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-b683442779510716d73beba1d8c6dcb0b6847b48ce156a9c1c0df6f50c3a04d2 | You have four marbles. Three have a radius of 1, and the other has a radius of 2. You form a pyramid by placing the three marbles of radius 1 tangent to each other and setting the radius 2 marble on top. How tall is the figure? | [
"\\( 3 + \\sqrt{\\frac{23}{3}} \\)"
] | [
"You have four marbles. Three have a radius of 1, and the other has a radius of 2. You form a pyramid by placing the three marbles of radius 1 tangent to each other and setting the radius 2 marble on top. How tall is the figure?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 31
} | 79 | 0 | 0 |
omr-ws-2fdefb1b2100d07ef0e51bb383cdf5ebe22d702c2a6f65c9ecbd1b38c144ccbe | You have many $1 \times 1$ squares. You may color their edges with one of four colors and glue them together along edges of the same color. Your aim is to get an $m \times n$ rectangle. For which values of $m$ and $n$ is this possible? | [
"Any \\( m \\) and \\( n \\)"
] | [
"You have many $1 \\times 1$ squares. You may color their edges with one of four colors and glue them together along edges of the same color. Your aim is to get an $m \\times n$ rectangle. For which values of $m$ and $n$ is this possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 19
} | 67 | 0 | 0 |
omr-ws-00a0caeb530b3a40d194a45935da47b8e6319d62715d84f4301a881615801455 | You have many 1.1 squares. You may color their edges with one of four colors and glue them together along edges of the same color. Your aim is to get an m x n rectangle. For which values of m and n is this possible? | [
"\\( m \\) and \\( n \\) have the same parity."
] | [
"You have many 1.1 squares. You may color their edges with one of four colors and glue them together along edges of the same color. Your aim is to get an m x n rectangle. For which values of m and n is this possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-33ac91e92901be372f848b6a423a2fca602c76dc2ce54b5c0c420388667ebb06 | You have one fair coin and one biased coin with $\text{Prob(heads)} = \frac{2}{3}$. One of the coins is tossed once, resulting in heads. The other one is tossed three times, resulting in two heads and one tail. Find the probability of the more likely event. | [
"\\dfrac{9}{17}"
] | [
"You have one fair coin and one biased coin with $\\text{Prob(heads)} = \\frac{2}{3}$. One of the coins is tossed once, resulting in heads. The other one is tossed three times, resulting in two heads and one tail. Find the probability of the more likely event."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4746a95a92de74a6004bb80178e116fe93dec99df55f641967522c25f983764e | You have received a $2,000 invoice from the Acme Swamp Water Company with terms 3/10, net 40. If you fail to take advantage of the discount, what interest rate are you effectively paying? | [
"37.11%",
"37.63\\%"
] | [
"You have received a $2,000 invoice from the Acme Swamp Water Company with terms 3/10, net 40. If you fail to take advantage of the discount, what interest rate are you effectively paying?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
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"tir": 8
} | 25 | 0 | 0 |
omr-ws-f03b21a70ddb5eeb65a2022272c782b031e1b3727c1c457f959f9f3e26ae324e | You have six dice. Two dice have the configuration of 2 sides blue and 4 sides red, while the other four dice have the configuration of 2 sides red and 4 sides blue. A die is chosen at random and rolled six times, resulting in 4 blue and 2 red outcomes. What are the odds that the chosen die had the configuration of 2 s... | [
"\\(\\frac{1}{8}\\)"
] | [
"You have six dice. Two dice have the configuration of 2 sides blue and 4 sides red, while the other four dice have the configuration of 2 sides red and 4 sides blue. A die is chosen at random and rolled six times, resulting in 4 blue and 2 red outcomes. What are the odds that the chosen die had the configuration o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95e56c211ae5b4889303c57a9815d94e53b3f4b151fabcc5de3e5173852cbd40 | You have the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. If 1 and 10 cannot be at the ends, 5 cannot be in the middle, and 8 must be next to 4, how many ways can you arrange these numbers? | [
"465600"
] | [
"You have the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. If 1 and 10 cannot be at the ends, 5 cannot be in the middle, and 8 must be next to 4, how many ways can you arrange these numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-785de925fa4490b604443cccbe5674d10eeeca0e7b150cbb2cd8fb8fa7658289 | You have three bowls, each containing 6 balls. A move consists of selecting a random bowl and then a different random bowl and moving a ball from the first bowl to the second. What is the probability that after 5 such moves, all bowls will have 6 balls again? | [
"\\(\\frac{5}{108}\\)"
] | [
"You have three bowls, each containing 6 balls. A move consists of selecting a random bowl and then a different random bowl and moving a ball from the first bowl to the second. What is the probability that after 5 such moves, all bowls will have 6 balls again?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-d05058617e059c4d479665b804f2be26ffb816a75fbe44e52925d684de52338a | You have three boxes, each containing some coins. At each step, you choose two boxes with A and B coins, where A is greater than or equal to B, and transfer B coins from the box with A coins to the box with B coins (so now the boxes hold A-B and 2B coins). Given that the initial number of coins in the boxes are 100, 20... | [
"2"
] | [
"You have three boxes, each containing some coins. At each step, you choose two boxes with A and B coins, where A is greater than or equal to B, and transfer B coins from the box with A coins to the box with B coins (so now the boxes hold A-B and 2B coins). Given that the initial number of coins in the boxes are 10... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a738c55c9fe598598405aba8e0c8f63a9eed9277700301756e4f7a85bccc0c1d | You have three coins A, B, and C. Coin A has a 40% chance of getting heads (you lose) and a 60% chance of getting tails (you win). If you get heads on A, you flip A again. If you get tails on A, you flip B. Coin B has a 60% chance of getting heads (you lose) and a 40% chance of getting tails (you win). If you get heads... | [
"0.062"
] | [
"You have three coins A, B, and C. Coin A has a 40% chance of getting heads (you lose) and a 60% chance of getting tails (you win). If you get heads on A, you flip A again. If you get tails on A, you flip B. Coin B has a 60% chance of getting heads (you lose) and a 40% chance of getting tails (you win). If you get ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-61999f9202e28766f7edf5787d0373fda3a5ce6564ca4b130f9714289d558cb8 | You have three shirts and four pairs of pants. How many outfits consisting of one shirt and one pair of pants can you make? | [
"12"
] | [
"You have three shirts and four pairs of pants. How many outfits consisting of one shirt and one pair of pants can you make?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1d12f300859b81750e00ae9fd3ff807322de5ddd86630a9e189522cb2973f608 | You have to guess a number chosen from the set $\{0,1,2,\cdots,15\}$. You can ask 7 questions, each of which will be answered with "YES" or "NO". The responder can lie exactly once. Determine the chosen number using these questions. | [
"7"
] | [
"You have to guess a number chosen from the set $\\{0,1,2,\\cdots,15\\}$. You can ask 7 questions, each of which will be answered with \"YES\" or \"NO\". The responder can lie exactly once. Determine the chosen number using these questions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-f9467a01458dcc7b8f9260857af0d78f9b951ccb811240e8261bf873c30153cb | You have to pick one door out of three. Behind one door is a treasure, and behind the other two doors, there is nothing of value. After you pick a door, one of the remaining doors with no treasure is opened. You are then given the option to switch your choice to the other unopened door. What is the probability of winni... | [
"Probability of winning if you switch is \\( \\frac{2}{3} \\)."
] | [
"You have to pick one door out of three. Behind one door is a treasure, and behind the other two doors, there is nothing of value. After you pick a door, one of the remaining doors with no treasure is opened. You are then given the option to switch your choice to the other unopened door. What is the probability of ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 26
} | 26 | 0 | 0 |
omr-ws-416880275f174b57e70acaff96f65297fb1f71fa8e5edf6af4361929765ff300 | You have twelve one-foot tiles to tile a three-foot by four-foot stoop. If nine tiles are white, two are black, and one is red, how many distinguishable tilings are possible? | [
"660"
] | [
"You have twelve one-foot tiles to tile a three-foot by four-foot stoop. If nine tiles are white, two are black, and one is red, how many distinguishable tilings are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 16
} | 27 | 0 | 0 |
omr-ws-ba43236000d17896395e79a1b3ab25ae8a4440767c65261832e0673a8c9eb7ba | You have two blackboards $A$ and $B$. Write some integers from $2$ to $20$ on them such that each number on blackboard $A$ is co-prime with each number on blackboard $B$. Determine the maximum possible value of multiplying the number of numbers written in $A$ by the number of numbers written in $B$. | [
"65"
] | [
"You have two blackboards $A$ and $B$. Write some integers from $2$ to $20$ on them such that each number on blackboard $A$ is co-prime with each number on blackboard $B$. Determine the maximum possible value of multiplying the number of numbers written in $A$ by the number of numbers written in $B$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 19
} | 40 | 0 | 0 |
omr-ws-b2549118616e91eafe088df2cf14b2477b690ea5e64276ad32e81a8b3120155e | You have two blue socks, two red socks, and two yellow socks. On Monday, you randomly choose two socks. On Tuesday, you randomly choose two of the remaining socks. On Wednesday, you randomly choose two of the remaining socks. What is the probability that you didn't wear a matching pair on any of the three days? | [
"\\(\\frac{8}{15}\\)"
] | [
"You have two blue socks, two red socks, and two yellow socks. On Monday, you randomly choose two socks. On Tuesday, you randomly choose two of the remaining socks. On Wednesday, you randomly choose two of the remaining socks. What is the probability that you didn't wear a matching pair on any of the three days?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-b6ff382e79c1b8d391a0b5fc6bf5dc5389b5ae79ba432e011dd9eecddd583c20 | You have two bombs, each of which does \( \frac{100}{r^2} \) damage to each point in a 2D surface, where \( r \) is the distance from the point of impact. A point is "destroyed" when it takes 100 damage. Determine the maximum area that can be destroyed by placing the two bombs optimally. | [
"8.05"
] | [
"You have two bombs, each of which does \\( \\frac{100}{r^2} \\) damage to each point in a 2D surface, where \\( r \\) is the distance from the point of impact. A point is \"destroyed\" when it takes 100 damage. Determine the maximum area that can be destroyed by placing the two bombs optimally."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-82ffacad71e4308a2c94c54ac37290fa1a27877ee3bc5484e5b524116dd1f14a | You have two boxes and cards numbered 1-100. One by one, you place each card into one of the two boxes, chosen independently and uniformly at random. Find the expected value of the smallest card in the box containing card 100. | [
"2 - \\frac{1}{2^{99}}"
] | [
"You have two boxes and cards numbered 1-100. One by one, you place each card into one of the two boxes, chosen independently and uniformly at random. Find the expected value of the smallest card in the box containing card 100."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9036cfa4e219cfa041132cab5708689240d07a681ae0f219c9b004ee2ccf9c7c | You have two decks of 26 cards each, with each card having a different letter of the alphabet. You pick one card from each deck. A vowel is worth 3 points and a consonant is worth 0 points. Let \( X \) be the sum of the values of the two cards picked. Find \( E(X) \), \( V(X) \), and the standard deviation of \( X \). | [
"\\( E(X) = \\frac{174}{169} \\), \\( V(X) = \\frac{85320}{28561} \\), \\( \\sigma(X) = \\sqrt{\\frac{85320}{28561}} \\)",
"\\dfrac{3\\sqrt{210}}{26}"
] | [
"You have two decks of 26 cards each, with each card having a different letter of the alphabet. You pick one card from each deck. A vowel is worth 3 points and a consonant is worth 0 points. Let \\( X \\) be the sum of the values of the two cards picked. Find \\( E(X) \\), \\( V(X) \\), and the standard deviation o... | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-bd71c9625b38dca4c9c7854b548f85e4d4184c4164a3171478285f620882929d | You have two fair 6-sided dice and roll them 2 times. Find the probability of getting a sum of 6 exactly once. | [
"\\(\\frac{335}{1296}\\)"
] | [
"You have two fair 6-sided dice and roll them 2 times. Find the probability of getting a sum of 6 exactly once."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2b6623e51c9d0e1c62976a96f0e4bf3b68ad2495a82795ccd57b3f6482d48746 | You have two opponents, A and B, with whom you alternate play. The probability of winning against A is \( P_a \) and against B is \( P_b \), where \( P_b > P_a \). Your objective is to minimize the expected number of games you need to play to win two in a row. Should you start with A or B? | [
"A"
] | [
"You have two opponents, A and B, with whom you alternate play. The probability of winning against A is \\( P_a \\) and against B is \\( P_b \\), where \\( P_b > P_a \\). Your objective is to minimize the expected number of games you need to play to win two in a row. Should you start with A or B?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-31f67fb54b2fb7ed6b87dda0edbdc8246d6fb3decb437845ff25c5eb7edc8f06 | You have two urns, each containing both black and white balls. The probability of drawing a white ball from the first urn is $\frac{1}{3}$ and from the second urn is $\frac{1}{2}$. You start by choosing a ball from the first urn with probability $\frac{1}{4}$ and from the second urn with probability $\frac{3}{4}$. If y... | [
"\\dfrac{2221}{5184}"
] | [
"You have two urns, each containing both black and white balls. The probability of drawing a white ball from the first urn is $\\frac{1}{3}$ and from the second urn is $\\frac{1}{2}$. You start by choosing a ball from the first urn with probability $\\frac{1}{4}$ and from the second urn with probability $\\frac{3}{... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4b2db0f04016f631598e52db2063b2cb4a043c62c4db2cccad20045928b9fd12 | You have two watering cans. One holds five litres and the other three. How can you measure exactly one litre? | [
"1 litre"
] | [
"You have two watering cans. One holds five litres and the other three. How can you measure exactly one litre?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-05ecf7ceadd5bbbb2124fe97ada742c894d07bb7af08e212d22e4b0f878d0d2d | You hit randomly at a triangular dashboard with sides in geometric progression with a common ratio of 1.1. You win when the ratio of the distance between the point of strike and the orthocenter to the distance between the point of strike and the incenter is less than 1.1. What is your probability of winning? | [
"\\dfrac{1}{2}"
] | [
"You hit randomly at a triangular dashboard with sides in geometric progression with a common ratio of 1.1. You win when the ratio of the distance between the point of strike and the orthocenter to the distance between the point of strike and the incenter is less than 1.1. What is your probability of winning?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-88838de22a7cf5ddf519907dbddc5aa90a1763727e3c206fe27a780aab40d309 | You lay out a $100$ m rope flat on the ground in a straight line, anchored at both ends. You then add $1$ m of slack, making the rope $101$ m long, but still anchored at both ends. How high can the rope be lifted at the $50$ m mark? | [
"\\(\\frac{\\sqrt{201}}{2}\\)"
] | [
"You lay out a $100$ m rope flat on the ground in a straight line, anchored at both ends. You then add $1$ m of slack, making the rope $101$ m long, but still anchored at both ends. How high can the rope be lifted at the $50$ m mark?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-3454b616a87e728f2baa597116957b9557c54a2bea16f41bc047756f4f124873 | You make all the six-digit numbers you can with the digits 1, 2, 3, 4, 5, 6, 7, 7 using each digit at most the number of times it is in the list. Let \( x \) denote the median of the list of six-digit numbers. Let \( y \) denote the 35th number of the list of six-digit numbers when they are written in increasing order.... | [
"0"
] | [
"You make all the six-digit numbers you can with the digits 1, 2, 3, 4, 5, 6, 7, 7 using each digit at most the number of times it is in the list. Let \\( x \\) denote the median of the list of six-digit numbers. Let \\( y \\) denote the 35th number of the list of six-digit numbers when they are written in increasi... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-0a303854c4af0f76af6adfac370a46573c028e0a340e3b3e6b95a72053e63a10 | You may choose any five-digit number with distinct digits (excluding 0). You can switch the positions of two adjacent digits up to five times. What is the largest possible difference between the initial number and the final number obtained after these switches? | [
"85230"
] | [
"You may choose any five-digit number with distinct digits (excluding 0). You can switch the positions of two adjacent digits up to five times. What is the largest possible difference between the initial number and the final number obtained after these switches?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 7,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-a63e5c7c9016ec367dcb6d85db0f0012b761bde1987b5233278dad7d2e5de02a | You need to design a rectangular garden along an existing wall using 20 sections of 1-meter long fence pieces. How many sections should be used along the width and the length of the garden to maximize the area? | [
"10 sections along the length (parallel to the wall) and 5 sections along the width."
] | [
"You need to design a rectangular garden along an existing wall using 20 sections of 1-meter long fence pieces. How many sections should be used along the width and the length of the garden to maximize the area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-410784ff47df5036bbdb2ee4c0026c4b1bbea8dd33879ead68cf6218e5cee457 | You need to run a total of 265 km. Initially, your speed is 1 km/s. You can choose to wait for 1 second to double your speed, but you must run at least 1 km between each doubling. What is the minimum time required to complete the 265 km run? | [
"\\(8 + \\frac{767}{256}\\)"
] | [
"You need to run a total of 265 km. Initially, your speed is 1 km/s. You can choose to wait for 1 second to double your speed, but you must run at least 1 km between each doubling. What is the minimum time required to complete the 265 km run?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 18,
"genselect": 7,
"tir": 23
} | 48 | 0 | 0 |
omr-ws-5fc89bd8c47a98e8e609a38423c5a6174a439421bd541b03678441436e563c69 | You own 100 cats. Each cat returns to the house with probability \( \frac{p}{k} \), where \( p \) is a constant and \( k \) is the number of cats currently in the house (if there are 0 cats, the probability is \( p \)). What is the expected value of the number of cats in the house after asking all 100 cats to return? | [
"10\\sqrt{p}"
] | [
"You own 100 cats. Each cat returns to the house with probability \\( \\frac{p}{k} \\), where \\( p \\) is a constant and \\( k \\) is the number of cats currently in the house (if there are 0 cats, the probability is \\( p \\)). What is the expected value of the number of cats in the house after asking all 100 cat... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-49e3a28012b3758cfa34bfaa0daa2c4a364e5057899fc81d528b9e1f73f92c62 | You paint the faces of a cube randomly with 3 colors, with two sides painted with each color. What is the probability that some pair of opposite faces will be painted with the same color? | [
"\\(\\frac{7}{15}\\)"
] | [
"You paint the faces of a cube randomly with 3 colors, with two sides painted with each color. What is the probability that some pair of opposite faces will be painted with the same color?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-504035aef38cae6b5369136534b20ff17ecd611f6cc068aa350b6fd5ad3ff0c5 | You pass through five traffic signals on your way to work. Each signal is either red, yellow, or green. A red signal is always immediately followed by a yellow signal; a green signal is never followed immediately by another green signal. How many sequences of colors are possible for the five signals? | [
"41"
] | [
"You pass through five traffic signals on your way to work. Each signal is either red, yellow, or green. A red signal is always immediately followed by a yellow signal; a green signal is never followed immediately by another green signal. How many sequences of colors are possible for the five signals?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-ba13bfdb99585ca926d68c9a60e5cfee73ea27bafc120ac5dbd27c9060c956cc | You pay $1$ dollar to play a game where you flip a fair coin until a TAILS appears. You earn $1$ dollar for each HEAD that appears before the TAILS. What is the expected value for your earnings? | [
"0"
] | [
"You pay $1$ dollar to play a game where you flip a fair coin until a TAILS appears. You earn $1$ dollar for each HEAD that appears before the TAILS. What is the expected value for your earnings?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1440ea62c0e36149253c87f9022fa16f0bd28ab742c17fb7c73c0c424dcff3d6 | You place 15 pool balls on a standard billiards table with 6 pockets. Each ball randomly goes into a pocket, with each pocket being equally likely. After each round, if a pocket contains 4 or more balls, all balls in that pocket are removed and placed back on the table. The game ends when no pocket contains more than 3... | [
"2"
] | [
"You place 15 pool balls on a standard billiards table with 6 pockets. Each ball randomly goes into a pocket, with each pocket being equally likely. After each round, if a pocket contains 4 or more balls, all balls in that pocket are removed and placed back on the table. The game ends when no pocket contains more t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-a2f8e36be8f4cea64634aacda871a81cd4cb7a57fc244d5bbce93bded8391a9f | You place 50 cards into each of two boxes randomly. Find the expected value of the smallest card in the box containing card 100. | [
"2"
] | [
"You place 50 cards into each of two boxes randomly. Find the expected value of the smallest card in the box containing card 100."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2051b63192248406aea9aa8c265e7d5d20e9c027f5b8575b3d5e9a14ea2b6d5b | You plan on manufacturing wrenches where the interior consists of three sides of a square. The nuts are regular octagons varying in size from 0.5 to 2 centimeters. Each wrench can handle nuts varying from some size \( a \) to some size \( b \). What is the minimum number of wrenches you need to manufacture to cover all... | [
"3"
] | [
"You plan on manufacturing wrenches where the interior consists of three sides of a square. The nuts are regular octagons varying in size from 0.5 to 2 centimeters. Each wrench can handle nuts varying from some size \\( a \\) to some size \\( b \\). What is the minimum number of wrenches you need to manufacture to ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dfc4ea604f4c45f0158c2884a221ec4f4dff5e11f3300e4a6815c8fa7ad05372 | You play a carnival game as follows: The carnival worker has a circular mat of radius 20 cm, and on top of that is a square mat of side length $10$ cm, placed so that the centers of the two mats coincide. The carnival worker also has three disks, one each of radius $1$ cm, $2$ cm, and $3$ cm. You start by paying the wo... | [
"96\\pi"
] | [
"You play a carnival game as follows: The carnival worker has a circular mat of radius 20 cm, and on top of that is a square mat of side length $10$ cm, placed so that the centers of the two mats coincide. The carnival worker also has three disks, one each of radius $1$ cm, $2$ cm, and $3$ cm. You start by paying t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9f5ef13622ec8bb736a81a500811b2af851887c2031fc32855c92f7280b6e1ef | You play a game where you flip a coin repeatedly up to 10 times. If at any point you flip 2 consecutive tails, the game is over. Otherwise, you continue flipping until you have flipped the coin 10 times. How many different ways can the game end? | [
"232"
] | [
"You play a game where you flip a coin repeatedly up to 10 times. If at any point you flip 2 consecutive tails, the game is over. Otherwise, you continue flipping until you have flipped the coin 10 times. How many different ways can the game end?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-150c2e826bf495d7cf24667dc991111f495a1713fd305932c791f517ff7d7711 | You play a game where you roll a fair $6$-sided die repeatedly until the cumulative sum of your rolls is exactly a perfect cube, at which point the game ends. What is the expected number of rolls before the game ends? | [
"6"
] | [
"You play a game where you roll a fair $6$-sided die repeatedly until the cumulative sum of your rolls is exactly a perfect cube, at which point the game ends. What is the expected number of rolls before the game ends?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-17c697e2b4156cf40f0c71740f0118c686777fa5b00e17719c5a0383fc4353d0 | You play a game where you throw a six-sided fair die repeatedly. You may choose to stop after any throw, except that you must stop if you throw a 1. Your score is the number obtained on your last throw. Determine the strategy that maximizes your expected score. | [
"4"
] | [
"You play a game where you throw a six-sided fair die repeatedly. You may choose to stop after any throw, except that you must stop if you throw a 1. Your score is the number obtained on your last throw. Determine the strategy that maximizes your expected score."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 6
} | 26 | 0 | 0 |
omr-ws-b9859078a625bb2bfbf23bc8c244ed8c6db494cfc3205c44b5486818df27d2cc | You play a game with a winning probability \( p \). At the beginning, you may play only one round. For every round you win, you gain 1 Point and 2 additional rounds. What is the expected value of Points if you play until you have 0 rounds left? | [
"\\( \\frac{p}{1-2p} \\)"
] | [
"You play a game with a winning probability \\( p \\). At the beginning, you may play only one round. For every round you win, you gain 1 Point and 2 additional rounds. What is the expected value of Points if you play until you have 0 rounds left?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-b1dceef0ec171a62eab7b3932456ea60f5ecb4e58fdef11cdddb8553e0f2b628 | You randomly choose two integers in the interval [1,9] without repetition, and then multiply them together. The system does the same. What is the probability that the number you pick will have (and only have) two prime common factors with the number that the system picks? | [
"\\dfrac{241}{1296}"
] | [
"You randomly choose two integers in the interval [1,9] without repetition, and then multiply them together. The system does the same. What is the probability that the number you pick will have (and only have) two prime common factors with the number that the system picks?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-ec646aa9c4cba0d65702feaf8fd57cc112b2afc9b78b5b33f052ff4e837b99f1 | You randomly pick a real number $R$ in the interval $(0,1)$. You then construct a match of length $R$ cm. You drop the match on a 1cm by 1cm piece of cloth. Given that the match head landed on top of the cloth, what is the probability that the entire match is entirely on top of the cloth? | [
"\\(\\frac{\\pi-1}{\\pi+1}\\)"
] | [
"You randomly pick a real number $R$ in the interval $(0,1)$. You then construct a match of length $R$ cm. You drop the match on a 1cm by 1cm piece of cloth. Given that the match head landed on top of the cloth, what is the probability that the entire match is entirely on top of the cloth?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6efbdda119058e0259757da93327da5b1257ebd24abea76b34205aeede21233d | You randomly pick from $n$ elements (say, Wikipedia pages) one at a time, with replacement. The probability of picking each element is $1/n$. What is the expected number of trials before any element is repeated? | [
"\\sum_{k=0}^n \\frac{n!}{n^k (n - k)!}"
] | [
"You randomly pick from $n$ elements (say, Wikipedia pages) one at a time, with replacement. The probability of picking each element is $1/n$. What is the expected number of trials before any element is repeated?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cf99091b21e8ac3bf9fe07d296dc017d2880e62c15d618a3a819fd94f0f0217b | You recently bought more than 100 eggs. The eggs are sorted in containers that can store exactly 12 eggs. However, upon inspecting the containers, you realize that two containers each hold only 11 eggs, while all the other containers hold 12 eggs. What is the smallest number of eggs you could have right now? | [
"106"
] | [
"You recently bought more than 100 eggs. The eggs are sorted in containers that can store exactly 12 eggs. However, upon inspecting the containers, you realize that two containers each hold only 11 eggs, while all the other containers hold 12 eggs. What is the smallest number of eggs you could have right now?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5cbfaafabc14cae60b97318147f56f5014876fa453b43b1ddf6218d3a971d429 | You recently came into 20,000 dollars and want to place part of your money in a savings account paying 7% per year and part in a certificate of deposit paying 9% per year. If you wish to obtain an overall return of 1700 dollars per year, how much would you place in each investment? | [
"15000"
] | [
"You recently came into 20,000 dollars and want to place part of your money in a savings account paying 7% per year and part in a certificate of deposit paying 9% per year. If you wish to obtain an overall return of 1700 dollars per year, how much would you place in each investment?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-fc0065fc8ac0a21c9580987a1cd9c407c88a038dbc23caf1fe6c45af93d1586a | You repeatedly draw any two balls from an urn containing $n$ balls, each initially painted a different color. You paint the first ball the color of the second ball and return them to the urn. What is the expected number of draws before all balls are the same color? | [
"(n-1)^2"
] | [
"You repeatedly draw any two balls from an urn containing $n$ balls, each initially painted a different color. You paint the first ball the color of the second ball and return them to the urn. What is the expected number of draws before all balls are the same color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-afc91409bc35cc88cd4322bc7de487a4751756d0883e4a73237d0a584545e34d | You repeatedly draw two balls of different colors from an urn containing $n$ balls, each initially painted a different color. You paint the first ball the color of the second ball and return them to the urn. What is the expected number of draws before all balls are the same color? | [
"\\( n - 1 \\)"
] | [
"You repeatedly draw two balls of different colors from an urn containing $n$ balls, each initially painted a different color. You paint the first ball the color of the second ball and return them to the urn. What is the expected number of draws before all balls are the same color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f16438ea5975c756792a3967d5ec9c957975e53110a9acbad94aa41f3b2d85e9 | You repeatedly flip a coin with probability $p$ of heads and $q=1-p$ of tails. Flips are independent. Let $X$ denote the number of flips until you get two heads in a row. By considering the partition $A_1, A_2, A_3$ where $A_1$ is the event {tail on the first flip}, $A_2$ is the event {head on the first flip, tail on t... | [
"\\(\\frac{p+1}{p^2}\\)"
] | [
"You repeatedly flip a coin with probability $p$ of heads and $q=1-p$ of tails. Flips are independent. Let $X$ denote the number of flips until you get two heads in a row. By considering the partition $A_1, A_2, A_3$ where $A_1$ is the event {tail on the first flip}, $A_2$ is the event {head on the first flip, tail... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-b09d14fb72b08f4d92d1f70f4229ea4766af0cab506feb39069e89d52097a0a5 | You repeatedly toss two fair coins until both coins show heads. Let $X$ and $Y$ denote the number of heads resulting from the tosses of the first and the second coin respectively. What are the marginal distributions of $X$ and $Y$? | [
"\\text{Geometric}\\left(\\frac{1}{2}\\right)"
] | [
"You repeatedly toss two fair coins until both coins show heads. Let $X$ and $Y$ denote the number of heads resulting from the tosses of the first and the second coin respectively. What are the marginal distributions of $X$ and $Y$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3265e2efd8755e7a602d476d0e6685e3ef5e4b123dcdfbfce724ab5e573047fd | You repeatedly toss two fair coins until both coins show heads. Let $X$ and $Y$ denote the number of heads resulting from the tosses of the first and the second coin respectively. What is $P(X = Y)$? | [
"\\dfrac{\\sqrt{5}}{5}"
] | [
"You repeatedly toss two fair coins until both coins show heads. Let $X$ and $Y$ denote the number of heads resulting from the tosses of the first and the second coin respectively. What is $P(X = Y)$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-c618a22a5c34823039bc4b459e54cbd6090deb491d79325d7b948476d225d3fd | You repeatedly toss two fair coins until both coins show heads. Let $X$ and $Y$ denote the number of heads resulting from the tosses of the first and the second coin respectively. What is the joint probability mass function of $X$ and $Y$? | [
"\\(\\mathbb{P}[X=x,Y=y] = \\frac{1}{3^{x+y-1}} \\binom{x+y-2}{x-1}\\)"
] | [
"You repeatedly toss two fair coins until both coins show heads. Let $X$ and $Y$ denote the number of heads resulting from the tosses of the first and the second coin respectively. What is the joint probability mass function of $X$ and $Y$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 3,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-9e245b7db9dc9fddad4dea4b68f0d04c1724a23fd2da677495bdc4e787bac96f | You roll 6 fair six-sided dice. Given that 2 of them are 6's, what is the probability that exactly 4 of them are 6's? | [
"\\(\\frac{375}{12281}\\)"
] | [
"You roll 6 fair six-sided dice. Given that 2 of them are 6's, what is the probability that exactly 4 of them are 6's?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 2,
"tir": 25
} | 38 | 0 | 0 |
omr-ws-ba21aa2eb02b193f4bc04d3b8b7fa1c8b4af02cc22fab35dd4b62f8a02698876 | You roll a die 5 times. What is the probability that the die comes up as a 6 more than or equal to 3 times? | [
"\\dfrac{23}{648}"
] | [
"You roll a die 5 times. What is the probability that the die comes up as a 6 more than or equal to 3 times?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff08d5d476cdbb7ab3ceebe120856a2e914944f964a1b5d77f5c06fbe7d92c46 | You roll a die, and I roll a die. You win if the number showing on your die is strictly greater than the one on mine. If we play the game five times, what is the probability that you win at least 4 times? | [
"\\(\\frac{59375}{248832}\\)"
] | [
"You roll a die, and I roll a die. You win if the number showing on your die is strictly greater than the one on mine. If we play the game five times, what is the probability that you win at least 4 times?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-43719d24712c83357782da8036c01a018e09d2df15d81eb49e716b197694b6b8 | You roll a fair 12-sided die repeatedly. Find the probability that all prime numbers show up at least once before seeing any other numbers. | [
"\\(\\frac{1}{792}\\)"
] | [
"You roll a fair 12-sided die repeatedly. Find the probability that all prime numbers show up at least once before seeing any other numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 25
} | 72 | 0 | 0 |
omr-ws-8e38809334ef649b8247ba4c4f625eb09a88889ab6472e96e438aaec5229d043 | You roll a fair 12-sided die repeatedly. The probability that all primes show up at least once before seeing any of the other numbers can be expressed as a fraction $p/q$ in lowest terms. What is $p+q$? | [
"793"
] | [
"You roll a fair 12-sided die repeatedly. The probability that all primes show up at least once before seeing any of the other numbers can be expressed as a fraction $p/q$ in lowest terms. What is $p+q$?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-b0b1ebc3f25c0d174bb073a79922475ddc6b75f1a62f45e8cffb4915dc20a7eb | You roll a fair die and can choose to either win the number of dollars rolled or reroll. After your second roll, you get the same decision, but you must take whatever your third roll is. What is the expected value of this game under optimal play? | [
"3"
] | [
"You roll a fair die and can choose to either win the number of dollars rolled or reroll. After your second roll, you get the same decision, but you must take whatever your third roll is. What is the expected value of this game under optimal play?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0093b5661d168f5cf722125d23824b55b5cc00a2a6459fe5e24642fb866aa092 | You roll a fair die five times and add the numbers that come up. What is the probability that the sum is $10$? | [
"126"
] | [
"You roll a fair die five times and add the numbers that come up. What is the probability that the sum is $10$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c20b28b8a66d4255e6796757190faf25d12aec93033aab20215ca451d7a1ce71 | You roll a fair six-sided die repeatedly until the sum is at least $n$. As $n$ grows large, what is the expectation of the final sum? | [
"\\( n + 3.5 \\)"
] | [
"You roll a fair six-sided die repeatedly until the sum is at least $n$. As $n$ grows large, what is the expectation of the final sum?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 7,
"tir": 3
} | 13 | 0 | 0 |
omr-ws-b7e20aae34e3caebebc5d07af0dd34fd849e20fb72e9e8314b169b3a114df008 | You roll a fair six-sided die repeatedly until the sum is at least $n$. As $n$ grows large, what is the probability that the final roll is 1? 2? ... 6? What is the expectation of the final roll? | [
"Probability for each roll (1 through 6) is \\( \\frac{1}{6} \\), and the expectation of the final roll is \\( 3.5 \\)."
] | [
"You roll a fair six-sided die repeatedly until the sum is at least $n$. As $n$ grows large, what is the probability that the final roll is 1? 2? ... 6? What is the expectation of the final roll?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-beaf8f1ca81da538efbaa5c40b8711337ef9796d2ba39427f3d554617059e0ec | You roll a fair six-sided die repeatedly until the sum is at least $n$. As $n$ grows large, what is the probability that the sum you end on is $n$? $n+1$? etc. | [
"\\dfrac{1}{21}"
] | [
"You roll a fair six-sided die repeatedly until the sum is at least $n$. As $n$ grows large, what is the probability that the sum you end on is $n$? $n+1$? etc."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-78455b6c5e52620bc859d5f24009c7f0fc3605e277ec195ebbbc90cc6a30caf8 | You roll four dice until you have at least two equal numbers. Then, you win if at least one of the four dice shows a $3$. What is your chance of winning? (Represent the answer as a fraction.) | [
"\\(\\frac{431}{936}\\)"
] | [
"You roll four dice until you have at least two equal numbers. Then, you win if at least one of the four dice shows a $3$. What is your chance of winning? (Represent the answer as a fraction.)"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-aa3b2bcb75e3c1522f183fc71375912eeb7ab7fe88128b9adfe3e4487a3d3973 | You roll four fair dice. You keep rolling until at least two of the dice show equal numbers. Then, you win if at least one of the dice shows a 1. What is the probability you win? | [
"\\(\\frac{671}{1176}\\)"
] | [
"You roll four fair dice. You keep rolling until at least two of the dice show equal numbers. Then, you win if at least one of the dice shows a 1. What is the probability you win?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f14583e2cf818f9a634cacd30ca76027de44b4f1db471f7cc0cc247cf5c5bb19 | You roll four fair six-sided dice: (A) The probability that the numbers on the faces multiply to 36 can be expressed as \(\frac{m}{n}\) for relatively prime integers \(m\) and \(n\). Find \(m + n\). (B) The probability that the numbers on the faces add to 21 can be expressed as \(\frac{m}{n}\) for relatively prime inte... | [
"5"
] | [
"You roll four fair six-sided dice:\n(A) The probability that the numbers on the faces multiply to 36 can be expressed as \\(\\frac{m}{n}\\) for relatively prime integers \\(m\\) and \\(n\\). Find \\(m + n\\).\n(B) The probability that the numbers on the faces add to 21 can be expressed as \\(\\frac{m}{n}\\) for re... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8424ef96dd7751b90c89d1b8f624f28bb677c3c41462d8f44bf7aaeda5f7c4cf | You roll three fair six-sided dice. Given that the highest number you rolled is a $5$, the expected value of the sum of the three dice can be written as $\frac{a}{b}$ in simplest form. Find $a + b$. | [
"706"
] | [
"You roll three fair six-sided dice. Given that the highest number you rolled is a $5$, the expected value of the sum of the three dice can be written as $\\frac{a}{b}$ in simplest form. Find $a + b$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-ab11fdf3b257ce19af640d22c7ae559d17bbb0411b54c06cac0f542163b72c4e | You roll two fair dice. If the sum of the two numbers obtained is 2, 3, 4, 9, 10, 11, or 12, Scott will pay you $20. However, if the sum of the two numbers is 5, 6, 7, or 8, you will pay Scott $20. Is this game fair to you? Should you accept this offer? Support your conclusion with appropriate calculations. | [
"-2.22"
] | [
"You roll two fair dice. If the sum of the two numbers obtained is 2, 3, 4, 9, 10, 11, or 12, Scott will pay you $20. However, if the sum of the two numbers is 5, 6, 7, or 8, you will pay Scott $20. Is this game fair to you? Should you accept this offer? Support your conclusion with appropriate calculations."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 19
} | 21 | 0 | 0 |
omr-ws-ab0c290d81ea70cac09c76428096e5fd923fb639409c2eaa1d63b211738c5525 | You shoot \( n \) free throws in a row, with the probability of making the \( i \)th shot being equal to the fraction of free throws made in the previous \( i - 1 \) shots. Given that you made the first free throw and missed the second free throw (assuming \( n > 2 \)), calculate the sum of all possible values of \( x ... | [
"\\dfrac{n(n-1)}{2}"
] | [
"You shoot \\( n \\) free throws in a row, with the probability of making the \\( i \\)th shot being equal to the fraction of free throws made in the previous \\( i - 1 \\) shots. Given that you made the first free throw and missed the second free throw (assuming \\( n > 2 \\)), calculate the sum of all possible va... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f44838744d236e985575b2c67b1c2f7d5be91c541d6ac85a15b53199f94b6da9 | You start at 1 on the number line. If the number you're at can be factored as \(a \times b\) (where \(a\) and \(b\) are natural numbers), then you are allowed to move exactly \(a\) units forward (in the positive direction), but doing so would cost you \(b\) points. Let \(C(n)\) be the minimum cost of getting to \(n\). ... | [
"10"
] | [
"You start at 1 on the number line. If the number you're at can be factored as \\(a \\times b\\) (where \\(a\\) and \\(b\\) are natural numbers), then you are allowed to move exactly \\(a\\) units forward (in the positive direction), but doing so would cost you \\(b\\) points. Let \\(C(n)\\) be the minimum cost of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-da54637cee91915a22aab6d54454ccb95893be6c65621e18f9df0af84121ef6f | You start at one corner of a square with side length 10 inches and leap to one of the non-adjacent sides. Your path is tangent to a circle of radius 1 inch concentric to the square. You continue leaping, each time ensuring your path is tangent to the circle and not going backwards, until you arrive at another corner. W... | [
"\\( 16\\sqrt{2} \\)"
] | [
"You start at one corner of a square with side length 10 inches and leap to one of the non-adjacent sides. Your path is tangent to a circle of radius 1 inch concentric to the square. You continue leaping, each time ensuring your path is tangent to the circle and not going backwards, until you arrive at another corn... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a9fa52a5ecac3dab63a9f8d8126351a3b56c0210974410a933e3d678bef5edeb | You start at step 1 on a ladder. If you reach step 0, you die. The movement rules are as follows: - On steps 0, 2 (mod 3), you can either take one step up with probability $\beta$ or take two steps down with probability $1-\beta$. - On steps 1 (mod 3), you can go up one step with probability $\alpha \beta$, go down two... | [
"\\frac{1 + \\beta}{(1 - \\beta)(1 - \\alpha \\beta)}"
] | [
"You start at step 1 on a ladder. If you reach step 0, you die. The movement rules are as follows:\n- On steps 0, 2 (mod 3), you can either take one step up with probability $\\beta$ or take two steps down with probability $1-\\beta$.\n- On steps 1 (mod 3), you can go up one step with probability $\\alpha \\beta$, ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-cbfcffef69e45d646ecccb2067d256b2be0f6506052db7d794a4193793b3273a | You start at the bottom of a Ferris wheel, 3 feet off the ground. The Ferris wheel takes 8 seconds to reach the highest point, which is 63 feet off the ground. How high off the ground will you be 5 seconds after the ride starts? Answer to the nearest tenth of a foot. | [
"44.5"
] | [
"You start at the bottom of a Ferris wheel, 3 feet off the ground. The Ferris wheel takes 8 seconds to reach the highest point, which is 63 feet off the ground. How high off the ground will you be 5 seconds after the ride starts? Answer to the nearest tenth of a foot."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-90ddc0d6b0b245e88f6926a397c85582990ea1b7adafe3c4509ca90ba7a04ba2 | You start at the origin on a plane. At step $1,$ you flip a fair coin. If you get a heads, you move one unit right, to the point $(1,0).$ If tails, move one unit up, to $(0,1).$ At step $2,$ you flip the coin again and move from your current position either $\frac{1}{2}$ unit up or $\frac{1}{2}$ unit to the right. At s... | [
"\\(1+\\dfrac{\\text{sinh}^{-1}(1)}{\\sqrt{2}}\\)"
] | [
"You start at the origin on a plane. At step $1,$ you flip a fair coin. If you get a heads, you move one unit right, to the point $(1,0).$ If tails, move one unit up, to $(0,1).$ At step $2,$ you flip the coin again and move from your current position either $\\frac{1}{2}$ unit up or $\\frac{1}{2}$ unit to the righ... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-80978b5580dd276be2a03c1d506e2e68882ce9e9323efa02e3b5a4beac99b5a2 | You start from $(0, 0)$ and run north at $a$ units per second. A mountain lion starts at $(n, 0)$, with $n>0$, and runs directly towards your current position at $b>a$ units per second. What is the path of the mountain lion's movement and how long will it take for the mountain lion to catch you? | [
"\\( T = \\frac{n}{b \\sqrt{1 - \\frac{a^2}{b^2}}} \\)"
] | [
"You start from $(0, 0)$ and run north at $a$ units per second. A mountain lion starts at $(n, 0)$, with $n>0$, and runs directly towards your current position at $b>a$ units per second. What is the path of the mountain lion's movement and how long will it take for the mountain lion to catch you?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 8
} | 20 | 0 | 0 |
omr-ws-7d82f9d92e768953f60815da1c4efefc265f3ebd3d89590e602ac2d088e854a0 | You start running away to the northeast at $10 \text{m/s}$ and manage a three-second head start over a velociraptor. The raptor runs at $15\sqrt{2} \text{m/s}$ but only runs either north or east at any given time. How many seconds do you have until the raptor catches up to you? | [
"6"
] | [
"You start running away to the northeast at $10 \\text{m/s}$ and manage a three-second head start over a velociraptor. The raptor runs at $15\\sqrt{2} \\text{m/s}$ but only runs either north or east at any given time. How many seconds do you have until the raptor catches up to you?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 1,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-d9ccde0ed4f1764b90adb6d66576beb1edf0d1d5c3f471ab5586ca1e580dfc2b | You start with $100$ dollars in your bank account. Each month, you can either double your money or add $100$ dollars to your account. What is the minimum number of months required to have at least $20000$ dollars in your bank account? | [
"8"
] | [
"You start with $100$ dollars in your bank account. Each month, you can either double your money or add $100$ dollars to your account. What is the minimum number of months required to have at least $20000$ dollars in your bank account?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-b4d55a5251080230013505fdb4f8b2a17d8e005ca2cc0b1a30e6e4cfb2894021 | You start with $N$ dollars and aim to reach at least $M > N$ dollars. On each turn, you choose a nonzero integer amount to bet. With probability $p$, you win the wager and receive that amount; otherwise, you lose that amount. You lose the game if you run out of money. What is the optimal betting strategy to maximize th... | [
"If $p \\geq 1/2$, bet $1$ each time; if $p < 1/2$, bet everything each time except for the last bet when you bet just enough to get to $M$."
] | [
"You start with $N$ dollars and aim to reach at least $M > N$ dollars. On each turn, you choose a nonzero integer amount to bet. With probability $p$, you win the wager and receive that amount; otherwise, you lose that amount. You lose the game if you run out of money. What is the optimal betting strategy to maximi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 2
} | 14 | 0 | 0 |
omr-ws-e810c903602947cc1ce07f6dfa69e385d1f61fdd507d028ae61011484ea6348d | You start with 50 dollars. Each time you flip a coin, if it's heads, you win 2 dollars, and if it's tails, you lose 1 dollar. What is the probability that at some point you will have 0 dollars? | [
"\\( \\left[ \\frac{\\sqrt{5}-1}{2} \\right]^{50} \\) or approximately \\( 3.55 \\cdot 10^{-11} \\)"
] | [
"You start with 50 dollars. Each time you flip a coin, if it's heads, you win 2 dollars, and if it's tails, you lose 1 dollar. What is the probability that at some point you will have 0 dollars?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-601524724a8ce10e87cfa872367cdcd66be70abeaa047ee33248297da2e0d9f1 | You start with 500 dollars and I start with 1000 dollars. We flip a fair coin repeatedly. Each time it comes up heads, I give you a dollar. Each time it comes up tails, you give me a dollar. We continue playing until one of us has all the money. What is the probability you will win this game? | [
"\\(\\frac{1}{3}\\)"
] | [
"You start with 500 dollars and I start with 1000 dollars. We flip a fair coin repeatedly. Each time it comes up heads, I give you a dollar. Each time it comes up tails, you give me a dollar. We continue playing until one of us has all the money. What is the probability you will win this game?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-8985ec8bc9d0e6e6cf299bd300fe007dcd194a73b22261f37f07a673a23490ec | You start with \( n \geq 5 \) worthless bronze coins. A machine allows you to trade in five bronze coins for one silver coin with a success probability \( p \). If the trade fails, the machine takes away 1 to 4 coins with equal probability. If you keep trading until you can't anymore, what is the probability that you g... | [
"\\( 1 - P_n \\) where \\( P_n \\) is computed recursively as described."
] | [
"You start with \\( n \\geq 5 \\) worthless bronze coins. A machine allows you to trade in five bronze coins for one silver coin with a success probability \\( p \\). If the trade fails, the machine takes away 1 to 4 coins with equal probability. If you keep trading until you can't anymore, what is the probability ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 21
} | 21 | 0 | 0 |
omr-ws-3f36b693b643d86b9d79d1d723f4eb427518b069284d486c5e892b108f24a0ac | You start with a pile of 300 tokens, of which 99 are red, 100 are blue, and 101 are green. In each move, you take two tokens of different colors off the pile and put one token of the third color back onto the pile. After performing a sequence of moves until only one token remains, calculate the number of possible color... | [
"1"
] | [
"You start with a pile of 300 tokens, of which 99 are red, 100 are blue, and 101 are green. In each move, you take two tokens of different colors off the pile and put one token of the third color back onto the pile. After performing a sequence of moves until only one token remains, calculate the number of possible ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2093a61e2bc6798d35fa281d28a723e206edbd77474d5ea22597740d9ccbfe8b | You start with a single 0 on your screen, which is already copied. At each step, you can either paste (1 move) or perform a sequence of select all, copy all, and paste all (4 moves). Determine an optimal method to maximize the number of 0's on the screen while minimizing the total number of moves. | [
"Perform \"select all, copy all, paste all\" once, followed by two paste operations."
] | [
"You start with a single 0 on your screen, which is already copied. At each step, you can either paste (1 move) or perform a sequence of select all, copy all, and paste all (4 moves). Determine an optimal method to maximize the number of 0's on the screen while minimizing the total number of moves."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 2
} | 2 | 0 | 0 |
omr-ws-137eda00408ccd50dfe9942959aaa3bbfcc6f6821c8cd76b3a238d128f0fcd03 | You start with four congruent right triangles. In one step, you may take any triangle and cut it into two with the altitude from the right angle. After performing this process $n$ times, let $T(n)$ be the minimum number of congruent triangles that must remain. Find $T(10)$. | [
"4"
] | [
"You start with four congruent right triangles. In one step, you may take any triangle and cut it into two with the altitude from the right angle. After performing this process $n$ times, let $T(n)$ be the minimum number of congruent triangles that must remain. Find $T(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-dbd4117418aea425c6532c8dd46f9f3f016415c957903909f0974d1c48d8b64b | You start with four pencils on a table. You flip a fair coin. If you flip heads, you put another pencil on the table. If you flip tails, you take one pencil off the table. You keep flipping the coin until all the pencils are off the table. What is the expected number of times you will flip the coin? | [
"\\infty"
] | [
"You start with four pencils on a table. You flip a fair coin. If you flip heads, you put another pencil on the table. If you flip tails, you take one pencil off the table. You keep flipping the coin until all the pencils are off the table. What is the expected number of times you will flip the coin?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-6476eea960f2dfdd5d8b403c4b0351a50d291a79447f8b8c135c0326fcc60dfe | You throw a dart randomly at a circular dartboard of radius 1 centered at (0,0). Two points inside the dartboard are given at coordinates (1/2, 3/4) and (-3/4, -1/2). You win an amount equal to the ratio of the area of the triangle formed by the dart's landing point and the two given points to the perimeter of that tri... | [
"0"
] | [
"You throw a dart randomly at a circular dartboard of radius 1 centered at (0,0). Two points inside the dartboard are given at coordinates (1/2, 3/4) and (-3/4, -1/2). You win an amount equal to the ratio of the area of the triangle formed by the dart's landing point and the two given points to the perimeter of tha... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-81c3d8e78045e1b88f72dfbc27cfea00f3866287a0695543c78401ff5f97a0b2 | You toss a fair coin and after each toss, you can either stop and collect your winnings (equal to the ratio of heads to total flips) or toss again. What is the expected value of this game? | [
"0.793"
] | [
"You toss a fair coin and after each toss, you can either stop and collect your winnings (equal to the ratio of heads to total flips) or toss again. What is the expected value of this game?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8fc72a60f8376e49b3fc1a1e2f3676f01d05c6dfca504379d78432724346ca00 | You travel for 1 mile at 20 mph, another mile at 28 mph, and finally 1 more mile at $K$ mph ($K$ is finite). If your average (mean) rate for the entire trip is $x$ mph, compute the smallest positive integral value of $x$ that is unattainable. | [
"35"
] | [
"You travel for 1 mile at 20 mph, another mile at 28 mph, and finally 1 more mile at $K$ mph ($K$ is finite). If your average (mean) rate for the entire trip is $x$ mph, compute the smallest positive integral value of $x$ that is unattainable."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-cc0625b51d659a9292096b0e564a4262cef7a889f214ad204d789d0ce46cdaf7 | You turn on a calculator and the screen reads ‘$O$’. The calculator can only display numbers smaller than $1 \times 10^{100}$. When you push the exponential button $e^x$ the calculator computes and displays the exponential of whatever is on the calculator screen and similarly when you push the natural logarithm button ... | [
"\\(\\frac{5}{8}\\)"
] | [
"You turn on a calculator and the screen reads ‘$O$’. The calculator can only display numbers smaller than $1 \\times 10^{100}$. When you push the exponential button $e^x$ the calculator computes and displays the exponential of whatever is on the calculator screen and similarly when you push the natural logarithm b... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6c3715369ccb9c7f98e7c19571283758b75284f764a2b4d8edaf96e568be4760 | You wait at a bus stop where Bus $A$ comes every 12 minutes and Bus $B$ comes every 30 minutes. What is the expected value of the time you'll have to wait for a bus? | [
"\\(\\frac{360}{41}\\) minutes"
] | [
"You wait at a bus stop where Bus $A$ comes every 12 minutes and Bus $B$ comes every 30 minutes. What is the expected value of the time you'll have to wait for a bus?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-cef76341656a2b32c30aac5c6a173b926a6f8fe88b5108a99b1e46e0b4450af1 | You walk 1 mile east, then \(\sqrt{2}\) miles northeast, and finally 2 miles south. What is the distance from your final position to your starting point (campground)? | [
"\\( \\sqrt{5} \\)"
] | [
"You walk 1 mile east, then \\(\\sqrt{2}\\) miles northeast, and finally 2 miles south. What is the distance from your final position to your starting point (campground)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-6f0250890eed9634269556da2d0688adf8645cc15cc4bd2f7bad850856899147 | You want to arrange 8 biologists of 8 different heights in two rows for a photograph. Each row must have 4 biologists. Height must increase from left to right in each row. Each person in back must be taller than the person directly in front of him. How many different arrangements are possible? | [
"14"
] | [
"You want to arrange 8 biologists of 8 different heights in two rows for a photograph. Each row must have 4 biologists. Height must increase from left to right in each row. Each person in back must be taller than the person directly in front of him. How many different arrangements are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4c646cc5295ce983253623beda98e7e969d12b0b91271524b84426700c7db741 | You want to enclose a rectangular vegetable garden with 100 feet of fence, with one side of the garden being your garage. How should you lay out the fence to maximize the area of the garden? | [
"25 feet for the sides perpendicular to the garage and 50 feet for the side opposite the garage."
] | [
"You want to enclose a rectangular vegetable garden with 100 feet of fence, with one side of the garden being your garage. How should you lay out the fence to maximize the area of the garden?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 22
} | 44 | 0 | 0 |
omr-ws-b9639ac9ac25137f7285b7036bb39f95f166436b37c51253af352b1cd26b31d4 | You want to mix cashews and walnuts to sell for $2 per pound. Walnuts cost $1.50 per pound and cashews cost $3 per pound. If you want to have a total of 60 pounds of your mixture, how many pounds of cashews and how many pounds of walnuts must you buy to sell your mixture at $2 per pound? | [
"40 pounds of walnuts and 20 pounds of cashews"
] | [
"You want to mix cashews and walnuts to sell for $2 per pound. Walnuts cost $1.50 per pound and cashews cost $3 per pound. If you want to have a total of 60 pounds of your mixture, how many pounds of cashews and how many pounds of walnuts must you buy to sell your mixture at $2 per pound?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 30
} | 39 | 0 | 0 |
omr-ws-d28fd1727692dce675cb16d6b7cd286e378916a62eabace983c61671ebba8439 | You want to shade a 4*4 grid completely. You can shade an unshaded square if that square has 2 adjacent squares (diagonal squares aren't considered adjacent) which are already shaded. What is the minimum number of initially shaded squares needed to accomplish the task? | [
"4"
] | [
"You want to shade a 4*4 grid completely. You can shade an unshaded square if that square has 2 adjacent squares (diagonal squares aren't considered adjacent) which are already shaded. What is the minimum number of initially shaded squares needed to accomplish the task?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc1dcde468dfcfd8d3322ca6ca87e72f052ec399875334071095cb829265a493 | You will generate 5-digit codes using the numbers 1 to 5 once. Any adjacent three digits used in one code cannot be used in the same way in another code. What is the maximum number of codes that can be produced according to this rule? | [
"20"
] | [
"You will generate 5-digit codes using the numbers 1 to 5 once. Any adjacent three digits used in one code cannot be used in the same way in another code. What is the maximum number of codes that can be produced according to this rule?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 44,
"genselect": 8
} | 52 | 0 | 0 |
omr-ws-8f3bcd370d51e3832be184fcdf8f55aaaeab4d9801d70b2808ec277dcd2db828 | You wish to tile an n × 1 walkway with blue and red tiles each of dimension 1 × 1. A blue tile can be followed by either a red or blue tile. However, only a red tile can follow a red tile. Determine a formula in terms of n for the number of ways in which an n × 1 walkway may be tiled. | [
"\\( n + 1 \\)"
] | [
"You wish to tile an n × 1 walkway with blue and red tiles each of dimension 1 × 1. A blue tile can be followed by either a red or blue tile. However, only a red tile can follow a red tile. Determine a formula in terms of n for the number of ways in which an n × 1 walkway may be tiled."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 2
} | 25 | 0 | 0 |
omr-ws-38a987f2f931602394ac962c9b0d9ecdb90d0ae27eb51a38ab8dd9f1d3be1a94 | You won a math contest and can choose between two prizes: 1) For every day for the rest of your life, you will receive $\frac{1}{10}$th of the money you received the previous day, starting with \$900,000. 2) For every day for the rest of your life, you will receive \$\frac{10}{n}$ on the $n$th day after you won the pri... | [
"First option"
] | [
"You won a math contest and can choose between two prizes:\n1) For every day for the rest of your life, you will receive $\\frac{1}{10}$th of the money you received the previous day, starting with \\$900,000.\n2) For every day for the rest of your life, you will receive \\$\\frac{10}{n}$ on the $n$th day after you ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 2,
"tir": 26
} | 39 | 0 | 0 |
omr-ws-77efe2647df3be251eaacb1214e65cdcb75bcb817ac8308f53e340deb456b9d7 | You're given $n$ blocks each with weight at least $1$ and total weight $2n$. Let $S(r)$ denote the number of subsets of the blocks whose total weight is between $r$ and $r+2$ inclusive, for every real number $r \in [0, 2n-2]$. Calculate the minimum possible value of $S(r)$ over all such sets of blocks and all $r \in [0... | [
"2"
] | [
"You're given $n$ blocks each with weight at least $1$ and total weight $2n$. Let $S(r)$ denote the number of subsets of the blocks whose total weight is between $r$ and $r+2$ inclusive, for every real number $r \\in [0, 2n-2]$. Calculate the minimum possible value of $S(r)$ over all such sets of blocks and all $r ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-af0f0771187d5ffd554e7305e97ff3f734b1431df3aa69f50de30d1de5a8c475 | You're playing a game in which the prize pool starts at $\$1$. On every turn, you flip a fair coin. If you flip heads, the prize pool doubles, and the game continues. If you flip tails, the game ends, and you win the prize pool. What is the expected value of your winnings? | [
"\\(\\infty\\)"
] | [
"You're playing a game in which the prize pool starts at $\\$1$. On every turn, you flip a fair coin. If you flip heads, the prize pool doubles, and the game continues. If you flip tails, the game ends, and you win the prize pool. What is the expected value of your winnings?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-7e8c9045cc17b148aac1a2806487f8831bbc35a539ffbb7e20e4c1c1384ade35 | You, Zan, Winnie, Xavier, and Yahan play a game where each person rolls two dice in alphabetical order from W to Z. The first person to roll a sum of 8 or 9 must do all the chores. If no one rolls a sum of 8 or 9, Yahan must do all the chores. What is the probability that you won't have to do all the work? | [
"\\(\\frac{229}{256}\\)"
] | [
"You, Zan, Winnie, Xavier, and Yahan play a game where each person rolls two dice in alphabetical order from W to Z. The first person to roll a sum of 8 or 9 must do all the chores. If no one rolls a sum of 8 or 9, Yahan must do all the chores. What is the probability that you won't have to do all the work?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 5
} | 19 | 0 | 0 |
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