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-ca1365221a68a72dd55472c43fbd67aedf5498e66b1250b8e29591d5f4e0f05b | What is the minimum number of jumps required for a grasshopper to reach the 22nd step of a 30-step ladder, given that it can jump up 3 steps at a time or down 4 steps at a time? | [
"12"
] | [
"What is the minimum number of jumps required for a grasshopper to reach the 22nd step of a 30-step ladder, given that it can jump up 3 steps at a time or down 4 steps at a time?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bae78ce65c594a14278b036230de2b033dd131f275fd47d5a4c4c80f593966ba | What is the minimum number of jumps that the flea must make from the origin $(0, 0)$ to reach the point $(100, 1)$, where the flea can jump from point $(x, y)$ to $(x + p, y + p^2)$ for any positive real number $p$? | [
"3"
] | [
"What is the minimum number of jumps that the flea must make from the origin $(0, 0)$ to reach the point $(100, 1)$, where the flea can jump from point $(x, y)$ to $(x + p, y + p^2)$ for any positive real number $p$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba710de1e635d6758e2d999d3283237d14d6374925725162c0cc2dd288398456 | What is the minimum number of kangaroos needed to visit each square of a $2015 \times 2015$ chessboard, given that kangaroos in the northwest corner can jump either south or east, and kangaroos in the southwest corner can jump either north or east? | [
"1344"
] | [
"What is the minimum number of kangaroos needed to visit each square of a $2015 \\times 2015$ chessboard, given that kangaroos in the northwest corner can jump either south or east, and kangaroos in the southwest corner can jump either north or east?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bf80f98c9e112ed170424ae1efbdc2e36e95594e0b902388a7463041751e4478 | What is the minimum number of knights needed on an $n \times n$ chessboard so that every square is either occupied by a knight or can be reached by a knight in one move? | [
"12"
] | [
"What is the minimum number of knights needed on an $n \\times n$ chessboard so that every square is either occupied by a knight or can be reached by a knight in one move?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-b12c37d09c0e73bd256d389155bb253a029190cb0a979d61cc0888bd78c5ddc9 | What is the minimum number of knights so that every cell on a chessboard is controlled by a knight (cells occupied by a knight do not count as controlled)? | [
"12"
] | [
"What is the minimum number of knights so that every cell on a chessboard is controlled by a knight (cells occupied by a knight do not count as controlled)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f27cbccb0b12cfe8aaa888fa229eb797089f9403f974f78eaeb59b195693e39 | What is the minimum number of knights so that every cell on a chessboard is either occupied by a knight or controlled by one? | [
"12"
] | [
"What is the minimum number of knights so that every cell on a chessboard is either occupied by a knight or controlled by one?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6665dbe29ab5e70314213aa83aaa68ab708becc0a3adf71d9426cab6b327c053 | What is the minimum number of locks required such that any three out of five people can open a locker, but any two cannot? | [
"10"
] | [
"What is the minimum number of locks required such that any three out of five people can open a locker, but any two cannot?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-e72f8228e516f6e016d94d4f6052818da751a046aa652ff181a8d6fc7ab53603 | What is the minimum number of locks that must be placed in the safe to ensure that: - The Director can open the safe by himself. - The Deputy Director can only open the safe with one of the Heads of Sector. - Heads of Sector can only open the vault in groups of three. | [
"7"
] | [
"What is the minimum number of locks that must be placed in the safe to ensure that:\n- The Director can open the safe by himself.\n- The Deputy Director can only open the safe with one of the Heads of Sector.\n- Heads of Sector can only open the vault in groups of three."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-431c9ddf56a9319b1908b4b7b4997ac586e4e6a3e923fd59b78d57301b3731f2 | What is the minimum number of monsters needed to complete 1000000 movies, given that each movie has at least 2 monsters and no two movies have the same set of monsters? | [
"20"
] | [
"What is the minimum number of monsters needed to complete 1000000 movies, given that each movie has at least 2 monsters and no two movies have the same set of monsters?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-df56b337b0448df0544e6e6d4f52ddd1ff13234ff76afc9cc6aa84080a825e56 | What is the minimum number of moves needed to reach a point $(p, q)$ from the origin if each move is a diagonal step, consisting of one unit to the right or left followed by one unit either up or down? | [
"\\(\\max(|p|, |q|)\\)"
] | [
"What is the minimum number of moves needed to reach a point $(p, q)$ from the origin if each move is a diagonal step, consisting of one unit to the right or left followed by one unit either up or down?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-8330e612d7bfda3db9b1606ffb72e84b38bb88181aa09df0ffecde037e493014 | What is the minimum number of moves needed to reach a point $(p,q)$ from the origin if each move is a diagonal step, consisting of one unit to the right or left followed by one unit either up or down? | [
"\\(\\max(|p|, |q|)\\)"
] | [
"What is the minimum number of moves needed to reach a point $(p,q)$ from the origin if each move is a diagonal step, consisting of one unit to the right or left followed by one unit either up or down?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-be1296d274b9d2c63ef2c3d2873756c7a04fc691d433cf3532654cdcca247e6a | What is the minimum number of moves needed to reach a point (p,q) from the origin by moving diagonally, where each move consists of one unit to the right or left followed by one unit either up or down? | [
"\\(\\frac{|p| + |q|}{2}\\)"
] | [
"What is the minimum number of moves needed to reach a point (p,q) from the origin by moving diagonally, where each move consists of one unit to the right or left followed by one unit either up or down?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e08ef1c77629a27557181fcb775f15d288e5d225502f8d2b8b7ebab9cf8a2140 | What is the minimum number of moves needed to reach a point \((p, q)\) from the origin \((0, 0)\) if each move consists of one unit to the right or left followed by one unit either up or down? | [
"\\(\\text{max}(|p|, |q|)\\)"
] | [
"What is the minimum number of moves needed to reach a point \\((p, q)\\) from the origin \\((0, 0)\\) if each move consists of one unit to the right or left followed by one unit either up or down?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-ead2b729bec0eac3d0cbee5c087d3dd5450b39699e0c8d73bdd7ac58160678cb | What is the minimum number of moves required to reverse the order of a deck of 13 different cards, where a move consists of taking out a stack of consecutive cards and placing it elsewhere in the deck without flipping or changing the order within the stack? | [
"12"
] | [
"What is the minimum number of moves required to reverse the order of a deck of 13 different cards, where a move consists of taking out a stack of consecutive cards and placing it elsewhere in the deck without flipping or changing the order within the stack?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-f847fb5e2faeee0ed25448f657fffd1d9b742f65f7131611eb4d356755efbdc4 | What is the minimum number of moves required to reverse the order of a deck of 5 different cards, where a move consists of taking out a stack of consecutive cards and placing it elsewhere in the deck without flipping or changing the order within the stack? | [
"4"
] | [
"What is the minimum number of moves required to reverse the order of a deck of 5 different cards, where a move consists of taking out a stack of consecutive cards and placing it elsewhere in the deck without flipping or changing the order within the stack?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-f234da4d833d4a44bf7d859b0dde7726de80b93cdefc7ef013235dcdedf22663 | What is the minimum number of moves required to reverse the order of a deck of 9 different cards, where a move consists of taking out a stack of consecutive cards and placing it elsewhere in the deck without flipping or changing the order within the stack? | [
"4"
] | [
"What is the minimum number of moves required to reverse the order of a deck of 9 different cards, where a move consists of taking out a stack of consecutive cards and placing it elsewhere in the deck without flipping or changing the order within the stack?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-c64a3e4c1ee7f7df50cf26684433af1a247b15d234ca1acfe46adc6d0fba3ce4 | What is the minimum number of nails needed to nail the corners of a $2017 \times 2017$ board such that every unit square has at least 2 nailed corners? | [
"\\(1018081\\)"
] | [
"What is the minimum number of nails needed to nail the corners of a $2017 \\times 2017$ board such that every unit square has at least 2 nailed corners?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8181f5462e42dfa518017e57257516dd3d4204cb06761921a9e8d8fedb98f1e4 | What is the minimum number of nails required to block any convex polygon $P$ from moving on a flat wooden table, given that the nails must not go through $P$ but may touch its boundary? | [
"4"
] | [
"What is the minimum number of nails required to block any convex polygon $P$ from moving on a flat wooden table, given that the nails must not go through $P$ but may touch its boundary?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-154a1085ee3dcb5c2837fabe38d536654b37575d01a8b12485dc35f9b34016a3 | What is the minimum number of non-overlapping triangles needed to form a hexagon? | [
"6"
] | [
"What is the minimum number of non-overlapping triangles needed to form a hexagon?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-000f657796132c2961c97cd83feda4d79d06be991a18df992332ba35d0a872b9 | What is the minimum number of non-zero terms that can result from squaring a polynomial \( P(x) \) with integer coefficients and a given degree, such that \( P(x) \) has the fewest possible non-zero terms? | [
"1"
] | [
"What is the minimum number of non-zero terms that can result from squaring a polynomial \\( P(x) \\) with integer coefficients and a given degree, such that \\( P(x) \\) has the fewest possible non-zero terms?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-50a601a031d7e3968ee367773743ea74a20b8479a04e88e3f582c2f18c7df4f6 | What is the minimum number of obtuse angles in a convex 100-gon? | [
"97"
] | [
"What is the minimum number of obtuse angles in a convex 100-gon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-2f48f9ac3af5a3ff4b1fc29fec9e6758d9cf66954faf3bf46080e5bf706a2cba | What is the minimum number of obtuse triangles into which a square can be divided? Provide the configuration that achieves this minimum. | [
"7"
] | [
"What is the minimum number of obtuse triangles into which a square can be divided? Provide the configuration that achieves this minimum."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-60129df68c49942329c0bae104575ccc0a35bd8981eabecf49119bbd44490518 | What is the minimum number of operations needed to get 47 pieces of 46-sided polygons by repeatedly cutting a rectangular piece of paper into two pieces with each operation? | [
"46"
] | [
"What is the minimum number of operations needed to get 47 pieces of 46-sided polygons by repeatedly cutting a rectangular piece of paper into two pieces with each operation?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-99ce029f294b5d0beb9fce1952f1a5c1b0659ab7fbaf2a8ec8a628343f8b738c | What is the minimum number of operations needed to get at least 47 pieces of 46-sided polygons by repeatedly cutting a rectangular piece of paper into two pieces with each operation? | [
"2020"
] | [
"What is the minimum number of operations needed to get at least 47 pieces of 46-sided polygons by repeatedly cutting a rectangular piece of paper into two pieces with each operation?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-1159a9f1e8ae8fc4ba207e6f27ab462df60edc981e4b4c37e20ba6c81897e7b2 | What is the minimum number of ordinary tubs of size $1 \times 1$ needed to block the view around the entire perimeter of a square garden of size $n \times n$, where the central cell contains a golden tub of size $1 \times 1$? What is the minimum garden size? | [
"4 ordinary tubs, minimum garden size is \\(3 \\times 3\\)."
] | [
"What is the minimum number of ordinary tubs of size $1 \\times 1$ needed to block the view around the entire perimeter of a square garden of size $n \\times n$, where the central cell contains a golden tub of size $1 \\times 1$? What is the minimum garden size?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-201da59f907cfa402afa9caf9e8b1ae3215aa79068d410b43bf3c48e25d4d71c | What is the minimum number of ordinary tubs of size $1 \times 2$ needed to block the view around the entire perimeter of a square garden of size $n \times n$, where the central cell contains a golden tub of size $1 \times 1$? What is the minimum garden size? | [
"8"
] | [
"What is the minimum number of ordinary tubs of size $1 \\times 2$ needed to block the view around the entire perimeter of a square garden of size $n \\times n$, where the central cell contains a golden tub of size $1 \\times 1$? What is the minimum garden size?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-87fe353f8cf61d228e271317d8c3a875b842c788fdf26e7511919a31bf2d3995 | What is the minimum number of ordinary tubs of size $1 \times 2$ needed to block the view around the entire perimeter of a square garden of size $n \times n$, where the central cell contains a golden tub of size $1 \times 2$? What is the minimum garden size? | [
"8 ordinary tubs, minimum garden size \\(4 \\times 4\\)."
] | [
"What is the minimum number of ordinary tubs of size $1 \\times 2$ needed to block the view around the entire perimeter of a square garden of size $n \\times n$, where the central cell contains a golden tub of size $1 \\times 2$? What is the minimum garden size?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17518d42442526444b800d5aa570690c6da7357775a0366cfa60ab37ddc4a6a9 | What is the minimum number of ordinary tubs of size $1 \times n$ needed to block the view around the entire perimeter of a square garden of size $n \times n$, where the central cell contains a golden tub of size $1 \times 1$? What is the minimum garden size? (Start exploring the problem at least with small values of $n... | [
"\\(4(n-1)\\)"
] | [
"What is the minimum number of ordinary tubs of size $1 \\times n$ needed to block the view around the entire perimeter of a square garden of size $n \\times n$, where the central cell contains a golden tub of size $1 \\times 1$? What is the minimum garden size? (Start exploring the problem at least with small valu... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac6e0a0f361d38b0b82757f44d37da8f9a0037c9c6732ad5a465a4e80a72dd3c | What is the minimum number of ordinary tubs of size $1 \times n$ needed to block the view around the entire perimeter of a square garden of size $n \times n$, where the central cell contains a golden tub of size $1 \times m$? What is the minimum garden size? (Begin to explore the problem at least with small values of $... | [
"2m - 1"
] | [
"What is the minimum number of ordinary tubs of size $1 \\times n$ needed to block the view around the entire perimeter of a square garden of size $n \\times n$, where the central cell contains a golden tub of size $1 \\times m$? What is the minimum garden size? (Begin to explore the problem at least with small val... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-7be5c35aaf35dfd27b81a0a06d8c779a0f14293fd82c663725c571438772129d | What is the minimum number of ordinary tubs of size $1 \times n$ needed to block the view around the entire perimeter of a square garden of size $n \times n$, where the central cell contains a golden tub of size $1 \times n$? What is the minimum garden size? (Explore the problem for various $n \in \mathbb{N}$.) | [
"3"
] | [
"What is the minimum number of ordinary tubs of size $1 \\times n$ needed to block the view around the entire perimeter of a square garden of size $n \\times n$, where the central cell contains a golden tub of size $1 \\times n$? What is the minimum garden size? (Explore the problem for various $n \\in \\mathbb{N}$... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-b4368bc96734d913619f2188737dbc6eeea7ba4c948f313cf999b12fa2bf1202 | What is the minimum number of pairwise comparisons needed to identify the heaviest and the second heaviest amongst 128 objects? | [
"133"
] | [
"What is the minimum number of pairwise comparisons needed to identify the heaviest and the second heaviest amongst 128 objects?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f4e3be9d6f5029fcc3492de98ad1c55c95e5f94f55febc0fe40069fe229382b6 | What is the minimum number of people in a group so that you would expect at least two people in the group to be born on the same day of the week? The expectation is based on the probability being $\geq \frac{1}{2}$. | [
"4"
] | [
"What is the minimum number of people in a group so that you would expect at least two people in the group to be born on the same day of the week? The expectation is based on the probability being $\\geq \\frac{1}{2}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9290cc9e223d6890aba3322d4ef5680368d491075af932a37e5415488f6f3b2a | What is the minimum number of people in a group such that either 4 of the people all know each other or all do not know each other? | [
"18"
] | [
"What is the minimum number of people in a group such that either 4 of the people all know each other or all do not know each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-89881142716bceaf213b5d0c2d264a5c299edbb28c8df3d2a778c4ad4396c00e | What is the minimum number of people in a single room to ensure that there are at least two people whose birthday is in February? | [
"13"
] | [
"What is the minimum number of people in a single room to ensure that there are at least two people whose birthday is in February?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-727d659db8fd51e7e96c0280e932d70b918af75b0b1838311e68fbac9385e314 | What is the minimum number of people needed in a room to guarantee that at least 2 of them will share birthdays in January? | [
"337"
] | [
"What is the minimum number of people needed in a room to guarantee that at least 2 of them will share birthdays in January?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-921400c46a366e8d64509c4c295925ad44fee8aede2f8e1bf3d94dd0812777b9 | What is the minimum number of people that need to be in a random Facebook group to ensure that there are 4 people who are either all friends with each other or all strangers to each other? Provide a counterexample for the next smaller case. | [
"18"
] | [
"What is the minimum number of people that need to be in a random Facebook group to ensure that there are 4 people who are either all friends with each other or all strangers to each other? Provide a counterexample for the next smaller case."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-985e363a7e1079e741141a0e7b0bcdd73e9d37a602b3e146c28cacffe74c6b57 | What is the minimum number of people who responded to Carlos’s survey, given that the box plot displays the minimum, first quartile, median, third quartile, and maximum of his data, with the vertical lines at values of 18, 20.5, 23.5, 24.5, and 31? | [
"8"
] | [
"What is the minimum number of people who responded to Carlos’s survey, given that the box plot displays the minimum, first quartile, median, third quartile, and maximum of his data, with the vertical lines at values of 18, 20.5, 23.5, 24.5, and 31?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-8dd46d72d772a930ee09a8b136cac642a04886753d1a0eb3d08ea7ca8f3f129d | What is the minimum number of planes belonging to three different directions required to divide space such that the number of unbounded parts is double the number of bounded parts? | [
"17"
] | [
"What is the minimum number of planes belonging to three different directions required to divide space such that the number of unbounded parts is double the number of bounded parts?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-f91a7eb26a90cad64565e6673dfda5aac93ad38463b20947c7c651df3afebac0 | What is the minimum number of planes determined by 6 points in space, given that the points are not all coplanar and no three points are collinear? | [
"11"
] | [
"What is the minimum number of planes determined by 6 points in space, given that the points are not all coplanar and no three points are collinear?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2811a62d94789a31fac5cacf17e7c153b45dbd4df2b928e95af665fae76bddea | What is the minimum number of points that can be chosen in a $2$ by $5$ rectangle such that the distance between any two points is less than $\sqrt{2}$? | [
"11"
] | [
"What is the minimum number of points that can be chosen in a $2$ by $5$ rectangle such that the distance between any two points is less than $\\sqrt{2}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aaedaa87221d028a5e09fc5099ec83e13ba428610d6e5ed95d9bad8b6ddddbaf | What is the minimum number of points that must be taken from a square with a side length of $2$, so that it is guaranteed to always pick two points whose distance between them is not more than $\frac12 \sqrt2$? | [
"17"
] | [
"What is the minimum number of points that must be taken from a square with a side length of $2$, so that it is guaranteed to always pick two points whose distance between them is not more than $\\frac12 \\sqrt2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-f9edaf0d981ae45015f436110f3678102cdd8dfaa023f7d2b639e4f66d22fe1f | What is the minimum number of primes in 20 consecutive numbers? | [
"0"
] | [
"What is the minimum number of primes in 20 consecutive numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-580501b092228b36022a915ae7b6f28e65cb93390b950e430753704edb1846e1 | What is the minimum number of queens that can be placed on a chessboard such that no two queens capture each other and no more queens can be added to the board? | [
"5"
] | [
"What is the minimum number of queens that can be placed on a chessboard such that no two queens capture each other and no more queens can be added to the board?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-984ab9e643046eccf0327f2eb499be3cf4888f1a076d22e5aa35ef4e2e3f8e4d | What is the minimum number of queens that can be placed on an 8x8 chessboard such that no two queens can capture each other and no additional queens can be placed on the board? | [
"5"
] | [
"What is the minimum number of queens that can be placed on an 8x8 chessboard such that no two queens can capture each other and no additional queens can be placed on the board?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-6a90b7eb355d2620486e3a50f5abf7eaf546a3b9d48daa79602ab01a8d9f13c6 | What is the minimum number of questions required to determine which people speak the truth and which people always lie, when that number is finite, given that Arnaldo can ask questions about the parity of truth-tellers in any subset $A$ of size $k$ from a group of $n$ people? | [
"\\( n \\)"
] | [
"What is the minimum number of questions required to determine which people speak the truth and which people always lie, when that number is finite, given that Arnaldo can ask questions about the parity of truth-tellers in any subset $A$ of size $k$ from a group of $n$ people?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d330682e33a13d34ff91fd39ba7e7597ee7a8d7df0fe55c1aee6770a05b8d7b4 | What is the minimum number of questions you need to ask, where each question can be answered with Yes, No, or Maybe, to determine a randomly selected card from a standard 52-card deck? | [
"4"
] | [
"What is the minimum number of questions you need to ask, where each question can be answered with Yes, No, or Maybe, to determine a randomly selected card from a standard 52-card deck?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-38497a61c9ebc68f8ed7b8e5eb760cc78667bcbe09c8946588dff41bce6541e4 | What is the minimum number of satellites that must be launched over a spherical planet so that at some moment signals from at least two satellites are available from each point on the planet's surface? A satellite is considered accessible from point $A$ of the planet's surface if it is located relative to the tangent p... | [
"4"
] | [
"What is the minimum number of satellites that must be launched over a spherical planet so that at some moment signals from at least two satellites are available from each point on the planet's surface? A satellite is considered accessible from point $A$ of the planet's surface if it is located relative to the tang... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-44495f162be96e57d8d3a5d5a892e8a2d64727aac2ed97990fb605fa28b15434 | What is the minimum number of shots needed for the probability of Rex hitting the target exactly 5 times to be more than 25%, given that his probability of hitting the target in a single shot is 0.6? | [
"7"
] | [
"What is the minimum number of shots needed for the probability of Rex hitting the target exactly 5 times to be more than 25%, given that his probability of hitting the target in a single shot is 0.6?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-d294cab106b5b62918cd2466711443aa49729e2761284496fda3e2113282e539 | What is the minimum number of shots needed to always find a $2 \times 2$ ship located on an $8 \times 8$ field, where each shot reveals whether a specified cell is part of the ship or not? | [
"16"
] | [
"What is the minimum number of shots needed to always find a $2 \\times 2$ ship located on an $8 \\times 8$ field, where each shot reveals whether a specified cell is part of the ship or not?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-b3b4ea78191a7e081cba212740201a60d55ef128f4682e81188a05e6c33ac8b2 | What is the minimum number of sides a polygon can have if its smallest internal angle is $120^\circ$ and each subsequent internal angle increases by $5^\circ$? | [
"9"
] | [
"What is the minimum number of sides a polygon can have if its smallest internal angle is $120^\\circ$ and each subsequent internal angle increases by $5^\\circ$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cc9c615c2015b9107a735138a4dd7de630d2b707d9ef69e861aca54bbb53583a | What is the minimum number of smaller equilateral triangles with side 1cm needed to fully cover an equilateral triangle $ABC$ with sides 2.1cm? | [
"6"
] | [
"What is the minimum number of smaller equilateral triangles with side 1cm needed to fully cover an equilateral triangle $ABC$ with sides 2.1cm?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-f41e616bf3c521c4652b5f289d6c5bff580699b727dec65e88011a7cc4164f21 | What is the minimum number of smaller equilateral triangles with side length $1$cm needed to fully cover an equilateral triangle with side length $2.1$cm? | [
"7"
] | [
"What is the minimum number of smaller equilateral triangles with side length $1$cm needed to fully cover an equilateral triangle with side length $2.1$cm?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 5
} | 9 | 0 | 0 |
omr-ws-6908ba048170f0462b9cd82a2ea665086c10957859f5043373a93808849da400 | What is the minimum number of smaller equilateral triangles, each with a side length of 1cm, needed to fully cover an equilateral triangle with a side length of 2.1cm? | [
"7"
] | [
"What is the minimum number of smaller equilateral triangles, each with a side length of 1cm, needed to fully cover an equilateral triangle with a side length of 2.1cm?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 5
} | 8 | 0 | 0 |
omr-ws-2e7af4117f213450d1f4be7cc6c2ffdf800c42290aa97db39cdb6a28b6a30268 | What is the minimum number of smaller squares required to break a pXp square into smaller squares, where p is a prime number? | [
"6"
] | [
"What is the minimum number of smaller squares required to break a pXp square into smaller squares, where p is a prime number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-3e30c9d30ac109b309c454bfda97fa5cc374b2d5deba070cde373343abddfa59 | What is the minimum number of spheres (not necessarily of the same size) needed in 3D space to satisfy the following conditions: - Two spheres may touch but not intersect. - Spheres may not touch or contain the origin. - Every possible ray starting at the origin must touch or go through at least one sphere. | [
"4"
] | [
"What is the minimum number of spheres (not necessarily of the same size) needed in 3D space to satisfy the following conditions:\n- Two spheres may touch but not intersect.\n- Spheres may not touch or contain the origin.\n- Every possible ray starting at the origin must touch or go through at least one sphere."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-fe0e8166bf0880e22e20f6d99e448368b9c9b5509cfcb0964aa98102d252cf52 | What is the minimum number of squares needed to cover a complete $n \times n$ grid? | [
"\\(2(n - 1)\\)"
] | [
"What is the minimum number of squares needed to cover a complete $n \\times n$ grid?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40d76e527142490fe746f7ca8247d39d3f986ef070ddb3d2a9dca6917ea712e8 | What is the minimum number of squares needed to draw on a white sheet to obtain a square grid of side \( n \)? | [
"2n - 1"
] | [
"What is the minimum number of squares needed to draw on a white sheet to obtain a square grid of side \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-301aafdca9801b7eb3d9d4060e4a0832038fedff11f7dc6b801696b4f7ab0b2c | What is the minimum number of squares that is necessary to draw on a white sheet to obtain a square grid of side $n$? | [
"\\( n^2 - \\frac{(n-2)^2}{2} \\) if \\( n \\) is even, and \\( n^2 - \\frac{(n-2)^2 + 1}{2} \\) if \\( n \\) is odd."
] | [
"What is the minimum number of squares that is necessary to draw on a white sheet to obtain a square grid of side $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-2a263c4eab46ecac81e909d0a867a385c965bfeb9e344180d42bb23544aafac8 | What is the minimum number of squares that one can color yellow on a 5 by 5 checkerboard so that every non-yellow square is adjacent to a yellow square? | [
"7"
] | [
"What is the minimum number of squares that one can color yellow on a 5 by 5 checkerboard so that every non-yellow square is adjacent to a yellow square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 6
} | 19 | 0 | 0 |
omr-ws-b74da2b781e20610b7c125692e55692f154d3980d6d9ac6ae963d9a5cab3dbab | What is the minimum number of squares that one needs to draw on a white sheet in order to obtain a complete grid with n squares on a side? | [
"\\( 2n - 1 \\)"
] | [
"What is the minimum number of squares that one needs to draw on a white sheet in order to obtain a complete grid with n squares on a side?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 2,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-42d41abf9ebd8a4aa5314ef5f204a8e811d3422a92e86c076d23bb0970558bc5 | What is the minimum number of stops that a tram network can have if it must meet the following conditions: - There are at least two distinct tram lines. - Each tram line serves exactly three stops (including the start and terminus). - For each pair of tram stops, there is exactly one tram line that serves both stops. | [
"7"
] | [
"What is the minimum number of stops that a tram network can have if it must meet the following conditions:\n- There are at least two distinct tram lines.\n- Each tram line serves exactly three stops (including the start and terminus).\n- For each pair of tram stops, there is exactly one tram line that serves both ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-7fc17c55aca05d70a7e5d71b8679676141f6b4ebea3d4901f9c2386328327eff | What is the minimum number of strictly acute triangles required to triangulate a strictly obtuse triangle? | [
"7"
] | [
"What is the minimum number of strictly acute triangles required to triangulate a strictly obtuse triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-1481cf748462036effeceeec4ab2b4c03518d3a28f411dfcaa9324b3fa3f3c16 | What is the minimum number of students in a class that ensures at least two of them went swimming on the same set of dates during March 11th to March 15, 2019? | [
"33"
] | [
"What is the minimum number of students in a class that ensures at least two of them went swimming on the same set of dates during March 11th to March 15, 2019?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-12f115b5e77ce2084f2c0510d0e0c03155cfe53a08bee87d6f0bd52d33eb4bb0 | What is the minimum number of students in a class to ensure that at least 3 students went swimming on the same set of days from March 11 to March 15? | [
"65"
] | [
"What is the minimum number of students in a class to ensure that at least 3 students went swimming on the same set of days from March 11 to March 15?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-3a199f5aa0c580ceea6fa0414230ad2a8378bec8746ca196fb8deef5d0613508 | What is the minimum number of students that must be in a class to guarantee that at least 3 paper sheets are identical on a multiple choice quiz with 10 questions, each having 4 possible answers? (Assume that no answers are left blank.) | [
"\\(2^{21} + 1\\)"
] | [
"What is the minimum number of students that must be in a class to guarantee that at least 3 paper sheets are identical on a multiple choice quiz with 10 questions, each having 4 possible answers? (Assume that no answers are left blank.)"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-e3457de24a0c256164c97aaf6e8128bac60213643f486fe814fa78a5765dcc07 | What is the minimum number of successive swaps of adjacent letters in the string ABCDEF that are needed to change the string to FEDCBA? | [
"15"
] | [
"What is the minimum number of successive swaps of adjacent letters in the string ABCDEF that are needed to change the string to FEDCBA?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-ec6f6f287b992ecc0357e411147eba73ca35245b83f146ac0200749f356050b3 | What is the minimum number of terms required to write 2011 as a "special" sum, where a "special" sum is defined as a sum of positive integers starting and ending with 1, and each term differs from its neighbor by at most 1? | [
"89"
] | [
"What is the minimum number of terms required to write 2011 as a \"special\" sum, where a \"special\" sum is defined as a sum of positive integers starting and ending with 1, and each term differs from its neighbor by at most 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 2,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-4e2366ef20657d71ef66fee07e115610e76b79673b125783d2ee777c8f463395 | What is the minimum number of times you must throw three fair six-sided dice to ensure that the same sum is rolled twice? | [
"17"
] | [
"What is the minimum number of times you must throw three fair six-sided dice to ensure that the same sum is rolled twice?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-49668ca475127dcacc8802a0e0b1548931ca30ffb23f4da59616a9be9bc8e80e | What is the minimum number of total button presses Alex needs to guarantee opening a lock whose code is a sequence of three letters, each being one of A, B, or C, possibly repeated, with three buttons labeled A, B, and C? | [
"29"
] | [
"What is the minimum number of total button presses Alex needs to guarantee opening a lock whose code is a sequence of three letters, each being one of A, B, or C, possibly repeated, with three buttons labeled A, B, and C?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 17
} | 17 | 0 | 0 |
omr-ws-2f672dc20f9a2e2cfa667ce1ebd9b3bd291b2876f8ddb85e0f6bf740f120388b | What is the minimum number of total grades required so that every tenth of a GPA can be represented without rounding, given the formula for unweighted GPA is $\frac{4x + 3y + 2z + 1t + 0f}{x + y + z + t + f}$, where $x, y, z, t, f$ represent the number of A's, B's, C's, D's, and F's respectively? | [
"10"
] | [
"What is the minimum number of total grades required so that every tenth of a GPA can be represented without rounding, given the formula for unweighted GPA is $\\frac{4x + 3y + 2z + 1t + 0f}{x + y + z + t + f}$, where $x, y, z, t, f$ represent the number of A's, B's, C's, D's, and F's respectively?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-378fdc3b7351d6ff2cc72bd32c7c3c13a3f19803086584d61b99494b0655a555 | What is the minimum number of triangle faces a polyhedron can have if it has no faces that are either quadrilaterals or pentagons? | [
"4"
] | [
"What is the minimum number of triangle faces a polyhedron can have if it has no faces that are either quadrilaterals or pentagons?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90dba31a3a49de85cef88959fb66b46992051349461aed3ea14fdf5b55de5bb8 | What is the minimum number of triangles in a 72-degree triangulation of a square, where every angle in the triangles is no more than 72 degrees? | [
"8"
] | [
"What is the minimum number of triangles in a 72-degree triangulation of a square, where every angle in the triangles is no more than 72 degrees?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-29ae8e83d5e22c4b8cbf2d10b2d490326b5e5bf635e9e25e01a45392fbe71d33 | What is the minimum number of turns needed to move through every single cube at least once in an $a$ by $b$ by $c$ cuboid? You can choose which 1 by 1 by 1 cube to start at. | [
"2(a + b + c - 3)"
] | [
"What is the minimum number of turns needed to move through every single cube at least once in an $a$ by $b$ by $c$ cuboid? You can choose which 1 by 1 by 1 cube to start at."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-e5a0f4b397a6f93ef78209f0170193a58b29dd892857f2a57a4bee27f13fa589 | What is the minimum number of turns needed to move through every single cube at least once in an $n$ by $n$ by $n$ cube? You can choose which 1 by 1 by 1 cube to start at. | [
"3n(n - 1)"
] | [
"What is the minimum number of turns needed to move through every single cube at least once in an $n$ by $n$ by $n$ cube? You can choose which 1 by 1 by 1 cube to start at."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-a2e0477de2b24d3cc245caf7f5dfa901e9c0f2ce12fd1cddb3ba2d2b50e96743 | What is the minimum number of turns needed to move through every single cube exactly once in an $a$ by $b$ by $c$ cuboid? You can choose which 1 by 1 by 1 cube to start at. | [
"2(a + b + c - 3)"
] | [
"What is the minimum number of turns needed to move through every single cube exactly once in an $a$ by $b$ by $c$ cuboid? You can choose which 1 by 1 by 1 cube to start at."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-cd0cfc6de4c3e02ec28b1fbe06b24bad4fa9efa7d0e52bec0880d74cca0df8c4 | What is the minimum number of turns needed to move through every single cube exactly once in an $n$ by $n$ by $n$ cube? You can choose which 1 by 1 by 1 cube to start at. | [
"2(n^2 - 1)"
] | [
"What is the minimum number of turns needed to move through every single cube exactly once in an $n$ by $n$ by $n$ cube? You can choose which 1 by 1 by 1 cube to start at."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-1e72ef8c8bbf74220924749fe883281f2d085f089423d2f23ba830ed9f377f65 | What is the minimum number of types of fruit needed to provide at least 2018 different fruit flavor combinations? | [
"11"
] | [
"What is the minimum number of types of fruit needed to provide at least 2018 different fruit flavor combinations?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-d67a3c8bbfaa7b93fc34e49bd55902161b842d68a1ada89f66ddca4ff3a31200 | What is the minimum number of unit squares required to completely cover a square of side length 2.1 units? Provide your answer as an integer. | [
"9"
] | [
"What is the minimum number of unit squares required to completely cover a square of side length 2.1 units? Provide your answer as an integer."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-2f329d4c97177fdcec59d3e1aedcce848bfb98ba02264d277bc9b4e5fc44e21c | What is the minimum number of unit squares required to cover an equilateral triangle with a side length of 2 units? | [
"3"
] | [
"What is the minimum number of unit squares required to cover an equilateral triangle with a side length of 2 units?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-874312097e9d7b51dfb918bb5620f5d9d692ba2a124a3be91e4372d0b607afe3 | What is the minimum number of vertices Peter can paint on a $2 \times n$ board so that every square has exactly one painted vertex? | [
"\\( \\lceil \\frac{n}{2} \\rceil \\)"
] | [
"What is the minimum number of vertices Peter can paint on a $2 \\times n$ board so that every square has exactly one painted vertex?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 4,
"tir": 19
} | 61 | 0 | 0 |
omr-ws-500f236533ff4f0ca9d6be2e44f2fdb598a3df7e3a1c6d2b72b00348e52afc8c | What is the minimum percentage of the population that agreed with the government on all three decisions, given that $93\%$ agreed with the first decision, $84\%$ with the second, and $74\%$ with the third? | [
"47%"
] | [
"What is the minimum percentage of the population that agreed with the government on all three decisions, given that $93\\%$ agreed with the first decision, $84\\%$ with the second, and $74\\%$ with the third?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2daf05c403efadb0de384428de3fcc2151654b38de84a0ee02bebb55eeeb251e | What is the minimum perimeter of a shape that separates two areas, one with area $9$ and one with area $4$? The two areas can share a common boundary. | [
"$10\\sqrt{\\pi}$"
] | [
"What is the minimum perimeter of a shape that separates two areas, one with area $9$ and one with area $4$? The two areas can share a common boundary."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-579139b259b6cd0b34bc09fa69d7cf36bb03dcea242c37b32a1265cac4c20408 | What is the minimum perimeter of a shape that separates two areas, one with area $a$ and the other with area $b$? | [
"\\( 2\\sqrt{\\pi}(\\sqrt{a} + \\sqrt{b}) \\)"
] | [
"What is the minimum perimeter of a shape that separates two areas, one with area $a$ and the other with area $b$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 8
} | 14 | 0 | 0 |
omr-ws-b9c35423bc94c24f49d9b5e67d081e685383d76617a00398e64476c52f0e0e36 | What is the minimum positive integer \( f(b) \) which cannot be expressed as the arithmetic mean of different nonnegative powers of \( b \)? | [
"\\( f(b) = 4 \\) if \\( b = 3 \\) and \\( f(b) = 2 \\) if \\( b > 3 \\)."
] | [
"What is the minimum positive integer \\( f(b) \\) which cannot be expressed as the arithmetic mean of different nonnegative powers of \\( b \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22e7dbe1153921f84a1252cefe80cdca1990086789d4364266cd4dd51258e8c4 | What is the minimum positive integer that is divisible by $2021$ and has an equal number of each digit $0, 1, 2, \ldots, 9$? | [
"11223344556677889900"
] | [
"What is the minimum positive integer that is divisible by $2021$ and has an equal number of each digit $0, 1, 2, \\ldots, 9$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-2c93941ffa73d76a34d6709c7e981ddc18e17af313ad94989601642cbec61904 | What is the minimum positive integer value of \( n \) such that \( 5^n - 1 \) is divisible by \( 2^{2001} \)? | [
"\\( 2^{2000} \\)"
] | [
"What is the minimum positive integer value of \\( n \\) such that \\( 5^n - 1 \\) is divisible by \\( 2^{2001} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-69d3fb70a509bd7094ae9fee5b1a01ed4b40e21b79f5005c6414023d2cd4281e | What is the minimum positive integer value of \(\frac{(x + y^2)(x^2 - y)}{xy}\) where \(x\) and \(y\) are two distinct positive integers? | [
"14"
] | [
"What is the minimum positive integer value of \\(\\frac{(x + y^2)(x^2 - y)}{xy}\\) where \\(x\\) and \\(y\\) are two distinct positive integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-ff6960c2211457163631b4ba9e6376e7d442407edf8d641db5cccb009a502ab5 | What is the minimum possible area of a parallelogram with side lengths 16 and 36? | [
"0"
] | [
"What is the minimum possible area of a parallelogram with side lengths 16 and 36?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f2447e2859f8b1d9dea954e585309d6d5414a88e7ce583c3d43bf442c11c70ab | What is the minimum possible area of a simple polygon with an odd number of sides, each of unit length? | [
"\\dfrac{\\sqrt{3}}{4}"
] | [
"What is the minimum possible area of a simple polygon with an odd number of sides, each of unit length?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-508896c041c67d917b3454f737be584d85f71eaebb265bb0d98b8f31c16ff48d | What is the minimum possible area of a triangle that contains a unit square? | [
"2"
] | [
"What is the minimum possible area of a triangle that contains a unit square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2d5201156774c4a8bc70e5651304f7e5faf67280b9abb188b0b3aac9058528d7 | What is the minimum possible number $k$ such that for every tree $T$ with 2024 vertices and a diameter of 100, one can reach a tree of diameter smaller than 100 in at most $k$ steps? | [
"50"
] | [
"What is the minimum possible number $k$ such that for every tree $T$ with 2024 vertices and a diameter of 100, one can reach a tree of diameter smaller than 100 in at most $k$ steps?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-61de1d852ef6d61ba0eda808f38f120ba214f0aca89f8ea54d9fdfbacfbefd5b | What is the minimum possible number of boys playing at least three games, given that each boy plays at least two games out of five, with the distribution as follows: Cricket 18, Football 40, Volleyball 42, Basketball 45, Hockey 55? | [
"0"
] | [
"What is the minimum possible number of boys playing at least three games, given that each boy plays at least two games out of five, with the distribution as follows: Cricket 18, Football 40, Volleyball 42, Basketball 45, Hockey 55?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-dc4a59a3d030056738695e92ef66c24268a679a8cc20214fd877361263a65566 | What is the minimum possible number of positive roots of the equation \(x^2 - (1+k)x + k - 2 = 0\), where \(k\) is a real number? | [
"1"
] | [
"What is the minimum possible number of positive roots of the equation \\(x^2 - (1+k)x + k - 2 = 0\\), where \\(k\\) is a real number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3494ab0585ed2851e40ff255b7a60684d11ef50c0634f732fba69ff5b41dd19c | What is the minimum possible product of three different numbers from the set $\{-8, -6, -4, 0, 3, 5, 7\}$? | [
"-280"
] | [
"What is the minimum possible product of three different numbers from the set $\\{-8, -6, -4, 0, 3, 5, 7\\}$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-2b148688b7d8715f5a4cfe0ad41ecc0c5e4207f821ae2cb5416713de8affaaf5 | What is the minimum possible value for $y$ in the equation $y = x^2 + 12x + 5$? | [
"-31"
] | [
"What is the minimum possible value for $y$ in the equation $y = x^2 + 12x + 5$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-60a9edf315616782e07d8e807b4e7226557fc5a814b9ad201b960e7c1e85f250 | What is the minimum possible value of $|Z|^2 + |Z - 3|^2 + |Z - 6i|^2$ where $Z$ is a complex number and $i = \sqrt{-1}$? | [
"30"
] | [
"What is the minimum possible value of $|Z|^2 + |Z - 3|^2 + |Z - 6i|^2$ where $Z$ is a complex number and $i = \\sqrt{-1}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-11ef02ba80ecdf3974c73490a90cc78d9d112788d906071ad8c2f70de20e18b9 | What is the minimum possible value of Aaron's age, to the nearest year, if he has been alive for all or part of three separate decades? | [
"10"
] | [
"What is the minimum possible value of Aaron's age, to the nearest year, if he has been alive for all or part of three separate decades?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-ec565754bdfe664a13b23b640fd89ab6c57784ed8bce664a0c86bfd9a24b5d2b | What is the minimum possible value of \( \frac{a}{2b} + \frac{b}{4c} + \frac{c}{8a} \), where \( a \), \( b \), and \( c \) are positive real numbers? | [
"\\(\\frac{3}{4}\\)"
] | [
"What is the minimum possible value of \\( \\frac{a}{2b} + \\frac{b}{4c} + \\frac{c}{8a} \\), where \\( a \\), \\( b \\), and \\( c \\) are positive real numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-220c93aa761fa5ea7d1329d05e4d6d303be0ec2012a86a6957b0c43d5673fc72 | What is the minimum possible value of \( \frac{a}{2b} + \frac{b}{4c} + \frac{c}{8a} \), where \( a, b, c \) are positive real numbers? | [
"\\(\\frac{3}{4}\\)"
] | [
"What is the minimum possible value of \\( \\frac{a}{2b} + \\frac{b}{4c} + \\frac{c}{8a} \\), where \\( a, b, c \\) are positive real numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-75731a7ab99ac91133a4b622352dd4702da980f764b898f1ba79fda3d466a4df | What is the minimum possible value of \( n \in \mathbb{N} \) such that there exist strictly positive reals \( a_1, a_2, \ldots, a_n \) satisfying: \[ \frac{1}{a_1} + \cdots + \frac{1}{a_n} = 45 \quad \text{and} \quad \sqrt{a_1} + \cdots + \sqrt{a_n} = 90? \] | [
"30"
] | [
"What is the minimum possible value of \\( n \\in \\mathbb{N} \\) such that there exist strictly positive reals \\( a_1, a_2, \\ldots, a_n \\) satisfying:\n\\[\n\\frac{1}{a_1} + \\cdots + \\frac{1}{a_n} = 45 \\quad \\text{and} \\quad \\sqrt{a_1} + \\cdots + \\sqrt{a_n} = 90?\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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