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-6e0a38c6cdd14f92ecced38b4923e6310bbb55f1fc659f2aad4c4a64476ffce9 | You and your (n-1) friends (n > 1) are on a boat that is about to sink. To lessen the weight of the boat, one of you must jump off. You all pick a straw from a randomly assorted set of n straws. The one with the short end will be sacrificed. Determine when you should draw a straw (as in first, second, etc.) to minimize... | [
"\\text{All positions have equal probability}"
] | [
"You and your (n-1) friends (n > 1) are on a boat that is about to sink. To lessen the weight of the boat, one of you must jump off. You all pick a straw from a randomly assorted set of n straws. The one with the short end will be sacrificed. Determine when you should draw a straw (as in first, second, etc.) to min... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c18408b96e30d5ddfb5b514ddbaa434942db4005f56f7011efa28e168cfb3b7d | You and your coworker together make $16 per hour. Your coworker earns 10 percent more than you do. What is your hourly wage? | [
"7.62"
] | [
"You and your coworker together make $16 per hour. Your coworker earns 10 percent more than you do. What is your hourly wage?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8f1fcef66a71e1b28032aafac10cdc3a7a66059bf077b956bc960423fb21d565 | You and your enemy will play a game consisting of an even number of turns, where each turn is worth one point. You score with probability $p < \frac{1}{2}$ and your enemy scores with probability $1-p$. To win the game, you must have more than half the points. Given that you know the probability $p$ of winning a round, ... | [
"\\(2n\\) where \\(\\frac{n-1}{2n-1} < p < \\frac{n}{2n+1}\\)"
] | [
"You and your enemy will play a game consisting of an even number of turns, where each turn is worth one point. You score with probability $p < \\frac{1}{2}$ and your enemy scores with probability $1-p$. To win the game, you must have more than half the points. Given that you know the probability $p$ of winning a r... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-8ee4faa11b46b24ce0b9ab9e9e447a747f4ddba212d4e22da4b6225d6f98bceb | You and your friend are having a water gun duel. You will start back-to-back and each take ten steps. Your friend will turn around and shoot you with a 25% chance on any step, in which case you will always be soaked. If not, on the tenth step, you will each turn around and fire once. You will fire at your opponent with... | [
"\\(\\frac{1033174611}{1048576000}\\)"
] | [
"You and your friend are having a water gun duel. You will start back-to-back and each take ten steps. Your friend will turn around and shoot you with a 25% chance on any step, in which case you will always be soaked. If not, on the tenth step, you will each turn around and fire once. You will fire at your opponent... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 19
} | 31 | 0 | 0 |
omr-ws-e3920a811b4b5f86bd48e93e61adf692370cda3bc4cc11ef55f7200b7baf23cc | You and your friend are in two adjacent compartments of a train with an infinite number of compartments. Neither of you knows which compartment the other is in. Both of you move towards the front or back compartment with the intention of meeting each other. A move consists of a person moving from one compartment to ano... | [
"\\infty"
] | [
"You and your friend are in two adjacent compartments of a train with an infinite number of compartments. Neither of you knows which compartment the other is in. Both of you move towards the front or back compartment with the intention of meeting each other. A move consists of a person moving from one compartment t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-6c47ec7eb93dc2d55ad2fddf798c2514fda372cff902e299a3674ff943638e2e | You and your friend are standing in opposite corners of a $10$-by-$30$ m rectangular pool. You can walk $25$ m/min and swim $20$ m/min. What is the least amount of time you need to reach your friend? | [
"1.5"
] | [
"You and your friend are standing in opposite corners of a $10$-by-$30$ m rectangular pool. You can walk $25$ m/min and swim $20$ m/min. What is the least amount of time you need to reach your friend?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-af67d24a4390f4d37c36ac05dea2020e21dd8a8673be452879f4bf6c6fe95d84 | You and your friend each have a piece of cord of different integer lengths. Your task is to cut the cord into any number of pieces such that the product of the lengths of the pieces is maximized. The length of your cord is 6 times the length of your friend's cord, and the number of pieces you obtain is 5 times the numb... | [
"42"
] | [
"You and your friend each have a piece of cord of different integer lengths. Your task is to cut the cord into any number of pieces such that the product of the lengths of the pieces is maximized. The length of your cord is 6 times the length of your friend's cord, and the number of pieces you obtain is 5 times the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-8db13c5f5c2a3de33a05e98ebc15ad9b86c662704ffe6a57c8b45fed439e8a18 | You and your friend play a game on a $7 \times 7$ grid of buckets. Your friend chooses 5 "lucky" buckets that form either a vertical or horizontal line of length 5. The possible sets of buckets are $\{(a, b),(a, b + 1),(a, b + 2),(a, b + 3),(a, b + 4)\}$ or $\{(b, a),(b + 1, a),(b + 2, a),(b + 3, a),(b + 4, a)\}$, with... | [
"9"
] | [
"You and your friend play a game on a $7 \\times 7$ grid of buckets. Your friend chooses 5 \"lucky\" buckets that form either a vertical or horizontal line of length 5. The possible sets of buckets are $\\{(a, b),(a, b + 1),(a, b + 2),(a, b + 3),(a, b + 4)\\}$ or $\\{(b, a),(b + 1, a),(b + 2, a),(b + 3, a),(b + 4, ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ef96acf0ee261237f36785e7268ff369dbc9e6f3680f50561571570bb27da0dc | You and your friend play the following dangerous game. You two start off at some point \((x, y)\) on the plane, where \( x \) and \( y \) are nonnegative integers. When it is player \( A \)'s turn, \( A \) tells his opponent \( B \) to move to another point on the plane. Then \( A \) waits for a while. If \( B \) is no... | [
"90133"
] | [
"You and your friend play the following dangerous game. You two start off at some point \\((x, y)\\) on the plane, where \\( x \\) and \\( y \\) are nonnegative integers.\nWhen it is player \\( A \\)'s turn, \\( A \\) tells his opponent \\( B \\) to move to another point on the plane. Then \\( A \\) waits for a whi... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-445a4f8f98777d4d91a042ac9f50a07d906d2235dd4ccfe691bee1463c9efd75 | You and your friends have been tasked with building a cardboard castle in the two-dimensional Cartesian plane. The castle is built by the following rules: 1. There is a tower of height $2^n$ at the origin. 2. From towers of height $2^i \ge 2$, a wall of length $2^{i-1}$ can be constructed between the aforementioned tow... | [
"(2^{1000} - 1)(2^{1001} - 1) + 1"
] | [
"You and your friends have been tasked with building a cardboard castle in the two-dimensional Cartesian plane. The castle is built by the following rules:\n1. There is a tower of height $2^n$ at the origin.\n2. From towers of height $2^i \\ge 2$, a wall of length $2^{i-1}$ can be constructed between the aforementi... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cde69ec8a889f5b95b39c6cded9ef47064ed322af705b8d7b7b1bb6cb47571a3 | You and your mom have a total of $20 to spend on dinner. Each of you orders a meal that costs the same amount. What is the maximum amount you can spend on the meals if you must leave a tip that is 25% of the total meal cost? | [
"16"
] | [
"You and your mom have a total of $20 to spend on dinner. Each of you orders a meal that costs the same amount. What is the maximum amount you can spend on the meals if you must leave a tip that is 25% of the total meal cost?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 15
} | 28 | 0 | 0 |
omr-ws-3ada9f8e1a344f52721a329f7a7d742ae58438b8b82e3b3d5bb0e08d529dffef | You are 10 miles off a straight highway and your destination is 100 miles ahead, also 10 miles off the highway. You can drive 40 miles per hour off-road and 80 miles per hour on the highway. What is the fewest number of minutes in which you can get to your destination? Round your answer to the nearest minute. | [
"101"
] | [
"You are 10 miles off a straight highway and your destination is 100 miles ahead, also 10 miles off the highway. You can drive 40 miles per hour off-road and 80 miles per hour on the highway. What is the fewest number of minutes in which you can get to your destination? Round your answer to the nearest minute."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 5
} | 26 | 0 | 0 |
omr-ws-7a07dfd6314a1aa9ec547e1b1e5fea6b82a349452bc6c083a699fd10782b800f | You are 200 feet from the river, your tent is 240 feet downstream from your current location, and the tent is 120 feet away from the river bank. You must run to the river to fill a bucket and then run to your tent. How far did you run with the bucket full and how far did you run with the bucket empty? | [
"250 feet (empty), 150 feet (full)"
] | [
"You are 200 feet from the river, your tent is 240 feet downstream from your current location, and the tent is 120 feet away from the river bank. You must run to the river to fill a bucket and then run to your tent. How far did you run with the bucket full and how far did you run with the bucket empty?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 30
} | 31 | 0 | 0 |
omr-ws-e3afc4a234120af8f24a83dd5acf227a05052f65223b3c5562403e594feaf989 | You are 200 feet from the river, your tent is 240 feet downstream from your current location, and the tent is 120 feet away from the river bank. You must run to the river to fill a bucket and then run to your tent. What is the shortest distance you must run to accomplish this? | [
"400 feet"
] | [
"You are 200 feet from the river, your tent is 240 feet downstream from your current location, and the tent is 120 feet away from the river bank. You must run to the river to fill a bucket and then run to your tent. What is the shortest distance you must run to accomplish this?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-57d58bfb4f199b5e3ede81571638335c2f6dbcc39025f93ae5e0e0adfaa07997 | You are about to take a test that contains questions of type A worth 4 points and type B worth 7 points. You must answer at least 5 of type A and 3 of type B, but no more than 10 of either type. In total, you can answer no more than 18 questions. How many questions of each type must you answer to maximize your score? W... | [
"102"
] | [
"You are about to take a test that contains questions of type A worth 4 points and type B worth 7 points. You must answer at least 5 of type A and 3 of type B, but no more than 10 of either type. In total, you can answer no more than 18 questions. How many questions of each type must you answer to maximize your sco... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 11
} | 17 | 0 | 0 |
omr-ws-8a36133997ddced4e009fa0dc1a83b0d310ca37202124c7af13df53fb3cdbd15 | You are allowed to choose a number $n$ and then toss a fair coin $n$ times. You will get a prize if you can get either seven or nine heads. What is your best choice of the number $n$? | [
"16"
] | [
"You are allowed to choose a number $n$ and then toss a fair coin $n$ times. You will get a prize if you can get either seven or nine heads. What is your best choice of the number $n$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-4bd87036bdcbbd06d13650c584931b85c22c83252d580602f547dba78d6f34ae | You are allowed to traverse the following 4-by-4 array of numbers by taking upward and downward steps, as well as steps to the left and to the right. However, you are not allowed to go diagonally, nor are you allowed to go to the same position twice. How many different ways can you get the sum of the numbers to add up ... | [
"104"
] | [
"You are allowed to traverse the following 4-by-4 array of numbers by taking upward and downward steps, as well as steps to the left and to the right. However, you are not allowed to go diagonally, nor are you allowed to go to the same position twice. How many different ways can you get the sum of the numbers to ad... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-9f762ef0ab336f3125afd7c794dc407a40ebe1ad9b7cc90ce72fb48b6721c259 | You are at Point A and the angle of elevation to the top of a vertical building is $35^\circ$. You walk 90 feet directly towards the building to Point B, where the angle of elevation is $48^\circ$. How much farther do you have to walk to reach the building? | [
"153.6"
] | [
"You are at Point A and the angle of elevation to the top of a vertical building is $35^\\circ$. You walk 90 feet directly towards the building to Point B, where the angle of elevation is $48^\\circ$. How much farther do you have to walk to reach the building?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4da7320cdfabe09e7cb06d3eb48b0209572ba9ad93688f76484e7205bbc8cc33 | You are at a camp and need to get to a doctor by either crossing a river with a current or a lake with no current. Both routes are the same distance. Which route should you take in order to get to the doctor and back in the shortest amount of time? (Solve this algebraically.) | [
"lake"
] | [
"You are at a camp and need to get to a doctor by either crossing a river with a current or a lake with no current. Both routes are the same distance. Which route should you take in order to get to the doctor and back in the shortest amount of time? (Solve this algebraically.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-4dcbbaf4507e5d07c2d3b9612e15f819c6b6b1d263e0f91163536088ead4c447 | You are at a point \((a, b)\) and need to reach \((c, d)\), where both points are below the line \(x = y\). You can move in a straight line of unit length either up or to the right, but you may not move onto any point on the line \(x = y\). How many paths are there? | [
"\\dbinom{(c - a) + (d - b)}{c - a} - \\dbinom{(c - b) + (d - a)}{c - b}"
] | [
"You are at a point \\((a, b)\\) and need to reach \\((c, d)\\), where both points are below the line \\(x = y\\). You can move in a straight line of unit length either up or to the right, but you may not move onto any point on the line \\(x = y\\). How many paths are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-89e31cf0527823b5d8f3a29b475dc977adc1aa8018dade9c18245a5f5962f68f | You are at a point \((a,b)\) and need to reach another point \((c,d)\). Both points are below the line \(x = y\) and have integer coordinates. You can move in steps of length 1, either upwards or to the right, but you may not move to a point on the line \(x = y\). How many different paths are there? | [
"\\(\\binom{(c-a)+(d-b)}{c-a} - \\binom{(c-a)+(d-b)}{c-a+1}\\)"
] | [
"You are at a point \\((a,b)\\) and need to reach another point \\((c,d)\\). Both points are below the line \\(x = y\\) and have integer coordinates. You can move in steps of length 1, either upwards or to the right, but you may not move to a point on the line \\(x = y\\). How many different paths are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 20,
"genselect": 5,
"tir": 6
} | 31 | 0 | 0 |
omr-ws-a76e4c35723b9596008d1ebb3ccd5ac28c506817bb2bcc3da0666c2363d53cf8 | You are at one vertex of an equilateral triangle with side length $1$. The laser beam bounces off exactly $137$ edges and returns to the original vertex without touching any other vertices. Let $M$ be the maximum possible distance the beam could have traveled, and $m$ be the minimum possible distance the beam could hav... | [
"\\(\\sqrt{4831} - \\sqrt{3679}\\)"
] | [
"You are at one vertex of an equilateral triangle with side length $1$. The laser beam bounces off exactly $137$ edges and returns to the original vertex without touching any other vertices. Let $M$ be the maximum possible distance the beam could have traveled, and $m$ be the minimum possible distance the beam coul... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8af4172c6467fc1a830f8090fe6e3b9f0dd0688a27cce4c16259564abf1463f3 | You are at one vertex of an equilateral triangle with side length 1. The laser beam bounces off exactly 137 edges and returns to the original vertex without touching any other vertices. Let \( M \) be the maximum possible distance the beam could have traveled, and \( m \) be the minimum possible distance the beam could... | [
"864"
] | [
"You are at one vertex of an equilateral triangle with side length 1. The laser beam bounces off exactly 137 edges and returns to the original vertex without touching any other vertices. Let \\( M \\) be the maximum possible distance the beam could have traveled, and \\( m \\) be the minimum possible distance the b... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-87d8857cdb7fe30828a7e419060760ba553054087cd680e6f13c0438d17b6a3f | You are at point $A$ at the bottom-left corner of a $7 \times 7$ grid, and point $B$ is at the top-right corner. Every second, you can move up one unit or right one unit. There are two teleports $T_1$ and $T_2$ positioned at two unoccupied lattice points. When you land on a teleport, you can either ignore it or instant... | [
"$T_1 = (4,3)$, $T_2 = (3,4)$"
] | [
"You are at point $A$ at the bottom-left corner of a $7 \\times 7$ grid, and point $B$ is at the top-right corner. Every second, you can move up one unit or right one unit. There are two teleports $T_1$ and $T_2$ positioned at two unoccupied lattice points. When you land on a teleport, you can either ignore it or i... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 4
} | 10 | 0 | 0 |
omr-ws-6b4c77aef48916c92dc902f42dc4cf3ab4be3a6e006749906963422bb052321a | You are at point \( A(2,2) \) on land and need to reach point \( P(6,-2) \) in the water. The edge of the bank is the x-axis. If your speed on land is \( k \) times your speed in water, find the optimal path to minimize the total time taken to reach point \( P \). | [
"\\left( x, 0 \\right)"
] | [
"You are at point \\( A(2,2) \\) on land and need to reach point \\( P(6,-2) \\) in the water. The edge of the bank is the x-axis. If your speed on land is \\( k \\) times your speed in water, find the optimal path to minimize the total time taken to reach point \\( P \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-96935ab980ca596ef801183f1d7fe878934c70100fdc51c32f802d7db9f44eab | You are at the center of a 20m equilateral triangle with a hungry velociraptor at each corner. The velociraptors accelerate at 4 m/s^2, reaching a top speed of 25 m/s, but the top raptor is limited to 10 m/s due to a wounded leg. You have an infinitely large acceleration and immediately reach your top speed of 6 m/s. A... | [
"90^\\circ"
] | [
"You are at the center of a 20m equilateral triangle with a hungry velociraptor at each corner. The velociraptors accelerate at 4 m/s^2, reaching a top speed of 25 m/s, but the top raptor is limited to 10 m/s due to a wounded leg. You have an infinitely large acceleration and immediately reach your top speed of 6 m... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-d5361b4dc07a9495f713cc63f3e7c3b4f9098ae84da19509daa68cbd714164cc | You are at the leftmost bottom point of an $n \times n$ grid and need to reach the rightmost topmost point. You can only move upward or to the right, and you cannot move to any squares along the diagonal due to mines. Determine the number of ways to reach the destination. | [
"\\(\\frac{2}{n} \\cdot \\binom{2n-2}{n-1}\\)"
] | [
"You are at the leftmost bottom point of an $n \\times n$ grid and need to reach the rightmost topmost point. You can only move upward or to the right, and you cannot move to any squares along the diagonal due to mines. Determine the number of ways to reach the destination."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-0d442adef98a80247da6cfcdde948295ed13443e43282e90430e00a83fd49a74 | You are blind and cannot feel the difference between a coin that is heads up or tails up. There are 100 coins in front of you and are told that exactly 10 of them are heads up. Explain how you can split the otherwise indistinguishable coins into two groups so that both groups have the same number of heads. | [
"Take any 10 coins and flip them."
] | [
"You are blind and cannot feel the difference between a coin that is heads up or tails up. There are 100 coins in front of you and are told that exactly 10 of them are heads up. Explain how you can split the otherwise indistinguishable coins into two groups so that both groups have the same number of heads."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-900be29ccdaba363c6e18cc06df82938550e2607009661b847025c5c95e9f506 | You are blindfolded and told there are at least 128 pennies on a table, with 128 heads and some number of tails. Create an algorithm to partition all the pennies into two sets such that each set has an equal number of heads. The algorithm must work for any number of pennies, as long as the total number of pennies is gr... | [
"The algorithm is to put 128 coins in one pile and the rest in another pile, then flip all the coins in the first pile."
] | [
"You are blindfolded and told there are at least 128 pennies on a table, with 128 heads and some number of tails. Create an algorithm to partition all the pennies into two sets such that each set has an equal number of heads. The algorithm must work for any number of pennies, as long as the total number of pennies ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-6d9b751c9c6a6d9b0f997d2c3e7c603b41a6b1b55d1f4356f3cd280b9fdb4f12 | You are constructing a cardboard box from a piece of cardboard with dimensions 2 m by 4 m by cutting equal-size squares from each corner and folding up the edges. What are the dimensions of the box with the largest volume? | [
"\\( 2 + \\frac{2\\sqrt{3}}{3} \\), \\( \\frac{2\\sqrt{3}}{3} \\), \\( \\frac{3 - \\sqrt{3}}{3} \\)"
] | [
"You are constructing a cardboard box from a piece of cardboard with dimensions 2 m by 4 m by cutting equal-size squares from each corner and folding up the edges. What are the dimensions of the box with the largest volume?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-0ce4969216eb4c169053eb0d2c84d0db54d55b27961f8c9660f6732ebaa7c2b9 | You are constructing a window with the shape of a rectangle of height \( h \) surmounted by a semicircle of radius \( r \). You must construct the exterior edge of the window, as well as the three internal struts shown, using only 21 feet of edging. Find the value of \( r \) that maximizes the area of the window. | [
"\\( \\frac{42}{24 + \\pi} \\)"
] | [
"You are constructing a window with the shape of a rectangle of height \\( h \\) surmounted by a semicircle of radius \\( r \\). You must construct the exterior edge of the window, as well as the three internal struts shown, using only 21 feet of edging. Find the value of \\( r \\) that maximizes the area of the wi... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-636dc624bc117a68ce0f705b18137add60e7e6b6949486ee3ede5446067b38a1 | You are creating a test with 89 questions, where each column must have between 4 and 13 entries. How many items should be in each row to maximize the number of prime number chains (2 or more primes in a row)? Additionally, compute the maximum number of dollars you can earn if you receive 1 dollar for every 2 primes in ... | [
"32768"
] | [
"You are creating a test with 89 questions, where each column must have between 4 and 13 entries. How many items should be in each row to maximize the number of prime number chains (2 or more primes in a row)? Additionally, compute the maximum number of dollars you can earn if you receive 1 dollar for every 2 prime... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-7230308fa3befbec01219241c4b864da33a512c2c383e9b666c1ba2e201b95cd | You are distributing 87 candies to 4 kids. Kid #1 must receive a number of candies which is not a multiple of 3. Kid #2 must receive an even number of candies. Kid #3 can receive any number of candies. Kid #4 must receive a number of candies that is a multiple of 7. How many ways are there to distribute the candies, gi... | [
"4366"
] | [
"You are distributing 87 candies to 4 kids. Kid #1 must receive a number of candies which is not a multiple of 3. Kid #2 must receive an even number of candies. Kid #3 can receive any number of candies. Kid #4 must receive a number of candies that is a multiple of 7. How many ways are there to distribute the candie... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-c5dbd06a5ea74c3fedb47516fdab0fe942f56c375a15d8b4c2ec48c15cb7cd1f | You are doing a quest in the game of Run Escape. You are standing at point \( A \), in between two parallel rivers (River Lum and River Salve) that are 10 kilometers apart. You are 5 kilometers away from both rivers. To complete the quest, you must get a bucket of water from River Lum, get a bucket of water from River ... | [
"29"
] | [
"You are doing a quest in the game of Run Escape. You are standing at point \\( A \\), in between two parallel rivers (River Lum and River Salve) that are 10 kilometers apart. You are 5 kilometers away from both rivers. To complete the quest, you must get a bucket of water from River Lum, get a bucket of water from... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e2f51df4615fb8322fecacbc45e2c3a5d59193a8c14860742296499edea0fdc9 | You are drawing balls from a bag. The balls can be labeled A, B, or C and can be either red, green, or blue. The probability of drawing a ball with each letter label is unknown but fixed and balls are replaced after each draw. A ball labeled A is half as likely to be green as the other colors. A ball labeled B is half ... | [
"\\(\\frac{1}{4}\\)"
] | [
"You are drawing balls from a bag. The balls can be labeled A, B, or C and can be either red, green, or blue. The probability of drawing a ball with each letter label is unknown but fixed and balls are replaced after each draw. A ball labeled A is half as likely to be green as the other colors. A ball labeled B is ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-eb4901ec2c78ccbe89660160edf1a80d153f5177d171ad13530b7c95d374f5b6 | You are gambling with the Devil with 3 dollars in your pocket. The Devil will play five games with you. In each game, you give the Devil an integral number of dollars, from 0 up to what you have at the time. If you win, you get back from the Devil double the amount of what you pay. If you lose, the Devil just keeps wha... | [
"16"
] | [
"You are gambling with the Devil with 3 dollars in your pocket. The Devil will play five games with you. In each game, you give the Devil an integral number of dollars, from 0 up to what you have at the time. If you win, you get back from the Devil double the amount of what you pay. If you lose, the Devil just keep... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 20,
"genselect": 8,
"tir": 5
} | 33 | 0 | 0 |
omr-ws-22d426d9280afca4bb4b58f2d0276d35d8ebd980b943b8d7b05823aac22500d5 | You are given $2024$ yellow and $2024$ blue points on the plane, with no three points collinear. A pair of nonnegative integers $(a, b)$ is called [i]good[/i] if there exists a half-plane containing exactly $a$ yellow and $b$ blue points, with points on the boundary line considered outside the half-plane. Find the smal... | [
"6073"
] | [
"You are given $2024$ yellow and $2024$ blue points on the plane, with no three points collinear. A pair of nonnegative integers $(a, b)$ is called [i]good[/i] if there exists a half-plane containing exactly $a$ yellow and $b$ blue points, with points on the boundary line considered outside the half-plane. Find the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-c7f2bf6b55abbadeb8715349fbcff655923395dafcfc4a621d94b160cf9da513 | You are given 10 cards. Eight of them are numbered 1 through 8, one is one of the numbers 1 through 8, and the other is marked with an X. If you reveal a pair of numbers, you win; otherwise, you lose. What is the probability of winning? | [
"\\(\\frac{1}{2}\\)"
] | [
"You are given 10 cards. Eight of them are numbered 1 through 8, one is one of the numbers 1 through 8, and the other is marked with an X. If you reveal a pair of numbers, you win; otherwise, you lose. What is the probability of winning?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-412b6cd59eedc5daa16ffe3baa216069a89ec804dccd4e3198c9ede5c07704ae | You are given 10 letters, A through J, each representing a unique number. Each day, you can ask the jailkeeper to compare any two of these numbers. In the worst-case scenario, how many days will it take to determine the order of all 10 numbers from least to greatest? | [
"25"
] | [
"You are given 10 letters, A through J, each representing a unique number. Each day, you can ask the jailkeeper to compare any two of these numbers. In the worst-case scenario, how many days will it take to determine the order of all 10 numbers from least to greatest?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-178a5ed44d37a8fe3d74178f378a8d5d942fed57e02e7d0ff13db29796d204ce | You are given 100 cards numbered $100, 99, \ldots, 3, 2, 1$ in that order. You want to put them in the order $1, 2, 3, \ldots, 99, 100$ by switching only two adjacent cards at a time. What is the minimum number of switches necessary? | [
"4950"
] | [
"You are given 100 cards numbered $100, 99, \\ldots, 3, 2, 1$ in that order. You want to put them in the order $1, 2, 3, \\ldots, 99, 100$ by switching only two adjacent cards at a time. What is the minimum number of switches necessary?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-38f021dc1ecaa996ef3c0165a435ed38b231b90f46f9a4235ec8967d7a0e0c9f | You are given 100 cards numbered 100, 99, 98, ..., 3, 2, 1 in that order. You want to put them in the order 1, 2, 3, ..., 98, 99, 100 by switching only two adjacent cards at a time. What is the minimum number of switches necessary? | [
"4950"
] | [
"You are given 100 cards numbered 100, 99, 98, ..., 3, 2, 1 in that order. You want to put them in the order 1, 2, 3, ..., 98, 99, 100 by switching only two adjacent cards at a time. What is the minimum number of switches necessary?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-e143ee8d554ce5488cda3d51e9d7ee1c9ffec5ae820997f0f584f07a4ed457f0 | You are given 1000 identical coins, two of which are fake. One fake coin is lighter than all the rest, and the other fake coin is heavier than all the rest. You have a balance and can weigh any two piles of coins. How many times must you use the balance to find the fake coins in the worst-case scenario? | [
"13"
] | [
"You are given 1000 identical coins, two of which are fake. One fake coin is lighter than all the rest, and the other fake coin is heavier than all the rest. You have a balance and can weigh any two piles of coins. How many times must you use the balance to find the fake coins in the worst-case scenario?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8d4faa714bfac5ac4f201b0523375b08179809002aa9e98a27f52a16de386559 | You are given 16 pieces of paper numbered $16, 15, \ldots, 2, 1$ in that order. You want to put them in the order $1, 2, \ldots, 15, 16$ by switching only two adjacent pieces of paper at a time. What is the minimum number of switches necessary? | [
"120"
] | [
"You are given 16 pieces of paper numbered $16, 15, \\ldots, 2, 1$ in that order. You want to put them in the order $1, 2, \\ldots, 15, 16$ by switching only two adjacent pieces of paper at a time. What is the minimum number of switches necessary?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-69078ac51937111aa91b90f55e479e52875054279c30b47ed77d6c943f3c2b46 | You are given 2009 stones, each with a positive integer weight. If any one stone is removed, the remaining stones can be split into two heaps with the same total weight, each containing exactly 1004 stones. Let the weights of the stones be represented by $w_1, w_2, \ldots, w_{2009}$. Calculate the sum of all possible v... | [
"0"
] | [
"You are given 2009 stones, each with a positive integer weight. If any one stone is removed, the remaining stones can be split into two heaps with the same total weight, each containing exactly 1004 stones. Let the weights of the stones be represented by $w_1, w_2, \\ldots, w_{2009}$. Calculate the sum of all poss... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b94a9a4f0ee621e8f0ddc9ea8e53c7f300425caac068eee8ef384fd41d8bcfe3 | You are given 26 pieces of paper numbered 26, 25, 24, ..., 2, 1. You want to rearrange them in the order 1, 2, 3, ..., 25, 26 by switching only two adjacent pieces of paper at a time. Find the minimum number of switches required. | [
"325"
] | [
"You are given 26 pieces of paper numbered 26, 25, 24, ..., 2, 1. You want to rearrange them in the order 1, 2, 3, ..., 25, 26 by switching only two adjacent pieces of paper at a time. Find the minimum number of switches required."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-002a69646a13a697c8f2af738cc1bf6e90982e0cb19d9bbec2c8cae88ccfa6aa | You are given 5 red and 8 blue discs in a box. Peter, Ron, and Betty, in that order, select a disc without replacement. The first to select a red disc wins. What is the probability that Ron wins? | [
"0.3147"
] | [
"You are given 5 red and 8 blue discs in a box. Peter, Ron, and Betty, in that order, select a disc without replacement. The first to select a red disc wins. What is the probability that Ron wins?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-2d159e23b54d21fda80ddc7cd9c010f2383e13e351567b05aff0c571d3754307 | You are given 50 black tokens and 50 white tokens, which you are free to place in two identical containers in any way you wish. After distributing the tokens, you will be blindfolded and asked to choose one of the two containers and draw a single token from the chosen container. If you draw a white token, you will win ... | [
"\\(\\frac{74}{99}\\)"
] | [
"You are given 50 black tokens and 50 white tokens, which you are free to place in two identical containers in any way you wish. After distributing the tokens, you will be blindfolded and asked to choose one of the two containers and draw a single token from the chosen container. If you draw a white token, you will... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 6,
"tir": 18
} | 29 | 0 | 0 |
omr-ws-1b5f02d14d3fdfd9de7f545dabd8aee4d019b084e26087dd5b8a105c677751b7 | You are given a $100 \times 100$ grid and a 7-letter word, such as "Peanuts". Each letter in the word must be in a square on the board and must be in the correct order "Peanuts". A move is defined as randomly selecting a letter from the word "Peanuts" and rotating the entire word about that letter either $90$ degrees c... | [
"72944"
] | [
"You are given a $100 \\times 100$ grid and a 7-letter word, such as \"Peanuts\". Each letter in the word must be in a square on the board and must be in the correct order \"Peanuts\". A move is defined as randomly selecting a letter from the word \"Peanuts\" and rotating the entire word about that letter either $9... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-0fc199162edb1f68019e9b441cbd4e7bd94c13c8d53cf0500258e02e505d04b4 | You are given a bag with $n$ balls of $k$ different types. One of these types (say type $1$) is the desired type. The probability of drawing a ball of type $x$ is $\frac{|x|}{n}$, where $|x|$ is the number of balls of type $x$. In one turn, you draw a ball of type $x$, and all balls of type $x$ are removed from the bag... | [
"1 + \\sum_{x \\neq 1} \\frac{|x|}{|x| + |1|}"
] | [
"You are given a bag with $n$ balls of $k$ different types. One of these types (say type $1$) is the desired type. The probability of drawing a ball of type $x$ is $\\frac{|x|}{n}$, where $|x|$ is the number of balls of type $x$. In one turn, you draw a ball of type $x$, and all balls of type $x$ are removed from t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6d958ce0ce1a489376c8ad957dfd7f618bbedb548140d030e8564594f573f37e | You are given a bag with tiles numbered from $0$ to $9$ and a decimal point tile. The tiles are pulled from the bag one at a time and placed on the table to form a number. Compute the expected value of this number. | [
"505050505"
] | [
"You are given a bag with tiles numbered from $0$ to $9$ and a decimal point tile. The tiles are pulled from the bag one at a time and placed on the table to form a number. Compute the expected value of this number."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f002fddcf29619635fbea01b571254f925756dbd88c4037093583c2d5642ffaf | You are given a basket filled with balls marked with numbers. There are 2020 balls marked "1", 2019 balls marked "2", 2018 balls marked "3", and so on, until 1 ball marked "2020". What is the size of the largest set of balls in the basket such that no two are marked with numbers that sum to a multiple of 4? | [
"511062"
] | [
"You are given a basket filled with balls marked with numbers. There are 2020 balls marked \"1\", 2019 balls marked \"2\", 2018 balls marked \"3\", and so on, until 1 ball marked \"2020\". What is the size of the largest set of balls in the basket such that no two are marked with numbers that sum to a multiple of 4... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8fd15f58e5fe84b94a1dfc3b4f498664fe5ec61d3b92c2a35460a788f944c1a6 | You are given a circle in the xy-plane with a radius that is a positive integer at most $2008$. The origin lies somewhere inside the circle. You can ask up to sixty questions of the form "Is the point $(x,y)$ inside your circle?" After each question, you receive a truthful answer of "yes" or "no". Determine the radius ... | [
"2017036"
] | [
"You are given a circle in the xy-plane with a radius that is a positive integer at most $2008$. The origin lies somewhere inside the circle. You can ask up to sixty questions of the form \"Is the point $(x,y)$ inside your circle?\" After each question, you receive a truthful answer of \"yes\" or \"no\". Determine ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-11e1745bd65c507e15646d9e3da9f2822c1f1baf05d8a82449c3259cb186b425 | You are given a number, and round it to the nearest thousandth, then round this result to the nearest hundredth, and finally round this result to the nearest tenth. If the final result is $0.7$, what is the smallest number you could have been given? As is customary, $5$’s are always rounded up. Give the answer as a dec... | [
"0.6445"
] | [
"You are given a number, and round it to the nearest thousandth, then round this result to the nearest hundredth, and finally round this result to the nearest tenth. If the final result is $0.7$, what is the smallest number you could have been given? As is customary, $5$’s are always rounded up. Give the answer as ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a535f637832a2f7d6430ad27dc0558b81611a70b3df61ec159d5c58cba02b5e | You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2 \times 1$ and $3 \times 1$. The playing field should be seamlessly covered with such dominoes without overlapping, and no domino should protrude beyond the playing field. Furthermore, all dominoes must be aligned in t... | [
"257"
] | [
"You are given a rectangular playing field of size $13 \\times 2$ and any number of dominoes of sizes $2 \\times 1$ and $3 \\times 1$. The playing field should be seamlessly covered with such dominoes without overlapping, and no domino should protrude beyond the playing field. Furthermore, all dominoes must be alig... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 6,
"tir": 25
} | 78 | 0 | 0 |
omr-ws-dd188e4dc5b29f01d80e43f83818f6f664c58f84569055f1449b2daa9f35991a | You are given an $N \times N$ grid. Place a digit from $1$ to $N$ in each cell such that every row and every column contains the digits $1, 2, \ldots, N$ in some order. Each number in the grid represents the height of a building, and the clues on the outside of the grid indicate how many buildings can be "seen" when lo... | [
"\n\\begin{array}{ccccc}\n2 & 3 & 4 & 5 & 1 \\\\\n3 & 4 & 5 & 1 & 2 \\\\\n4 & 5 & 1 & 2 & 3 \\\\\n5 & 1 & 2 & 3 & 4 \\\\\n1 & 2 & 3 & 4 & 5 \\\\\n\\end{array}\n"
] | [
"You are given an $N \\times N$ grid. Place a digit from $1$ to $N$ in each cell such that every row and every column contains the digits $1, 2, \\ldots, N$ in some order. Each number in the grid represents the height of a building, and the clues on the outside of the grid indicate how many buildings can be \"seen\... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-83e946abb70107871b6384402455f338b3f23991b3a15d579b9b9a690c0e542c | You are given an $N \times N$ grid. Place a digit from $1$ to $N$ in each cell such that every row and every column contains the digits $1, 2, \ldots, N$ in some order. Each number in the grid represents the height of a building. The clues on the outside of the grid indicate how many buildings can be "seen" when lookin... | [
"1"
] | [
"You are given an $N \\times N$ grid. Place a digit from $1$ to $N$ in each cell such that every row and every column contains the digits $1, 2, \\ldots, N$ in some order. Each number in the grid represents the height of a building. The clues on the outside of the grid indicate how many buildings can be \"seen\" wh... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-71a882b84e9798642d47385dd748fd6d3198fdb5414b7d9d8a0a5b7f0e24d956 | You are given an unfair $n$-sided die, labeled from $1$ to $n$, with each side weighted according to its value. For example, the $3$ side is three times as likely to be rolled than is the $1$ side. What is the expected value of the sum of three rolls? | [
"$2n+1$"
] | [
"You are given an unfair $n$-sided die, labeled from $1$ to $n$, with each side weighted according to its value. For example, the $3$ side is three times as likely to be rolled than is the $1$ side. What is the expected value of the sum of three rolls?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-d308335406535ad6f36b720264812286ae661e2f9ef4e1175df5c27f9d0dfd9b | You are given eight identical $1 \times 1 \times 1$ cubes. Each cube has a pair of opposite faces colored red, a pair colored green, and a pair colored blue. You put the cubes in a bag, shake them up, and, blindfolded, assemble them into one $2 \times 2 \times 2$ cube. What is the probability that the big cube has each... | [
"\\(\\frac{3}{256}\\)"
] | [
"You are given eight identical $1 \\times 1 \\times 1$ cubes. Each cube has a pair of opposite faces colored red, a pair colored green, and a pair colored blue. You put the cubes in a bag, shake them up, and, blindfolded, assemble them into one $2 \\times 2 \\times 2$ cube. What is the probability that the big cube... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-234876e2bc1da8b08eb18ca87df09358f31afbd0048fa32c133b208285eb3a62 | You are given that $3^{400}\equiv 1\pmod{1000}$. What are the last three digits of $3^{12000}$? | [
"001"
] | [
"You are given that $3^{400}\\equiv 1\\pmod{1000}$. What are the last three digits of $3^{12000}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-592b8f2ccecaac8cc4a0bec9ea1f7871063c8408679fa31734ab7dbc41b0831b | You are given that $\log_{10} 2 \approx 0.3010$ and that the first (leftmost) two digits of $2^{1000}$ are 10. Compute the number of integers $n$ with $1000 \le n \le 2000$ such that $2^n$ starts with either the digit $8$ or $9$ (in base $10$). | [
"97"
] | [
"You are given that $\\log_{10} 2 \\approx 0.3010$ and that the first (leftmost) two digits of $2^{1000}$ are 10. Compute the number of integers $n$ with $1000 \\le n \\le 2000$ such that $2^n$ starts with either the digit $8$ or $9$ (in base $10$)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-1d5e8d9c49b0545f983ba910246d2b8deaa76f26df8871382ba1c1e0845ab9a2 | You are given that $x$ is directly proportional to $y^3$, and $y$ is inversely proportional to $\sqrt{z}$. If the value of $x$ is 3 when $z$ is $12$, what is the value of $x$ when $z$ is equal to $75$? Express your answer as a common fraction. | [
"\\frac{24}{125}"
] | [
"You are given that $x$ is directly proportional to $y^3$, and $y$ is inversely proportional to $\\sqrt{z}$. If the value of $x$ is 3 when $z$ is $12$, what is the value of $x$ when $z$ is equal to $75$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-760e7b547be2fbddd34e9d2da2de2e79b29ed9d280be16d1d53af162240c9076 | You are given the $2n$ integers from $1$ to $2n$. Randomly choose $n$ numbers to make set $A$, and the remaining numbers to make set $B$. Form an increasing sequence of the differences of each pairwise integers in $A \pmod{n}$. Do the same for set $B$. Let $S(n)$ be the sum of all elements in the resulting sequence for... | [
""
] | [
"You are given the $2n$ integers from $1$ to $2n$. Randomly choose $n$ numbers to make set $A$, and the remaining numbers to make set $B$. Form an increasing sequence of the differences of each pairwise integers in $A \\pmod{n}$. Do the same for set $B$. Let $S(n)$ be the sum of all elements in the resulting sequen... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-fab62fbd34dd6b41fcbcca78a7c05f7143f6e01593980a6a995ccbd1c8fd5bd5 | You are given three buckets with a capacity to hold $8$, $5$, and $3$ quarts of water, respectively. Initially, the first bucket is filled with $8$ quarts of water, while the remaining two buckets are empty. There are no markings on the buckets, so you are only allowed to empty a bucket into another one or to fill a bu... | [
"4"
] | [
"You are given three buckets with a capacity to hold $8$, $5$, and $3$ quarts of water, respectively. Initially, the first bucket is filled with $8$ quarts of water, while the remaining two buckets are empty. There are no markings on the buckets, so you are only allowed to empty a bucket into another one or to fill... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-5a1971dddabebda2517bb1b7fcb38600c77ede65d73095a7f64ae62741f31a44 | You are given two diameters $AB$ and $CD$ of circle $\omega$ with radius 1. A circle is drawn in one of the smaller sectors formed such that it is tangent to $AB$ at $E$, tangent to $CD$ at $F$, and tangent to $\omega$ at $P$. Lines $PE$ and $PF$ intersect $\omega$ at $X$ and $Y$. What is the length of $XY$, given that... | [
"\\(\\frac{4\\sqrt{2}}{3}\\)"
] | [
"You are given two diameters $AB$ and $CD$ of circle $\\omega$ with radius 1. A circle is drawn in one of the smaller sectors formed such that it is tangent to $AB$ at $E$, tangent to $CD$ at $F$, and tangent to $\\omega$ at $P$. Lines $PE$ and $PF$ intersect $\\omega$ at $X$ and $Y$. What is the length of $XY$, gi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 18
} | 35 | 0 | 0 |
omr-ws-e77590fc69f0ec191c7da60a199803f8a265420236e833a22ebf52dd4ab79905 | You are given two sealed envelopes, one of which contains twice as much money as the other. You choose an envelope and open it, at which point you can either keep the money inside or switch to the other envelope. What is the expected value of your winnings if you switch to the other envelope? | [
"\\(\\frac{5}{4}x\\)"
] | [
"You are given two sealed envelopes, one of which contains twice as much money as the other. You choose an envelope and open it, at which point you can either keep the money inside or switch to the other envelope. What is the expected value of your winnings if you switch to the other envelope?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 3
} | 11 | 0 | 0 |
omr-ws-bce16aea8dc81550c0fbfd12a8f8695dc7c7acbcd3acbf64f0cce9d94784349e | You are going to play 2 games of chess against an opponent who is equally likely to be a master, an intermediate, or a beginner. Your probability of winning any single game is 90%, 50%, and 30%, against a beginner, intermediate, and master, respectively. (a) What is the probability of winning the first game? (b) You wo... | [
"\\dfrac{9}{115}"
] | [
"You are going to play 2 games of chess against an opponent who is equally likely to be a master, an intermediate, or a beginner. Your probability of winning any single game is 90%, 50%, and 30%, against a beginner, intermediate, and master, respectively.\n(a) What is the probability of winning the first game?\n(b)... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f29b018fe6a034ad27c3888091031fd018161427a85fedc60fd0c5429bb7bf2e | You are going to play a board game four times in succession. Against opponent A, you win $\frac{2}{3}$ of the time, and against opponent B, you win $\frac{1}{3}$ of the time. Your goal is to win at least two games in a row. You can choose to play in the order A - A - B - A or B - B - A - B. Determine if there is an adv... | [
"A - A - B - A"
] | [
"You are going to play a board game four times in succession. Against opponent A, you win $\\frac{2}{3}$ of the time, and against opponent B, you win $\\frac{1}{3}$ of the time. Your goal is to win at least two games in a row. You can choose to play in the order A - A - B - A or B - B - A - B. Determine if there is... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-0e2b08fedf2d37725445c88a8235120c8f1d02ca32ce982e13f330a722fd2d22 | You are going to play a game. First, you roll a standard six-sided die. Then, using the number on the die, you toss that number of coins. Determine all the possible outcomes for this game. | [
"126"
] | [
"You are going to play a game. First, you roll a standard six-sided die. Then, using the number on the die, you toss that number of coins. Determine all the possible outcomes for this game."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-3a60b3dbd68eca7a340b1b63cd42bed3b1b16fe080d9b5b221e8ad1c28cd3c31 | You are going to play a game. First, you roll a standard six-sided die. Then, using the number on the die, you toss that number of coins. Suppose you win 1 dollar for each head that you toss. Determine the probability distribution for the amount of money won. | [
"\\dfrac{1}{384}"
] | [
"You are going to play a game. First, you roll a standard six-sided die. Then, using the number on the die, you toss that number of coins. Suppose you win 1 dollar for each head that you toss. Determine the probability distribution for the amount of money won."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-5bab9c7584430ea91a219828f521306ccd70ab87021656ac27aaa4a389bc9ba4 | You are in a cube-shaped room with each side measuring 5 inches. You are located 1 inch from the top and 2 inches from the side of one wall. The opening is on the opposite wall, 3 inches from the top and 1 inch from the side. You can crawl at a speed of 2 inches per second and can hold your breath for 5 more seconds. C... | [
"3"
] | [
"You are in a cube-shaped room with each side measuring 5 inches. You are located 1 inch from the top and 2 inches from the side of one wall. The opening is on the opposite wall, 3 inches from the top and 1 inch from the side. You can crawl at a speed of 2 inches per second and can hold your breath for 5 more secon... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b9a1da05789da7768c37d356d0dd4680ffc02a24433819a74f2bbe4c695f17c7 | You are in a room with $n$ supernatural guardians (cats), each speaking a unique language with two words. You cannot distinguish between the cats but can ask yes-or-no questions. Determine the minimum number of questions, $Q(n)$, needed to guarantee identifying which of two doors leads to heaven. Find $Q(5)$. | [
"3"
] | [
"You are in a room with $n$ supernatural guardians (cats), each speaking a unique language with two words. You cannot distinguish between the cats but can ask yes-or-no questions. Determine the minimum number of questions, $Q(n)$, needed to guarantee identifying which of two doors leads to heaven. Find $Q(5)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-ed43301408b57d16eaac0cf2ca42cc525a26e566bcb296ba787305c0fcd5aa86 | You are in a swimming pool 100 meters south of a line marked "Finish" that encircles the globe. Land is 25 meters to your right, where you can move at 2 meters per second, compared to 1 meter per second in water. What angle should you approach the land to reach the "Finish" line in the shortest amount of time? | [
"\\(30^\\circ\\)"
] | [
"You are in a swimming pool 100 meters south of a line marked \"Finish\" that encircles the globe. Land is 25 meters to your right, where you can move at 2 meters per second, compared to 1 meter per second in water. What angle should you approach the land to reach the \"Finish\" line in the shortest amount of time?... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 13
} | 18 | 0 | 0 |
omr-ws-1a932e8cda5c17f40bd359ac1de5a88d550b27bc23108827408578967f49e162 | You are in a swimming pool 100 meters south of a line marked "Finish" that encircles the globe. Land is 25 meters to your right, where you can move at 3 meters per second, compared to 1 meter per second in water. What angle should you approach the land to reach the "Finish" line in the shortest amount of time? | [
"\\(\\arctan\\left(\\frac{1}{2\\sqrt{2}}\\right) \\approx 19.471\\) degrees"
] | [
"You are in a swimming pool 100 meters south of a line marked \"Finish\" that encircles the globe. Land is 25 meters to your right, where you can move at 3 meters per second, compared to 1 meter per second in water. What angle should you approach the land to reach the \"Finish\" line in the shortest amount of time?... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 12
} | 13 | 0 | 0 |
omr-ws-2d7c590b99cfbb40b3a37b1b9512ae809110e3406475c3433870251d76b4d196 | You are in a tower with an infinite number of floors, each initially having 0 boxes. You can perform two operations: 1. For a floor $i$, place 2 boxes on floor $i+5$, 6 on floor $i+4$, 13 on floor $i+3$, 12 on floor $i+2$, 8 on floor $i+1$, and 1 on floor $i$, or remove the corresponding number of boxes from each floor... | [
"8190"
] | [
"You are in a tower with an infinite number of floors, each initially having 0 boxes. You can perform two operations:\n1. For a floor $i$, place 2 boxes on floor $i+5$, 6 on floor $i+4$, 13 on floor $i+3$, 12 on floor $i+2$, 8 on floor $i+1$, and 1 on floor $i$, or remove the corresponding number of boxes from each... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-fdd675d53151f3342c7a038b416bc1860c37865de21adfa3d9647df2f8a792dc | You are in an above-ground pool shaped like an inverted triangular prism with a height of 6 feet, length of 10 feet, and base of 3 feet. Water is leaking out of a hole at the base at a rate of 1 empty beer bottle (16 fluid ounces) every 10 seconds. At what rate (in feet per second) is the depth of the pool decreasing w... | [
"\\(-\\frac{1}{1800}\\)"
] | [
"You are in an above-ground pool shaped like an inverted triangular prism with a height of 6 feet, length of 10 feet, and base of 3 feet. Water is leaking out of a hole at the base at a rate of 1 empty beer bottle (16 fluid ounces) every 10 seconds. At what rate (in feet per second) is the depth of the pool decreas... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 4
} | 10 | 0 | 0 |
omr-ws-c33520d31faad0359fae841e1bec023d75f5850f7c149747839f1271138994d8 | You are in line to get into a club. The bouncers admit $\lfloor \frac{n}{2} \rfloor$ people every 5 minutes, and then 5 more people join the line. There are 99 people in line when you join (right after letting some people in, and no one else joins the line at this moment). How many minutes will you have to wait to get ... | [
"20"
] | [
"You are in line to get into a club. The bouncers admit $\\lfloor \\frac{n}{2} \\rfloor$ people every 5 minutes, and then 5 more people join the line. There are 99 people in line when you join (right after letting some people in, and no one else joins the line at this moment). How many minutes will you have to wait... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-41d76e2471e90aa697f1256c30c1fc6a35883de7eb53e7e852681061d7fa6590 | You are in the middle of a circular swimming pool with radius \( r \). Your dad is running around the outside of the pool at a speed of \( m \). What is the minimum swimming speed you need to have to reach the edge of the pool before your dad can catch you? | [
"\\( a \\approx 0.217233628211221 \\)"
] | [
"You are in the middle of a circular swimming pool with radius \\( r \\). Your dad is running around the outside of the pool at a speed of \\( m \\). What is the minimum swimming speed you need to have to reach the edge of the pool before your dad can catch you?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-00e6025b2f5796e466ae20b329e38a60f546b53924b419dd13718f40e9741b87 | You are initially standing at $P_0=5+12i$ on the complex plane. To walk to the origin, you walk in a straight line to $P_1$, then in a straight line to $P_2$, and so on, where for each $n \geq 1$, $P_n$ satisfies the recurrence relation $P_n = \left(\frac{1}{2} + \frac{1}{2}i\right)P_{n-1}$. What is the total distance ... | [
"\\( 13(\\sqrt{2} + 1) \\)"
] | [
"You are initially standing at $P_0=5+12i$ on the complex plane. To walk to the origin, you walk in a straight line to $P_1$, then in a straight line to $P_2$, and so on, where for each $n \\geq 1$, $P_n$ satisfies the recurrence relation $P_n = \\left(\\frac{1}{2} + \\frac{1}{2}i\\right)P_{n-1}$. What is the total... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-77bebb1ea772cc542e2dbc69165e5a54e35e6d9e19f9516d90e182a08c74dac6 | You are lost in a forest shaped like a strip with a width of 1 unit length and infinite length. What is the least distance you must travel to ensure that you are out of the forest? Ignore any objects that might block your path. | [
"2.278"
] | [
"You are lost in a forest shaped like a strip with a width of 1 unit length and infinite length. What is the least distance you must travel to ensure that you are out of the forest? Ignore any objects that might block your path."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d24097ff1a7f252f951e321ef58ffec38070e28915073994ab1969a795cce46a | You are next to a river with 2024 wolverines labeled from 1 to 2024, inclusive. They are all initially on the left side of the river. You spend the next 1012 days moving the wolverines back and forth across the river. On the first day, you take all the wolverines to the right side of the river. On the second day, you t... | [
"75"
] | [
"You are next to a river with 2024 wolverines labeled from 1 to 2024, inclusive. They are all initially on the left side of the river. You spend the next 1012 days moving the wolverines back and forth across the river. On the first day, you take all the wolverines to the right side of the river. On the second day, ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-4bc751e3ab5f9ab255fb16bbe4137ccebd446f212b045fe1d9a3228b34108a2e | You are offered two jobs. The first pays $30,000 the first year with annual raises of $2,000 each year. The second offers a starting salary of $27,000 and raises of 8% each year. Over a ten-year period, how much would you earn from (a) the first job? (b) the second job? | [
"(a) $390,000, (b) $391,137.19"
] | [
"You are offered two jobs. The first pays $30,000 the first year with annual raises of $2,000 each year. The second offers a starting salary of $27,000 and raises of 8% each year. Over a ten-year period, how much would you earn from (a) the first job? (b) the second job?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-bf3c2fc696427fd081530432e09758f53d551d1d203902b360f132b364279d47 | You are on a 1000-rung escape ladder, starting at the first rung. Every second, a standard 6-sided die is rolled. If the roll is 1 or 2, you move down one rung; if the roll is 3, 4, 5, or 6, you move up one rung. If you move down from the first rung, you fall into the fire. If you move up from the 1000th rung, you esca... | [
"\\(\\frac{2^{1000}}{2^{1001} - 1}\\)"
] | [
"You are on a 1000-rung escape ladder, starting at the first rung. Every second, a standard 6-sided die is rolled. If the roll is 1 or 2, you move down one rung; if the roll is 3, 4, 5, or 6, you move up one rung. If you move down from the first rung, you fall into the fire. If you move up from the 1000th rung, you... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 5,
"tir": 3
} | 13 | 0 | 0 |
omr-ws-aab4d628dcaa79fb369927c8915a2f8fc6eb3220a12aee59d1c27f8478e77ba3 | You are on a 2D grid starting at point $(0,0)$ and need to reach point $(10,0)$. From any point $(x,y)$, you can move to $(x+1,y+1)$, $(x+1,y)$, or $(x+1,y-1)$, provided the new point is within the grid. How many different paths exist from $(0,0)$ to $(10,0)$? | [
"8953"
] | [
"You are on a 2D grid starting at point $(0,0)$ and need to reach point $(10,0)$. From any point $(x,y)$, you can move to $(x+1,y+1)$, $(x+1,y)$, or $(x+1,y-1)$, provided the new point is within the grid. How many different paths exist from $(0,0)$ to $(10,0)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-86bc7c776308883e7cea7fd80a95da80f15fd8edd36a932796df8b423f101192 | You are on a 2D grid starting at point (0,0) and need to reach point (10,0). From any point (x,y), you can move to (x+1,y+1), (x+1,y), or (x+1,y-1) as long as the destination point is within the grid. How many different paths exist from (0,0) to (10,0)? | [
"3363"
] | [
"You are on a 2D grid starting at point (0,0) and need to reach point (10,0). From any point (x,y), you can move to (x+1,y+1), (x+1,y), or (x+1,y-1) as long as the destination point is within the grid. How many different paths exist from (0,0) to (10,0)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-02466f05f1a542915e288c6c6304f93161d164801bd5db6d42f48ab639223be0 | You are on a flat planet with 100 cities at points \(x = 1, \ldots, 100\) along the line \(y = -1\) and another 100 cities at points \(x = 1, \ldots, 100\) along the line \(y = 1\). There are archways between all pairs of cities with different \(y\) coordinates. The effort to cross an archway equals the square of the d... | [
"397"
] | [
"You are on a flat planet with 100 cities at points \\(x = 1, \\ldots, 100\\) along the line \\(y = -1\\) and another 100 cities at points \\(x = 1, \\ldots, 100\\) along the line \\(y = 1\\). There are archways between all pairs of cities with different \\(y\\) coordinates. The effort to cross an archway equals th... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-68f590354bac67d07d0ca82cad7b2d43c5a3e204fcb275b5a8b0881dacceb244 | You are on a pogo stick on the coordinate plane and can jump from $(x,y)$ to one of the points $(x,y+2x)$, $(x,y-2x)$, $(x+2y,y)$, or $(x-2y,y)$. When you leave a point, you cannot return to it on the next jump. Suppose that you start at $(1,\sqrt{2})$. Let $N$ be the number of distinct points you can visit after exact... | [
"324"
] | [
"You are on a pogo stick on the coordinate plane and can jump from $(x,y)$ to one of the points $(x,y+2x)$, $(x,y-2x)$, $(x+2y,y)$, or $(x-2y,y)$. When you leave a point, you cannot return to it on the next jump. Suppose that you start at $(1,\\sqrt{2})$. Let $N$ be the number of distinct points you can visit after... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1193423f22620d08f09a012ec4b77237ce15d1bf7d88247035c2de98f5081ee8 | You are on a train moving at 80 km/h. An oncoming train of length 150 meters passes by the window in 3 seconds. How fast, in km/h, is the other train going? | [
"100"
] | [
"You are on a train moving at 80 km/h. An oncoming train of length 150 meters passes by the window in 3 seconds. How fast, in km/h, is the other train going?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-7acceb9858ebd73221d0e5d4a35c33a154a9c501a81eef04c8a7d3c8616ca998 | You are on an integer on the number line and, with equal probability, will jump to a number on the right or left of you. Find \( k \), the expected number of moves it will take for you to reach the number to your right, given that you can take an infinite number of jumps. | [
"\\( \\infty \\)"
] | [
"You are on an integer on the number line and, with equal probability, will jump to a number on the right or left of you. Find \\( k \\), the expected number of moves it will take for you to reach the number to your right, given that you can take an infinite number of jumps."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-775f827ffa7348438cbf6ca092dff0347138472f741b4ac7affc1e1f8037ca69 | You are on rock 1 and need to get to rock 12 by jumping from rock to rock. You can either jump from rock $n$ to $n + 1$ or from rock $n$ to $n + 2$, but you cannot jump from rock $n$ to $n + 2$ twice in a row. How many different sequences of jumps will take you to rock 12? | [
"60"
] | [
"You are on rock 1 and need to get to rock 12 by jumping from rock to rock. You can either jump from rock $n$ to $n + 1$ or from rock $n$ to $n + 2$, but you cannot jump from rock $n$ to $n + 2$ twice in a row. How many different sequences of jumps will take you to rock 12?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-d7fb343a723027aab015bad40cfd7e3415609ce2c976d2b1ce64dd96deadad0c | You are on the first rung of a 1000-rung ladder. If you roll a 1 or 2, you move down one rung. If you roll a 3, 4, 5, or 6, you move up one rung. If you fall off the first rung, you die, but if you move up from the 1000th rung, you escape. What is the probability that you escape? | [
"\\(\\frac{2^{1000}}{2^{1001}-1}\\)"
] | [
"You are on the first rung of a 1000-rung ladder. If you roll a 1 or 2, you move down one rung. If you roll a 3, 4, 5, or 6, you move up one rung. If you fall off the first rung, you die, but if you move up from the 1000th rung, you escape. What is the probability that you escape?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 7
} | 24 | 0 | 0 |
omr-ws-1ebeb7af0885296defd448762702c7aa17fae5fb5a2d42b51f29ad91da8364dd | You are on the ground floor of a 100-floor building and play a game where you roll a 6-faced die. The rules are: - If the result is 1 or 2, you go down 1 floor (but not below the ground floor). - If the result is 3, 4, or 5, you go up 1 floor. - If the result is 6, you roll again and go up the number of floors equal to... | [
"0.8259"
] | [
"You are on the ground floor of a 100-floor building and play a game where you roll a 6-faced die. The rules are:\n- If the result is 1 or 2, you go down 1 floor (but not below the ground floor).\n- If the result is 3, 4, or 5, you go up 1 floor.\n- If the result is 6, you roll again and go up the number of floors ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0f67fbbf92cdca717f9cc26a60ea19609ca688876d628b2b8af1c593fe4c9fee | You are on the ground floor of a 100-floor building and play a game where you roll a 6-faced die. The rules are: - If the result is 1 or 2, you go down 1 floor. - If the result is 3, 4, or 5, you go up 1 floor. - If the result is 6, you roll again and go up the number of floors equal to the second roll. - There is a 0.... | [
"0.6"
] | [
"You are on the ground floor of a 100-floor building and play a game where you roll a 6-faced die. The rules are:\n- If the result is 1 or 2, you go down 1 floor.\n- If the result is 3, 4, or 5, you go up 1 floor.\n- If the result is 6, you roll again and go up the number of floors equal to the second roll.\n- Ther... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-9d52cb3eb1dc4cc094c0ac38fec1c97f7f00aca5f73def5e47af8f953bb9c2ae | You are playing a game in which you flip 3 fair coins. It costs \$1 to play the game, which must be subtracted from your winnings. If all coins show the same (all heads or all tails), you win \$5, otherwise you lose your \$1. Calculate the expected value for the game. | [
"0.25"
] | [
"You are playing a game in which you flip 3 fair coins. It costs \\$1 to play the game, which must be subtracted from your winnings. If all coins show the same (all heads or all tails), you win \\$5, otherwise you lose your \\$1. Calculate the expected value for the game."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bf338fb3eeeb9986d98be8c6b65a59c8dc8cb95ff069effec28a8d166da3bd18 | You are playing a game on the number line. At the beginning of the game, every real number on \([0, 4)\) is uncovered, and the rest are covered. A turn consists of picking a real number \(r\) such that, for all \(x\) where \(r \le x < r + 1\), \(x\) is uncovered. The turn ends by covering all such \(x\). At the beginni... | [
"\\dfrac{11 - 4 \\ln 2}{3}"
] | [
"You are playing a game on the number line. At the beginning of the game, every real number on \\([0, 4)\\) is uncovered, and the rest are covered. A turn consists of picking a real number \\(r\\) such that, for all \\(x\\) where \\(r \\le x < r + 1\\), \\(x\\) is uncovered. The turn ends by covering all such \\(x\... | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-c6eba21eb7bd590e079436231c6b2407b5efcf64a6f5053d083bbad41c623543 | You are playing a game where every $a$ seconds, where $0 < a < 1$, you have a chance to win a prize with probability $p$. The probability $p$ increases by $0.03$ every second until it reaches $1$, at which point you are guaranteed to win a prize on the next chance. Once you win the prize, $p$ resets to the original val... | [
"\\(\\frac{1 - p}{0.03a}\\)"
] | [
"You are playing a game where every $a$ seconds, where $0 < a < 1$, you have a chance to win a prize with probability $p$. The probability $p$ increases by $0.03$ every second until it reaches $1$, at which point you are guaranteed to win a prize on the next chance. Once you win the prize, $p$ resets to the origina... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc7690944e1c7b4463cc9c8f829c1e988289f183f035839a25e61b9f9d4f5845 | You are playing a game where there is a pile of money on the table, starting with a single dollar. On each turn, you can either take the money and end the game or roll a die. If you roll a 6, the game ends with no money. Otherwise, a dollar is added to the pile and the game continues. What is the optimal strategy for t... | [
"Roll the die 4 or 5 times before taking the money."
] | [
"You are playing a game where there is a pile of money on the table, starting with a single dollar. On each turn, you can either take the money and end the game or roll a die. If you roll a 6, the game ends with no money. Otherwise, a dollar is added to the pile and the game continues. What is the optimal strategy ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-b1bc86e565de7ee2c959bc029af6e8191f46a23e15322fb7a5e636cbfd4de621 | You are playing a game where you roll five fair six-sided dice. Each die has three sides with a picture of a car and three sides without. You roll all five dice. The ones that show a car are put aside, and you roll the remaining dice again. Any that show a car are put aside, and if there are any other dice, you roll th... | [
"\\(\\frac{16807}{32768}\\)"
] | [
"You are playing a game where you roll five fair six-sided dice. Each die has three sides with a picture of a car and three sides without. You roll all five dice. The ones that show a car are put aside, and you roll the remaining dice again. Any that show a car are put aside, and if there are any other dice, you ro... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-c30fc1795e2f6ea902cbb15efefcaf001b42c03688563b6b31a4a88ed16a5e71 | You are playing a game with an $n$-sided fair die. After your first roll, you can either get paid the face value of the roll or pay one dollar and roll again. You can continue to do this until you decide to stop. What is the expected value of this game? | [
"\\frac{n+1}{2}"
] | [
"You are playing a game with an $n$-sided fair die. After your first roll, you can either get paid the face value of the roll or pay one dollar and roll again. You can continue to do this until you decide to stop. What is the expected value of this game?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-9dd032200fb859bcfef8c694bd9e2caa07a0eaaecd08f24d458f84b7b6fa95c8 | You are playing a game with coins with your friends Alice and Bob. When all three of you flip your respective coins, the majority side wins. Alice makes a biased coin that lands on heads with probability $p$, and Bob makes a biased coin that lands on heads with probability $1 - p$. You design your own biased coin to ma... | [
"\\dfrac{3}{4}"
] | [
"You are playing a game with coins with your friends Alice and Bob. When all three of you flip your respective coins, the majority side wins. Alice makes a biased coin that lands on heads with probability $p$, and Bob makes a biased coin that lands on heads with probability $1 - p$. You design your own biased coin ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4413418ecf7cf46c32084cfc77c819fdbd14422ca480b2e1a938c52b3068c012 | You are playing a match against an opponent in which at each point either you serve or your opponent does. If you serve, you win the point with probability $p_1$; if your opponent serves, you win the point with probability $p_2$. Consider two possible conventions for serving: (i) Serves alternate. (ii) The player servi... | [
"\\dfrac{80}{19683}"
] | [
"You are playing a match against an opponent in which at each point either you serve or your opponent does. If you serve, you win the point with probability $p_1$; if your opponent serves, you win the point with probability $p_2$. Consider two possible conventions for serving:\n(i) Serves alternate.\n(ii) The playe... | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
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