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-3ec83c8d76fe480a95f7616b7194521455994a4d15806396b77db0881dac1afb | Bob and Jim use hoses from both houses to fill Bob's swimming pool. Together, they can fill the pool in 18 hours. Bob's hose alone can fill the pool in 6 hours less than Jim's hose. How much time is required by each hose alone to fill the pool? | [
"33.25 hours (Bob's hose), 39.25 hours (Jim's hose)"
] | [
"Bob and Jim use hoses from both houses to fill Bob's swimming pool. Together, they can fill the pool in 18 hours. Bob's hose alone can fill the pool in 6 hours less than Jim's hose. How much time is required by each hose alone to fill the pool?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-c219f278a397c3e7ef2f0ec6e0c12ac842ec32cf222433026430e4757dc02674 | Bob and Joe are playing a game on an $8 \times 8$ grid. They take turns writing a single-digit number in a box, with the restriction that no $0$'s can be written in the first row or column. Bob wins if at any point, an 8-digit number read top-to-bottom or left-to-right is a multiple of $137$. Determine who wins the gam... | [
"Bob"
] | [
"Bob and Joe are playing a game on an $8 \\times 8$ grid. They take turns writing a single-digit number in a box, with the restriction that no $0$'s can be written in the first row or column. Bob wins if at any point, an 8-digit number read top-to-bottom or left-to-right is a multiple of $137$. Determine who wins t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e832ad01796a1ee95d381ea1e45fcbc5028a67004c003b6089728114da0ac3c3 | Bob and Joe are playing a game with a stack of $n$ coins, all head side up. Starting with Joe, they take turns taking some amount of coins from the top of the stack and flipping all of them over to form a new stack (taking $0$ coins or flipping over the entire stack are not allowed). They continue doing this until they... | [
"n \\text{ is odd and } n \\geq 3"
] | [
"Bob and Joe are playing a game with a stack of $n$ coins, all head side up. Starting with Joe, they take turns taking some amount of coins from the top of the stack and flipping all of them over to form a new stack (taking $0$ coins or flipping over the entire stack are not allowed). They continue doing this until... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-036453df4591f11bb82c66076f22f3e3d401495af7b37eacd382597808d572b6 | Bob and Joe are playing a game with chips. Each starts with 2 chips. They keep flipping an unfair coin until one of them runs out of chips. If the coin comes up heads (with probability $\frac{3}{5}$), Joe gives a chip to Bob. If it comes up tails (with probability $\frac{2}{5}$), Bob gives a chip to Joe. What is the pr... | [
"\\(\\frac{9}{13}\\)"
] | [
"Bob and Joe are playing a game with chips. Each starts with 2 chips. They keep flipping an unfair coin until one of them runs out of chips. If the coin comes up heads (with probability $\\frac{3}{5}$), Joe gives a chip to Bob. If it comes up tails (with probability $\\frac{2}{5}$), Bob gives a chip to Joe. What is... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-dd545b8a0c937ec23169c5981842059f8af612797de0b3e4326bf9c758f627fe | Bob and Judy are running at constant rates. Bob's rate is an integral number of feet per minute, and Judy's rate is 50 feet more per minute than Bob's. Both can run a mile in more than 5 minutes but less than 5.5 minutes. Judy is x feet ahead of Bob, and point A is 4000 feet ahead of Bob. Judy will reach point A at the... | [
"1025"
] | [
"Bob and Judy are running at constant rates. Bob's rate is an integral number of feet per minute, and Judy's rate is 50 feet more per minute than Bob's. Both can run a mile in more than 5 minutes but less than 5.5 minutes. Judy is x feet ahead of Bob, and point A is 4000 feet ahead of Bob. Judy will reach point A a... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-7c22a65d7f64183f3abd6deccc9bd5a02eca4e3cd048b5b6c167af049ec0ee64 | Bob and Larry are driving an 800-mile road trip to Washington D.C. Bob leaves at 6:30am and drives at 50 mph, stopping for 10 minutes at 10:30am for gas. Larry leaves at 8:00am and drives at 65 mph. Both stay in Washington D.C. for 30 minutes before returning home at their original speeds. Who arrives home first and by... | [
"Larry by \\(\\frac{236}{39}\\) hours."
] | [
"Bob and Larry are driving an 800-mile road trip to Washington D.C. Bob leaves at 6:30am and drives at 50 mph, stopping for 10 minutes at 10:30am for gas. Larry leaves at 8:00am and drives at 65 mph. Both stay in Washington D.C. for 30 minutes before returning home at their original speeds. Who arrives home first a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 6,
"tir": 21
} | 35 | 0 | 0 |
omr-ws-8828e2682e35b4365910959e63f7447098cdc72238b40dfb767777d201f263c7 | Bob and Larry are on different escalators. Bob's escalator moves up at 3 mph, and he walks down at 6 mph. Larry's escalator moves up at 5 mph, and he walks down at 7 mph. The escalators are 400 feet long. Who gets down first and by how much time? | [
"Bob by 4.5454... minutes."
] | [
"Bob and Larry are on different escalators. Bob's escalator moves up at 3 mph, and he walks down at 6 mph. Larry's escalator moves up at 5 mph, and he walks down at 7 mph. The escalators are 400 feet long. Who gets down first and by how much time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 18
} | 37 | 0 | 0 |
omr-ws-426de83da148a2232c184429a57b07b4e52ab7fd9cfa21637b49137d80c263b0 | Bob and Nick are playing a game. Bob rolls a die and continues rolling until he gets a 2, summing all outcomes and calling this sum \(X\). Nick rolls a die and continues rolling until he gets a 5, summing all outcomes and calling this sum \(Y\). At the end of the game, Bob pays Nick \(Y\) dollars and Nick pays Bob \(X\... | [
"Neither Bob nor Nick has an advantage."
] | [
"Bob and Nick are playing a game. Bob rolls a die and continues rolling until he gets a 2, summing all outcomes and calling this sum \\(X\\). Nick rolls a die and continues rolling until he gets a 5, summing all outcomes and calling this sum \\(Y\\). At the end of the game, Bob pays Nick \\(Y\\) dollars and Nick pa... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-c1117b4383eab4ed7bc062fc0c1f90aedef99bfa790f6749292b95b1f6e541c9 | Bob and Rob play a game where they toss an unfair coin 3 times. The coin has a $\frac{2}{3}$ chance of landing heads and a $\frac{1}{3}$ chance of landing tails. If exactly 2 out of 3 tosses land heads, Bob gets a point; otherwise, Rob gets a point. They play until one of them has a total of 3 points. What is the proba... | [
"\\(\\frac{80}{243}\\)"
] | [
"Bob and Rob play a game where they toss an unfair coin 3 times. The coin has a $\\frac{2}{3}$ chance of landing heads and a $\\frac{1}{3}$ chance of landing tails. If exactly 2 out of 3 tosses land heads, Bob gets a point; otherwise, Rob gets a point. They play until one of them has a total of 3 points. What is th... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c4302fc5a87a03535867b2b4f892f32ce46cdc52a1e288d8022c9baa306d6d86 | Bob and Tom each have a cube and can paint each side either red or blue. After painting both cubes, what is the probability that the two cubes are indistinguishable? | [
"\\(\\frac{147}{1024}\\)"
] | [
"Bob and Tom each have a cube and can paint each side either red or blue. After painting both cubes, what is the probability that the two cubes are indistinguishable?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0167d2158c73a06ad193ea9ff6f39b689e71666cc8f4f788c62167d1e2e6f298 | Bob and Tom each have a cube and some red and blue paint. They paint each side of their cubes either red or blue. What is the probability that the two painted cubes are indistinguishable? | [
"\\(\\frac{147}{1024}\\)"
] | [
"Bob and Tom each have a cube and some red and blue paint. They paint each side of their cubes either red or blue. What is the probability that the two painted cubes are indistinguishable?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 4,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-98cc64ccca66de2c98fe2356907bd764f0e8f2162349ce67de9686a968ab3759 | Bob can make 100 sweaters in 8 hours, and Jane can make the same amount in 6 hours. How long does it take them to make 200 sweaters working together? | [
"\\(\\frac{48}{7}\\) hrs."
] | [
"Bob can make 100 sweaters in 8 hours, and Jane can make the same amount in 6 hours. How long does it take them to make 200 sweaters working together?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-dd80328f721991cfb0e73f41385b21f113053d1e73aeb7ec40775b7ca7897564 | Bob chooses a 4-digit binary string uniformly at random and examines an infinite sequence of uniformly and independently random binary bits. If \( N \) is the least number of bits Bob has to examine in order to find his chosen string, find the expected value of \( N \). For example, if Bob’s string is \( 0000 \) and th... | [
"19"
] | [
"Bob chooses a 4-digit binary string uniformly at random and examines an infinite sequence of uniformly and independently random binary bits. If \\( N \\) is the least number of bits Bob has to examine in order to find his chosen string, find the expected value of \\( N \\). For example, if Bob’s string is \\( 0000... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-3167b47375b38529786848182ff3d6a5bdbdc5719de66d77586259eb4b731024 | Bob chooses a 4-element subset $\mathcal{T}$ of $\{1, 2, 3, \ldots, 10\}$. He places the four elements randomly around the vertices of a square $ABCD$. Denote $V_I$ to be the number at vertex $I$. How many ways can Bob do this if the following conditions are satisfied: $$|V_{A}-V_{B}|\le 4$$ $$|V_{B}-V_{C}|\le 4$$ $$|V... | [
"600"
] | [
"Bob chooses a 4-element subset $\\mathcal{T}$ of $\\{1, 2, 3, \\ldots, 10\\}$. He places the four elements randomly around the vertices of a square $ABCD$. Denote $V_I$ to be the number at vertex $I$. How many ways can Bob do this if the following conditions are satisfied:\n$$|V_{A}-V_{B}|\\le 4$$ \n$$|V_{B}-V_{C}... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-fb5f4d638589d76713aad5641f06236f0b41082a8ac81ea81785a92f2f2d6a14 | Bob chose four distinct digits from the set $\{1,2,3,4,5,6,7,8,9\}$. With them, he formed all possible four-digit numbers and added all the numbers formed. The result was 193,314. Find the four digits Bob chose. | [
"5, 7, 8, 9"
] | [
"Bob chose four distinct digits from the set $\\{1,2,3,4,5,6,7,8,9\\}$. With them, he formed all possible four-digit numbers and added all the numbers formed. The result was 193,314. Find the four digits Bob chose."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 30
} | 43 | 0 | 0 |
omr-ws-1d13363510ec3cf4604fa2aef21a09fc261071f904b89278cbd8b0ea014b8b95 | Bob dips the vertices of a regular dodecahedron with black paint and rolls it without slipping on an infinitely large canvas, creating black dots at the points where the vertices touch the canvas. Given that the side length of the dodecahedron is 1 unit, determine the minimum number of distinct dots that can be created... | [
"20"
] | [
"Bob dips the vertices of a regular dodecahedron with black paint and rolls it without slipping on an infinitely large canvas, creating black dots at the points where the vertices touch the canvas. Given that the side length of the dodecahedron is 1 unit, determine the minimum number of distinct dots that can be cr... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-84608cbdcb961ef6245f95b21b9410e0941cf75203625493a0a0f3a67c3554d6 | Bob draws a unit quarter-circle $ABC$ with $B$ at the center and $A$ and $C$ as endpoints of the arc. He randomly chooses points $P$ and $Q$ on segments $AB$ and $BC$, respectively. Determine the probability that Bob can draw two right triangles $PQR$ with $R$ lying on arc $AC$. | [
"\\(1 - \\frac{\\pi}{16}\\)"
] | [
"Bob draws a unit quarter-circle $ABC$ with $B$ at the center and $A$ and $C$ as endpoints of the arc. He randomly chooses points $P$ and $Q$ on segments $AB$ and $BC$, respectively. Determine the probability that Bob can draw two right triangles $PQR$ with $R$ lying on arc $AC$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-865c1850809ac34ae40e05aef7b526b71a41a72ae972ec3ae8d6211a535613f0 | Bob draws a unit square \(A_1B_1C_1D_1\). After drawing square \(A_nB_nC_nD_n\), he defines the next square \(A_{n+1}B_{n+1}C_{n+1}D_{n+1}\) with the points defined as follows: - \(A_{n+1}\) is \(\frac{3}{7}\)ths of the way from \(A_n\) to \(B_n\). - \(B_{n+1}\) is \(\frac{3}{7}\)ths of the way from \(B_n\) to \(C_n\).... | [
"14"
] | [
"Bob draws a unit square \\(A_1B_1C_1D_1\\). After drawing square \\(A_nB_nC_nD_n\\), he defines the next square \\(A_{n+1}B_{n+1}C_{n+1}D_{n+1}\\) with the points defined as follows:\n\n- \\(A_{n+1}\\) is \\(\\frac{3}{7}\\)ths of the way from \\(A_n\\) to \\(B_n\\).\n- \\(B_{n+1}\\) is \\(\\frac{3}{7}\\)ths of the... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 2
} | 48 | 0 | 0 |
omr-ws-5625a3ad89d5b6e8a5eaceb3e5fe0753a25c211624700bb6adf1235b16288f82 | Bob draws an isosceles trapezoid $\text{ABCD}$ such that $AB \parallel CD$ and $\overline{AD} = \overline{BC}$. It is known that $\overline{AB} = 6$, $\overline{CD} = 6 + 6\sqrt{3}$, and $\angle B = 120^\circ$. Bob then draws a perpendicular from line $AB$ at $B$ and extends it until it intersects line $CD$ at point $E... | [
"84"
] | [
"Bob draws an isosceles trapezoid $\\text{ABCD}$ such that $AB \\parallel CD$ and $\\overline{AD} = \\overline{BC}$. It is known that $\\overline{AB} = 6$, $\\overline{CD} = 6 + 6\\sqrt{3}$, and $\\angle B = 120^\\circ$. Bob then draws a perpendicular from line $AB$ at $B$ and extends it until it intersects line $C... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-24537491b2ba04fd11dc7bd12f86ace7f5aa7f0d6ae8441606101e2f28d0bd19 | Bob flips two fair coins, while Dan flips $L$ fair coins. Let $p$ be the probability that the number of heads showing on Bob's coins is greater than the number of heads showing on Dan's coins. If $p=\frac{q}{r}$, where $q$ and $r$ are relatively prime positive integers, find $q+r$. | [
"9"
] | [
"Bob flips two fair coins, while Dan flips $L$ fair coins. Let $p$ be the probability that the number of heads showing on Bob's coins is greater than the number of heads showing on Dan's coins. If $p=\\frac{q}{r}$, where $q$ and $r$ are relatively prime positive integers, find $q+r$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e514ceedfdb5a38ef6f086c02f91a642988f9b25dd423181a797e4d2c8badd11 | Bob goes one mile north, turns 90 degrees, goes forward 2 miles, turns 90 degrees, goes forward 3 miles, and continues this pattern until he goes forward n miles and stops. Find the distance from his original starting place to where he is now, in terms of n. | [
"\\(\\sqrt{\\left(\\sum_{n=1}^{\\lceil \\frac{n}{2} \\rceil}(-1)^n (2n-1)\\right)^2 + \\left(\\sum_{n=1}^{\\lfloor \\frac{n}{2} \\rfloor}(-1)^n (2n)\\right)^2}\\)"
] | [
"Bob goes one mile north, turns 90 degrees, goes forward 2 miles, turns 90 degrees, goes forward 3 miles, and continues this pattern until he goes forward n miles and stops. Find the distance from his original starting place to where he is now, in terms of n."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9cec5fa7ae8589f1ec95f1ded1846f492125a19715c6ae607f9fd4b90e5d978a | Bob goes through an obstacle course of 5 total obstacles, where he must successfully complete all 5 obstacles in a row to pass. The probability he successfully completes each obstacle is $1$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, and $\frac{1}{5}$, respectively, and each attempt takes 1 second. He is allowed to ... | [
"153"
] | [
"Bob goes through an obstacle course of 5 total obstacles, where he must successfully complete all 5 obstacles in a row to pass. The probability he successfully completes each obstacle is $1$, $\\frac{1}{2}$, $\\frac{1}{3}$, $\\frac{1}{4}$, and $\\frac{1}{5}$, respectively, and each attempt takes 1 second. He is al... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 7
} | 26 | 0 | 0 |
omr-ws-5fb91bacef52647e45a5f65028fc216d7345625c2fe3e278639c1db81cdf6c66 | Bob has $2$ sets of all the integers from $1$ through $14$, inclusive. He first picks a number $n$ out of the first set and then chooses a number $m \ne n$ uniformly at random from the second set subject to the following conditions: - If $n$ is greater than $7$, then $m$ is less than $n$, - If $n$ is less than $7$, the... | [
"2867"
] | [
"Bob has $2$ sets of all the integers from $1$ through $14$, inclusive. He first picks a number $n$ out of the first set and then chooses a number $m \\ne n$ uniformly at random from the second set subject to the following conditions:\n- If $n$ is greater than $7$, then $m$ is less than $n$,\n- If $n$ is less than ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8d86e9746e058eaec9ad0ce6d1d3a0f566a1be6ef87b579a3fef822f894c1e4e | Bob has $k$ apple trees, $8$ peach trees, and $k - 5$ money trees. When planting these trees in a row, the number of ways to plant them such that no two apple trees are next to each other is $2380$. What is the smallest possible value for $k$? | [
"13"
] | [
"Bob has $k$ apple trees, $8$ peach trees, and $k - 5$ money trees. When planting these trees in a row, the number of ways to plant them such that no two apple trees are next to each other is $2380$. What is the smallest possible value for $k$?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-8a39771d226631ee7303ee1a36c5023a2e9ae78bbdcb6a8cd236896e80533eb1 | Bob has 11 socks in his drawer: 3 different matched pairs, and 5 socks that do not match with any others. Suppose he pulls socks from the drawer one at a time until he manages to get a matched pair. What is the probability he will need to draw exactly 9 socks? | [
"\\dfrac{8}{165}"
] | [
"Bob has 11 socks in his drawer: 3 different matched pairs, and 5 socks that do not match with any others. Suppose he pulls socks from the drawer one at a time until he manages to get a matched pair. What is the probability he will need to draw exactly 9 socks?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5565d9b50dc4b79c8b54b046280281045d9c27514769899429afac94ee07fd7c | Bob has 11 socks in his drawer: 3 different matched pairs, and 5 socks that don’t match with any others. Suppose he pulls socks from the drawer one at a time until he manages to get a matched pair. What is the probability he will need to draw exactly 9 socks? | [
"\\dfrac{8}{165}"
] | [
"Bob has 11 socks in his drawer: 3 different matched pairs, and 5 socks that don’t match with any others. Suppose he pulls socks from the drawer one at a time until he manages to get a matched pair. What is the probability he will need to draw exactly 9 socks?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c0fe1093bad826f41456a7e627e04ee2061eb040af1446487db8ebfa448712e6 | Bob has 120 small cubes: 80 blue and 40 white. He wants to build a large rectangular parallelepiped using these cubes. What is the minimum number of visible faces of the small cubes that will be blue? | [
"60"
] | [
"Bob has 120 small cubes: 80 blue and 40 white. He wants to build a large rectangular parallelepiped using these cubes. What is the minimum number of visible faces of the small cubes that will be blue?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 12
} | 34 | 0 | 0 |
omr-ws-84111079bdc8a6a0554228a7673405747c2662a37e3e9ff061ba9cfec5b4fd79 | Bob has 120 small identical cubes: 80 blue and 40 white. He wants to use them to build a large rectangular parallelepiped with glue. Its surface will be entirely made up of the faces of the small cubes. At a minimum, how many visible faces of the small cubes will be blue? | [
"60"
] | [
"Bob has 120 small identical cubes: 80 blue and 40 white. He wants to use them to build a large rectangular parallelepiped with glue. Its surface will be entirely made up of the faces of the small cubes. At a minimum, how many visible faces of the small cubes will be blue?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b3823a0a2a3e95f7710ee4180a333a6aec01df9d762b5e7263da17080ef8c23 | Bob has 2 identical red coins and 2 identical blue coins, as well as 4 distinguishable buckets. He places some, but not necessarily all, of the coins into the buckets such that no bucket contains two coins of the same color, and at least one bucket is not empty. In how many ways can he do this? | [
"120"
] | [
"Bob has 2 identical red coins and 2 identical blue coins, as well as 4 distinguishable buckets. He places some, but not necessarily all, of the coins into the buckets such that no bucket contains two coins of the same color, and at least one bucket is not empty. In how many ways can he do this?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c469c080f955685cec6833ea4f4c132a498568d448ae6994cec75b9cbc84057e | Bob has 4 types of coins: pennies (1 cent), nickels (5 cents), dimes (10 cents), and quarters (25 cents). How many ways can Bob use exactly 20 coins to make 1 dollar (100 cents)? | [
"4"
] | [
"Bob has 4 types of coins: pennies (1 cent), nickels (5 cents), dimes (10 cents), and quarters (25 cents). How many ways can Bob use exactly 20 coins to make 1 dollar (100 cents)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48ea429f90d6620e2daab9768d2374a2973d812e4a44f94d953bdd8d35909495 | Bob has B blue cards, O orange cards, and W white cards. He scores points as he plays each card. If he plays a blue card, his score is the number of white cards remaining in his hand. If he plays an orange card, his score is three times the number of blue cards remaining in his hand. If he plays a white card, his score... | [
"B! \\cdot O! \\cdot W!"
] | [
"Bob has B blue cards, O orange cards, and W white cards. He scores points as he plays each card. If he plays a blue card, his score is the number of white cards remaining in his hand. If he plays an orange card, his score is three times the number of blue cards remaining in his hand. If he plays a white card, his ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-d29c75fba1c8a966bd22c297a16c3d334222acf7b18fdddd8dd515ec1d9d3ed2 | Bob has B blue cards, O orange cards, and W white cards. He scores points as he plays each card. If he plays a blue card, his score is the number of white cards remaining in his hand. If he plays an orange card, his score is three times the number of blue cards remaining in his hand. If he plays a white card, his score... | [
"\\(\\min(BW, 3BR, 2RW)\\)"
] | [
"Bob has B blue cards, O orange cards, and W white cards. He scores points as he plays each card. If he plays a blue card, his score is the number of white cards remaining in his hand. If he plays an orange card, his score is three times the number of blue cards remaining in his hand. If he plays a white card, his ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 5,
"tir": 7
} | 33 | 0 | 0 |
omr-ws-8dce7bd1c4e1a413cc3fcf5c57493c61f61a4e5ba6a234a3811427784eda2b5b | Bob has \( n \) stacks of rocks in a row, each with heights randomly and uniformly selected from the set \(\{1,2,3,4,5\}\). In each move, he picks a group of consecutive stacks with positive heights and removes 1 rock from each stack. Find, in terms of \( n \), the expected value of the minimum number of moves he must ... | [
"\\(\\frac{4n + 11}{5}\\)"
] | [
"Bob has \\( n \\) stacks of rocks in a row, each with heights randomly and uniformly selected from the set \\(\\{1,2,3,4,5\\}\\). In each move, he picks a group of consecutive stacks with positive heights and removes 1 rock from each stack. Find, in terms of \\( n \\), the expected value of the minimum number of m... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 7,
"tir": 9
} | 63 | 0 | 0 |
omr-ws-1c7deb31ce8aff196605e90d69c8ea0b1ef65602c4d8a7519677ecc584f93a23 | Bob has a 10 percent chance of blundering, a 30 percent chance of making a mistake, and a 60 percent chance of doing the best move. Sob has a 30 percent chance of blundering, a 20 percent chance of making a mistake, and a 50 percent chance of making the best move. A blunder counts as 2 mistakes, and a player loses if t... | [
"\\dfrac{3}{7}"
] | [
"Bob has a 10 percent chance of blundering, a 30 percent chance of making a mistake, and a 60 percent chance of doing the best move. Sob has a 30 percent chance of blundering, a 20 percent chance of making a mistake, and a 50 percent chance of making the best move. A blunder counts as 2 mistakes, and a player loses... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-f07f019eb9264aba2989040c8dce316ca789f230bb4e747e21b375d507ddd09a | Bob has a broken calculator that can only display two numbers, one on the left and one on the right. It has two buttons: one that switches the left and right numbers and randomly flips the sign of one of the numbers, and another that adds the number on the right to the number on the left. Initially, the calculator disp... | [
"\\dfrac{1}{256}"
] | [
"Bob has a broken calculator that can only display two numbers, one on the left and one on the right. It has two buttons: one that switches the left and right numbers and randomly flips the sign of one of the numbers, and another that adds the number on the right to the number on the left. Initially, the calculator... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-d850bc6ed762f345ab401bbafdb4910a91880c2dffa58ef03c154f920ad6463b | Bob has a pile of 100 rocks. Each turn, he splits a pile into two smaller piles until he has 100 piles, each with one rock. Let $N$ be the number of times during this process when there are exactly 30 piles with a total of 60 rocks. Calculate the minimum possible value of $N$. | [
"1"
] | [
"Bob has a pile of 100 rocks. Each turn, he splits a pile into two smaller piles until he has 100 piles, each with one rock. Let $N$ be the number of times during this process when there are exactly 30 piles with a total of 60 rocks. Calculate the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-d9aa9d6338ef66746ef38a4464e88b8af8477a8b1c3f7b21e4daafc9d7cfa69c | Bob has a positive integer $N$. In one move, Bob can split $N$ into two positive integers $a$ and $b$ such that $a + b = N$, and then replace $N$ with $a + b$. What is the maximum number of moves Bob needs to turn any positive integer $N$ into a one-digit number? | [
"8"
] | [
"Bob has a positive integer $N$. In one move, Bob can split $N$ into two positive integers $a$ and $b$ such that $a + b = N$, and then replace $N$ with $a + b$. What is the maximum number of moves Bob needs to turn any positive integer $N$ into a one-digit number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7387fb2a813cf08a79774882edf844429b87462692eed95197f3aa298311e2e8 | Bob has a regular dodecahedron with its vertices dipped in black paint. He rolls the dodecahedron on an infinitely big canvas. Let $N$ be the minimum number of distinct dots that can be painted within a circular patch of radius $1$ unit on the canvas. Find $N$. | [
"5"
] | [
"Bob has a regular dodecahedron with its vertices dipped in black paint. He rolls the dodecahedron on an infinitely big canvas. Let $N$ be the minimum number of distinct dots that can be painted within a circular patch of radius $1$ unit on the canvas. Find $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-f8a7fafa8ae0e6828650979f840c562dabd5af28a9119150b57d8e0eef6215b2 | Bob has a semicircular plot of land and wants to divide it into six parts by building two triangular fences whose common base is the diameter of the semicircle (AB) and whose other vertices (C and D) are on the semicircle. The triangles ABC and ABD are non-congruent and triangle ABP is not isosceles. Alice and Bob buil... | [
"56"
] | [
"Bob has a semicircular plot of land and wants to divide it into six parts by building two triangular fences whose common base is the diameter of the semicircle (AB) and whose other vertices (C and D) are on the semicircle. The triangles ABC and ABD are non-congruent and triangle ABP is not isosceles. Alice and Bob... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-6d084bea5557fd63c78b861ccddb26e5011bbc76ff6b9303ad98f0c3166fff74 | Bob has a six-sided die with a number written on each face such that the sums of the numbers written on each pair of opposite faces are equal to each other. Suppose that the numbers $109$, $131$, and $135$ are written on three faces which share a corner. Determine the maximum possible sum of the numbers on the three re... | [
"39"
] | [
"Bob has a six-sided die with a number written on each face such that the sums of the numbers written on each pair of opposite faces are equal to each other. Suppose that the numbers $109$, $131$, and $135$ are written on three faces which share a corner. Determine the maximum possible sum of the numbers on the thr... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-75d56bbe7b5004176a7da994b5b2d97598afa061a682115825501b313c3653ce | Bob has a spinner with four equal sections numbered 1 through 4, and a section three times as big as the others numbered 5 (so it is three times as likely to land on 5 as each other number). Bob spins the spinner until he has written each of the numbers 1 through 5 at least once. Find the expected number of times Bob w... | [
"\\dfrac{293}{20}"
] | [
"Bob has a spinner with four equal sections numbered 1 through 4, and a section three times as big as the others numbered 5 (so it is three times as likely to land on 5 as each other number). Bob spins the spinner until he has written each of the numbers 1 through 5 at least once. Find the expected number of times ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ca63a705bb4929619761f7051cc2a4e089370ab584997a33d1401298fab4aa43 | Bob has a targeted value of 6,000 at the end of a 30-month payment period. The first payment is 60, and each subsequent payment increases by 175% of the previous payment. How much must Bob pay each period to reach the targeted value? | [
"60"
] | [
"Bob has a targeted value of 6,000 at the end of a 30-month payment period. The first payment is 60, and each subsequent payment increases by 175% of the previous payment. How much must Bob pay each period to reach the targeted value?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-99ef78a6a5af6430cc9225084a22cf17b9b2b12450e8d6f48a62f73cb6b4f56d | Bob has four children (3 boys and 1 girl) and 5 nephews and 6 nieces. He plans to take 5 out of the 15 to an event. He will take at least 2 of his children and will not take all boys or all girls. How many different choices of the five he takes are possible? | [
"1196"
] | [
"Bob has four children (3 boys and 1 girl) and 5 nephews and 6 nieces. He plans to take 5 out of the 15 to an event. He will take at least 2 of his children and will not take all boys or all girls. How many different choices of the five he takes are possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5112b49bd61d18e0c8c115f8c62cb84924569fd729442d6ab1e315b02cc8cb10 | Bob has picked a positive integer $1 < N < 100$. Alice tells Bob some integers, and Bob replies with the remainder of the division of each integer by $N$. What is the smallest number of integers Alice should tell Bob to determine $N$ for sure? | [
"1"
] | [
"Bob has picked a positive integer $1 < N < 100$. Alice tells Bob some integers, and Bob replies with the remainder of the division of each integer by $N$. What is the smallest number of integers Alice should tell Bob to determine $N$ for sure?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-870fc3fd68209dbe2f25ac5647f4d7e45ba12c25ed1ade8dde0c3adf708910eb | Bob has picked a positive integer \(1 < N < 100\). Alice tells him some integers, and Bob replies with the remainder of the division of these integers by \(N\). What is the smallest number of integers Alice should tell Bob to determine \(N\) for sure? | [
"3"
] | [
"Bob has picked a positive integer \\(1 < N < 100\\). Alice tells him some integers, and Bob replies with the remainder of the division of these integers by \\(N\\). What is the smallest number of integers Alice should tell Bob to determine \\(N\\) for sure?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-aea10ba8fedf2e13ffe772c3a92fca8106b527aee99cc339f93f71631a5e33e9 | Bob has red, blue, green, yellow, purple, and orange paint. If he wants to paint each face of his Rubik's cube a different color, in how many different ways can he do this if two colorings are considered the same when one can be obtained from the other by rotating the cube? | [
"30"
] | [
"Bob has red, blue, green, yellow, purple, and orange paint. If he wants to paint each face of his Rubik's cube a different color, in how many different ways can he do this if two colorings are considered the same when one can be obtained from the other by rotating the cube?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b437f648c883c3b298f84c6c2ccdabc9061944b37b216f8f7d3a1a893f60fbc7 | Bob has some integral number of wine bottles less than 1000. If he organizes them in groups of 5, there is 1 bottle left over. If he organizes them in groups of 4, there are 3 wine bottles left over. If they are put in groups of 7, there are 4 bottles left over. Find the sum of all possible values of the number of wine... | [
"4008"
] | [
"Bob has some integral number of wine bottles less than 1000. If he organizes them in groups of 5, there is 1 bottle left over. If he organizes them in groups of 4, there are 3 wine bottles left over. If they are put in groups of 7, there are 4 bottles left over. Find the sum of all possible values of the number of... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-2d0c58e68b2bdc3bf1235dc62fd577e960bebad9d034c71139ec103b6c5817f7 | Bob has two bags. Bag A contains 8 green marbles and 3 red marbles, and Bag B contains 5 green marbles and 7 red marbles. If Bob chooses a bag at random, picks out a marble, and sees it's red, what is the probability that the marble came from Bag B? | [
"\\(\\frac{77}{113}\\)"
] | [
"Bob has two bags. Bag A contains 8 green marbles and 3 red marbles, and Bag B contains 5 green marbles and 7 red marbles. If Bob chooses a bag at random, picks out a marble, and sees it's red, what is the probability that the marble came from Bag B?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-4c865c8f77ede127394f5ec7a19f46e742e6c42bf640ddd5901a70614023bd00 | Bob invests 100 dollars. During the first year he suffers a 15% loss, but during the second year the remaining investment shows a 20% gain. Over the full two-year period, what was the percent change in Bob's investment? | [
"2\\%"
] | [
"Bob invests 100 dollars. During the first year he suffers a 15% loss, but during the second year the remaining investment shows a 20% gain. Over the full two-year period, what was the percent change in Bob's investment?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b28a11fdcc37706a6c319d83e3c74d22d8d12c39d33a6f774d49aee05bae5643 | Bob is a weatherman for “The World Daily”. He is reporting about the temperature in Orlando, Florida, as many tourists are flocking over to Orlando. He states that there is a $60\%$ chance of rain on Friday, and states if it does rain on Friday, there is a $70\%$ chance of rain on Saturday. If there is no rain on Frida... | [
"169"
] | [
"Bob is a weatherman for “The World Daily”. He is reporting about the temperature in Orlando, Florida, as many tourists are flocking over to Orlando. He states that there is a $60\\%$ chance of rain on Friday, and states if it does rain on Friday, there is a $70\\%$ chance of rain on Saturday. If there is no rain o... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb57d34ea653e3a4ad243375f7772fa742cf39b189be46ed051c43d4c824d8de | Bob is about to hang 8 shirts in the wardrobe. He has 4 different styles of shirt, 2 identical ones of each particular style. How many different arrangements are possible if no 2 identical shirts are next to one another? | [
"864"
] | [
"Bob is about to hang 8 shirts in the wardrobe. He has 4 different styles of shirt, 2 identical ones of each particular style. How many different arrangements are possible if no 2 identical shirts are next to one another?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6b2dc5b264bdc54cc15e3e3bb1813e1b2f10db89e2f38787429f582dd395bc40 | Bob is about to hang 8 shirts in the wardrobe. He has 4 different styles of shirt, with 2 identical ones of each particular style. How many different arrangements are possible if no 2 identical shirts are next to one another? | [
"864"
] | [
"Bob is about to hang 8 shirts in the wardrobe. He has 4 different styles of shirt, with 2 identical ones of each particular style. How many different arrangements are possible if no 2 identical shirts are next to one another?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb0beeee64a6e7e8f093e29a2fe3c08ddd81dda250725530565ae2b616569d93 | Bob is at the origin on a Cartesian coordinate grid. Every second, he randomly hops to a neighboring point. What is the probability that he will reach (-1,0) before (1,1)? What is the probability that he will reach (3,4) before (-1,0)? Is there a general method for these kinds of problems? | [
"\\dfrac{7}{8}"
] | [
"Bob is at the origin on a Cartesian coordinate grid. Every second, he randomly hops to a neighboring point. What is the probability that he will reach (-1,0) before (1,1)? What is the probability that he will reach (3,4) before (-1,0)? Is there a general method for these kinds of problems?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-a1f7a23963c4d455d10008951ce1dff7ddaf6e343459e899065ce80865e08930 | Bob is at the origin on a Cartesian coordinate grid. Every second, he randomly takes a step to a neighboring point (i.e., he can go to (±1,0) or (0,±1)). What is the probability that he will never reach the point (1,1)? What is the probability he will never reach (3,4)? Is there a general method for these kinds of prob... | [
"1"
] | [
"Bob is at the origin on a Cartesian coordinate grid. Every second, he randomly takes a step to a neighboring point (i.e., he can go to (±1,0) or (0,±1)). What is the probability that he will never reach the point (1,1)? What is the probability he will never reach (3,4)? Is there a general method for these kinds of... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-190b02076c285878b1b255178b63eea044768cad8aa1eddcc50f110077f67ed1 | Bob is flipping bottles. Each time he flips the bottle, he has a $0.25$ probability of landing it. After successfully flipping a bottle, he has a $0.8$ probability of landing his next flip. What is the expected value of the number of times he has to flip the bottle in order to flip it twice in a row? | [
"\\(\\frac{25}{4}\\)"
] | [
"Bob is flipping bottles. Each time he flips the bottle, he has a $0.25$ probability of landing it. After successfully flipping a bottle, he has a $0.8$ probability of landing his next flip. What is the expected value of the number of times he has to flip the bottle in order to flip it twice in a row?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-bdefda311d020f81890d5911bc8906bb64bd4f0a2baffc5076406f64dba58735 | Bob is in detention and repeatedly draws line segments from \((t, 0)\) to \((0, 1-t)\) for \(t\) between 0 and 1. What is the equation of the curve bounded by these segments? | [
"\\sqrt{x} + \\sqrt{y} = 1"
] | [
"Bob is in detention and repeatedly draws line segments from \\((t, 0)\\) to \\((0, 1-t)\\) for \\(t\\) between 0 and 1. What is the equation of the curve bounded by these segments?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-471aec6cd0d839dacf9d68e7626513ef575af5b3453d90ebe4010229a1d8696d | Bob is playing Bingo alone on a 5x5 board, including a free space in the middle. On each turn, he spins the spinner and places a chip on the corresponding space. To win, he needs to acquire 5 chips that form a straight line (diagonal, vertical, or horizontal). What is the probability that Bob will win on his 7th turn? ... | [
"\\(\\frac{5}{483}\\)"
] | [
"Bob is playing Bingo alone on a 5x5 board, including a free space in the middle. On each turn, he spins the spinner and places a chip on the corresponding space. To win, he needs to acquire 5 chips that form a straight line (diagonal, vertical, or horizontal). What is the probability that Bob will win on his 7th t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-6a91425442013bfef09586587bad6cdc43fef1085d4b056c5245699d3108b193 | Bob is playing a game on an $8 \times 8$ grid. He can place any non-negative integer in each cell of the grid. After filling the grid, he calculates the sum $S$ of the smallest numbers in each column. What is the minimum positive integer value that Bob can ensure $S$ to be, regardless of how he fills the grid, if he wa... | [
"2"
] | [
"Bob is playing a game on an $8 \\times 8$ grid. He can place any non-negative integer in each cell of the grid. After filling the grid, he calculates the sum $S$ of the smallest numbers in each column. What is the minimum positive integer value that Bob can ensure $S$ to be, regardless of how he fills the grid, if... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8493d464b650121cd97918bf323796adca92934002ef68a7a44a451cba119b0a | Bob is punished by his math teacher and has to write all perfect squares, one after another. His teacher's blackboard has space for exactly 2012 digits. He can stop when he cannot fit the next perfect square on the board. If \( n^2 \) is the largest perfect square he writes, what is \( n \)? | [
"411"
] | [
"Bob is punished by his math teacher and has to write all perfect squares, one after another. His teacher's blackboard has space for exactly 2012 digits. He can stop when he cannot fit the next perfect square on the board. If \\( n^2 \\) is the largest perfect square he writes, what is \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fc1e76cb56fafbdd16efc3dfc449d60b5c1b5e1ef0ed3ebc4c443b6ea4ea368e | Bob is sitting on a vertex of a regular octahedron with side length $1$. The probability that Bob is at his original position after $6$ moves can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. (Each move is moving from a single vertex to an adjacent vertex.) | [
"75"
] | [
"Bob is sitting on a vertex of a regular octahedron with side length $1$. The probability that Bob is at his original position after $6$ moves can be expressed as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. (Each move is moving from a single vertex to an adjacent vertex.)... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-74ae80d8b9d58b878ad340c997b0c7aec9b7254f061bca7ef34cf5b157d19e26 | Bob is standing at the number $1$ on the number line. If Bob is standing at the number $n,$ he can move to $n+1, n+2,$ or $n+4.$ In how many different ways can he move to the number $10$? | [
"96"
] | [
"Bob is standing at the number $1$ on the number line. If Bob is standing at the number $n,$ he can move to $n+1, n+2,$ or $n+4.$ In how many different ways can he move to the number $10$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-26dc1a884e406b3628d50aa84ada64bcf23834edeb919246878f1a10f3385f01 | Bob is taking a 10-question test. The probability that he gets each problem right decreases by 10% for each subsequent problem, starting at 90% for the first problem and ending at 0% for the tenth problem. What is the probability that Bob passes the test, where passing requires answering at least 70% of the questions c... | [
"0.0556"
] | [
"Bob is taking a 10-question test. The probability that he gets each problem right decreases by 10% for each subsequent problem, starting at 90% for the first problem and ending at 0% for the tenth problem. What is the probability that Bob passes the test, where passing requires answering at least 70% of the questi... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2b465a92255a9804490d9c1b063f252f78afbc06515a1d65af53d6394792cccb | Bob is taking the AMC10. He can't solve any of the problems and must randomly guess to maximize the probability that he scores 132 or higher. How many problems should he guess? | [
"25"
] | [
"Bob is taking the AMC10. He can't solve any of the problems and must randomly guess to maximize the probability that he scores 132 or higher. How many problems should he guess?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-b4db963f0d69780400f4435417db7e64ef0abfb4b2e670a166d7e5503f2bc273 | Bob is trying to decide between two cell phone plans. Plan A has no flat fee, but the user must pay $10$ cents per minute on the phone. Plan B requires a one-time fee of $\$20$, but only requires a payment of $5$ cents per minute on the phone. What is the minimum whole number of minutes Bob has to use the phone for to ... | [
"401"
] | [
"Bob is trying to decide between two cell phone plans. Plan A has no flat fee, but the user must pay $10$ cents per minute on the phone. Plan B requires a one-time fee of $\\$20$, but only requires a payment of $5$ cents per minute on the phone. What is the minimum whole number of minutes Bob has to use the phone f... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b7332d1efe38b0eeb7ba95f2b9563b4fcaca0c6893dc876570ba70543f5638ad | Bob is trying to guess a list of $n$ numbers, each between $1$ and $k$. He can ask yes-or-no questions of the form "Is the $i$-th number in the list less than or equal to $x$?" for any $i$ and $x$. Let $Q(n, k)$ be the minimum number of questions Bob needs to ask to guarantee he can determine the list. Find $Q(5, 3)$. | [
"10"
] | [
"Bob is trying to guess a list of $n$ numbers, each between $1$ and $k$. He can ask yes-or-no questions of the form \"Is the $i$-th number in the list less than or equal to $x$?\" for any $i$ and $x$. Let $Q(n, k)$ be the minimum number of questions Bob needs to ask to guarantee he can determine the list. Find $Q(5... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dcffcb1899cd6dde956ab9e89ea4f6defc54c9ab8979ae326c1cf64f468a7a72 | Bob is using 5 colors to color a $5 \times 5$ chessboard such that each color is used to color exactly 5 cells. Let $S$ be the sum of the maximum number of distinct colors found in any row and the maximum number of distinct colors found in any column. Find the minimum possible value of $S$. | [
"6"
] | [
"Bob is using 5 colors to color a $5 \\times 5$ chessboard such that each color is used to color exactly 5 cells. Let $S$ be the sum of the maximum number of distinct colors found in any row and the maximum number of distinct colors found in any column. Find the minimum possible value of $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-831efaf574b12fe9ce8a3f68816d1b6202bdca61ace4ea4e99b40d4c35743e0f | Bob is visiting Japan and he wants to buy a coffee for $200$ yen. If one U.S. dollar is worth $108$ yen, how much money, in USD to the nearest hundredth, does he have to spend for the coffee? (You may use a calculator on this problem.) | [
"1.85\\ \\text{USD}"
] | [
"Bob is visiting Japan and he wants to buy a coffee for $200$ yen. If one U.S. dollar is worth $108$ yen, how much money, in USD to the nearest hundredth, does he have to spend for the coffee? (You may use a calculator on this problem.)"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e83322ae93e15854a0f2474c790842ffd0edebba1775e19ef9694d2bf84ca31b | Bob lives at $(0, 0)$ and works at $(6, 6)$. Every day, he walks down 12 blocks, each block either increasing his $x$-coordinate or his $y$-coordinate. How many ways are there for him to get to work if the intersections $(2, 1)$ and $(2, 4)$ are closed off? | [
"366"
] | [
"Bob lives at $(0, 0)$ and works at $(6, 6)$. Every day, he walks down 12 blocks, each block either increasing his $x$-coordinate or his $y$-coordinate. How many ways are there for him to get to work if the intersections $(2, 1)$ and $(2, 4)$ are closed off?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-d0d53c45f41167d9a52048c6bb30441f4ad90fb727d701ec20c85cfbb772e1dc | Bob lives at $(0, 0)$ and works at $(6, 6)$. To get to work, he walks $12$ blocks, each either increasing his $x-$ or $y-$ coordinate. In how many ways can he get to work if he avoids the points $(2, 1)$ and $(2, 4)$? | [
"366"
] | [
"Bob lives at $(0, 0)$ and works at $(6, 6)$. To get to work, he walks $12$ blocks, each either increasing his $x-$ or $y-$ coordinate. In how many ways can he get to work if he avoids the points $(2, 1)$ and $(2, 4)$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-3bfce55a7afea34a233273d0382fbf6eb018626ccc385dd865730bf734310289 | Bob makes problems each day such that the number of problems forms a strictly increasing arithmetic sequence. The ratio of the rejected problems to the total number of problems he made forms a strictly decreasing geometric sequence. If he cannot make an integer number of problems that satisfy both conditions, he stops.... | [
"4",
"6"
] | [
"Bob makes problems each day such that the number of problems forms a strictly increasing arithmetic sequence. The ratio of the rejected problems to the total number of problems he made forms a strictly decreasing geometric sequence. If he cannot make an integer number of problems that satisfy both conditions, he s... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-6347e60fe5e0fd3aa39d985bf1cfcd90e6bb8bba9ab8be9b167dd0533b65e54c | Bob orders Alice to draw a circle with a diameter and mark the numbers $1$ and $2$ on the endpoints of the diameter. Alice then places $100$ points in each of the semicircles (not on the endpoints of the diameter) and marks those points randomly with numbers $1$ and $2$. Alice paints red all segments with different mar... | [
"201"
] | [
"Bob orders Alice to draw a circle with a diameter and mark the numbers $1$ and $2$ on the endpoints of the diameter. Alice then places $100$ points in each of the semicircles (not on the endpoints of the diameter) and marks those points randomly with numbers $1$ and $2$. Alice paints red all segments with differen... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-aa480bca97255056eec3af386e7edce554cf393745395ef0cf5a4321c0bb6bd6 | Bob performs a trick with a deck of 26 cards. A spectator picks a number, and Bob can choose to count from either end of the row to discard the card at that position. Bob claims he can guarantee that the final remaining card will be the 2 of hearts. For which starting positions of the 2 of hearts can Bob guarantee this... | [
"13, 14"
] | [
"Bob performs a trick with a deck of 26 cards. A spectator picks a number, and Bob can choose to count from either end of the row to discard the card at that position. Bob claims he can guarantee that the final remaining card will be the 2 of hearts. For which starting positions of the 2 of hearts can Bob guarantee... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-672560d9d845d5bf5a1c5d67b2143a3df178105dbc97682d29f5c548e2ef055d | Bob picks two integers from 1 to 100, inclusive, with replacement. Let the probability that the product of his two numbers is odd be $\frac{a}{b}$, a common fraction. John does the same thing, but on the second pick, it cannot be the same number as before. Let the probability that the product of his numbers is odd be $... | [
"252"
] | [
"Bob picks two integers from 1 to 100, inclusive, with replacement. Let the probability that the product of his two numbers is odd be $\\frac{a}{b}$, a common fraction. John does the same thing, but on the second pick, it cannot be the same number as before. Let the probability that the product of his numbers is od... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-da2377417af097c4bf465b23fe6ed5503aa188efea4c1f034e0f52cc4c64c22a | Bob plays a dice game with a fair 8-sided die where he wins if he gets to 3 points. He starts at 0 points and gains 3 points if he rolls a 1, 2, 3, or 4. He loses 3 points for rolling a 5 or a 6, and he loses 4 points for rolling a 7 or an 8. What is the probability that Bob never wins the game? | [
"\\dfrac{1}{2}"
] | [
"Bob plays a dice game with a fair 8-sided die where he wins if he gets to 3 points. He starts at 0 points and gains 3 points if he rolls a 1, 2, 3, or 4. He loses 3 points for rolling a 5 or a 6, and he loses 4 points for rolling a 7 or an 8. What is the probability that Bob never wins the game?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-996b714815f991b596a408d554c8786c3071eac12cbd1fbd2fd22ccde3d97e85 | Bob plots \( n > 4 \) points in the plane such that the distance between any two points is an integer. Let \( C(n) \) be the minimum number of collinear points among all possible configurations of \( n \) points. Find \( C(5) \). | [
"3"
] | [
"Bob plots \\( n > 4 \\) points in the plane such that the distance between any two points is an integer. Let \\( C(n) \\) be the minimum number of collinear points among all possible configurations of \\( n \\) points. Find \\( C(5) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc30ec4123b91a06476fec1c54eca6aeaac9eb58d27c39d271b87e1737e6d893 | Bob randomly guesses the answers to an exam with 10 questions worth 2 marks each, 10 questions worth 3 marks each, and 10 questions worth 5 marks each. The passing mark is 10. What is the probability that Bob passes the exam? | [
"1"
] | [
"Bob randomly guesses the answers to an exam with 10 questions worth 2 marks each, 10 questions worth 3 marks each, and 10 questions worth 5 marks each. The passing mark is 10. What is the probability that Bob passes the exam?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-5a5b9488bfdf34784f596b3d39200b20e376656c8ddd262f766bc9f510e4744d | Bob randomly picks an integer \( n \) from 65 to 127 inclusive. Consider the set \( S = \{1, 2, 3, \ldots, n\} \). Bob randomly picks a subset of \( S \) and writes all numbers in that subset in binary. He then lines them up so that all digits match up. For each column, if there is an even number of 1's, Bob writes the... | [
"\\(\\frac{7}{128}\\)"
] | [
"Bob randomly picks an integer \\( n \\) from 65 to 127 inclusive. Consider the set \\( S = \\{1, 2, 3, \\ldots, n\\} \\). Bob randomly picks a subset of \\( S \\) and writes all numbers in that subset in binary. He then lines them up so that all digits match up. For each column, if there is an even number of 1's, ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-fe6bc056b8bfdd3d2af2e4436379c2687b1e2bff2c76ae4116e973cb5edefb5b | Bob randomly places 3 math, 2 science, and 2 English books on a shelf. Find the probability that the 3 math books are together on the shelf. | [
"$\\frac{1}{7}$"
] | [
"Bob randomly places 3 math, 2 science, and 2 English books on a shelf. Find the probability that the 3 math books are together on the shelf."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-1cf6303888eb69ec97c370a3909118ad7e09103fc50be8b0dd990437d20d36c3 | Bob rolls a fair 6-sided die until the sum of his rolls is greater than or equal to 700. What is the expected number of rolls needed? Any answer within .0001 of the correct answer will be accepted. | [
"200.7619"
] | [
"Bob rolls a fair 6-sided die until the sum of his rolls is greater than or equal to 700. What is the expected number of rolls needed? Any answer within .0001 of the correct answer will be accepted."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-9a94a07f774a471f9b9fec06c333451a93d1dd7529f1edbf6fe263cb4ee9e814 | Bob rolls a fair six-sided die each morning. If Bob rolls a composite number, he eats sweetened cereal. If he rolls a prime number, he eats unsweetened cereal. If he rolls a 1, then he rolls again. In a non-leap year, what is the expected number of times Bob will roll his die? | [
"438"
] | [
"Bob rolls a fair six-sided die each morning. If Bob rolls a composite number, he eats sweetened cereal. If he rolls a prime number, he eats unsweetened cereal. If he rolls a 1, then he rolls again. In a non-leap year, what is the expected number of times Bob will roll his die?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-10a04ebd85012ff0c6212ef7938c89ffa47949286efa2fd308d70d951fd07757 | Bob rolls a fair six-sided die each morning. If Bob rolls a composite number, he eats sweetened cereal. If he rolls a prime number, he eats unsweetened cereal. If he rolls a 1, then he rolls again. In a non-leap year, what is the expected value of the difference between the number of days Bob eats unsweetened cereal an... | [
"73"
] | [
"Bob rolls a fair six-sided die each morning. If Bob rolls a composite number, he eats sweetened cereal. If he rolls a prime number, he eats unsweetened cereal. If he rolls a 1, then he rolls again. In a non-leap year, what is the expected value of the difference between the number of days Bob eats unsweetened cere... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-808bd2a83580449e34d0170107deaf27c6d94bc4503fb20c577a369d2ec56fab | Bob rows at a rate of 4 kilometers per hour in a stream without a current. He rows under one bridge and then rows under a second bridge 10 kilometers upstream. As he rows under the second bridge, his hat falls off. After rowing 5 more kilometers, he realizes his hat is gone and turns around, catching the hat exactly as... | [
"2"
] | [
"Bob rows at a rate of 4 kilometers per hour in a stream without a current. He rows under one bridge and then rows under a second bridge 10 kilometers upstream. As he rows under the second bridge, his hat falls off. After rowing 5 more kilometers, he realizes his hat is gone and turns around, catching the hat exact... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5bd7b01527f9940b84d103b77af72049400c1ac1f8808630d97aa470345c71f3 | Bob sets his watch to the correct time at noon. At the actual time of 1:00 PM, his watch reads 12:57 and 36 seconds. Assuming that his watch loses at a constant rate, what will be the actual time when his watch reads 10:00 PM? | [
"10:25 PM\n\nExplanation:\nThe solution provided in Post 3 explains that the watch loses 2 minutes and 24 seconds (144 seconds) in an hour. This means that for every 3600 seconds, the watch loses 144 seconds, resulting in a ratio of 25:24 between actual time and the watch's time. When the watch reads 10:00 PM, the ... | [
"Bob sets his watch to the correct time at noon. At the actual time of 1:00 PM, his watch reads 12:57 and 36 seconds. Assuming that his watch loses at a constant rate, what will be the actual time when his watch reads 10:00 PM?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-d17568a443a938749d9055dbc154eabec183054e43c4e2b2185f3d9762144bcb | Bob starts at 0 on the number line and can move 1 unit left, 2 units right, or 1 unit right every second. What is the probability that after 5 seconds, Bob is on the number 5 or a larger number? | [
"\\(\\frac{29}{81}\\)"
] | [
"Bob starts at 0 on the number line and can move 1 unit left, 2 units right, or 1 unit right every second. What is the probability that after 5 seconds, Bob is on the number 5 or a larger number?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 29
} | 40 | 0 | 0 |
omr-ws-259eb2b5b8448b613fad31722c5e802b95f67e8030a8b75578bb3752b769423d | Bob starts at 1 kg and eats 1 apple a day. For every apple he eats, he gains 1/7 of his current weight. As Bob grows in weight, he eats $\lfloor x \rfloor$ apples when he weighs $x$ kg. After how many days will Bob weigh 100 kg? | [
"14"
] | [
"Bob starts at 1 kg and eats 1 apple a day. For every apple he eats, he gains 1/7 of his current weight. As Bob grows in weight, he eats $\\lfloor x \\rfloor$ apples when he weighs $x$ kg. After how many days will Bob weigh 100 kg?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b06c518b0dfcc1e447826da3fefe3db143c03f1f134038140598554b4d17d3b5 | Bob starts at coordinates (1,1) and wishes to visit Alice at coordinates (6,5) by taking the diagonal path. If Bob travels 3 meters along this path, what are his coordinates after traveling these 3 meters? | [
"\\(\\left( 1+\\frac{15}{ \\sqrt{41}}, 1+\\frac{12}{ \\sqrt{41}}\\right)\\)"
] | [
"Bob starts at coordinates (1,1) and wishes to visit Alice at coordinates (6,5) by taking the diagonal path. If Bob travels 3 meters along this path, what are his coordinates after traveling these 3 meters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-99d543dad469b32929d3bfc2a91ba5e6ac6ac5224fd56305a9ddd9bcfe5a1d0a | Bob starts on the first of 10 stones and can move either forwards or backwards one stone at a time. He cannot move behind the first stone. In how many ways can he reach the 10th stone and take one more step forward by taking exactly $k$ moves, where $k \geq 10$ and $k$ is even? | [
"\\dbinom{k}{\\frac{k - 10}{2}} - \\dbinom{k}{\\frac{k - 12}{2}}"
] | [
"Bob starts on the first of 10 stones and can move either forwards or backwards one stone at a time. He cannot move behind the first stone. In how many ways can he reach the 10th stone and take one more step forward by taking exactly $k$ moves, where $k \\geq 10$ and $k$ is even?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-092b0841255c160b5b7ab546f649ef25eb2813e9b80ac9011f881e8ea13439b2 | Bob starts to climb a mountain at 9 AM and walks up at a rate of 6 kilometers per hour. After reaching the top, he rests for 1.5 hours and then descends at a rate of 7.5 kilometers per hour. He reaches the bottom at 1:30 PM. How many kilometers did he walk that day climbing up and down the mountain? | [
"20 km"
] | [
"Bob starts to climb a mountain at 9 AM and walks up at a rate of 6 kilometers per hour. After reaching the top, he rests for 1.5 hours and then descends at a rate of 7.5 kilometers per hour. He reaches the bottom at 1:30 PM. How many kilometers did he walk that day climbing up and down the mountain?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-84c9aaf5cd33c98a140ef1340b42b2b01959b9f00154ce09fff3308d3858d261 | Bob starts with the number $0$ and adds $111$ (i.e., $7$ in base $2$) to it one hundred times to obtain the number $1010111100$ (i.e., $700$ in base $2$). How many carries occur in total during these one hundred calculations? | [
"294"
] | [
"Bob starts with the number $0$ and adds $111$ (i.e., $7$ in base $2$) to it one hundred times to obtain the number $1010111100$ (i.e., $700$ in base $2$). How many carries occur in total during these one hundred calculations?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 22,
"genselect": 6,
"tir": 9
} | 37 | 0 | 0 |
omr-ws-3f52bb0ac53702c63cecbac6eb9467e655ea2f475e03981479b54f9d16934bc6 | Bob starts with the set $\{1\}$. Each turn, he adds, multiplies, subtracts, or divides two (not necessarily distinct) numbers in the set and puts the result in the set. What is the least number of turns Bob needs until the set includes $2021$? | [
"7"
] | [
"Bob starts with the set $\\{1\\}$. Each turn, he adds, multiplies, subtracts, or divides two (not necessarily distinct) numbers in the set and puts the result in the set. What is the least number of turns Bob needs until the set includes $2021$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-5e20b3f7328a0ea8abc45f6c4f0dec969eeb0cd2868ed9322f72d0b5a8238962 | Bob tells Alice that his favorite prime number leaves a remainder of 2 when divided by 5 and a remainder of 5 when divided by 11. He also tells Alice that it is less than 150. What is Bob's favorite prime number? | [
"137"
] | [
"Bob tells Alice that his favorite prime number leaves a remainder of 2 when divided by 5 and a remainder of 5 when divided by 11. He also tells Alice that it is less than 150. What is Bob's favorite prime number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e6733197520a3c06be2ad2f4a38f84ec967cd9794d8a39a01f03171f1a3da201 | Bob the Bunny is hopping around the sides of a regular pentagon ABCDE. During each minute, Bob hops along one of the two adjacent sides with equal probability. After seven minutes, what is the probability that Bob is at point B? | [
"\\(\\frac{1}{5}\\)"
] | [
"Bob the Bunny is hopping around the sides of a regular pentagon ABCDE. During each minute, Bob hops along one of the two adjacent sides with equal probability. After seven minutes, what is the probability that Bob is at point B?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8358cc419c656ce1fc7fcb7e8b31c82e8e13aae516d863b9c5943ef3a132ca52 | Bob the ant lives on the Cartesian plane and only steps on lattice points. A lattice point is "acceptable" if its distance to the origin is less than $\sqrt[4]{2\pi^2}$. Bob moves to another lattice point if it is 1 unit away from his current position and is an "acceptable" lattice point. If there are multiple directio... | [
"5"
] | [
"Bob the ant lives on the Cartesian plane and only steps on lattice points. A lattice point is \"acceptable\" if its distance to the origin is less than $\\sqrt[4]{2\\pi^2}$. Bob moves to another lattice point if it is 1 unit away from his current position and is an \"acceptable\" lattice point. If there are multip... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2d851f7d9651fe5d3c19e14cb9f53b70a53081dbeb9ca1b1a4e807a51af0c6ca | Bob thinks of a number between 1 and 32. Alice can ask Bob any yes-or-no questions she wants. How can Alice figure out Bob's number with at most five questions? | [
"Alice can determine Bob's number by asking five yes-or-no questions about the binary representation of (N-1)."
] | [
"Bob thinks of a number between 1 and 32. Alice can ask Bob any yes-or-no questions she wants. How can Alice figure out Bob's number with at most five questions?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 23
} | 42 | 0 | 0 |
omr-ws-839c89ab1f548ab2a0ed6506c37179065ea451b6d6b2b8328c3ee9a093cda892 | Bob thinks of a polynomial \( p(x) \) with all coefficients as positive integers. John asks for the numerical values of \( p(1) \) and \( p(p(1) + 1) \). Given that \( p(1) = 10 \) and \( p(p(1) + 1) = 1234 \), find the sum of all coefficients of \( p(x) \). | [
"10"
] | [
"Bob thinks of a polynomial \\( p(x) \\) with all coefficients as positive integers. John asks for the numerical values of \\( p(1) \\) and \\( p(p(1) + 1) \\). Given that \\( p(1) = 10 \\) and \\( p(p(1) + 1) = 1234 \\), find the sum of all coefficients of \\( p(x) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ee8ef357978d5067eee93be9e03c56e40ae78645da0001bc3baea8a578dfdbbb | Bob thinks of four positive integers \(a, b, c, d\). He makes an expression by inserting some of the operators \(+\), \(-\), \(\times\), or \(\div\) between each pair of adjacent numbers and then encloses some parts of the expression with parentheses. He finds out that any expression made gives a positive integer. Find... | [
"4"
] | [
"Bob thinks of four positive integers \\(a, b, c, d\\). He makes an expression by inserting some of the operators \\(+\\), \\(-\\), \\(\\times\\), or \\(\\div\\) between each pair of adjacent numbers and then encloses some parts of the expression with parentheses. He finds out that any expression made gives a posit... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-f2700ebf40c98179fec7fd4457d50fb89d7e0d3a06c1ae4d27dd7b5f33ef1081 | Bob walks a path starting from $(0, 0, 0)$, where each move, he walks from $(x, y, z)$ to $(x+1, y, z)$, $(x, y+1, z)$, or $(x, y, z+1)$. Let $X$ be the number of ways to get to $(5, 5, 5)$ given that each move, he cannot move in the same direction as the previous move. Find the remainder when $X$ is divided by $1000$. | [
"188"
] | [
"Bob walks a path starting from $(0, 0, 0)$, where each move, he walks from $(x, y, z)$ to $(x+1, y, z)$, $(x, y+1, z)$, or $(x, y, z+1)$. Let $X$ be the number of ways to get to $(5, 5, 5)$ given that each move, he cannot move in the same direction as the previous move. Find the remainder when $X$ is divided by $1... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 16
} | 17 | 0 | 0 |
omr-ws-a16e92687a7f798c316e914adc1fffc231be1908ab1589ca61965c6fc265ffb4 | Bob wants to build an arch bridge that reaches down 10 feet on each side and crosses a 20-foot gap. The arch is a semi-circle with a radius of 9 feet, and the bridge is 5 feet wide. How many cubic feet of material does Bob need? | [
"\\(\\frac{405\\pi}{2}\\)"
] | [
"Bob wants to build an arch bridge that reaches down 10 feet on each side and crosses a 20-foot gap. The arch is a semi-circle with a radius of 9 feet, and the bridge is 5 feet wide. How many cubic feet of material does Bob need?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-2891daeb12988480f0b88a4a13d3f25e1ae4aae5a117d19bef69c74a9db39902 | Bob writes the numbers 1, 2, 3, ..., 1024 on a blackboard. They are divided into pairs. Then each pair is wiped off the board and he writes the non-negative difference of its numbers on the board instead. This process is repeated until one number remains on the blackboard after ten such operations. Determine the sum of... | [
"261632"
] | [
"Bob writes the numbers 1, 2, 3, ..., 1024 on a blackboard. They are divided into pairs. Then each pair is wiped off the board and he writes the non-negative difference of its numbers on the board instead. This process is repeated until one number remains on the blackboard after ten such operations. Determine the s... | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-104d951d0650560807621299a69efea95d306ec4d750bc2bc1bbb12461ebe4c8 | Bob wrote a 4-digit number on the blackboard. Jason erased the final digit and created a number. The difference between the initial number and the new number is 2019. Which number did Bob write? | [
"2243"
] | [
"Bob wrote a 4-digit number on the blackboard. Jason erased the final digit and created a number. The difference between the initial number and the new number is 2019. Which number did Bob write?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
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