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-a933d596c7f16ce7becb6f60e25f79bfd81e4aecb6eadedd9c28e843dce323c5 | Alice places a coin, heads up, on a table, then turns off the light and leaves the room. Bill enters the room with 2 coins and flips them onto the table and leaves. Carl enters the room, in the dark, and removes a coin at random. Alice reenters the room, turns on the light, and notices that both coins are heads. What i... | [
"\\(\\frac{3}{5}\\)"
] | [
"Alice places a coin, heads up, on a table, then turns off the light and leaves the room. Bill enters the room with 2 coins and flips them onto the table and leaves. Carl enters the room, in the dark, and removes a coin at random. Alice reenters the room, turns on the light, and notices that both coins are heads. W... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-dba8d44c58a6ba885f6471c842235154606eb3552152153acb2dc92e572b3694 | Alice places counters on an $8 \times 8$ board such that: a) There is at least one counter in any $2 \times 1$ or $1 \times 2$ rectangle. b) There are at least two neighboring counters in any $7 \times 1$ or $1 \times 7$ rectangle. Find the minimum number of counters Alice needs to satisfy both conditions. | [
"32"
] | [
"Alice places counters on an $8 \\times 8$ board such that:\na) There is at least one counter in any $2 \\times 1$ or $1 \\times 2$ rectangle.\nb) There are at least two neighboring counters in any $7 \\times 1$ or $1 \\times 7$ rectangle.\n\nFind the minimum number of counters Alice needs to satisfy both condition... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2fd0df969fd9782b8ac04e5bff7c3e88b3d58b604c54e5113b5415a68fa4bf73 | Alice plays a game where she has 4 standard fair 6-sided dice and rolls all of them. Once she observes the values, she can choose any subset of the dice to reroll, paying \$1 per die per reroll. There are no restrictions on how many times Alice can reroll the dice. Once Alice decides to no longer reroll any dice, she r... | [
"14"
] | [
"Alice plays a game where she has 4 standard fair 6-sided dice and rolls all of them. Once she observes the values, she can choose any subset of the dice to reroll, paying \\$1 per die per reroll. There are no restrictions on how many times Alice can reroll the dice. Once Alice decides to no longer reroll any dice,... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-f6fcf4fbd43c9f1a1c41cf437f06f2854f99cc78ee15ba9233ff03834f4355cb | Alice plays a game with a standard deck of 52 cards. She assigns values to the cards where Aces are 1's, royal cards are 10's, and all other cards are their face values. Each turn, she flips the top card and creates a new pile with $12 - v$ cards on top of the flipped card, where $v$ is the value of the flipped card. I... | [
"17"
] | [
"Alice plays a game with a standard deck of 52 cards. She assigns values to the cards where Aces are 1's, royal cards are 10's, and all other cards are their face values. Each turn, she flips the top card and creates a new pile with $12 - v$ cards on top of the flipped card, where $v$ is the value of the flipped ca... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-22a409e648e70cf0d9c96bc951d2920689e27872c05fee1faca74c457500c2b9 | Alice puts counters in some cells of an $8\times8$ board such that: a) There is at least one counter in any rectangle of size $2\times1$ or $1\times2$. b) There are at least two adjacent counters in any rectangle of size $7\times1$ or $1\times7$. Find the minimum number of counters that Alice needs to satisfy both cond... | [
"32"
] | [
"Alice puts counters in some cells of an $8\\times8$ board such that:\na) There is at least one counter in any rectangle of size $2\\times1$ or $1\\times2$.\nb) There are at least two adjacent counters in any rectangle of size $7\\times1$ or $1\\times7$.\nFind the minimum number of counters that Alice needs to sati... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-0c59a31c1ea7c30e5b89555c2c13f752b30bd53623cccf90abeba33e74a60a78 | Alice randomly picks real numbers \( x_1 \) from \([0,1]\), \( x_2 \) from \([0,2]\), ..., and \( x_n \) from \([0,n]\). What is the probability that \( x_1 < x_2 < \cdots < x_n \)? | [
"\\dfrac{(n+1)^{n-1}}{(n!)^2}"
] | [
"Alice randomly picks real numbers \\( x_1 \\) from \\([0,1]\\), \\( x_2 \\) from \\([0,2]\\), ..., and \\( x_n \\) from \\([0,n]\\). What is the probability that \\( x_1 < x_2 < \\cdots < x_n \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b3c17797fef88546b2a2c4968c09cce7e0fef1552fc41d7384568f0ce499bcb9 | Alice refuses to sit next to either Bob or Carla, and Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions? | [
"28"
] | [
"Alice refuses to sit next to either Bob or Carla, and Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 13
} | 44 | 0 | 0 |
omr-ws-c68d3f0289bfc222dd452517eb433f5d0e35968ee23f56ad241cc31e838b7077 | Alice rolls a 6-sided die repeatedly until she gets two consecutive rolls whose product is 10. Using a Markov chain, compute the expected number of rolls. | [
"21"
] | [
"Alice rolls a 6-sided die repeatedly until she gets two consecutive rolls whose product is 10. Using a Markov chain, compute the expected number of rolls."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-3816af2637495ab41290a83cd71f65b26e00f55bdb2c8e876974a54257b9ce53 | Alice rolls a fair six-sided die with the numbers $1$ through $6$, and Bob rolls a fair eight-sided die with the numbers $1$ through $8$. Alice wins if her number divides Bob’s number, and Bob wins otherwise. What is the probability that Alice wins? | [
"\\(\\frac{3}{8}\\)"
] | [
"Alice rolls a fair six-sided die with the numbers $1$ through $6$, and Bob rolls a fair eight-sided die with the numbers $1$ through $8$. Alice wins if her number divides Bob’s number, and Bob wins otherwise. What is the probability that Alice wins?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f065f9d97912e7a5723c240f6be64fa973868cc158343f0d273f4bff05067bf7 | Alice searches for her term paper in a filling cabinet with several drawers. She knows that the term paper is in drawer $j$ with probability $p_j > 0$. The probability that she finds the term paper in drawer $i$ if it is there is $d_i$. Given that her search in drawer $i$ was unsuccessful, what is the probability that ... | [
"\\(\\begin{cases} \n\\frac{(1 - d_i)p_i}{(1 - d_i)p_i + \\sum_{k \\neq i} p_k} & \\text{if } j = i \\\\\n\\frac{p_j}{(1 - d_i)p_i + \\sum_{k \\neq i} p_k} & \\text{if } j \\neq i \n\\end{cases}\\)"
] | [
"Alice searches for her term paper in a filling cabinet with several drawers. She knows that the term paper is in drawer $j$ with probability $p_j > 0$. The probability that she finds the term paper in drawer $i$ if it is there is $d_i$. Given that her search in drawer $i$ was unsuccessful, what is the probability ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 3
} | 9 | 0 | 0 |
omr-ws-cbc61302f7eb4addafebd42cdd04fe3d29a8990b9774b878e5e189e666dee58f | Alice starts at Riverbank City and wants to end up at Conscious City in Wanderland, which consists of a finite number of cities connected by two-way trains such that any two cities are reachable from each other. Every day, Alice picks a train going out of her current city uniformly at random and boards it. Let $E$ be t... | [
"18"
] | [
"Alice starts at Riverbank City and wants to end up at Conscious City in Wanderland, which consists of a finite number of cities connected by two-way trains such that any two cities are reachable from each other. Every day, Alice picks a train going out of her current city uniformly at random and boards it. Let $E$... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-5d305283d407b747876fa3b4f00d3f16b65d4ced8eb7916497a8ae14a823962c | Alice starts on a unit square of a \(2n \times 2n\) chessboard. She should visit every single unit square exactly once and return to the starting square. If Alice moves from unit square \(P\) to unit square \(Q\), then we connect \(PQ\). Determine the maximum number of edges in the polygon formed by Alice's path. | [
"(2n)^2"
] | [
"Alice starts on a unit square of a \\(2n \\times 2n\\) chessboard. She should visit every single unit square exactly once and return to the starting square. If Alice moves from unit square \\(P\\) to unit square \\(Q\\), then we connect \\(PQ\\). Determine the maximum number of edges in the polygon formed by Alice... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-051c632f668f7bf470721debfd7bb0362b6dce9904901b88f6de7519643d5cea | Alice starts with the number $0$. At each step, she either adds $1$ to her number or subtracts $2$ from her number with equal probability, and she stops when the absolute value of her number exceeds $5$. Find the probability that when she stops, her number is positive. | [
"\\dfrac{5}{94}"
] | [
"Alice starts with the number $0$. At each step, she either adds $1$ to her number or subtracts $2$ from her number with equal probability, and she stops when the absolute value of her number exceeds $5$. Find the probability that when she stops, her number is positive."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-be02c53fb0da18ef95b5aa92956bc7a5e104b5451e67473cd31ea16cf834214e | Alice takes a number from the set {$1,2,\cdots, 144$}, and Bob wants to know the number taken. Bob can take a subset $T$ from the set {$1,2,\cdots, 144$} each time and ask Alice: "Does the number you have taken belong to $T$?" If the answer is YES, then Bob pays Alice two yuan; otherwise, pay one yuan. On the premise t... | [
"11"
] | [
"Alice takes a number from the set {$1,2,\\cdots, 144$}, and Bob wants to know the number taken. Bob can take a subset $T$ from the set {$1,2,\\cdots, 144$} each time and ask Alice: \"Does the number you have taken belong to $T$?\" If the answer is YES, then Bob pays Alice two yuan; otherwise, pay one yuan. On the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 40,
"genselect": 8,
"tir": 23
} | 71 | 0 | 0 |
omr-ws-55abcc7615836dc45bcc5cec8afbc8c8abebe00bef0c0a68fe3c0b384dacb44a | Alice the ant starts at vertex $A$ of a regular hexagon $ABCDEF$ and moves either right or left each move with equal probability. After 35 moves, what is the probability that she is on either vertex $A$ or $C$? | [
"0"
] | [
"Alice the ant starts at vertex $A$ of a regular hexagon $ABCDEF$ and moves either right or left each move with equal probability. After 35 moves, what is the probability that she is on either vertex $A$ or $C$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f398dab340db4f8f134ca80c8bb6eae9307776c48d6d58e453f4881f7bc7d47 | Alice thinks of a natural number. Bob asks if the number is divisible by each of the integers from 1 to 10. Alice answers "yes" to all questions except one, and for the question she answers "no", she adds that the greatest common factor of the number and the divisor is 1. To which question did Alice answer "no"? | [
"7"
] | [
"Alice thinks of a natural number. Bob asks if the number is divisible by each of the integers from 1 to 10. Alice answers \"yes\" to all questions except one, and for the question she answers \"no\", she adds that the greatest common factor of the number and the divisor is 1. To which question did Alice answer \"n... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-fadf6cba2b6448ae77060bc6c527cebecbce082e165c29a627d3c6bd699ec7d5 | Alice thinks of a secret polynomial \( p(x) \) with non-negative integer coefficients. Bob chooses an integer \( a \) and Alice tells Bob the value \( p(a) \). Then Bob chooses an integer \( b \) and Alice tells Bob the value \( p(b) \). Bob can then determine the polynomial \( p(x) \) correctly. What is Bob’s strategy... | [
"Bob's strategy is to first choose \\( a = 1 \\) and then choose \\( b = 10^{p(1)} \\)."
] | [
"Alice thinks of a secret polynomial \\( p(x) \\) with non-negative integer coefficients. Bob chooses an integer \\( a \\) and Alice tells Bob the value \\( p(a) \\). Then Bob chooses an integer \\( b \\) and Alice tells Bob the value \\( p(b) \\). Bob can then determine the polynomial \\( p(x) \\) correctly. What ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 2,
"tir": 3
} | 7 | 0 | 0 |
omr-ws-804201fc257fd306f9919309b61a447286554648b4d26b5afca2df54d8f8e1aa | Alice tiles a $2016 \times 2016$ board with $2 \times 1$ dominoes. Bob then colours each domino with one of $k$ colours such that each domino touches the side of at most 1 other domino with the same colour. Find the minimum value of $k$ such that Bob can guarantee a win. | [
"3"
] | [
"Alice tiles a $2016 \\times 2016$ board with $2 \\times 1$ dominoes. Bob then colours each domino with one of $k$ colours such that each domino touches the side of at most 1 other domino with the same colour. Find the minimum value of $k$ such that Bob can guarantee a win."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-c53d7a6ae7c93a9fe7e65c08ab157c666406bf6a0ff54150541db53d142ca014 | Alice tiles a $2016 \times 2016$ board with $2 \times 1$ dominoes. Bob then colours each domino with one of $k$ colours such that each domino touches the side of at most 2 other dominoes with the same colour. Find the minimum value of $k$ such that Bob can guarantee a win. | [
"3"
] | [
"Alice tiles a $2016 \\times 2016$ board with $2 \\times 1$ dominoes. Bob then colours each domino with one of $k$ colours such that each domino touches the side of at most 2 other dominoes with the same colour. Find the minimum value of $k$ such that Bob can guarantee a win."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-bab0ffa6062a4a1a308e7d6649a4b35f64aa35290ae7ebfaea0f63b3be066a03 | Alice tosses a coin until it lands heads and counts the number of trials it takes. Bob tosses a die until it lands 6 and counts the number of trials it takes. What is the probability that the number of trials Alice requires is fewer than the number of trials Bob requires? | [
"\\dfrac{5}{7}"
] | [
"Alice tosses a coin until it lands heads and counts the number of trials it takes. Bob tosses a die until it lands 6 and counts the number of trials it takes. What is the probability that the number of trials Alice requires is fewer than the number of trials Bob requires?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f1870594ca221529f02a1a79d66f4aa7116068f5bd3c31c37a6b2846c0696a09 | Alice tosses two biased coins, each with a probability \( p \) of obtaining a head, simultaneously and repeatedly until she gets two heads. Given that there is a 36% chance that the number of tosses \( r \) is even, what is the value of \( p \)? | [
"\\(\\frac{\\sqrt{7}}{4}\\)"
] | [
"Alice tosses two biased coins, each with a probability \\( p \\) of obtaining a head, simultaneously and repeatedly until she gets two heads. Given that there is a 36% chance that the number of tosses \\( r \\) is even, what is the value of \\( p \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-219d89d86618fa211aa1626f4d4c04d34c9cc3c223dc74821d2e4110114868bc | Alice uses a fair 6-sided die to decide what to have for dinner. She keeps rolling until she gets all the even numbers (2, 4, 6) before getting any odd number. Find the probability that Alice could have McDonald’s for dinner. | [
"\\(\\frac{1}{20}\\)"
] | [
"Alice uses a fair 6-sided die to decide what to have for dinner. She keeps rolling until she gets all the even numbers (2, 4, 6) before getting any odd number. Find the probability that Alice could have McDonald’s for dinner."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-1b5c7c7dad9a742a1c66fa11060c0c18f35876fd6cf255243e80a7a9a6f5b218 | Alice wants to paint each face of an octahedron either red or blue. She can paint any number of faces a particular color, including zero. Compute the number of ways in which she can do this. Two ways of painting the octahedron are considered the same if you can rotate the octahedron to get from one to the other. | [
"23"
] | [
"Alice wants to paint each face of an octahedron either red or blue. She can paint any number of faces a particular color, including zero. Compute the number of ways in which she can do this. Two ways of painting the octahedron are considered the same if you can rotate the octahedron to get from one to the other."
... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-a04771f2790f4d8d4788b5efadabe04dec438ccec72276ba1d0a026b195835a3 | Alice will write the positive integers from $1$ to $7$ on a whiteboard in some order. The first number she writes is $2$. For the $i$th number she writes, if $i$ has been written before, she picks a random integer from $1$ to $7$ that hasn't been written yet. Otherwise, she writes the number $i$. Let $A$ be the number ... | [
"91"
] | [
"Alice will write the positive integers from $1$ to $7$ on a whiteboard in some order. The first number she writes is $2$. For the $i$th number she writes, if $i$ has been written before, she picks a random integer from $1$ to $7$ that hasn't been written yet. Otherwise, she writes the number $i$. Let $A$ be the nu... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-66d607fe810d1a7a3dda744504491a5afb36dae5cb83feefbb8d24da362e045e | Alice wishes to walk from the point $(0, 0)$ to the point $(6, 4)$ in increments of $(1, 0)$ and $(0, 1)$, and Bob wishes to walk from the point $(0, 1)$ to the point $(6, 5)$ in increments of $(1, 0)$ and $(0, 1)$. How many ways are there for Alice and Bob to get to their destinations if their paths never pass through... | [
"5292"
] | [
"Alice wishes to walk from the point $(0, 0)$ to the point $(6, 4)$ in increments of $(1, 0)$ and $(0, 1)$, and Bob wishes to walk from the point $(0, 1)$ to the point $(6, 5)$ in increments of $(1, 0)$ and $(0, 1)$. How many ways are there for Alice and Bob to get to their destinations if their paths never pass th... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b6688df4f7a756e8dad7e5b61b5e58876e036e9d01a46f98a7dfeb2d8eadfaed | Alice works at amusement park \( A = (-3, -1) \) and Bruno works at bakery \( B = (3, 3) \). They want to find an apartment such that both have exactly the same distance to walk to work. Where should they look for an apartment? | [
"(0, 1)"
] | [
"Alice works at amusement park \\( A = (-3, -1) \\) and Bruno works at bakery \\( B = (3, 3) \\). They want to find an apartment such that both have exactly the same distance to walk to work. Where should they look for an apartment?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-586a35aae1abd28a6603fe0ced3a0d5eabc94ffc1bfe58200df1824dfa52ec5c | Alice works at amusement park \( A = (-3, -1) \) and Bruno works at bakery \( B = (3, 3) \). They want to find an apartment such that the sum of the distances Alice and Bruno have to walk to their respective workplaces is minimized, and Alice should not have to walk any farther than Bruno. Where should they look for an... | [
"on the line segment from \\((0, 1)\\) to \\((3, 3)\\)"
] | [
"Alice works at amusement park \\( A = (-3, -1) \\) and Bruno works at bakery \\( B = (3, 3) \\). They want to find an apartment such that the sum of the distances Alice and Bruno have to walk to their respective workplaces is minimized, and Alice should not have to walk any farther than Bruno. Where should they lo... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0803b8a4a79656cca47478852ea206001d80a1e043b98644c11ef2377414c8d7 | Alice works at amusement park \( A = (-3, -1) \) and Bruno works at bakery \( B = (3, 3) \). They want to find an apartment such that the sum of the distances Alice and Bruno have to walk to their respective workplaces is minimized. Where should they look for an apartment? | [
"anywhere on the line connecting \\((-3, -1)\\) and \\((3, 3)\\)"
] | [
"Alice works at amusement park \\( A = (-3, -1) \\) and Bruno works at bakery \\( B = (3, 3) \\). They want to find an apartment such that the sum of the distances Alice and Bruno have to walk to their respective workplaces is minimized. Where should they look for an apartment?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 17
} | 38 | 0 | 0 |
omr-ws-7aa5c33bf176d1245e271cc5b12ad31fdc17da96ef53c959e634ac03b14a2968 | Alice writes 1001 letters on a blackboard, each chosen independently and uniformly at random from the set \( S = \{a, b, c\} \). A move consists of erasing two distinct letters from the board and replacing them with the third letter in \( S \). What is the probability that Alice can perform a sequence of moves which re... | [
"\\dfrac{3^{1001} + 1}{4 \\cdot 3^{1000}}"
] | [
"Alice writes 1001 letters on a blackboard, each chosen independently and uniformly at random from the set \\( S = \\{a, b, c\\} \\). A move consists of erasing two distinct letters from the board and replacing them with the third letter in \\( S \\). What is the probability that Alice can perform a sequence of mov... | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-5f478b9604cd7c8d652eb8509521262d8b4ba8246862d31fdd3b9be8d203a8d3 | Alice writes 1001 letters on a blackboard, each one chosen independently and uniformly at random from the set \( S = \{a, b, c\} \). A move consists of erasing two distinct letters from the board and replacing them with the third letter in \( S \). What is the probability that Alice can perform a sequence of moves whic... | [
"\\(\\frac{3^{1001} + 1}{4 \\cdot 3^{1001}}\\)"
] | [
"Alice writes 1001 letters on a blackboard, each one chosen independently and uniformly at random from the set \\( S = \\{a, b, c\\} \\). A move consists of erasing two distinct letters from the board and replacing them with the third letter in \\( S \\). What is the probability that Alice can perform a sequence of... | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-ba9eaeae70a9a13b438c0a2473e9e0c3cd0a88f3da8e30d258ff761a03cad2d1 | Alice writes different real numbers on a board. If \(a, b, c\) are three numbers on the board, then at least one of the numbers \(a + b\), \(b + c\), or \(a + c\) is also on the board. What is the largest possible number of numbers that can be written on the board? | [
"7"
] | [
"Alice writes different real numbers on a board. If \\(a, b, c\\) are three numbers on the board, then at least one of the numbers \\(a + b\\), \\(b + c\\), or \\(a + c\\) is also on the board. What is the largest possible number of numbers that can be written on the board?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 40,
"genselect": 8,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-fa654494ef948cfee7e3b710b78df4f5d76bfd750456ea1cdab51b8e69b1bfd9 | Alice writes down a 37-digit number on the blackboard, with all digits greater than 0. Bob must erase some (but not all) digits so that the remaining number is a multiple of 271. Let $N$ be the smallest possible value of the 37-digit number Alice can write such that Bob cannot achieve his goal. Find the sum of the digi... | [
"181"
] | [
"Alice writes down a 37-digit number on the blackboard, with all digits greater than 0. Bob must erase some (but not all) digits so that the remaining number is a multiple of 271. Let $N$ be the smallest possible value of the 37-digit number Alice can write such that Bob cannot achieve his goal. Find the sum of the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-c40e455a1c18a5dbeb21e029410dcd3b527cdcb57be1bd1d6284b3c3ae328ba2 | Alice writes numbers from $1$ to $49$ in a $7 \times 7$ table, one number per unit square. Bob can ask Alice about the numbers in any $k \times k$ square with sides parallel to the sides of the big square. Alice will tell him all the numbers in that square but not their positions. Determine the minimum value of $k$ suc... | [
"4"
] | [
"Alice writes numbers from $1$ to $49$ in a $7 \\times 7$ table, one number per unit square. Bob can ask Alice about the numbers in any $k \\times k$ square with sides parallel to the sides of the big square. Alice will tell him all the numbers in that square but not their positions. Determine the minimum value of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-ae24803cc24d93cb3348ca7ee7d4a91a644814ca4bf8a2351d2731f3947a8dbf | Alice writes the numbers from $1$ to $15$ on a board. She deletes two numbers $a$ and $b$, and replaces them with $ab + 2a + 2b + 2$. After repeating this operation $14$ times, what number will be left on the board? | [
"\\(\\frac{17!}{2} - 2\\)"
] | [
"Alice writes the numbers from $1$ to $15$ on a board. She deletes two numbers $a$ and $b$, and replaces them with $ab + 2a + 2b + 2$. After repeating this operation $14$ times, what number will be left on the board?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 17,
"genselect": 1
} | 18 | 0 | 0 |
omr-ws-4f180b4a1e954978f8d3455bee80ccd94a8b37c9fbe6e031b01760897f2f9835 | Alice wrote a sequence of \( n > 2 \) nonzero nonequal numbers such that each is greater than the previous one by the same amount. Bob wrote the inverses of those \( n \) numbers in some order. It so happened that each number in his row also is greater than the previous one by the same amount, possibly not the same as ... | [
"\\( n = 3 \\)"
] | [
"Alice wrote a sequence of \\( n > 2 \\) nonzero nonequal numbers such that each is greater than the previous one by the same amount. Bob wrote the inverses of those \\( n \\) numbers in some order. It so happened that each number in his row also is greater than the previous one by the same amount, possibly not the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-ec5e8ce6bec63f858084be73880cf91a00af8d046743a55305566b0d47d9daf8 | Alice wrote a series of $n>2$ non-zero non-equal real numbers such that each is greater than the previous by a constant amount. Bob wrote the inverses (reciprocals) of those $n$ numbers in some order. Each of Bob's numbers was also greater than the previous by a constant amount (this may not be the same constant as Ali... | [
"4"
] | [
"Alice wrote a series of $n>2$ non-zero non-equal real numbers such that each is greater than the previous by a constant amount. Bob wrote the inverses (reciprocals) of those $n$ numbers in some order. Each of Bob's numbers was also greater than the previous by a constant amount (this may not be the same constant a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-cccbcbf5d20ac109f95feb16359bab09f114cc18cdc77f93b43986f9b7af8fed | Alice's favorite number has the following properties: It has 8 distinct digits. The digits are decreasing when read from left to right. It is divisible by 180. What is the sum of digits of Alice's favorite number? | [
"36"
] | [
"Alice's favorite number has the following properties: It has 8 distinct digits. The digits are decreasing when read from left to right. It is divisible by 180. What is the sum of digits of Alice's favorite number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-ec3c4b1c4b14b42c4d4f3ab1848d3b7883acedf90a333407bc38bae9ffde5cee | Alice, Beatrix, and Claire visit Daphne according to the following schedule: Alice and Beatrix visit together every 12 days, Alice and Claire visit together every 15 days, and Beatrix and Claire visit together every 20 days. They visit all together 6 times. Calculate the total number of days exactly two friends visit D... | [
"54"
] | [
"Alice, Beatrix, and Claire visit Daphne according to the following schedule: Alice and Beatrix visit together every 12 days, Alice and Claire visit together every 15 days, and Beatrix and Claire visit together every 20 days. They visit all together 6 times. Calculate the total number of days exactly two friends vi... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-d84fb05da23fb9835c8575e43a2a2831e4d4ce5f81bdc68c1559114f2b9fee82 | Alice, Belle, and Cathy had an arm-wrestling contest. In each game, two girls wrestled, while the third rested. After each game, the winner played the next game against the girl who rested. In total, Alice played $10$ times, Belle played $15$ times, and Cathy played $17$ times. Who lost the second game? | [
"Alice"
] | [
"Alice, Belle, and Cathy had an arm-wrestling contest. In each game, two girls wrestled, while the third rested. After each game, the winner played the next game against the girl who rested. In total, Alice played $10$ times, Belle played $15$ times, and Cathy played $17$ times. Who lost the second game?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-1627e22490b97c03023084e49bc80bb773e1b8a40bafbd49b98a90ae49dc4fa3 | Alice, Benedict, Calvin, Dennis, and Evelyn each randomly grabbed a left shoe and a right shoe at the end of a party. What is the probability that each person left with exactly one of their own shoes? | [
"\\dfrac{2}{225}"
] | [
"Alice, Benedict, Calvin, Dennis, and Evelyn each randomly grabbed a left shoe and a right shoe at the end of a party. What is the probability that each person left with exactly one of their own shoes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6ac120959198bf7503f18e0f702f0a573f30674caeff94fc92dbcc937e4e721a | Alice, Betty, and Chris each wear a hat with either a 1 or a 0 written on it, chosen at random. Each person can only see the numbers on the other two hats. Each person votes on whether there are more 1s than 0s based on what they observe. If unsure, they flip a coin. Determine whether the majority vote will be correct ... | [
"\\(\\frac{13}{16}\\)"
] | [
"Alice, Betty, and Chris each wear a hat with either a 1 or a 0 written on it, chosen at random. Each person can only see the numbers on the other two hats. Each person votes on whether there are more 1s than 0s based on what they observe. If unsure, they flip a coin. Determine whether the majority vote will be cor... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-69c22a37709b13fda90c89de8c76144c363dce5153e07865adaf5eb468d595fd | Alice, Bob, Cara, David, Eve, Fred, and Grace are sitting in a row. Alice and Bob like to pass notes to each other. However, anyone sitting between Alice and Bob can read the notes they pass. How many ways are there for the students to sit if Eve wants to be able to read Alice and Bob’s notes, assuming reflections are ... | [
"48"
] | [
"Alice, Bob, Cara, David, Eve, Fred, and Grace are sitting in a row. Alice and Bob like to pass notes to each other. However, anyone sitting between Alice and Bob can read the notes they pass. How many ways are there for the students to sit if Eve wants to be able to read Alice and Bob’s notes, assuming reflections... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e9d9f2753897ed7c43db7d6e310456e0baa3d310447689c358bba4650107e126 | Alice, Bob, Carol, and Daniel each pick a number from the set $\{1, 2, 3, 4, 5, 6\}$ independently and at random. A player wins if the number they pick divides the number of another person. Let the probability that there are exactly 2 winners be $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. F... | [
"65"
] | [
"Alice, Bob, Carol, and Daniel each pick a number from the set $\\{1, 2, 3, 4, 5, 6\\}$ independently and at random. A player wins if the number they pick divides the number of another person. Let the probability that there are exactly 2 winners be $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive int... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d9630f9c1a2e9b53c6bad5ed4aed539d7dd074e896e7bdc076475e698d27074f | Alice, Bob, Carol, and Dave stand in a circle. Simultaneously, each player selects another player at random and points at that person, who must then sit down. What is the probability that Alice is the only person who remains standing? | [
"$\\frac{4}{27}$"
] | [
"Alice, Bob, Carol, and Dave stand in a circle. Simultaneously, each player selects another player at random and points at that person, who must then sit down. What is the probability that Alice is the only person who remains standing?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 21
} | 40 | 0 | 0 |
omr-ws-3f4d98feee0a939dcd77c5c4e57c9a7cd70d08aeca5006b0dc9962a8ed33d3d6 | Alice, Bob, Carol, and David decide to share meals, with David cooking on 4 days of the week and Alice, Bob, and Carol each cooking on one day. If Alice, Bob, and Carol each choose their cooking day uniformly at random without overlap, what is the probability that David does not cook on three consecutive days? | [
"\\dfrac{19}{35}"
] | [
"Alice, Bob, Carol, and David decide to share meals, with David cooking on 4 days of the week and Alice, Bob, and Carol each cooking on one day. If Alice, Bob, and Carol each choose their cooking day uniformly at random without overlap, what is the probability that David does not cook on three consecutive days?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c37eeadc5bc281c299bf6dafc87635083f72d9ee0947dc1c563bc28446ba33ce | Alice, Bob, Cassie, Dave, and Ethan are going on a road trip and need to arrange themselves among a driver's seat, a passenger seat, and three distinguishable back row seats. Alice, Bob, and Cassie are not allowed to drive. Alice and Bob are also not allowed to sit in the front passenger seat. Find the number of possib... | [
"24"
] | [
"Alice, Bob, Cassie, Dave, and Ethan are going on a road trip and need to arrange themselves among a driver's seat, a passenger seat, and three distinguishable back row seats. Alice, Bob, and Cassie are not allowed to drive. Alice and Bob are also not allowed to sit in the front passenger seat. Find the number of p... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a362f9f26719f42feeb6da71f908d7815982025b92bfeabbd7ed57353703ac71 | Alice, Bob, Charlie, Diana, Emma, and Fred sit in a circle and each roll a six-sided die. Each person looks at their own roll and the roll of either the person to the right or to the left, chosen at random. They then state the expected sum of the dice rolls based on the information they have. Alice, Bob, Charlie, and D... | [
"1440"
] | [
"Alice, Bob, Charlie, Diana, Emma, and Fred sit in a circle and each roll a six-sided die. Each person looks at their own roll and the roll of either the person to the right or to the left, chosen at random. They then state the expected sum of the dice rolls based on the information they have. Alice, Bob, Charlie, ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-ddb2ba1ec2ca92258f1a6aa62853116f5d97ac36f16fb7359239d324d476018d | Alice, Bob, Charlie, and Diane play tennis in sets: two of them play a set and the winner stays on the court for the next set, with the loser replaced by the person who was idle the longest. At the end of the day, Alice played 61 sets, Bob played 22 sets, Charlie played 21 sets, and Diane played 20 sets. Who played in ... | [
"Alice and Diane"
] | [
"Alice, Bob, Charlie, and Diane play tennis in sets: two of them play a set and the winner stays on the court for the next set, with the loser replaced by the person who was idle the longest. At the end of the day, Alice played 61 sets, Bob played 22 sets, Charlie played 21 sets, and Diane played 20 sets. Who playe... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3e6ba6dd247252fa57357a9de82d78e64c898a12ca6d8f441b2366137c252334 | Alice, Bob, David, and Eve are sitting in a row on a couch and are passing back and forth a bag of chips. Whenever Bob gets the bag of chips, he passes the bag back to the person who gave it to him with probability $\frac{1}{3}$, and he passes it on in the same direction with probability $\frac{2}{3}$. Whenever David g... | [
"\\dfrac{21}{25}"
] | [
"Alice, Bob, David, and Eve are sitting in a row on a couch and are passing back and forth a bag of chips. Whenever Bob gets the bag of chips, he passes the bag back to the person who gave it to him with probability $\\frac{1}{3}$, and he passes it on in the same direction with probability $\\frac{2}{3}$. Whenever ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5e0b3cd96dd2acb27093ae29083b2a6b02ef4734e20e5f24df2070508e755f97 | Alice, Bob, and Carl are playing a game of Math Roulette in a circle. Each person has an equal chance to start. On their turn, players can shoot one paintball and hit their intended target (Alice and Bob always hit, Carl hits $70\%$ of the time). Play continues clockwise until only one person remains. Assuming they eac... | [
"\\dfrac{161}{300}"
] | [
"Alice, Bob, and Carl are playing a game of Math Roulette in a circle. Each person has an equal chance to start. On their turn, players can shoot one paintball and hit their intended target (Alice and Bob always hit, Carl hits $70\\%$ of the time). Play continues clockwise until only one person remains. Assuming th... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-58eee49b4a438a2f294f35e40819be32668663d5ffead0650782bdf2cdf46173 | Alice, Bob, and Carol are driving in three cars on the same highway, each at a constant speed. When Alice passed Bob, Carol was 7 miles behind them. When Carol caught up and passed Bob, Alice was 4 miles ahead of them. How far behind was Bob when Carol passed Alice? | [
"\\(\\frac{28}{3}\\) miles"
] | [
"Alice, Bob, and Carol are driving in three cars on the same highway, each at a constant speed. When Alice passed Bob, Carol was 7 miles behind them. When Carol caught up and passed Bob, Alice was 4 miles ahead of them. How far behind was Bob when Carol passed Alice?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-0519d893269db58d708a2173c4521b7555ec3fdcf454a802dfe1a55c59610730 | Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$, respectively. They then compute the polynomial $f(x)=x^{3}+p x^{2}+q x+r$ with roots $a, b, c$. If the expected value of the sum of the squares of the coefficients of $f(x)$ is $\frac{m}{n}$ for relatively prime positiv... | [
"551"
] | [
"Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$, respectively. They then compute the polynomial $f(x)=x^{3}+p x^{2}+q x+r$ with roots $a, b, c$. If the expected value of the sum of the squares of the coefficients of $f(x)$ is $\\frac{m}{n}$ for relatively prime p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 43,
"genselect": 8,
"tir": 14
} | 65 | 0 | 0 |
omr-ws-48d216e39e8b90fecfe6f9590d583b6d565d9128c8fdf564cbb29d000764fa62 | Alice, Bob, and Carol each think of an expression that is a fraction with 1 as the numerator and a constant integer times some power of \( x \) as the denominator. The simplest common denominator of Alice's and Bob's expressions is \( 4x^2 \). The simplest common denominator of Bob's and Carol's expressions is \( 12x^3... | [
"\\(\\frac{1}{3x^3}, \\frac{1}{6x^3}, \\frac{1}{12x^3}\\)"
] | [
"Alice, Bob, and Carol each think of an expression that is a fraction with 1 as the numerator and a constant integer times some power of \\( x \\) as the denominator. The simplest common denominator of Alice's and Bob's expressions is \\( 4x^2 \\). The simplest common denominator of Bob's and Carol's expressions is... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aba372fdeb5e16c361f7927cdf3a2ae403201c4448d3b6f43baa9dd205773801 | Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner is the one whose number is between the numbers chosen by the other two players. Alice chooses her number uniformly at random from all the numbers between 0 and 1, and Bob chooses his number uniformly at random from... | [
"\\(\\frac{13}{24}\\)"
] | [
"Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner is the one whose number is between the numbers chosen by the other two players. Alice chooses her number uniformly at random from all the numbers between 0 and 1, and Bob chooses his number uniformly at random... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 60
} | 92 | 0 | 0 |
omr-ws-8399ab914a21eda546986660fde50662755872d51a785e75f62fd314e4f6ebaf | Alice, Bob, and Carol repeatedly take turns tossing a die in alphabetical order by their names. Find the probability that Carol will be the first person to toss a 6. | [
"\\(\\frac{25}{91}\\)"
] | [
"Alice, Bob, and Carol repeatedly take turns tossing a die in alphabetical order by their names. Find the probability that Carol will be the first person to toss a 6."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-d8d7824835aae051148df55a3668e3f051131982dbf3fc16de8e6c908f023e70 | Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $\frac{1}{6}$, independent of the outcome o... | [
"\\(\\frac{25}{91}\\)"
] | [
"Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $\\frac{1}{6}$, independent of the out... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-7cfdc953cccb590efc8f3ae3797e83a2f94a50ab7a72d4b5a3a04d4c36296ae9 | Alice, Bob, and Charles play a game with a byte (8 bits). Alice writes a byte and designates one bit. Bob must change exactly one bit in the byte. After Bob leaves, Charles must designate one bit. If Bob and Charles agree on a strategy beforehand, find the number of distinct strategies they can use to ensure that Charl... | [
"40320"
] | [
"Alice, Bob, and Charles play a game with a byte (8 bits). Alice writes a byte and designates one bit. Bob must change exactly one bit in the byte. After Bob leaves, Charles must designate one bit. If Bob and Charles agree on a strategy beforehand, find the number of distinct strategies they can use to ensure that ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2cbb964d3d1c047038e4f1b16ea2d198682ae89467a9dc822a7a7bdf5e1c9408 | Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other 2 players, and each pair plays until a winner is decided (i.e., in the event of a tie, they play again). What is the probability that each player wins exactly once? | [
"\\dfrac{1}{4}"
] | [
"Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other 2 players, and each pair plays until a winner is decided (i.e., in the event of a tie, they play again). What is the probability that each player wins exactly once?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6575d3728e11c76971edc644e987f76069bf278a13cbc2d63025d87446b80662 | Alice’s favorite number is the smallest positive multiple of 6 that has 15 positive divisors. Bob’s favorite number is the smallest odd positive multiple of both 5 and 7 that has 8 positive divisors. Clarke’s favorite number is the product of Alice and Bob’s favorite numbers. How many positive divisors does Clarke’s fa... | [
"80"
] | [
"Alice’s favorite number is the smallest positive multiple of 6 that has 15 positive divisors. Bob’s favorite number is the smallest odd positive multiple of both 5 and 7 that has 8 positive divisors. Clarke’s favorite number is the product of Alice and Bob’s favorite numbers. How many positive divisors does Clarke... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-ad81a331b2a80d8f31e6db9da85d9a4458819616c9c3d974d31dc2d97aad842e | Alicia and Bob play a game on the plane. Alicia selects 10 points in the plane, no three of which are collinear. Bob then colors each point either red or blue. Let \( N \) be the number of monochromatic triangles (all vertices the same color) that are similar to a given non-isosceles triangle \( T \). Find the minimum ... | [
"2"
] | [
"Alicia and Bob play a game on the plane. Alicia selects 10 points in the plane, no three of which are collinear. Bob then colors each point either red or blue. Let \\( N \\) be the number of monochromatic triangles (all vertices the same color) that are similar to a given non-isosceles triangle \\( T \\). Find the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7758ca7b78992c01b9a84722aedd51c37f7c38bbf4c4ba48ce0bd12f2577e612 | Alicia and Bruno are on a row-boat at the center of a circular lake with a radius of 105 meters. They take turns rowing 10 meters either to the East or North, with Alicia starting. The first to reach the shore wins. Determine which of them has a winning strategy and describe it. | [
"Alicia has a winning strategy."
] | [
"Alicia and Bruno are on a row-boat at the center of a circular lake with a radius of 105 meters. They take turns rowing 10 meters either to the East or North, with Alicia starting. The first to reach the shore wins. Determine which of them has a winning strategy and describe it."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 1,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-f97289b8859d7d005e5e9f374b2e7d53ade029a821f214c1d6c1148b028c9b27 | Alicia and Bruno are on a row-boat at the center of a circular lake with a radius of 95 meters. They take turns rowing 10 meters either to the East or North, with Alicia starting. The first to reach the shore wins. Determine which of them has a winning strategy and describe it. | [
"Alicia has a winning strategy."
] | [
"Alicia and Bruno are on a row-boat at the center of a circular lake with a radius of 95 meters. They take turns rowing 10 meters either to the East or North, with Alicia starting. The first to reach the shore wins. Determine which of them has a winning strategy and describe it."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 13
} | 14 | 0 | 0 |
omr-ws-4c28791738b978a1f61f2eeecb5996b49fd5c28eee1ff51ee8f2c2164b168195 | Alicia has $n$ candies, where $n$ is a positive integer with three digits. If she buys $5$ more, she will have a multiple of $8$. If she loses $8$, she will have a multiple of $5$. What is the smallest possible value of $n$? | [
"123"
] | [
"Alicia has $n$ candies, where $n$ is a positive integer with three digits. If she buys $5$ more, she will have a multiple of $8$. If she loses $8$, she will have a multiple of $5$. What is the smallest possible value of $n$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f71f575333d650d89c616a260d8f48124270010f153849fba908bec213cbf783 | Aliens have abducted 100 human scientists and prepared 101 robes numbered 1 to 101. One robe is put aside, and the remaining 100 robes are randomly assigned to the scientists. The scientists stand in a line such that each scientist can see the numbers on the robes of the scientists in front of them but not their own or... | [
"1"
] | [
"Aliens have abducted 100 human scientists and prepared 101 robes numbered 1 to 101. One robe is put aside, and the remaining 100 robes are randomly assigned to the scientists. The scientists stand in a line such that each scientist can see the numbers on the robes of the scientists in front of them but not their o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-f90ece549fef9d484efb430ea6569d29896964f935209639de36c297bf01b496 | Alina writes the numbers $1, 2, \dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the numbers in each group will be the same. In how many ways can this be done? | [
"2"
] | [
"Alina writes the numbers $1, 2, \\dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f03bed6525ba004562e56b96ac925dd97915118d44d970fd52f83f93a572cf7 | Alina writes the numbers $1, 2, \dots, 9$ on separate cards, one number per card. She wishes to divide the cards into 3 groups of 3 cards so that the sum of the numbers in each group will be the same. In how many ways can this be done? | [
"8"
] | [
"Alina writes the numbers $1, 2, \\dots, 9$ on separate cards, one number per card. She wishes to divide the cards into 3 groups of 3 cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-95b6074b8b0454cf1c3f2ba4e42855219e275e41c6b8636da816a60c1ef29248 | Alireza and Amin play a game involving polynomials with integer coefficients. Alireza chooses 1400 such polynomials, and Amin selects 700 of them. Let the set of polynomials chosen by Alireza be \( B \) and the set chosen by Amin be \( A \). Amin wins if for all \( n \), the following inequality holds: \[ \max_{P \in A... | [
"Alireza has the winning strategy."
] | [
"Alireza and Amin play a game involving polynomials with integer coefficients. Alireza chooses 1400 such polynomials, and Amin selects 700 of them. Let the set of polynomials chosen by Alireza be \\( B \\) and the set chosen by Amin be \\( A \\). Amin wins if for all \\( n \\), the following inequality holds:\n\\[ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 3,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-d6fbad4b979837fc68c78de96bfce913e96d8dc936fe415412c4fe148d4f3916 | Alison has compiled a list of 20 hockey teams, ordered by her preference, but refuses to share it. Benjamin can ask Alison about any three teams and she will either identify the weakest or the strongest team among the three. Determine the largest N such that Benjamin can guarantee to find a sequence T1, T2,..., TN of t... | [
"10"
] | [
"Alison has compiled a list of 20 hockey teams, ordered by her preference, but refuses to share it. Benjamin can ask Alison about any three teams and she will either identify the weakest or the strongest team among the three. Determine the largest N such that Benjamin can guarantee to find a sequence T1, T2,..., TN... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-5b264cee8df3b3d170b39f14d6289eb5c0f34e2c9c1221a10c9a643c2ff7b587 | Alison has compiled a list of 20 hockey teams, ordered by how good she thinks they are. Benjamin can ask Alison about any three teams and she will either tell him which one she thinks is the weakest or which one she thinks is the strongest. Determine the largest \( N \) such that Benjamin can guarantee to find a sequen... | [
"10"
] | [
"Alison has compiled a list of 20 hockey teams, ordered by how good she thinks they are. Benjamin can ask Alison about any three teams and she will either tell him which one she thinks is the weakest or which one she thinks is the strongest. Determine the largest \\( N \\) such that Benjamin can guarantee to find a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 5,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-e751db53d81bf43e5b5e2a0d458fc6aa88cd9405191fb90ea5a7f9c19efb925f | Alison has compiled a list of 20 hockey teams, ordered by how good she thinks they are. Benjamin can ask her about any three teams and she will either tell him which she thinks is the weakest or which she thinks is the strongest. Determine the largest $N$ such that Benjamin can guarantee to find a sequence $T_1, T_2, .... | [
"7"
] | [
"Alison has compiled a list of 20 hockey teams, ordered by how good she thinks they are. Benjamin can ask her about any three teams and she will either tell him which she thinks is the weakest or which she thinks is the strongest. Determine the largest $N$ such that Benjamin can guarantee to find a sequence $T_1, T... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-2468bc5818ccc9f297283ef1d594951fd39fa4acc3e2c4d0f7adb716c07100e4 | Alison has compiled a list of 20 soccer teams ordered by how good she thinks they are. Benjamin can ask her about any three teams and she will either tell him the best or the worst among them. Determine the largest positive integer \( N \) such that Benjamin can find a sequence of teams \( T_1, T_2, T_3, \ldots, T_N \)... | [
"10"
] | [
"Alison has compiled a list of 20 soccer teams ordered by how good she thinks they are. Benjamin can ask her about any three teams and she will either tell him the best or the worst among them. Determine the largest positive integer \\( N \\) such that Benjamin can find a sequence of teams \\( T_1, T_2, T_3, \\ldot... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-b21c79eb53b7096c43630c4174fab25b32370e66b00da14be057cf0c8fbec9f2 | Alistar selects a number \( n \) uniformly from \([0,1]\) and eats all grass within \( n \) meters from where he's standing. He then moves 2 meters in a random direction and repeats the process, stopping if any of the grass he wants to eat is already eaten. Estimate the expected amount of grass Alistar is expected to e... | [
"\\pi"
] | [
"Alistar selects a number \\( n \\) uniformly from \\([0,1]\\) and eats all grass within \\( n \\) meters from where he's standing. He then moves 2 meters in a random direction and repeats the process, stopping if any of the grass he wants to eat is already eaten. Estimate the expected amount of grass Alistar is ex... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-6ed1ad3dbb3a7efe6ffdaa5db9ea4db36533e15f77a48858cbfa23fa78f4554c | Alka finds a number $n$ written on a board that ends in $5$. She performs a sequence of operations with the number on the board. At each step, she decides to carry out one of the following two operations: 1. Erase the written number $m$ and write its cube $m^3$. 2. Erase the written number $m$ and write the product $20... | [
"2"
] | [
"Alka finds a number $n$ written on a board that ends in $5$. She performs a sequence of operations with the number on the board. At each step, she decides to carry out one of the following two operations:\n\n1. Erase the written number $m$ and write its cube $m^3$.\n2. Erase the written number $m$ and write the pr... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f5ec0b884163ee40c400a091eec2cb9e2a6a1cbc0fe772a683b86db289361e64 | All $10$-digit numbers consisting of digits $1, 2,$ and $3$ are written one under the other. Each number has one more digit added to the right, which can be $1$, $2$, or $3$. It is given that the number $111\ldots11$ has $1$ added, the number $222\ldots22$ has $2$ added, and the number $333\ldots33$ has $3$ added. It i... | [
"59049"
] | [
"All $10$-digit numbers consisting of digits $1, 2,$ and $3$ are written one under the other. Each number has one more digit added to the right, which can be $1$, $2$, or $3$. It is given that the number $111\\ldots11$ has $1$ added, the number $222\\ldots22$ has $2$ added, and the number $333\\ldots33$ has $3$ add... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7903f613b7907a07bfac608e3ac219c9594363d24b3baab6ea4ce13f2acc9e7d | All $n \le 1000$ such that the number $9 + 99 + 999 + \ldots + \underbrace{99\ldots9}_{n \text{ digits}}$ has exactly $n$ digits equal to the number $1$ are ... | [
"999"
] | [
"All $n \\le 1000$ such that the number $9 + 99 + 999 + \\ldots + \\underbrace{99\\ldots9}_{n \\text{ digits}}$ has exactly $n$ digits equal to the number $1$ are ..."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-3adc8a05d43b99afec364a6511648d2696bc0a1b4594587f9dcb4cde72dcb00b | All arrangements of the letters $VNNWHTAAIE$ are listed in lexicographic (dictionary) order. If $AAEHINNTVW$ is the first entry, what entry number is $VANNAWHITE$? | [
"738826"
] | [
"All arrangements of the letters $VNNWHTAAIE$ are listed in lexicographic (dictionary) order. If $AAEHINNTVW$ is the first entry, what entry number is $VANNAWHITE$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-fcefedeacf0b06fc37691026a4df00070f1f0c2b0ba4a4905d2e2e6308d33d5b | All balls except the white one are positioned in a frame in the shape of an equilateral triangle, where the balls touch each other. A smaller ball $K_b$ is placed in this arrangement, touching the three larger balls $K_1$, $K_2$, and $K_3$ below it. The highest points of each small and large ball lie in one plane. Dete... | [
"\\dfrac{R}{3}"
] | [
"All balls except the white one are positioned in a frame in the shape of an equilateral triangle, where the balls touch each other. A smaller ball $K_b$ is placed in this arrangement, touching the three larger balls $K_1$, $K_2$, and $K_3$ below it. The highest points of each small and large ball lie in one plane.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-74e7d46a84f1752f273ac81c9b339ce8b2c1a282173ba68ab1b1f23a138b28f9 | All cells of an $n \times n$ board are colored using one of $k$ colors such that if for row $r$ and columns $i < j$, the cells $(r,i)$ and $(r,j)$ have the same color $C$, then no cells above $(r,i)$ contain color $C$ (i.e., for any $r' < r$, the cell $(r',i)$ does not contain color $C$). Find the minimum value of $k$. | [
"n"
] | [
"All cells of an $n \\times n$ board are colored using one of $k$ colors such that if for row $r$ and columns $i < j$, the cells $(r,i)$ and $(r,j)$ have the same color $C$, then no cells above $(r,i)$ contain color $C$ (i.e., for any $r' < r$, the cell $(r',i)$ does not contain color $C$). Find the minimum value o... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-c31d7422b0956088496e0b45f6962a81295de89152872db623828d5f57514de4 | All consecutive odd number triples $(a, b, c)$ with $a < b < c$ such that $a^2 + b^2 + c^2$ is a number with $4$ digits such that all the digits are the same are ... | [
"(41, 43, 45)"
] | [
"All consecutive odd number triples $(a, b, c)$ with $a < b < c$ such that $a^2 + b^2 + c^2$ is a number with $4$ digits such that all the digits are the same are ..."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-29a79897f60eb86ffaee58ee8f182006ba7a7312ea5e71b6d8703f1f1d2d93ff | All convex polygons with perimeter 1 have area less than $x$. What is the smallest possible value of $x$? | [
"\\dfrac{1}{4\\pi}"
] | [
"All convex polygons with perimeter 1 have area less than $x$. What is the smallest possible value of $x$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-392fa7d33fd7f5034dba887be53f54676edbad033081934220fcf312433291a3 | All digits of the positive integer \( a \) are distinct. The number \( b \) is obtained from \( a \) by rearranging its digits. If every digit of the difference \( a - b \) is a 1, what is the largest value of this difference? | [
"111,111,111"
] | [
"All digits of the positive integer \\( a \\) are distinct. The number \\( b \\) is obtained from \\( a \\) by rearranging its digits. If every digit of the difference \\( a - b \\) is a 1, what is the largest value of this difference?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-f0bff661ff5dc231f0348bed7c53f3f4a8c60055881a11f9cc077f51d9a768fe | All edges of a cube are expanding at a rate of 3 inches per second. How fast is the volume changing when each edge is (a) 1 inch and (b) 10 inches? | [
"900"
] | [
"All edges of a cube are expanding at a rate of 3 inches per second. How fast is the volume changing when each edge is (a) 1 inch and (b) 10 inches?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5f444f7bb962d93379f8eb50332c8f4cf1deaf134111afda4f18cde7b58fa16a | All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called excentric. The excentricity of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Consider a polyhedron with 8 vertices. Calculat... | [
"0"
] | [
"All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called excentric. The excentricity of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Consider a polyhedron with 8 vertices. Cal... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c52ca3c966248c4188e210bd755706494072b88c027804313543db701b8bd434 | All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$. | [
"\\( \\frac{1}{\\sqrt{3}} \\)"
] | [
"All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-9bc9877a09d5fe0d3297530fc04f07ea861fbf0696dc28d963b62c55100218ff | All edges of a tetrahedron $ABCD$ are equal. The tetrahedron $ABCD$ is inscribed in a sphere. $CC'$ and $DD'$ are diameters of the sphere. Find the angle between the planes $ABC'$ and $ACD'$. | [
"109.47 degrees"
] | [
"All edges of a tetrahedron $ABCD$ are equal. The tetrahedron $ABCD$ is inscribed in a sphere. $CC'$ and $DD'$ are diameters of the sphere. Find the angle between the planes $ABC'$ and $ACD'$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-66cf4d3a355810772f69cf50fe7dbaacd69b1314f44d057c628e1bf741a12950 | All edges of the triangular prism \(ABCA_1B_1C_1\) are equal to \(a\) and \(\angle AA_1C_1 = \angle AA_1B_1 = 60^\circ\). Find the total surface area of the prism. | [
"\\(a \\left( 1 + \\frac{3\\sqrt{3}}{2} \\right)\\)"
] | [
"All edges of the triangular prism \\(ABCA_1B_1C_1\\) are equal to \\(a\\) and \\(\\angle AA_1C_1 = \\angle AA_1B_1 = 60^\\circ\\). Find the total surface area of the prism."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-753890418a791e0b861b090a01668c8e0b977936afcd4a20f8bed83b3eb50da1 | All edges of the triangular pyramid $ABCD$ are equal in length. Let $M$ be the midpoint of $DB$, $N$ be the point on $\overline{AB}$ such that $2NA=NB$, and $P$ be a point on the altitude through point $D$ in $\triangle BCD$. Find $\angle MPD$ if the intersection of the pyramid with the plane $(NMP)$ is a trapezoid. | [
"90^\\circ"
] | [
"All edges of the triangular pyramid $ABCD$ are equal in length. Let $M$ be the midpoint of $DB$, $N$ be the point on $\\overline{AB}$ such that $2NA=NB$, and $P$ be a point on the altitude through point $D$ in $\\triangle BCD$. Find $\\angle MPD$ if the intersection of the pyramid with the plane $(NMP)$ is a trape... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-dffe273045be8bd943de4277c3bc204a5402e2e105072325b36801ab581ff74d | All even numbers from 2 to 14 are written in each circle exactly once. If the sum of the numbers in any three circles lying in a straight line is equal, how many different ways can the circles be filled? | [
"144"
] | [
"All even numbers from 2 to 14 are written in each circle exactly once. If the sum of the numbers in any three circles lying in a straight line is equal, how many different ways can the circles be filled?"
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 25,
"genselect": 8
} | 33 | 0 | 0 |
omr-ws-3fd221990bed506c07fee2d5f0a0ef06dc53791695bfb28149c17dac165529b4 | All expressions in this problem are expressed in base \( 5 \), including the answer and solution. Let \( \frac{1}{11} + \frac{1}{13} + \frac{1}{22} = \frac{a}{b} \), a fraction in lowest terms. Compute \( a^2 + b^2 \). | [
"178"
] | [
"All expressions in this problem are expressed in base \\( 5 \\), including the answer and solution. Let \\( \\frac{1}{11} + \\frac{1}{13} + \\frac{1}{22} = \\frac{a}{b} \\), a fraction in lowest terms. Compute \\( a^2 + b^2 \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-25088df3fdd1dcf762fe0c3be6e0b30a05972317b89fc2fa8cd2420eaa598555 | All faces of a parallelepiped $ABCDA_1B_1C_1D_1$ are rhombi, and their angles at $A$ are all equal to $\alpha$. Points $M,N,P,Q$ are selected on the edges $A_1B_1,DC,BC,A_1D_1$, respectively, such that $A_1M=BP$ and $DN=A_1Q$. Find the angle between the intersection lines of the plane $A_1BD$ with the planes $AMN$ and ... | [
"90^\\circ"
] | [
"All faces of a parallelepiped $ABCDA_1B_1C_1D_1$ are rhombi, and their angles at $A$ are all equal to $\\alpha$. Points $M,N,P,Q$ are selected on the edges $A_1B_1,DC,BC,A_1D_1$, respectively, such that $A_1M=BP$ and $DN=A_1Q$. Find the angle between the intersection lines of the plane $A_1BD$ with the planes $AMN... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-c678df67b9ea2e9f4a62addeba597aa3b209b3868c769a3c43cd8d831e722c7e | All faces of a polyhedron are triangles. Each vertex of this polyhedron is independently assigned one of three colors: green, white, or black. A face is called "Extremadura" if its three vertices are of different colors (one green, one white, and one black). Let $P$ be a polyhedron with $V$ vertices, $E$ edges, and $F$... | [
"\\dfrac{2}{3}(E_G + E_W + E_B)"
] | [
"All faces of a polyhedron are triangles. Each vertex of this polyhedron is independently assigned one of three colors: green, white, or black. A face is called \"Extremadura\" if its three vertices are of different colors (one green, one white, and one black). Let $P$ be a polyhedron with $V$ vertices, $E$ edges, ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0c50818a0d51b6464a1253681148d4d0934247e445ac2799372c93b8c125e9bd | All faces of a tetrahedron are right-angled triangles. It is known that three of its edges have the same length $s$. Find the volume of the tetrahedron. | [
"\\(\\frac{s^3}{6}\\)"
] | [
"All faces of a tetrahedron are right-angled triangles. It is known that three of its edges have the same length $s$. Find the volume of the tetrahedron."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 12
} | 33 | 0 | 0 |
omr-ws-5dc2607dcd650ea7d7338e3078c0435642456e853dab739cd1021c9028378a88 | All faces of the parallelepiped $ABCDA_1B_1C_1D_1$ are equal rhombuses, and plane angles at vertex $A$ are equal. Points $K$ and $M$ are taken on the edges $A_1B_1$ and $A_1D_1$ such that $A_1K = a$ and $A_1M = b$, with $a + b$ being the length of an edge of the parallelepiped. The plane $AKM$ touches the sphere inscri... | [
"\\dfrac{a}{b}"
] | [
"All faces of the parallelepiped $ABCDA_1B_1C_1D_1$ are equal rhombuses, and plane angles at vertex $A$ are equal. Points $K$ and $M$ are taken on the edges $A_1B_1$ and $A_1D_1$ such that $A_1K = a$ and $A_1M = b$, with $a + b$ being the length of an edge of the parallelepiped. The plane $AKM$ touches the sphere i... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-2856d6f2d01b861bbcc4d405c5146afaccf476d32b94fee5051a7f6dae6be21f | All integer numbers are colored in 3 colors in an arbitrary way. Let $S$ be the set of all perfect squares. Find the minimum value of $k$ such that there exist $k$ distinct integers $a_1, a_2, \ldots, a_k$ where $a_i - a_j \in S$ for some $i \neq j$ and $a_i$ and $a_j$ are colored in the same color. | [
"9"
] | [
"All integer numbers are colored in 3 colors in an arbitrary way. Let $S$ be the set of all perfect squares. Find the minimum value of $k$ such that there exist $k$ distinct integers $a_1, a_2, \\ldots, a_k$ where $a_i - a_j \\in S$ for some $i \\neq j$ and $a_i$ and $a_j$ are colored in the same color."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fa9f246da70d20fd00ec1d618884a015c4e674794049269fc96b452a2f3e21c0 | All integers from $1$ to one million are written on a tape in some arbitrary order. The tape is then cut into pieces containing two consecutive digits each. Let $N$ be the number of distinct two-digit integers among these pieces. Find the minimum possible value of $N$. | [
"90"
] | [
"All integers from $1$ to one million are written on a tape in some arbitrary order. The tape is then cut into pieces containing two consecutive digits each. Let $N$ be the number of distinct two-digit integers among these pieces. Find the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-b666e62ee9ccb83053212fe7785ad8a393d054cc65fd8af0b21b80451ac19015 | All integers from 1 to \( n \) are colored either red or blue such that each color is used at least once. Every red number is the sum of two distinct blue numbers, and every blue number is the difference of two distinct red numbers. Find the smallest \( n \) for which such a coloring is possible. | [
"6"
] | [
"All integers from 1 to \\( n \\) are colored either red or blue such that each color is used at least once. Every red number is the sum of two distinct blue numbers, and every blue number is the difference of two distinct red numbers. Find the smallest \\( n \\) for which such a coloring is possible."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f1599e2277395b1cf98e0a3aa8e7c3c92c5974ea24c11e3782f605516c133449 | All integers in the range $[2,99]$ are placed in a box, and two are drawn. Alice is given their sum and Bob is given their product. Alice says, "I can tell you don't know the numbers." Bob replies, "Now I do." Alice then says, "Now I do too!" What are the numbers? | [
"4, 13"
] | [
"All integers in the range $[2,99]$ are placed in a box, and two are drawn. Alice is given their sum and Bob is given their product. Alice says, \"I can tell you don't know the numbers.\" Bob replies, \"Now I do.\" Alice then says, \"Now I do too!\" What are the numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 11
} | 21 | 0 | 0 |
omr-ws-03156822838c2440d1d97ad2483e7d6bf02d11a525021725030374ef1ef4dfcd | All lattice points are colored either red or blue. Find the minimum number of lattice points $n$ such that any $n$-point subset must contain a triangle with lattice vertices and a lattice barycenter, where all these points are the same color. | [
"9"
] | [
"All lattice points are colored either red or blue. Find the minimum number of lattice points $n$ such that any $n$-point subset must contain a triangle with lattice vertices and a lattice barycenter, where all these points are the same color."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-c4b2578f1229068207571de0030a024e3d7b77a12c46b7d630cfd6d9bfc6d593 | All natural numbers from $1$ to $N^2$ are arranged in an $N \times N$ table. A number is said to be "large" if it is the largest in its row, and "small" if it is the smallest in its column. Find the smallest possible difference between the sum of all large numbers and the sum of all small numbers. | [
"\\(\\frac{N(2N+5)(N-1)}{6}\\)"
] | [
"All natural numbers from $1$ to $N^2$ are arranged in an $N \\times N$ table. A number is said to be \"large\" if it is the largest in its row, and \"small\" if it is the smallest in its column. Find the smallest possible difference between the sum of all large numbers and the sum of all small numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8d7b803acc04edc67f41cc550e8c27d4b05ccea46f3f18c0b7b044ecfb5b36e4 | All numbers $1, 2, 3, \cdots, 64$ are filled randomly on a chessboard, with each box containing one number. In every row, the third largest number is chosen, and these numbers are summed to get $A$. Let $S_i$ be the sum of the numbers in the $i$-th row. Find the maximum possible value of $S_i - A$ for any row $i$. | [
"254"
] | [
"All numbers $1, 2, 3, \\cdots, 64$ are filled randomly on a chessboard, with each box containing one number. In every row, the third largest number is chosen, and these numbers are summed to get $A$. Let $S_i$ be the sum of the numbers in the $i$-th row. Find the maximum possible value of $S_i - A$ for any row $i$... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.