$ people in each of the other two classes. Determine the maximum number of non-intersecting teams of $3$ students, each from different classes and who all know each other, that can be formed. Express your answer in terms o... | [
"M"
] | [
"A school has three senior classes of $M$ students each. Every student knows at least $\\frac{3}{4}M$ people in each of the other two classes. Determine the maximum number of non-intersecting teams of $3$ students, each from different classes and who all know each other, that can be formed. Express your answer in t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2fba3e25e360f9b659e62309cf872ce48b3c51c3b3878d3f3e19bd7c8e270ac1 | A school invests $5000 at the end of each year at an annual interest rate of 6%. How much will the school have after 10 years? | [
"6893.62"
] | [
"A school invests $5000 at the end of each year at an annual interest rate of 6%. How much will the school have after 10 years?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b9646f4f104508cd1a5a4bb7114b3edb8511a8c6dd59d885cf7b1e71672d012 | A school is arranging chairs in rows for an assembly. $11$ chairs make a complete row, and right now there are $110$ chairs total. The school wants to have as few empty seats as possible, but all rows of chairs must be complete. If $70$ students will attend the assembly, how many chairs should be removed? | [
"33"
] | [
"A school is arranging chairs in rows for an assembly. $11$ chairs make a complete row, and right now there are $110$ chairs total. The school wants to have as few empty seats as possible, but all rows of chairs must be complete. If $70$ students will attend the assembly, how many chairs should be removed?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1e74ab832b7a21429e62671f010d892bba5a6a85086bab99e4aba54dd6eebbc7 | A school is running a raffle for two prizes with tickets numbered 1 through 59. Two tickets are drawn randomly. What is the expected value of \( |i - j| \), where \( i \) and \( j \) are the numbers on the drawn tickets? | [
"19.661"
] | [
"A school is running a raffle for two prizes with tickets numbered 1 through 59. Two tickets are drawn randomly. What is the expected value of \\( |i - j| \\), where \\( i \\) and \\( j \\) are the numbers on the drawn tickets?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be9289ad19956b3805a351f8c18037c7b5b4a95c8cae87d1c07fbb30bdc793bc | A school library was attended by 5 students on Monday, 6 on Tuesday, 4 on Wednesday, 8 on Thursday, and 7 on Friday. None of the students attended the library on two consecutive days. What is the least possible number of students who visited the library during the week? | [
"15"
] | [
"A school library was attended by 5 students on Monday, 6 on Tuesday, 4 on Wednesday, 8 on Thursday, and 7 on Friday. None of the students attended the library on two consecutive days. What is the least possible number of students who visited the library during the week?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 44,
"tir": 2
} | 46 | 0 | 0 |
omr-ws-faaaa508651e74e9832ab449db7ef6d9f45cc413ae74fbed52b3e3fa30fa04fc | A school organizes 112 groups of students for social work, each group having 11 students, and any two groups having exactly 1 student in common. Calculate the minimum number of students that must be present in all groups. | [
"112"
] | [
"A school organizes 112 groups of students for social work, each group having 11 students, and any two groups having exactly 1 student in common. Calculate the minimum number of students that must be present in all groups."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-f785af9b5fd1a6e6891607039e36516672565cfc305bd30bf83c6173236736e8 | A school organizes 7 summer classes. Each student attends at least one class, and each class has exactly 40 students. Additionally, for any two classes, there are no more than 9 students attending both of them. Calculate the minimum number of students in the school. | [
"124"
] | [
"A school organizes 7 summer classes. Each student attends at least one class, and each class has exactly 40 students. Additionally, for any two classes, there are no more than 9 students attending both of them. Calculate the minimum number of students in the school."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-c7b85e9851fdc236cf3c281b6c7404b55ecef6e7b649898ad06b699e17c76037 | A school organizes optional lectures for 200 students. At least 10 students have signed up for each proposed lecture, and for any two students, there is at most one lecture that both of them have signed up for. Determine the minimum number of days required to hold all these lectures so that no student has to attend two... | [
"22"
] | [
"A school organizes optional lectures for 200 students. At least 10 students have signed up for each proposed lecture, and for any two students, there is at most one lecture that both of them have signed up for. Determine the minimum number of days required to hold all these lectures so that no student has to atten... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-95335b7fe4ba52f0f76996f7b849c11d3a350b0ce9dc7b884f3620b555d4ed7d | A school play ran for two nights, with a total of 1390 adults and students attending. The total admission fees collected were $4285. One adult ticket cost $4.00 and one student ticket cost $2.50. The ratio of adults to students was 3:5 on the first night and 2:3 on the second night. How many students attended each nigh... | [
"500 students on the first night and 354 students on the second night."
] | [
"A school play ran for two nights, with a total of 1390 adults and students attending. The total admission fees collected were $4285. One adult ticket cost $4.00 and one student ticket cost $2.50. The ratio of adults to students was 3:5 on the first night and 2:3 on the second night. How many students attended each... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f5dd903261eb853b263a067d6cce5bea42ac7c33761e5c2f7a59f65141d72ebd | A school purchased 4 peach trees, 4 apricot trees, and 6 cherry trees that they want to plant in a row on the school grounds. If the trees are planted in random order, what is the probability that no two cherry trees are planted next to each other? | [
"\\dfrac{4}{143}"
] | [
"A school purchased 4 peach trees, 4 apricot trees, and 6 cherry trees that they want to plant in a row on the school grounds. If the trees are planted in random order, what is the probability that no two cherry trees are planted next to each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9348a502d5d2139714fac9658be2a6c07a9976ca4ee592432dac258505638735 | A science test has 10 questions worth 5 points each, 7 questions worth 6 points each, and 4 questions worth 2 points each. None of these questions will be given partial credit. How many scores between 0 and 100 are impossible to achieve? | [
"4"
] | [
"A science test has 10 questions worth 5 points each, 7 questions worth 6 points each, and 4 questions worth 2 points each. None of these questions will be given partial credit. How many scores between 0 and 100 are impossible to achieve?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-e7bc3917c8630872d77afde24a4c63fc5a781c58df1ec36b2c17eabd6044c65f | A scientific conference is attended by \( k \) chemists and alchemists, with chemists being in the majority. A chemist always tells the truth, while an alchemist may tell the truth or lie. A visiting mathematician must determine which of the \( k \) members are chemists and which are alchemists by asking questions of t... | [
"38"
] | [
"A scientific conference is attended by \\( k \\) chemists and alchemists, with chemists being in the majority. A chemist always tells the truth, while an alchemist may tell the truth or lie. A visiting mathematician must determine which of the \\( k \\) members are chemists and which are alchemists by asking quest... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40fcad5a7ed11404c5adef77aa87ed32f930d7f2f25025ae5af2997d2f7fd905 | A scientist has a lump of pure $^{304}\text{Ubu}$ weighing exactly $1$ gram. After 1 ms, the sample contains $0.000005271$ grams of $^{304}\text{Ubu}$, $0.004917026$ grams of $^{300}\text{Uue}$, and $0.995077703$ grams of $^{296}\text{Ts}$. Find the half-lifes of $^{304}\text{Ubu}$ and $^{300}\text{Uue}$. Express your ... | [
"130.380"
] | [
"A scientist has a lump of pure $^{304}\\text{Ubu}$ weighing exactly $1$ gram. After 1 ms, the sample contains $0.000005271$ grams of $^{304}\\text{Ubu}$, $0.004917026$ grams of $^{300}\\text{Uue}$, and $0.995077703$ grams of $^{296}\\text{Ts}$. Find the half-lifes of $^{304}\\text{Ubu}$ and $^{300}\\text{Uue}$. Ex... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-38f9a0592d0f5c593db4a0c1f0c81e45723bf88beefbc226f991c39811afcf3e | A scientist observed the weather for \( x \) days and noted the following: 1. It rained on 7 days, either during the day or at night. 2. If it rained at night, it was clear during the day. 3. There were 4 clear nights and 5 clear days. Find the value of \( x \). | [
"8"
] | [
"A scientist observed the weather for \\( x \\) days and noted the following:\n1. It rained on 7 days, either during the day or at night.\n2. If it rained at night, it was clear during the day.\n3. There were 4 clear nights and 5 clear days.\nFind the value of \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1ab23244aae92cdeb2acfcb7aa80b295fbd19de6a5baacbf751a335a65ab4fb4 | A scientist recorded the heights of 5 trees standing in a row as integers. Each tree was either twice as tall or half as tall as the one to its right. The recorded heights are as follows: \begin{tabular}{|c|c|} \hline Tree 1 & \rule{0.2cm}{0.15mm} meters \\ Tree 2 & 11 meters \\ Tree 3 & \rule{0.2cm}{0.15mm} meters \\ ... | [
"24.2"
] | [
"A scientist recorded the heights of 5 trees standing in a row as integers. Each tree was either twice as tall or half as tall as the one to its right. The recorded heights are as follows:\n\\begin{tabular}{|c|c|}\n\\hline Tree 1 & \\rule{0.2cm}{0.15mm} meters \\\\\nTree 2 & 11 meters \\\\\nTree 3 & \\rule{0.2cm}{0... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-3bfbce9c036a7e4705fa359eacd4b99d0e30b714c968e081939cd72dea38a31a | A scientist recorded the heights of 5 trees standing in a row, where each tree was either twice as tall or half as tall as the one to its right. The heights are recorded as follows: \begin{tabular}{|c|c|} \hline Tree 1 & \rule{0.2cm}{0.15mm} meters \\ Tree 2 & 11 meters \\ Tree 3 & \rule{0.2cm}{0.15mm} meters \\ Tree 4... | [
"24.2"
] | [
"A scientist recorded the heights of 5 trees standing in a row, where each tree was either twice as tall or half as tall as the one to its right. The heights are recorded as follows:\n\\begin{tabular}{|c|c|}\n\\hline Tree 1 & \\rule{0.2cm}{0.15mm} meters \\\\\nTree 2 & 11 meters \\\\\nTree 3 & \\rule{0.2cm}{0.15mm}... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7742a38913a10ef29187969da4564e6a49a2e90080a085e2652c79b64ccc95ed | A scientist recorded the heights of 5 trees standing in a row, where each tree was either twice as tall or half as tall as the one to its right. The recorded heights are as follows: - Tree 1: \(\rule{0.4cm}{0.15mm}\) meters - Tree 2: 11 meters - Tree 3: \(\rule{0.5cm}{0.15mm}\) meters - Tree 4: \(\rule{0.5cm}{0.15mm}\)... | [
"24.2"
] | [
"A scientist recorded the heights of 5 trees standing in a row, where each tree was either twice as tall or half as tall as the one to its right. The recorded heights are as follows:\n\n- Tree 1: \\(\\rule{0.4cm}{0.15mm}\\) meters\n- Tree 2: 11 meters\n- Tree 3: \\(\\rule{0.5cm}{0.15mm}\\) meters\n- Tree 4: \\(\\ru... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 25
} | 73 | 0 | 0 |
omr-ws-2df92345ac25b885c052f15faf8c3af899cdad53538eca0910fe8f5f36bcc03e | A scientist walking through a forest recorded as integers the heights of 5 trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately, some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the... | [
"12",
"14"
] | [
"A scientist walking through a forest recorded as integers the heights of 5 trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately, some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicatin... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-623e207979e71ff29157f32dc7b1a0fc57b7ab7b4f57ef02efdc9a2028d40ef8 | A scientist walking through a forest recorded as integers the heights of 5 trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately, some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the... | [
"24.2"
] | [
"A scientist walking through a forest recorded as integers the heights of 5 trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately, some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicatin... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0f38f07a682f3f4572a009a73b2de388f0d6a195c9c15c5dd536da754cd7da04 | A scoop is in the shape of the parabolic surface defined by \( -1 \leq x \leq 1 \), \( y = x^2 \), and \( 0 \leq z \leq 2 \), with one end covered by the region \( -1 \leq x \leq 1 \), \( x^2 \leq y \leq 1 \), and \( z = 0 \). How much water can the scoop hold? | [
"\\dfrac{8}{3}"
] | [
"A scoop is in the shape of the parabolic surface defined by \\( -1 \\leq x \\leq 1 \\), \\( y = x^2 \\), and \\( 0 \\leq z \\leq 2 \\), with one end covered by the region \\( -1 \\leq x \\leq 1 \\), \\( x^2 \\leq y \\leq 1 \\), and \\( z = 0 \\). How much water can the scoop hold?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-784dae5e60f82af11e41c61235dca1288252e33c6346fbab38379f4b19f0b59f | A score sequence of a tournament is a non-decreasing sequence of the out-degrees of its vertices. Let $S_n$ denote the number of distinct score sequences for tournaments on $n$ vertices that do not uniquely determine a tournament. Find the value of $S_5 + S_6$. | [
"7"
] | [
"A score sequence of a tournament is a non-decreasing sequence of the out-degrees of its vertices. Let $S_n$ denote the number of distinct score sequences for tournaments on $n$ vertices that do not uniquely determine a tournament. Find the value of $S_5 + S_6$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
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