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-faec81b4bd9ca2ee75c620d598c337aa91ab9c6175953f6e9ea34dfd90b27bd8 | You are sequentially placing $m$ items among $m$ cells according to the following probability distribution: each item will be independently put in cell $j$ with probability $p_j$, $j = 1, 2, ..., m$. Let $X$ denote the number of collisions that occur, where a collision occurs whenever an item is put into a non-empty ce... | [
"\\dfrac{1024}{625}"
] | [
"You are sequentially placing $m$ items among $m$ cells according to the following probability distribution: each item will be independently put in cell $j$ with probability $p_j$, $j = 1, 2, ..., m$. Let $X$ denote the number of collisions that occur, where a collision occurs whenever an item is put into a non-emp... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cb7c362dc18b5b45db5a723a7f75068d8eecaf931bb62c352923fe43ba88f642 | You are standing 6 feet from a circular aquarium tank. The distance from you to a point of tangency on the tank is 17 feet. What is the radius of the tank? | [
"\\( \\frac{253}{12} \\)"
] | [
"You are standing 6 feet from a circular aquarium tank. The distance from you to a point of tangency on the tank is 17 feet. What is the radius of the tank?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-0bc18fb329428b35f13abf01433ec5cba94741395e18cf90292c7cab8f6a77b9 | You are standing at point A, which is 1000 feet from point B. You choose a random direction and walk 1000 feet in that direction. What is the probability that you end up within 1000 feet of point B? | [
"\\(\\frac{1}{3}\\)"
] | [
"You are standing at point A, which is 1000 feet from point B. You choose a random direction and walk 1000 feet in that direction. What is the probability that you end up within 1000 feet of point B?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-77c965787ccac700d41f94165c107e416f64fca4c0bed93c8ea0cc0901dd0b7f | You are standing at the edge of a river which is 1 km wide. The camp on the opposite bank is 1 km downstream from the point directly across from you. You can swim at 2 km/hr and walk at 3 km/hr. What is the shortest time you will take to reach your camp? (Ignore the speed of the river and assume that the river banks ar... | [
"\\(\\frac{2+\\sqrt{5}}{6}\\)"
] | [
"You are standing at the edge of a river which is 1 km wide. The camp on the opposite bank is 1 km downstream from the point directly across from you. You can swim at 2 km/hr and walk at 3 km/hr. What is the shortest time you will take to reach your camp? (Ignore the speed of the river and assume that the river ban... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-0bcea295dd4da6acc1aecb8d48f37de0e5debd856848692306df7bb3d2f07b77 | You are standing at the edge of a river which is 1 km wide. The distance to your camp on the opposite bank, directly across from you, is 1 km. You can swim at 2 km/h and walk at 3 km/h. What is the shortest time you will take to reach your camp? (Ignore the speed of the river and assume the river banks are straight and... | [
"\\(\\frac{2+\\sqrt{5}}{6}\\) hours"
] | [
"You are standing at the edge of a river which is 1 km wide. The distance to your camp on the opposite bank, directly across from you, is 1 km. You can swim at 2 km/h and walk at 3 km/h. What is the shortest time you will take to reach your camp? (Ignore the speed of the river and assume the river banks are straigh... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 21
} | 23 | 0 | 0 |
omr-ws-042d10c73d1a120fb8ae3a54bffeb08e0f85891ae657666a7e7ef0ab0835c68a | You are standing at the point $(3, 0)$ in an infinitely long hallway with sides given by the lines $x = 0$ and $x = 6$. You want to get to the point $(3, 6)$. At each instant, your distance to $(3, 6)$ must either decrease or stay the same. What is the area of the set of points that you could pass through on your journ... | [
"\\(\\frac{21\\pi}{2} + 9\\sqrt{3}\\)"
] | [
"You are standing at the point $(3, 0)$ in an infinitely long hallway with sides given by the lines $x = 0$ and $x = 6$. You want to get to the point $(3, 6)$. At each instant, your distance to $(3, 6)$ must either decrease or stay the same. What is the area of the set of points that you could pass through on your ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b662ddf5d4778e0a2d0d71489f3c70d025fb4d6b62c53788244e3c814d1656d | You are standing on a straight road in the middle of a snowy field. Your maximum walking speed is 4 mph on the road and 2 mph in the field. Let \( A \) be the area, in square miles, of the region you can reach in an hour. Find \( \lfloor 10A \rfloor \). Details and assumptions: - The width of the road is negligible. - ... | [
"180"
] | [
"You are standing on a straight road in the middle of a snowy field. Your maximum walking speed is 4 mph on the road and 2 mph in the field. Let \\( A \\) be the area, in square miles, of the region you can reach in an hour. Find \\( \\lfloor 10A \\rfloor \\).\n\nDetails and assumptions:\n- The width of the road is... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-070b95cfa8f3d3d0ec4eccce7dcf68bc6d838ba78941f42003060398dec48297 | You are standing on one of the faces of a cube. Each turn, you randomly choose another face that shares an edge with the face you are standing on with equal probability, and move to that face. Let $F(n)$ be the probability that you land on the starting face after $n$ turns. Given that $F(8) = \frac{43}{256}$, find $F(1... | [
"1153"
] | [
"You are standing on one of the faces of a cube. Each turn, you randomly choose another face that shares an edge with the face you are standing on with equal probability, and move to that face. Let $F(n)$ be the probability that you land on the starting face after $n$ turns. Given that $F(8) = \\frac{43}{256}$, fin... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-59f05da6c20b8372fb544237c8b8705cdb17fd1414df171483254ef5bff847b1 | You are standing on the integer $1$ on the number line and your goal is to reach the integer $4$. Each roll of a four-sided die (with integers $1$ through $4$) determines your movement: if you roll an integer greater than your current position $x$, you move to $x+1$; if you roll an integer less than $x$, you move to $x... | [
"$\\frac{40}{3}$"
] | [
"You are standing on the integer $1$ on the number line and your goal is to reach the integer $4$. Each roll of a four-sided die (with integers $1$ through $4$) determines your movement: if you roll an integer greater than your current position $x$, you move to $x+1$; if you roll an integer less than $x$, you move ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-af5aefeed2a9a1d4e4f44c982fe83e1ab340302c862b4e772f96ba952c7ea88e | You are tasked with opening a lock with a four-digit combination using decimal digits. Below each digit is a light that is supposed to light up green if the correct digit is dialed and red otherwise. However, the light indicator is defective: at each position where the correct digit is dialed, the light goes green with... | [
"\\text{At each step, guess the combination with the highest current probability based on Bayesian updates from previous feedback.}"
] | [
"You are tasked with opening a lock with a four-digit combination using decimal digits. Below each digit is a light that is supposed to light up green if the correct digit is dialed and red otherwise. However, the light indicator is defective: at each position where the correct digit is dialed, the light goes green... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc3631a4bb6d72ae2f294620317ee6a1964db926f4501abe83bff496c7b3bb52 | You are the manager of a company and need to hire the best secretary from a sequence of \( n \) secretaries who will interview one after another. You must decide immediately after each interview whether to hire the current secretary or not. What strategy should you use to maximize the probability of hiring the best sec... | [
"\\( 0.37 \\)"
] | [
"You are the manager of a company and need to hire the best secretary from a sequence of \\( n \\) secretaries who will interview one after another. You must decide immediately after each interview whether to hire the current secretary or not. What strategy should you use to maximize the probability of hiring the b... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-8b05892fc6c7c3b9b452eae19ab6b2ddbbba87d2b588eb3dcf86f92159b9418e | You are thinking of a number between 0 and 15. You get to ask seven yes-or-no questions, and I am allowed to lie at most once in answering your questions. What seven questions can you ask that will always allow you to determine the number? Note: The questions must be independent of the answers received. | [
"1. Is p equal to 1?\n2. Is q equal to 1?\n3. Is r equal to 1?\n4. Is s equal to 1?\n5. Is p + q + r odd?\n6. Is p + q + s odd?\n7. Is q + r + s odd?"
] | [
"You are thinking of a number between 0 and 15. You get to ask seven yes-or-no questions, and I am allowed to lie at most once in answering your questions. What seven questions can you ask that will always allow you to determine the number? Note: The questions must be independent of the answers received."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 6,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-0ec7c03c67a30000b89fe3e5340aaa07446ff05c4e22360673bf1baf725d84fa | You are to color a $5 \times 5$ grid with one of four colors: blue, red, green, and yellow. The value of a blue cell is $1$, the value of a red cell is $2$, the value of a green cell is $3$, and the value of a yellow cell is $4$. The following conditions must hold for a coloring to be valid: - Any cell can be colored b... | [
"60"
] | [
"You are to color a $5 \\times 5$ grid with one of four colors: blue, red, green, and yellow. The value of a blue cell is $1$, the value of a red cell is $2$, the value of a green cell is $3$, and the value of a yellow cell is $4$. The following conditions must hold for a coloring to be valid:\n- Any cell can be co... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a851359f7135748d8df2f6d82961d8260968a7c3fa91594c82e7f5cd06155838 | You are to give Mr. Arnold $2000. For each person you convince to also give him $2000, you receive $13000. How can you maximize your money if you can convince up to ten people? | [
"128000"
] | [
"You are to give Mr. Arnold $2000. For each person you convince to also give him $2000, you receive $13000. How can you maximize your money if you can convince up to ten people?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c0ca0f5830349cf8a0a931dadd01c217410d3757641f8eabc29abc9cce5abf1a | You are tossing an unbiased coin. The last 28 consecutive flips have all resulted in heads. Let \( x \) be the expected number of additional tosses you must make before you get 60 consecutive heads. Find the sum of all distinct prime factors in \( x \). | [
"65821"
] | [
"You are tossing an unbiased coin. The last 28 consecutive flips have all resulted in heads. Let \\( x \\) be the expected number of additional tosses you must make before you get 60 consecutive heads. Find the sum of all distinct prime factors in \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-301e3e9ce190b9e133ad6567852b239649681b0ffb78f8503bb2c2e276c7a37c | You are walking north on a straight path. Ahead of you and on the east side of the path is a billboard 54 feet across. The billboard is aligned perpendicularly to the path, and the west edge of the billboard is 18 feet from the path. At what point on the path would you see the billboard from the widest possible angle? | [
"36 feet"
] | [
"You are walking north on a straight path. Ahead of you and on the east side of the path is a billboard 54 feet across. The billboard is aligned perpendicularly to the path, and the west edge of the billboard is 18 feet from the path. At what point on the path would you see the billboard from the widest possible an... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-eddf560eebe3732e52ce3133f3dd508f7eaec77d877ea02535b2fd6b38482554 | You are walking up a staircase with stairs that are $1$ ft. tall and $1$ ft. deep. The number of stairs from the ground floor to the first floor is $1$. From the first floor to the second is $2$ stairs. From the 99th to the hundredth stair is $100$ steps. At each floor, the staircase makes a $90^\circ$ turn to the righ... | [
"\\(50\\sqrt{10203}\\)"
] | [
"You are walking up a staircase with stairs that are $1$ ft. tall and $1$ ft. deep. The number of stairs from the ground floor to the first floor is $1$. From the first floor to the second is $2$ stairs. From the 99th to the hundredth stair is $100$ steps. At each floor, the staircase makes a $90^\\circ$ turn to th... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 21
} | 39 | 0 | 0 |
omr-ws-f6260e8ee03eb06857f4dcd87881a95848b1f5469afdcd9e04dab4c701b4cde0 | You arrange 2n biologists of 2n different heights in two rows of n, with the same conditions on height as in part (a). Give a formula in terms of n for the number of possible arrangements. | [
"\\frac{1}{n+1} \\binom{2n}{n}"
] | [
"You arrange 2n biologists of 2n different heights in two rows of n, with the same conditions on height as in part (a). Give a formula in terms of n for the number of possible arrangements."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-22107661b2a0c1a0cc1fc6bac7c68cf2f7e7c845c30c8b610dad97f4f34119bb | You ask a friend to guess a number from 1 to 10. Each time the friend answers, you say "higher" if the friend's number is too low, "lower" if the friend's number is too high, or "you win" if the number is correct. The friend is given 3 attempts to guess the number. What is the probability that your friend wins? | [
"\\(\\frac{7}{10}\\)"
] | [
"You ask a friend to guess a number from 1 to 10. Each time the friend answers, you say \"higher\" if the friend's number is too low, \"lower\" if the friend's number is too high, or \"you win\" if the number is correct. The friend is given 3 attempts to guess the number. What is the probability that your friend wi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 10
} | 24 | 0 | 0 |
omr-ws-a622e6615ca9d0ce0f175747051730bdab1731973aedaf2a8c62004c877bd144 | You begin a journey at 80 degrees west longitude on the equator and fly steadily northwest. How far have you flown before reaching the North Pole? | [
"\\( 10007 \\sqrt{2} \\) km"
] | [
"You begin a journey at 80 degrees west longitude on the equator and fly steadily northwest. How far have you flown before reaching the North Pole?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-35b2dab755ccf4c89fd26403f67809ac5c67e11ad7f66fb9858f8f7d4bb959b3 | You begin by paying $1,000,000$ dollars. The host flips a coin until he gets tails. Your payoff is $2^n$, where $n$ is the number of coin flips. If you decide to play the game until you come out even or earn a profit, what is the expected number of games you will play? | [
"524288"
] | [
"You begin by paying $1,000,000$ dollars. The host flips a coin until he gets tails. Your payoff is $2^n$, where $n$ is the number of coin flips. If you decide to play the game until you come out even or earn a profit, what is the expected number of games you will play?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-f13e77828f4b09d5161323a995924386e3054046c6b2b43cab891c13f1772af4 | You bought \( m \) red pens and \( n \) yellow pens from each of two stores. The first store sold red pens at four for a dollar and yellow pens at three for a dollar. The second store sold red pens at four for a dollar and yellow pens at six for a dollar. You spent a total of ten dollars. How many pens did you buy? | [
"40"
] | [
"You bought \\( m \\) red pens and \\( n \\) yellow pens from each of two stores. The first store sold red pens at four for a dollar and yellow pens at three for a dollar. The second store sold red pens at four for a dollar and yellow pens at six for a dollar. You spent a total of ten dollars. How many pens did you... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ae850598ca764dcd0b26982aa2098b85b9bf4b5369d355b22f258f95d66d58ee | You buy a box of $n$ condoms, where the success rate of the box is a random variable uniformly distributed from 0 to 1. Given that you use all but one of the condoms and they all work, what is the probability that the very last condom will also work, in terms of $n$? | [
"\\(\\frac{n}{n+1}\\)"
] | [
"You buy a box of $n$ condoms, where the success rate of the box is a random variable uniformly distributed from 0 to 1. Given that you use all but one of the condoms and they all work, what is the probability that the very last condom will also work, in terms of $n$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-97889398e799afdb9c792b9a60a20e7fdaf7266233d9bfbdb8e5263095d354cc | You buy a house including the land it sits on for $155,000. The real estate agent tells you that the land costs $24,000 more than the house. What are the prices of the house and the land? | [
"89500"
] | [
"You buy a house including the land it sits on for $155,000. The real estate agent tells you that the land costs $24,000 more than the house. What are the prices of the house and the land?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-76475f12c8c21ffab49d4584f33eee95362ccfdddb73dae648dfc1d8e3fca9d8 | You buy a house including the land it sits on for $155,000. The real estate agent tells you that the land costs $24,000 more than the house. What is the price of the house and the price of the land? | [
"The price of the house is $65,500 and the price of the land is $89,500."
] | [
"You buy a house including the land it sits on for $155,000. The real estate agent tells you that the land costs $24,000 more than the house. What is the price of the house and the price of the land?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 18
} | 18 | 0 | 0 |
omr-ws-309e29f87176e40d3c7861e2dbcdce41b6ab727215acd0b1261c479fe557e030 | You can buy 3 chocolates for 1 unit of currency. If you return 3 chocolate wrappers, you get 1 chocolate. Given 10 units of currency, what is the maximum number of chocolates you can buy? | [
"44"
] | [
"You can buy 3 chocolates for 1 unit of currency. If you return 3 chocolate wrappers, you get 1 chocolate. Given 10 units of currency, what is the maximum number of chocolates you can buy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-b5b6c979d02c9a4a1bad9f387136f45eeabe23bf9bd5e20dcdab6b27f2feff89 | You can buy a toffee for $1$ dollar. If you return $3$ toffee wrappers, you get $1$ more toffee. If you have $n$ dollars, how many toffees can you buy? Provide a general method for any number $n$. | [
"\\( n + \\left\\lfloor \\frac{n}{3} \\right\\rfloor + \\left\\lfloor \\frac{n}{3^2} \\right\\rfloor + \\left\\lfloor \\frac{n}{3^3} \\right\\rfloor + \\cdots + \\left\\lfloor \\frac{n}{3^k} \\right\\rfloor \\)"
] | [
"You can buy a toffee for $1$ dollar. If you return $3$ toffee wrappers, you get $1$ more toffee. If you have $n$ dollars, how many toffees can you buy? Provide a general method for any number $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-6a12bb1e98e58b0d4a98fac1cf9a0a9f6e40fe61a4c876f84a3228dc19127148 | You can choose to play only one of the games. In game one, you roll a die and get the square of your roll in dollars. In game two, you roll a die 11 times and get the square of the median of those 11 numbers in dollars. Which game should you pick to play? | [
"Game 1"
] | [
"You can choose to play only one of the games. In game one, you roll a die and get the square of your roll in dollars. In game two, you roll a die 11 times and get the square of the median of those 11 numbers in dollars. Which game should you pick to play?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-5751c164a12fb23292ac752a5d647f5653f28b621057854ef0bedf57d37f4eee | You can either roll a 10-sided die once and win the value of that roll in dollars, or you can roll an 8-sided die three times and take the highest value. Which option has a higher expected value? | [
"Roll the 8-sided die three times"
] | [
"You can either roll a 10-sided die once and win the value of that roll in dollars, or you can roll an 8-sided die three times and take the highest value. Which option has a higher expected value?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-86aa07e5b85acbff1c205f2a63e5ce24dbf69fb321868331f6300ddb96a34979 | You can either roll a 10-sided die once and win the value of that roll in dollars, or you can roll an 8-sided die twice and take the higher value. Which option has a higher expected value? | [
"5.8125"
] | [
"You can either roll a 10-sided die once and win the value of that roll in dollars, or you can roll an 8-sided die twice and take the higher value. Which option has a higher expected value?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-4e69cce1d4a062b80d94c8e59b522e677f2ed6b1e0fc5e4f4d151c315d92892a | You can paint a room in 7 hours, and your brother can paint the same room in 6 hours. How long will it take both of you to paint the room together? | [
"\\dfrac{42}{13}"
] | [
"You can paint a room in 7 hours, and your brother can paint the same room in 6 hours. How long will it take both of you to paint the room together?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-be83886ffe28aed5b8e3add9ae53b8ca7bdd364029e0075116f14ad80aa28dae | You can roll a die as many times as you want, but each roll costs a dollar. When you decide to stop rolling, the casino pays you the value of your last roll in dollars. What strategy maximizes your expected winnings? | [
"3"
] | [
"You can roll a die as many times as you want, but each roll costs a dollar. When you decide to stop rolling, the casino pays you the value of your last roll in dollars. What strategy maximizes your expected winnings?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 7,
"tir": 9
} | 21 | 0 | 0 |
omr-ws-f4d54236e7d0d861c4e14e578d96b0cb04792d406f98fb9e09f30d7f64801afc | You choose a number from $1$ to $144$, inclusive. I can pick a subset of $\{1, 2, \ldots, 144\}$ and ask whether your number is in the subset. An answer of “yes” costs $2$ dollars, and an answer of “no” costs $1$ dollar. What is the smallest amount of money needed to be sure to find your number? | [
"14"
] | [
"You choose a number from $1$ to $144$, inclusive. I can pick a subset of $\\{1, 2, \\ldots, 144\\}$ and ask whether your number is in the subset. An answer of “yes” costs $2$ dollars, and an answer of “no” costs $1$ dollar. What is the smallest amount of money needed to be sure to find your number?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2b876479560433d97839b7710d72d5a3400b976691fb395ffca2d0221006b2c | You choose a subset from the set \(\{1, 2, \ldots, 144\}\) and ask if a chosen number from 1 to 144 is in your subset. If the answer is yes, you get $2, and if the answer is no, you get $1. What is the least amount of money you can get? | [
"144"
] | [
"You choose a subset from the set \\(\\{1, 2, \\ldots, 144\\}\\) and ask if a chosen number from 1 to 144 is in your subset. If the answer is yes, you get $2, and if the answer is no, you get $1. What is the least amount of money you can get?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-d8e0f458168656070a9884ef2a846406164c5981ca81b3ccb5e7c8acf65743bc | You choose three numbers without replacement from the set \(\{1, 2, \ldots, 1000\}\), say \(a\), \(b\), and \(c\). Then you choose three numbers from the remaining 997 numbers, say \(e\), \(f\), and \(g\). The probability that a box of dimension \(a \times b \times c\) can be put into a box of dimension \(e \times f \t... | [
"5"
] | [
"You choose three numbers without replacement from the set \\(\\{1, 2, \\ldots, 1000\\}\\), say \\(a\\), \\(b\\), and \\(c\\). Then you choose three numbers from the remaining 997 numbers, say \\(e\\), \\(f\\), and \\(g\\). The probability that a box of dimension \\(a \\times b \\times c\\) can be put into a box of... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-08868aa05283ee498857b6b3c92b088ab1da00dce7ef01c574fe71f24b2b16c4 | You draw 4 cards from a standard deck of 52 cards. What is the probability that all of these cards have different denominations and there are no neighboring denominations (e.g., an 8 and a 9, a 10 and a Jack, or a queen and a king are considered neighbors)? | [
"\\(\\frac{1536}{7735}\\)"
] | [
"You draw 4 cards from a standard deck of 52 cards. What is the probability that all of these cards have different denominations and there are no neighboring denominations (e.g., an 8 and a 9, a 10 and a Jack, or a queen and a king are considered neighbors)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-57f2a1b07668f6548556108ace1846c8b3252c92d962589ac924319ebe0c6c2d | You draw a 2x6 grid of cells with the word "DOITSU" written in each row. During each move, you may swap the letters in any two cells that share a common vertex. Find the number of ways to swap letters such that after 10 moves, the grid returns to its starting position. | [
"26^{10}"
] | [
"You draw a 2x6 grid of cells with the word \"DOITSU\" written in each row. During each move, you may swap the letters in any two cells that share a common vertex. Find the number of ways to swap letters such that after 10 moves, the grid returns to its starting position."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-96302a9d5a7b55d9ebcf3f5ffea555a414609a7704939b38f527f482cda8aa1a | You earn \$10 per hour working as a manager at a grocery store and need to work at least 8 hours per week. You also teach music lessons for \$15 per hour. You need to earn at least \$120 per week but do not want to work more than 20 hours per week. Write and graph a system of linear inequalities that represents this si... | [
"\\( x \\geq 8 \\), \\( x + y \\leq 20 \\), \\( 10x + 15y \\geq 120 \\)"
] | [
"You earn \\$10 per hour working as a manager at a grocery store and need to work at least 8 hours per week. You also teach music lessons for \\$15 per hour. You need to earn at least \\$120 per week but do not want to work more than 20 hours per week. Write and graph a system of linear inequalities that represents... | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 2
} | 34 | 0 | 0 |
omr-ws-206f230542e5bcc431a9542ea84e0e6c39f159c69d9dbcc46f543dd74cbf35aa | You enter a draw by picking a number between 1 and 100. The winning number is two-thirds of the average of all numbers picked. Which number should you pick? | [
"1"
] | [
"You enter a draw by picking a number between 1 and 100. The winning number is two-thirds of the average of all numbers picked. Which number should you pick?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7ac073b6ebac30ef67d30c85ac9b1692e1cbe2900fc94a736380ea9abb7f38f3 | You flip a fair coin 2018 times and form a sequence using the heads/tails. Denote \( n \) as the sum of the squares of the lengths of each streak in this sequence (for example, if a portion of the sequence was HHHTH, it would be \( 3^2 + 1^2 + 1^2 \)). Find the expected value of \( n \). | [
"\\( 6050 + \\frac{1}{2^{2016}} \\)"
] | [
"You flip a fair coin 2018 times and form a sequence using the heads/tails. Denote \\( n \\) as the sum of the squares of the lengths of each streak in this sequence (for example, if a portion of the sequence was HHHTH, it would be \\( 3^2 + 1^2 + 1^2 \\)). Find the expected value of \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 6
} | 18 | 0 | 0 |
omr-ws-9145742a83212d9963a5cc9ade65c40597f977fdff1a31efa0fc2ba5ffa624b4 | You flip a fair coin repeatedly until either four consecutive Heads (H) or six consecutive Tails (T) occur. What is the probability that the sequence HHHH occurs before the sequence TTTTTT? | [
"\\(\\frac{21}{26}\\)"
] | [
"You flip a fair coin repeatedly until either four consecutive Heads (H) or six consecutive Tails (T) occur. What is the probability that the sequence HHHH occurs before the sequence TTTTTT?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-c23ae0e318b6b009c19f36c7df9e5d9b1fdfcaf6fedd9cbacc6a436b5487efe1 | You flip a fair coin repeatedly until either four consecutive heads or six consecutive tails occur. What is the probability that the sequence HHHH occurs before the sequence TTTTTT? | [
"\\dfrac{21}{26}"
] | [
"You flip a fair coin repeatedly until either four consecutive heads or six consecutive tails occur. What is the probability that the sequence HHHH occurs before the sequence TTTTTT?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ce7834ccc33429c741064842b6ca671674e99be76b5dfd0eeb30e232c791ed6 | You flip coin A twice and coin B four times. What is the probability that the number of heads obtained from coin A is equal to the number of heads obtained from coin B? | [
"\\(\\frac{15}{64}\\)"
] | [
"You flip coin A twice and coin B four times. What is the probability that the number of heads obtained from coin A is equal to the number of heads obtained from coin B?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-c721c923377ce3a8dc077617b9c9dbe9612ed6157a5cd0cf3742fe112d3090cb | You give someone the following instructions (which they follow accurately): - Choose any $4$-digit number. - Rearrange the digits to form a new $4$-digit number. - Subtract the smaller number from the larger number. - Circle one of the nonzero digits. - Rearrange the remaining digits and tell me the resulting number. T... | [
"5"
] | [
"You give someone the following instructions (which they follow accurately):\n- Choose any $4$-digit number.\n- Rearrange the digits to form a new $4$-digit number.\n- Subtract the smaller number from the larger number.\n- Circle one of the nonzero digits.\n- Rearrange the remaining digits and tell me the resulting... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-86aa1ce84fdd8e2e0dc153659430ec37479a91d5378f46873eaebc4fd35d3c30 | You glue two pentagon prisms together on their regular pentagon bases. Each lateral face can be colored either red or blue. How many rotationally distinct colorings are there? | [
"120"
] | [
"You glue two pentagon prisms together on their regular pentagon bases. Each lateral face can be colored either red or blue. How many rotationally distinct colorings are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-8f126a06766f0c432d93b71b318fa75dacbde9275e7247c7b9863e81f5bc16b3 | You go on flipping two fair dice until you get a sum of $5$ or $7$. Find the probability of getting a sum of $5$ before a sum of $7$. | [
"\\(\\frac{2}{5}\\)"
] | [
"You go on flipping two fair dice until you get a sum of $5$ or $7$. Find the probability of getting a sum of $5$ before a sum of $7$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-9c52a64b6778d195344a978646e86585949c26cccad8e61ae58bb3db9557ffbd | You have $ m$ girls sitting around a round table, with $ m$ even. Initially, one girl has $ n$ cards. In every move, if a girl has at least $ 2$ cards, she gives one card to each of her neighbors. For which values of $ n$ do the girls eventually stop passing cards? | [
"\\( n \\leq m \\)"
] | [
"You have $ m$ girls sitting around a round table, with $ m$ even. Initially, one girl has $ n$ cards. In every move, if a girl has at least $ 2$ cards, she gives one card to each of her neighbors. For which values of $ n$ do the girls eventually stop passing cards?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d62e5d0ebbd30ee462ac9107ed02b55d4a031f18bb3bff77010e5a9606e202ad | You have $ m$ girls sitting around a round table, with $ m$ even. Initially, one girl has $ n$ cards. In every move, if a girl has at least $ 2$ cards, she gives one card to each of her neighbors. For which values of $ n$ must the girls pass cards forever? | [
"\\( n > m \\)"
] | [
"You have $ m$ girls sitting around a round table, with $ m$ even. Initially, one girl has $ n$ cards. In every move, if a girl has at least $ 2$ cards, she gives one card to each of her neighbors. For which values of $ n$ must the girls pass cards forever?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-34c70bfcc71441634461f422db7085ba0cdbe2c88d4033a2ac9c44abdd7b86a2 | You have $100$ dollars. You may place zero bets, or $n>0$ bets each worth $100/n$ dollars. Each time you bet, you double the amount of money you bet or lose it with a probability of $1/2$. What is the optimal number of bets $n$ to maximize your expected final amount of money? | [
"\\( n \\to \\infty \\) (or the largest feasible \\( n \\)).\n\nSince the problem does not specify a maximum \\( n \\), the answer is:"
] | [
"You have $100$ dollars. You may place zero bets, or $n>0$ bets each worth $100/n$ dollars. Each time you bet, you double the amount of money you bet or lose it with a probability of $1/2$. What is the optimal number of bets $n$ to maximize your expected final amount of money?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1dcda9f77a0823bee21a5abac4b70551cc918db26f711d6f2f5acbf58cfb63a0 | You have $1000 and want to buy 1000 animals. The animals available are cows ($10 each), pigs ($3 each), and chickens ($0.5 each). How many of each animal can you buy to use exactly $1000 and have exactly 1000 animals? | [
"50 cows, 10 pigs, 940 chickens"
] | [
"You have $1000 and want to buy 1000 animals. The animals available are cows ($10 each), pigs ($3 each), and chickens ($0.5 each). How many of each animal can you buy to use exactly $1000 and have exactly 1000 animals?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-3563d7f0f896abd5d36a25fa1638d53d70c227ab2d18f7e3ea20f7aeaa03f155 | You have $1000. Every year you earn 20% more than the previous year. How much money will you have after 30 years? | [
"237376"
] | [
"You have $1000. Every year you earn 20% more than the previous year. How much money will you have after 30 years?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4cd4b1be78c042a0cd99a2cea6ce6e83f37cd1dec827a6fd9a50c827b034cea2 | You have $13$ loaded coins with probabilities of heads $\cos\left( \frac{0\pi}{24 }\right)$, $\cos\left( \frac{1\pi}{24 }\right)$, $\cos\left( \frac{2\pi}{24 }\right)$, ..., $\cos\left( \frac{12\pi}{24 }\right)$. What is the probability that an even number of them come up as heads? | [
"\\dfrac{1}{2}"
] | [
"You have $13$ loaded coins with probabilities of heads $\\cos\\left( \\frac{0\\pi}{24 }\\right)$, $\\cos\\left( \\frac{1\\pi}{24 }\\right)$, $\\cos\\left( \\frac{2\\pi}{24 }\\right)$, ..., $\\cos\\left( \\frac{12\\pi}{24 }\\right)$. What is the probability that an even number of them come up as heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a3127a5b6757fcf3c888a71e23260b7d8e7dceb7986acf6716d9b1c7b3926b76 | You have $17$ apples and $7$ friends, and you want to distribute apples to your friends. The only requirement is that Steven, one of your friends, does not receive more than half of the apples. Given that apples are indistinguishable and friends are distinguishable, compute the number of ways the apples can be distribu... | [
"95942"
] | [
"You have $17$ apples and $7$ friends, and you want to distribute apples to your friends. The only requirement is that Steven, one of your friends, does not receive more than half of the apples. Given that apples are indistinguishable and friends are distinguishable, compute the number of ways the apples can be dis... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aab366d0a184a6d77eee078d343ba60f14053e84e65271b71270b2823b0341a5 | You have $27$ identical unit cubes, where on each cube the three faces adjacent to one vertex are colored red, and the three faces adjacent to the opposite vertex are colored blue. The cubes are assembled randomly into a single $3$ by $3$ by $3$ cube. The probability that the outside of this cube is entirely a single c... | [
"27"
] | [
"You have $27$ identical unit cubes, where on each cube the three faces adjacent to one vertex are colored red, and the three faces adjacent to the opposite vertex are colored blue. The cubes are assembled randomly into a single $3$ by $3$ by $3$ cube. The probability that the outside of this cube is entirely a sin... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7eeedb5215a83064ebc2d40e7c6ee449a03d478baffcbf17a0db0ab03900b7c9 | You have $2^k$ balls distributed in $n$ boxes. An operation consists of picking two non-empty boxes and doubling the balls in one box by taking them from the other box. After performing a series of these operations, you manage to put all the balls into one box. Calculate the minimum number of operations required to ach... | [
"no_valid_answer_found"
] | [
"You have $2^k$ balls distributed in $n$ boxes. An operation consists of picking two non-empty boxes and doubling the balls in one box by taking them from the other box. After performing a series of these operations, you manage to put all the balls into one box. Calculate the minimum number of operations required t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-684435b4787df8c55fec42442dff50fd85d617dcbfdad25bb3ea2b5716538ffc | You have $5$ red shirts, $5$ green shirts, $6$ pairs of pants, $8$ green hats and $8$ red hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat without having the same color of shirts and hats? | [
"480"
] | [
"You have $5$ red shirts, $5$ green shirts, $6$ pairs of pants, $8$ green hats and $8$ red hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat without having the same color of shirts and hats?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-777c87e661b89c409f818a2fa16a3d24e7c44d1931f95ca964089869a6be5cff | You have $n$ coins that are each worth a distinct, positive integer amount of cents. To hitch a ride with Charon, you must pay some unspecified integer amount between 10 and 20 cents inclusive, and Charon wants exact change paid with exactly two coins. What is the least possible value of $n$ such that you can be certai... | [
"6"
] | [
"You have $n$ coins that are each worth a distinct, positive integer amount of cents. To hitch a ride with Charon, you must pay some unspecified integer amount between 10 and 20 cents inclusive, and Charon wants exact change paid with exactly two coins. What is the least possible value of $n$ such that you can be c... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-c2e29b2333a311e68cacd6ccf48725f9088df9a2fd40154b9bf936a67ef6a9c3 | You have $n$ items in a list. You choose a number $p > 0$ and cross out every $p$-th item, continuing down the list and restarting from the top until only one item remains. The last item left is expressed as $n - d$, where $d \in \mathbb{N} \cup \{0\}$. Find $d$ in terms of $n$ and/or $p$. | [
"n - J(n, p) - 1"
] | [
"You have $n$ items in a list. You choose a number $p > 0$ and cross out every $p$-th item, continuing down the list and restarting from the top until only one item remains. The last item left is expressed as $n - d$, where $d \\in \\mathbb{N} \\cup \\{0\\}$. Find $d$ in terms of $n$ and/or $p$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-6fdd070fb4dc65143d6dc344460c9adabd7a3e205a9c8e1f07673a322c583c58 | You have $n$ particles of a radioactive substance. Every second, each particle has a $\frac{1}{2}$ chance of decaying. What is the expected number of seconds before all particles have decayed? | [
"\\(2^n\\)"
] | [
"You have $n$ particles of a radioactive substance. Every second, each particle has a $\\frac{1}{2}$ chance of decaying. What is the expected number of seconds before all particles have decayed?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e4fb63cbe7e3d858f9d84c612d4d3363178ad0f813c916700458550d4e243f9b | You have $n$ people sitting around a circular table numbered $1$ to $n$. Starting from person $2$, every other person (that is, person $4$, then $6$, etc.) gets up, skipping the people who have already got up. Find a formula in terms of $n$ for the number of the last person to be sitting. | [
"\\( 2(n - 2^p) + 1 \\)"
] | [
"You have $n$ people sitting around a circular table numbered $1$ to $n$. Starting from person $2$, every other person (that is, person $4$, then $6$, etc.) gets up, skipping the people who have already got up. Find a formula in terms of $n$ for the number of the last person to be sitting."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 5
} | 22 | 0 | 0 |
omr-ws-503f76e7a9dbe340e3d9f717dc23ace6258fd8d562ef5ed0207b299d7beb0332 | You have $n$ red and $n$ blue balls randomly arranged in a circle. What is the probability that no two red balls are next to each other? | [
"\\(\\frac{2}{\\binom{2n}{n}}\\)"
] | [
"You have $n$ red and $n$ blue balls randomly arranged in a circle. What is the probability that no two red balls are next to each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 5,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-39a02a4fa08e53592578950bd9516eba661e9ed43f9412c4aeb8398c9c331d8b | You have $n$ types of fruits, with $a_i$ fruits of the $i$-th type. In how many ways can you make salads if you do not choose two fruits of the same type and you must take at least one type of fruit? Example: If you have 2 types of fruits, say $a$ and $b$, with 3 fruits of type $a$ ($a_1, a_2, a_3$) and 2 fruits of typ... | [
"\\(\\left(\\displaystyle\\prod_{i=1}^n (a_i+1)\\right) - 1\\)"
] | [
"You have $n$ types of fruits, with $a_i$ fruits of the $i$-th type. In how many ways can you make salads if you do not choose two fruits of the same type and you must take at least one type of fruit?\n\nExample: If you have 2 types of fruits, say $a$ and $b$, with 3 fruits of type $a$ ($a_1, a_2, a_3$) and 2 fruit... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-d8db85335361691d686a772b238d8e3da42e788f5048229ba252fd51b87c18a5 | You have 10 dollars to bet on a fair coin flip (1/2 chance of getting heads and 1/2 chance of getting tails). If you lose the bet, you lose 1 dollar, but if you win the bet, you get a profit of 2 dollars. If you hit 0 dollars, you cannot bet again. What is the probability of you ever running out of money as the number ... | [
"1"
] | [
"You have 10 dollars to bet on a fair coin flip (1/2 chance of getting heads and 1/2 chance of getting tails). If you lose the bet, you lose 1 dollar, but if you win the bet, you get a profit of 2 dollars. If you hit 0 dollars, you cannot bet again. What is the probability of you ever running out of money as the nu... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-94109b3e5bcad1e58acd469d8009505ec753bf2d1079cff3dadf5abdea100265 | You have 100 cards numbered from 1 to 100 on a table. You also have a pile of 99 blank cards. You repeatedly remove two cards from the table, compute the difference between the numbers on those cards, write the difference on a blank card, and place the blank card back on the table. This process continues until only one... | [
"2550"
] | [
"You have 100 cards numbered from 1 to 100 on a table. You also have a pile of 99 blank cards. You repeatedly remove two cards from the table, compute the difference between the numbers on those cards, write the difference on a blank card, and place the blank card back on the table. This process continues until onl... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-6bd87c89bb1d2d56921c035ece5c6a7e98bb13f15767d07af5053c99552d8bc4 | You have 100 jars of candies, each containing 20 candies. All candies weigh 1 gram except for those in one jar, which weigh 1.1 grams each. Using an electronic scale that is accurate to infinite decimal places, what is the minimum number of weightings needed to determine the jar with the heavier candies? | [
"2"
] | [
"You have 100 jars of candies, each containing 20 candies. All candies weigh 1 gram except for those in one jar, which weigh 1.1 grams each. Using an electronic scale that is accurate to infinite decimal places, what is the minimum number of weightings needed to determine the jar with the heavier candies?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-ac9876cfb32095de18da7960a3102464ce34723df63bbcec8ed1cee005b88346 | You have 100 red cards, 200 blue cards, and 300 green cards. You draw one card at a time with equal probability and discard the card afterwards. What is the probability that all the red cards are drawn before all the blue or green cards? | [
"\\(\\frac{7}{12}\\)"
] | [
"You have 100 red cards, 200 blue cards, and 300 green cards. You draw one card at a time with equal probability and discard the card afterwards. What is the probability that all the red cards are drawn before all the blue or green cards?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 5,
"tir": 2
} | 13 | 0 | 0 |
omr-ws-0558713d3c4a3e2904bcd54c92d0ce9a0cc6070586cb3499f706709d6a1d3101 | You have 1000 coins, all initially face up. You flip every second coin (2, 4, 6, ...), then every third coin (3, 6, 9, ...), and continue this process until you flip the 1000th coin. How many coins are face up after this process is completed? | [
"31"
] | [
"You have 1000 coins, all initially face up. You flip every second coin (2, 4, 6, ...), then every third coin (3, 6, 9, ...), and continue this process until you flip the 1000th coin. How many coins are face up after this process is completed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-e5bc80102577027506447f61c67c7598a234f5748de4b2eacbb70f675dfb1fae | You have 1000 friends numbered 1 to 1000. Person 1 refuses to go to any party. Person 2 will go to a party only if one specific friend is there, but not if more than one additional person is present. Person 3 will go to a party only if two specific friends are there, but not if more than two additional people are prese... | [
"500"
] | [
"You have 1000 friends numbered 1 to 1000. Person 1 refuses to go to any party. Person 2 will go to a party only if one specific friend is there, but not if more than one additional person is present. Person 3 will go to a party only if two specific friends are there, but not if more than two additional people are ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-75422bf83e290efddb3189154526367bb6f7c07bbed98c7ac81a59245acf019b | You have 12 coins, 11 of which are exactly the same, and 1 of which has a different weight (either heavier or lighter). Using a balance scale that only indicates which side is heavier, determine which coin is the odd one out in exactly three weighings. | [
"The fake coin can be identified in three weighings by systematically dividing and comparing groups, tracking possible heavy or light candidates based on scale tilts."
] | [
"You have 12 coins, 11 of which are exactly the same, and 1 of which has a different weight (either heavier or lighter). Using a balance scale that only indicates which side is heavier, determine which coin is the odd one out in exactly three weighings."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-6ca9ed62b3ab44bfb5cc66b89528825ab3b5f2c53d64104d041189199ed2ad07 | You have 12 gold coins, one of which is counterfeit and has a different mass than the others. Using a scale at most three times, determine which coin is counterfeit and whether it is lighter or heavier than the real coins. | [
"\\text{The counterfeit coin can be identified and determined if lighter or heavier in three weighings by strategically comparing groups and analyzing the balance outcomes.}"
] | [
"You have 12 gold coins, one of which is counterfeit and has a different mass than the others. Using a scale at most three times, determine which coin is counterfeit and whether it is lighter or heavier than the real coins."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40a54bff63754f99c1c4942acc05a8b5e4b524d19cba31127c570c2a4de255c8 | You have 12 packs of soda cans, each containing 12 cans. One of these packs contains cans that weigh 10% less than the others. Given that you know the weight of one regular soda can as \( X \), how many times do you need to use an electronic scale to ensure you identify the pack with the lighter cans? | [
"1"
] | [
"You have 12 packs of soda cans, each containing 12 cans. One of these packs contains cans that weigh 10% less than the others. Given that you know the weight of one regular soda can as \\( X \\), how many times do you need to use an electronic scale to ensure you identify the pack with the lighter cans?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a5f025507fce16d92010ad62c9c5cada5cf02dd2303e2cf4d5b399c1cb8bc1fb | You have 14 bread rolls, 19 beef patties, and 16 cheese slices. Plain burgers require 1 beef patty and 1 bread roll, double cheeseburgers require 2 beef patties, 1 bread roll, and 2 slices of cheese, and regular cheeseburgers require 1 beef patty, 1 bread roll, and 1 slice of cheese. How many of each type of burger sho... | [
"3 plain burgers, 5 double cheeseburgers, and 6 regular cheeseburgers"
] | [
"You have 14 bread rolls, 19 beef patties, and 16 cheese slices. Plain burgers require 1 beef patty and 1 bread roll, double cheeseburgers require 2 beef patties, 1 bread roll, and 2 slices of cheese, and regular cheeseburgers require 1 beef patty, 1 bread roll, and 1 slice of cheese. How many of each type of burge... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-e85e6275cdf8261d5e0ae5d6b7a3b237f33f2b083a1e768780ff1a742139cefc | You have 20 coin machines, each producing coins of the same weight. One machine is defective and produces coins that weigh 1 ounce less than they should. Using one electronic weighing machine, how can you determine which machine is defective with only one weighing? | [
"190"
] | [
"You have 20 coin machines, each producing coins of the same weight. One machine is defective and produces coins that weigh 1 ounce less than they should. Using one electronic weighing machine, how can you determine which machine is defective with only one weighing?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-24418111503f97bc19df1573695df81abf54669356659ca20e99469fe18e1987 | You have 20 jars with at least 20 identical red candies. In 19 of the jars, each candy weighs 1 gram, but in one jar, each candy weighs 1.1 grams. Using an electronic scale that is accurate to the gram, how can you determine which jar has the heavier candies with only one weighing? | [
"7"
] | [
"You have 20 jars with at least 20 identical red candies. In 19 of the jars, each candy weighs 1 gram, but in one jar, each candy weighs 1.1 grams. Using an electronic scale that is accurate to the gram, how can you determine which jar has the heavier candies with only one weighing?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-2fd55a7d3e845bc41492f867d99281a220a25d5dfbff3db789d3605deb04014a | You have 20 western, 15 romance, and 25 mystery novels to choose from. What is the probability that you choose at least 1 of each type when randomly selecting 4 books? | [
"\\(0.4383\\)"
] | [
"You have 20 western, 15 romance, and 25 mystery novels to choose from. What is the probability that you choose at least 1 of each type when randomly selecting 4 books?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-7db7c9524d9f237c1051f990b405e9b37c0cbd7542a52059322379bff0cf794d | You have 2003 switches arranged in a circle, each initially either ON or OFF. For each switch \( S \), if the two adjacent switches were initially in the same position, set \( S \) to ON; otherwise, set \( S \) to OFF. What is the probability that all switches will now be ON? | [
"\\(\\left(\\frac{1}{2}\\right)^{2002}\\)"
] | [
"You have 2003 switches arranged in a circle, each initially either ON or OFF. For each switch \\( S \\), if the two adjacent switches were initially in the same position, set \\( S \\) to ON; otherwise, set \\( S \\) to OFF. What is the probability that all switches will now be ON?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-84bca2014e8c065491f3df56c55324974d7877dd56f1fe6b3877735959eea2e1 | You have 2017 handwritten letters, each signed by at least one family member. You want to organize these letters into folders such that each folder contains letters that are either all signed by the same family member or each signed by a different family member. What is the least number of folders necessary to organize... | [
"1009"
] | [
"You have 2017 handwritten letters, each signed by at least one family member. You want to organize these letters into folders such that each folder contains letters that are either all signed by the same family member or each signed by a different family member. What is the least number of folders necessary to org... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-74c421ad11492959521694d0a71f5553969b843e40ffb41d543857d2ffb78625 | You have 2048 letters numbered 1 to 2048, stacked in order with 1 on top. You address every other letter starting with 1, putting addressed letters in an outbox and stacking the skipped letters. After the first pass, you repeat the procedure with the remaining letters until only one letter is left. What is the number o... | [
"1366"
] | [
"You have 2048 letters numbered 1 to 2048, stacked in order with 1 on top. You address every other letter starting with 1, putting addressed letters in an outbox and stacking the skipped letters. After the first pass, you repeat the procedure with the remaining letters until only one letter is left. What is the num... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-750e86a925cb634fb6d672da7ec5bad29c9c908795853ca98183ca3fface8c8f | You have 2048 letters numbered 1 to 2048. You address every other letter starting with #1 and put the addressed letters in an outbox. The skipped letters are stacked in reverse order. You repeat this process until only one letter is left. What is the number of the last remaining letter? | [
"1366"
] | [
"You have 2048 letters numbered 1 to 2048. You address every other letter starting with #1 and put the addressed letters in an outbox. The skipped letters are stacked in reverse order. You repeat this process until only one letter is left. What is the number of the last remaining letter?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-907b3aaf374de03402e3e698a80a4d7f8f52ce713b6cc91d65b106257825d1bf | You have 21 jars, each containing 21 balls marked with numbers from 1 to 21. You pick one ball from each jar. What is the probability that the product of the numbers on the picked balls is completely divisible by 21? | [
"\\( 1 - \\frac{14^{21} + 18^{21} - 12^{21}}{21^{21}} \\)"
] | [
"You have 21 jars, each containing 21 balls marked with numbers from 1 to 21. You pick one ball from each jar. What is the probability that the product of the numbers on the picked balls is completely divisible by 21?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 13
} | 19 | 0 | 0 |
omr-ws-def5a1587284cb9c0e0ae0047bdaafa5042c774223e80dc4fd041d026b305508 | You have 3 bottles, 15 red pills, 15 black pills, and 15 white pills. The red pills and the black pills are both poisonous on their own, but neutralize each other if taken together. The white pills are harmless. You distribute the pills into the 3 bottles so that each pill is in a bottle and each bottle has at least on... | [
"\\(\\frac{73}{132}\\)"
] | [
"You have 3 bottles, 15 red pills, 15 black pills, and 15 white pills. The red pills and the black pills are both poisonous on their own, but neutralize each other if taken together. The white pills are harmless. You distribute the pills into the 3 bottles so that each pill is in a bottle and each bottle has at lea... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de3ae9c51ddb1710d29a5d435bf657f604f19bc0a85db01fbf22ed79f9857671 | You have 3 coins. You toss each coin 200 times. What is the probability that all three coins will land tails exactly 27 times? | [
"\\({{200}\\choose{27}}\\times \\left(\\frac 18\\right)^{27}\\times\\left(\\frac 78\\right)^{173}\\)"
] | [
"You have 3 coins. You toss each coin 200 times. What is the probability that all three coins will land tails exactly 27 times?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-11793c411410e52470ea1da9e32dd7dd304f1475777cf3a319354ea17162bc24 | You have 3 different colors to paint the squares of a 3x3 grid. The total number of squares to be painted is 5. Each painted square must be adjacent (horizontally or vertically) to another painted square. Adjacent squares (horizontally or vertically) should be painted in different colors. How many different ways can th... | [
"516"
] | [
"You have 3 different colors to paint the squares of a 3x3 grid. The total number of squares to be painted is 5. Each painted square must be adjacent (horizontally or vertically) to another painted square. Adjacent squares (horizontally or vertically) should be painted in different colors. How many different ways c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-db07a02907163d065adbfc81b4a95f6937f504d48912930d41ca9e5be41cb480 | You have 3 red balls, 4 green balls, and 5 blue balls. Without seeing, you choose 3 balls. What is the probability that you have picked 2 red and 1 blue ball? (Balls are identical.) | [
"\\(\\frac{3}{44}\\)"
] | [
"You have 3 red balls, 4 green balls, and 5 blue balls. Without seeing, you choose 3 balls. What is the probability that you have picked 2 red and 1 blue ball? (Balls are identical.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d9dfa3059b8f46782259d9a8e9e4a5ed9378378bb3d80b1e6cfb0286aa376d81 | You have 3 types of pearls: type 1 can be of color from 1 to \( x \), type 2 can be of color from \( x+1 \) to \( x+y \), and type 3 can be of color from \( x+y+1 \) to \( x+y+z \). The total number of pearls of type 1 and type 3 is \( A \), and the total number of pearls of type 2 and type 3 is \( B \). Two chains of ... | [
"\\sum_{k=0}^{\\min(A,B)} \\frac{(A + B - k)!}{(A - k)! \\, (B - k)! \\, k!} \\cdot x^{A - k} y^{B - k} z^{k}"
] | [
"You have 3 types of pearls: type 1 can be of color from 1 to \\( x \\), type 2 can be of color from \\( x+1 \\) to \\( x+y \\), and type 3 can be of color from \\( x+y+1 \\) to \\( x+y+z \\). The total number of pearls of type 1 and type 3 is \\( A \\), and the total number of pearls of type 2 and type 3 is \\( B ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-19674a085c56c34fa73523647deb2f93e01305386bd1c55acae224ab9de06dc5 | You have 30 bags of coins, each containing 100 coins. The weight of each coin is a one-digit number in grams, and the scale can weigh at most 999 grams in each weighing. Determine the maximum number of bags whose total weight can be uniquely determined with exactly 9 weighings. | [
"9"
] | [
"You have 30 bags of coins, each containing 100 coins. The weight of each coin is a one-digit number in grams, and the scale can weigh at most 999 grams in each weighing. Determine the maximum number of bags whose total weight can be uniquely determined with exactly 9 weighings."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68c82510ef7815cfdd8c739491eb62ef9abf25e76c1aa67f8be957db39dc8ae5 | You have 3000 bananas and need to transport them across a 1000-mile desert using a camel that can carry 1000 bananas at a time. Each mile traveled consumes one banana. What is the maximum number of bananas you can get to the other side of the desert? | [
"500"
] | [
"You have 3000 bananas and need to transport them across a 1000-mile desert using a camel that can carry 1000 bananas at a time. Each mile traveled consumes one banana. What is the maximum number of bananas you can get to the other side of the desert?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cb01e2b2bd7fa951a597edff827c8a36d1bac41ed419bf06db46420f3cd3152f | You have 4 dice to roll. The first roll is set as \( n \). If 1 of the last 3 rolls is the same as the first roll, you win \$1. If 2 of the last 3 rolls are the same as the first roll, you win \$2. If all 3 of the last rolls are the same as the first roll, you win \$3. Otherwise, if all 4 rolls are different, you lose ... | [
"\\(-\\frac{41}{108}\\)"
] | [
"You have 4 dice to roll. The first roll is set as \\( n \\). If 1 of the last 3 rolls is the same as the first roll, you win \\$1. If 2 of the last 3 rolls are the same as the first roll, you win \\$2. If all 3 of the last rolls are the same as the first roll, you win \\$3. Otherwise, if all 4 rolls are different,... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2375c31dc0a62c75040fa3e46fcd61b4cb5580605528deb6630cd09f2511b167 | You have 4 different two-digit numbers such that the product of any three of them is a perfect square. What is the least number among those 4? | [
"16"
] | [
"You have 4 different two-digit numbers such that the product of any three of them is a perfect square. What is the least number among those 4?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-005c5ced6ae1863e7378185276fd62545139855950d90ba60a5d2197f9d56838 | You have 4 identical black socks, 4 identical blue socks, and 4 identical white socks. If you place them in a row, the probability that 2 identical socks are beside each other is $\frac{1}{x}$. What is $x$? | [
"385"
] | [
"You have 4 identical black socks, 4 identical blue socks, and 4 identical white socks. If you place them in a row, the probability that 2 identical socks are beside each other is $\\frac{1}{x}$. What is $x$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e14d1d8ed82e64e739bc6627f90fbc0d2b1f1fd4290b9f951deba450b430cfb | You have 4 light bulbs whose lifetimes are independent normal random variables with mean 100 hours and standard deviation 5 hours. Use the central limit theorem to approximate the probability that the 4 light bulbs together live for at least 402 hours. The answer should be written as an explicit integral involving the ... | [
"\\int_{402}^{\\infty} \\frac{1}{10\\sqrt{2\\pi}} e^{-\\frac{(x - 400)^2}{200}} \\, dx"
] | [
"You have 4 light bulbs whose lifetimes are independent normal random variables with mean 100 hours and standard deviation 5 hours. Use the central limit theorem to approximate the probability that the 4 light bulbs together live for at least 402 hours. The answer should be written as an explicit integral involving... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-51758fa6e8d467b3f779fbb50870702274eeb08924b5bd8446dacb0f02e9ebcb | You have 4 red marbles and 8 blue marbles. Randomly, you place the 12 marbles into a circle. Find the probability that no two red marbles are next to each other. | [
"\\(\\frac{7}{33}\\)"
] | [
"You have 4 red marbles and 8 blue marbles. Randomly, you place the 12 marbles into a circle. Find the probability that no two red marbles are next to each other."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 15
} | 37 | 0 | 0 |
omr-ws-822588eebab744e428108851e3d43fd90c6d7d1049174f9468baf4baf29e756b | You have 5 balls, each of a different color, in a box. Each time, you draw two balls in order. The second ball is repainted to be the same color as the first ball that was drawn. What is the expected value of the number of draws required for all balls to become the same color? | [
"16"
] | [
"You have 5 balls, each of a different color, in a box. Each time, you draw two balls in order. The second ball is repainted to be the same color as the first ball that was drawn. What is the expected value of the number of draws required for all balls to become the same color?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-2a4e32343821c723859c5eb99f4bc79642763c7531420adf2849d44d2219204c | You have 5 boxes and 2005 balls, with 286 blue, 645 green, and 1074 red balls. You place balls in the boxes as follows: - Box 1: 1 blue, 3 green, 3 red - Box 2: 2 blue, 5 green, 7 red - Box n (for n = 3, 4, 5): n blue, 2n + 1 green, 4n − 1 red You repeat this process until you run out of one color of balls. How many re... | [
"821"
] | [
"You have 5 boxes and 2005 balls, with 286 blue, 645 green, and 1074 red balls. You place balls in the boxes as follows: \n- Box 1: 1 blue, 3 green, 3 red\n- Box 2: 2 blue, 5 green, 7 red\n- Box n (for n = 3, 4, 5): n blue, 2n + 1 green, 4n − 1 red\n\nYou repeat this process until you run out of one color of balls.... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-a164fcc31667d8770d7ce225f3b3f5e20972c8f316128b059b61d75b40feb4cf | You have 5 boxes and 2005 balls, with 286 blue, 645 green, and 1074 red balls. You place balls in the boxes as follows: in box $n$, you place $n$ blue balls, $2n + 1$ green balls, and $4n - 1$ red balls for $n = 1, 2, 3, 4, 5$. This process is repeated until you run out of one color of balls. How many red balls are in ... | [
"821"
] | [
"You have 5 boxes and 2005 balls, with 286 blue, 645 green, and 1074 red balls. You place balls in the boxes as follows: in box $n$, you place $n$ blue balls, $2n + 1$ green balls, and $4n - 1$ red balls for $n = 1, 2, 3, 4, 5$. This process is repeated until you run out of one color of balls. How many red balls ar... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-a9dfb64d2969a8dc724636e3b6763e1490c441ca20746315029ebe3a535c6533 | You have 5 coins labeled 1 through 5. The probability of flipping heads for each coin is as follows: coin 1 has a probability of 1, coin 2 has a probability of 0.9, coin 3 has a probability of 0.8, coin 4 has a probability of 0.7, and coin 5 has a probability of 0.6. If you flip a tails on coin \( k \), you must re-fli... | [
"\\dfrac{563}{63}"
] | [
"You have 5 coins labeled 1 through 5. The probability of flipping heads for each coin is as follows: coin 1 has a probability of 1, coin 2 has a probability of 0.9, coin 3 has a probability of 0.8, coin 4 has a probability of 0.7, and coin 5 has a probability of 0.6. If you flip a tails on coin \\( k \\), you must... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-de1c09cb0c9500d9bd67435b58793700256e34d280c54df7443f5107ca73a3d9 | You have 5 indistinguishable blue balls, 4 indistinguishable green balls, and 3 indistinguishable red balls. You want to arrange them in a line such that no two balls of the same color are next to each other. Find the number of such arrangements. | [
"1320"
] | [
"You have 5 indistinguishable blue balls, 4 indistinguishable green balls, and 3 indistinguishable red balls. You want to arrange them in a line such that no two balls of the same color are next to each other. Find the number of such arrangements."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-4dc103c858513ec2fb1110e4f7f2a1af2c2e74cea22da8e5503c2c99ec22b10a | You have 5 pumpkins and want to know the weight of each pumpkin. Taking two pumpkins at a time, you find the following weights: 16, 18, 19, 20, 21, 22, 23, 24, 26, 27. How much does each pumpkin weigh? | [
"15, 12, 11, 9, 7"
] | [
"You have 5 pumpkins and want to know the weight of each pumpkin. Taking two pumpkins at a time, you find the following weights: 16, 18, 19, 20, 21, 22, 23, 24, 26, 27. How much does each pumpkin weigh?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-3716424dc27986cbabcbad71baed2764794f125368997b1c5099a48fc87463c0 | You have 5 segments of lengths 1, 2, 4, 4, 5. The perimeter of a triangle is built from all of the segments, where each segment is a part of exactly one side of the triangle. What is the largest possible area of such a triangle? | [
"12"
] | [
"You have 5 segments of lengths 1, 2, 4, 4, 5. The perimeter of a triangle is built from all of the segments, where each segment is a part of exactly one side of the triangle. What is the largest possible area of such a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4507f5391c2001f46314ffe831fe4ad6a43e0dfaa67dffe0866178ff409a8f5d | You have 5 suspects for a crime. Each piece of evidence can absolve 2 people, and the evidence is memoryless (it can absolve the same people multiple times). The evidence only absolves people and does not prove guilt. How many pieces of evidence on average would it take to identify the guilty person? | [
"\\( \\frac{19}{5} \\)"
] | [
"You have 5 suspects for a crime. Each piece of evidence can absolve 2 people, and the evidence is memoryless (it can absolve the same people multiple times). The evidence only absolves people and does not prove guilt. How many pieces of evidence on average would it take to identify the guilty person?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
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