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-464cd31ac04613341b423a54ad036d29a1a6dd30d777b27b6879a97d66571193 | What is the minimal polynomial and characteristic polynomial of an $n \times n$ matrix where all entries are equal to 1? | [
"Minimal Polynomial: \\( \\lambda^2 - n\\lambda \\), Characteristic Polynomial: \\( \\lambda^{n-1}(\\lambda - n) \\)"
] | [
"What is the minimal polynomial and characteristic polynomial of an $n \\times n$ matrix where all entries are equal to 1?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 18
} | 18 | 0 | 0 |
omr-ws-8ec65c171860aa58392936a4e0a456b098aab8f4a9fe623e42d347123ddf305e | What is the minimal polynomial of $3^{1/4} + 3^{1/2}$ over $\mathbb{Q}(\sqrt{3})$? | [
"\\((x - \\sqrt{3})^2 - \\sqrt{3}\\)"
] | [
"What is the minimal polynomial of $3^{1/4} + 3^{1/2}$ over $\\mathbb{Q}(\\sqrt{3})$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-f681a4c65721445aab89eea86b946ff01be9a3b7b9b5c7a4b4351ba13854c07a | What is the minimal positive $k$ such that there exists a positive integer $x$ for which $6x^2 + 6kx + k(2k+1)$ is a perfect square? | [
"2"
] | [
"What is the minimal positive $k$ such that there exists a positive integer $x$ for which $6x^2 + 6kx + k(2k+1)$ is a perfect square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff326da7ee59bde01862ad390105072e1c2c48460875222a6263b55d90154670 | What is the minimal positive \( k \) such that there exists a positive integer \( x \) such that \( 6x^2 + 6kx + k(2k+1) \) is the square of an integer? | [
"2"
] | [
"What is the minimal positive \\( k \\) such that there exists a positive integer \\( x \\) such that \\( 6x^2 + 6kx + k(2k+1) \\) is the square of an integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-22bf07de0c18deb0dc15ece256045744c3341488c16a3e8cc737f8b0fc91aee3 | What is the minimal radius of a disc that can always cover a set of points in the plane, given that the distance between every pair of points does not exceed $d$? | [
"\\(\\frac{d}{2}\\)"
] | [
"What is the minimal radius of a disc that can always cover a set of points in the plane, given that the distance between every pair of points does not exceed $d$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f6f83c784d7c58c7760efb63beb69d421e24191eac3461b0fa533c5c845cb62b | What is the minimal size of an \( m \times m \) board such that 26 knights, each representing a distinct letter (A to Z), can be placed on the board and after at most \( k \) distinct moves (with \( k = 20 \) for example), no two knights reach the same position? Cyclic moves are allowed to be repeated, and each targete... | [
"24"
] | [
"What is the minimal size of an \\( m \\times m \\) board such that 26 knights, each representing a distinct letter (A to Z), can be placed on the board and after at most \\( k \\) distinct moves (with \\( k = 20 \\) for example), no two knights reach the same position? Cyclic moves are allowed to be repeated, and ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-029b8456ef26c4c963240a3ae0647e1b79cec3eab5d2f967717007bca6558717 | What is the minimal value of $\sqrt{2x+1}+\sqrt{3y+1}+\sqrt{4z+1}$, if $x$, $y$, and $z$ are nonnegative real numbers such that $x+y+z=4$? | [
"5"
] | [
"What is the minimal value of $\\sqrt{2x+1}+\\sqrt{3y+1}+\\sqrt{4z+1}$, if $x$, $y$, and $z$ are nonnegative real numbers such that $x+y+z=4$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fb476ef5829185c31e77eb571785c44adc6483a4563cc48ce3dc9c2c09f1c6d4 | What is the minimal value of $\sqrt{x+y}\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}\right)$ for positive real numbers $x$ and $y$? | [
"\\(2\\sqrt{2}\\)"
] | [
"What is the minimal value of $\\sqrt{x+y}\\left(\\frac{1}{\\sqrt{x}}+\\frac{1}{\\sqrt{y}}\\right)$ for positive real numbers $x$ and $y$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-11b8746f5d9598aa5d4430cfd0b42b49ef82897c0f0d700596dbc16120f7181e | What is the minimal value of $k$ (depending on $n$) such that one can find a $C^{\infty}$ periodic function $f: \mathbb{R}^n \to \mathbb{R}^k$ such that for the partial derivatives $D_i f$, the condition $<D_i f(x), D_j f(x)> = \delta_{ij}$ holds, where $<>$ denotes the scalar product and $\delta_{ij}$ is the Kronecker... | [
"2n"
] | [
"What is the minimal value of $k$ (depending on $n$) such that one can find a $C^{\\infty}$ periodic function $f: \\mathbb{R}^n \\to \\mathbb{R}^k$ such that for the partial derivatives $D_i f$, the condition $<D_i f(x), D_j f(x)> = \\delta_{ij}$ holds, where $<>$ denotes the scalar product and $\\delta_{ij}$ is th... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4c6126689df0df79fac20b6ae51c46444315e12180a819ad31829e180ee62794 | What is the minimal value of \( K \) such that every \( K \)-element subset of \(\{1, 2, \ldots, 50\}\) contains two elements \( a \) and \( b \) such that \( a + b \) divides \( ab \)? | [
"39"
] | [
"What is the minimal value of \\( K \\) such that every \\( K \\)-element subset of \\(\\{1, 2, \\ldots, 50\\}\\) contains two elements \\( a \\) and \\( b \\) such that \\( a + b \\) divides \\( ab \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-df698203e36ecf7176a22972aee3ca05c6dc3d3db34afcb542dfa886881725d4 | What is the minimal value of \( n \) such that a magician and their helper can successfully perform a trick where the helper turns all but one of \( n \) face-up cards face down (without changing their order), and the magician can then identify the face color of the one face-down card? The cards have one of 2017 possib... | [
"2018"
] | [
"What is the minimal value of \\( n \\) such that a magician and their helper can successfully perform a trick where the helper turns all but one of \\( n \\) face-up cards face down (without changing their order), and the magician can then identify the face color of the one face-down card? The cards have one of 20... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 20,
"genselect": 5,
"tir": 2
} | 27 | 0 | 0 |
omr-ws-1076f88adf4a6b949af8751c5ebfc4699c6d9bc4f10b9f4203a66be6d606991e | What is the minimal value of the function \( f(x) = (x - x_1)^2 + (x - x_2)^2 + \cdots + (x - x_n)^2 \) where \( x_1, x_2, \ldots, x_n \) are given numbers? | [
"\\( x_1^2 + x_2^2 + \\cdots + x_n^2 - \\frac{(x_1 + x_2 + \\cdots + x_n)^2}{n} \\)"
] | [
"What is the minimal value of the function \\( f(x) = (x - x_1)^2 + (x - x_2)^2 + \\cdots + (x - x_n)^2 \\) where \\( x_1, x_2, \\ldots, x_n \\) are given numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-15fbbdd055979924c9163a4ec610d61ee94c8398d9ddc946a7cb07dbebe0b2b3 | What is the minimum $k$ such that every $k$-element subset of $\mathbb{Z}_p$ (where $p$ is prime) has a subset for which the sum of the elements of that subset is $0$ modulo $p$? | [
"\\frac{p+1}{2}"
] | [
"What is the minimum $k$ such that every $k$-element subset of $\\mathbb{Z}_p$ (where $p$ is prime) has a subset for which the sum of the elements of that subset is $0$ modulo $p$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-c852c996315fafdd19bf88decf2c4b5b08883f41aa006c73c637e0a2c49111ce | What is the minimum $n$ such that an $n \times n$ grid can be covered with an equal number of $2 \times 2$ squares and L-triominoes (a $2 \times 2$ square with one square removed)? | [
"14"
] | [
"What is the minimum $n$ such that an $n \\times n$ grid can be covered with an equal number of $2 \\times 2$ squares and L-triominoes (a $2 \\times 2$ square with one square removed)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-44885a58f390345b2ae885ff33858b362bdc9b36d66c47140a0eefa784689c7c | What is the minimum Hamming distance between any two code words for a single bit parity check? | [
"2"
] | [
"What is the minimum Hamming distance between any two code words for a single bit parity check?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bdd1cefcfb9f7b76c5b45b83a315f7bc8b79e1958f3d90b226b6f0a353abf213 | What is the minimum \( n \) such that the set of numbers from \( 1 \) to \( 100 \) can be partitioned into \( n \) disjoint groups \( 5 \) different times, ensuring that no two elements are in the same group more than once? The partitions do not have to be of the same size. | [
"10"
] | [
"What is the minimum \\( n \\) such that the set of numbers from \\( 1 \\) to \\( 100 \\) can be partitioned into \\( n \\) disjoint groups \\( 5 \\) different times, ensuring that no two elements are in the same group more than once? The partitions do not have to be of the same size."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-89cf6c82ebdfb3a86f43dc95ac104f2567acf8933ea1a56e9ccb2ab39e3cff07 | What is the minimum amount that can be withdrawn in 30 different ways if the withdrawal units are 100, 200, and 500? | [
"2100"
] | [
"What is the minimum amount that can be withdrawn in 30 different ways if the withdrawal units are 100, 200, and 500?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-7733d27d1cc7fc6531c7b02d52ff68a5e4a39f61a63e5e876a1ed8c464ee6757 | What is the minimum and maximum number of primes in 1000 consecutive numbers? | [
"Minimum = 0, Maximum not found."
] | [
"What is the minimum and maximum number of primes in 1000 consecutive numbers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68b424a79eae3b946db1875f3738b0e30428dc85828487f12befe4e7ce3d6e9c | What is the minimum and maximum number of terms in a symmetric homogeneous polynomial of degree $k$ in $K[x_1, \cdots, x_n]$ for the cases $k < n$, $k = n$, and $k > n$? | [
"\\dbinom{n + k - 1}{k}"
] | [
"What is the minimum and maximum number of terms in a symmetric homogeneous polynomial of degree $k$ in $K[x_1, \\cdots, x_n]$ for the cases $k < n$, $k = n$, and $k > n$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-a47725d66b65276aaaa37a28b5c51ceda8b4f6428275987cb60379e760d4bda1 | What is the minimum and maximum of the function \( f(x) = \sqrt{x^2 + 9} + \sqrt{x^2 - 6x + 10} \) for \( x \in [0, 3] \)? | [
"Minimum = 5, Maximum = \\( 3 + \\sqrt{10} \\)"
] | [
"What is the minimum and maximum of the function \\( f(x) = \\sqrt{x^2 + 9} + \\sqrt{x^2 - 6x + 10} \\) for \\( x \\in [0, 3] \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 27
} | 29 | 0 | 0 |
omr-ws-44dd41fd1e13ae4b2b6b84143bf379aedceba83418ddce77f3c3fe53c6680f70 | What is the minimum and maximum value of the function \( f(x) = \dfrac{9x^2 \sin^2 x + 4}{x \sin x} \)? | [
"Minimum value is \\(-\\infty\\), Maximum value diverges to \\(\\infty\\). If considering positive values, the minimum value is 12."
] | [
"What is the minimum and maximum value of the function \\( f(x) = \\dfrac{9x^2 \\sin^2 x + 4}{x \\sin x} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-1463dad89f8aee03d70b96eb79c87a3cef977473cc25cebfe13c53f1a099be52 | What is the minimum area of a square hole that a regular octahedron with a side length of $1$ can pass through? | [
"1"
] | [
"What is the minimum area of a square hole that a regular octahedron with a side length of $1$ can pass through?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6a522fed70b251c4a299dd93ef3c550cb336612b87c29c46fd3d9532eaad8d28 | What is the minimum area of the surface bounded by the curves \( y = 0 \), \( f(x) = x^3 + 3x^2 + x + a \), and the vertical lines \( x_1 \) and \( x_2 \) such that \( f'(x_1) = f'(x_2) = 0 \), where \( a \in \mathbb{R} \)? | [
"\\(\\frac{4\\sqrt{6}}{9}\\)"
] | [
"What is the minimum area of the surface bounded by the curves \\( y = 0 \\), \\( f(x) = x^3 + 3x^2 + x + a \\), and the vertical lines \\( x_1 \\) and \\( x_2 \\) such that \\( f'(x_1) = f'(x_2) = 0 \\), where \\( a \\in \\mathbb{R} \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0c945db60617ca88a12003a14eb31e4208a15999fe5f23e60dcc0506a423b2fe | What is the minimum area of the triangle formed by any tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the coordinate axes? (Answer in terms of $a$ and $b$.) | [
"\\( ab \\)"
] | [
"What is the minimum area of the triangle formed by any tangent to the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ and the coordinate axes? (Answer in terms of $a$ and $b$.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-f1523b0c9f36e7412e89ffb6b54e190eddd9a2b3c95eea6f0321acd8d5e1202c | What is the minimum area that a square can have in the plane such that it contains exactly 3 integer coordinate points inside it? | [
"2"
] | [
"What is the minimum area that a square can have in the plane such that it contains exactly 3 integer coordinate points inside it?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-3118f6ef56e82645f589e44c043d868749b222e9c9a727319f60b9215aa728ec | What is the minimum constant $c$ that can replace the factor $2$ in the inequality \[ \sum_{k=1}^n \sum_{j=1}^k \sum_{i=1}^j x_i \leq c \sum_{k=1}^n \left( \sum_{j=1}^k x_j \right)^2 x_k^{-1}? \] | [
"1"
] | [
"What is the minimum constant $c$ that can replace the factor $2$ in the inequality\n\\[\n\\sum_{k=1}^n \\sum_{j=1}^k \\sum_{i=1}^j x_i \\leq c \\sum_{k=1}^n \\left( \\sum_{j=1}^k x_j \\right)^2 x_k^{-1}?\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-63ef95ca21303b90a3f77d1efddcb4798fb90f39f8e1e9d071f4ac9cad1128ee | What is the minimum degree of an irreducible polynomial over a field that has a Galois group isomorphic to $Q_8$? | [
"8"
] | [
"What is the minimum degree of an irreducible polynomial over a field that has a Galois group isomorphic to $Q_8$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3a03c3c75ae9110aceb072b4237c099f56e9fba887d3c1c5acf40612125be7d7 | What is the minimum distance a fly must travel when flying from one corner to the opposite corner of a rectangular box that measures 2 feet by 3 feet by 6 feet? | [
"7 ft"
] | [
"What is the minimum distance a fly must travel when flying from one corner to the opposite corner of a rectangular box that measures 2 feet by 3 feet by 6 feet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 5,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-d4406b7c26d97eeccf8fcbdc601f9d3cad92d462fbc6295d4c5f762c248fe7f0 | What is the minimum distance between a point on the curve \( y = e^x \) and a point on the curve \( y = \ln x \)? | [
"\\(\\sqrt{2}\\)"
] | [
"What is the minimum distance between a point on the curve \\( y = e^x \\) and a point on the curve \\( y = \\ln x \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-5f83d30059fad009b8b74c38aac00ca01b4000cf07fe3e8283bee9919bd79c26 | What is the minimum distance between any point on the circle $x^2 + y^2 = 25$ and the line $y = -\frac{3}{4}x + \frac{75}{4}$? | [
"10"
] | [
"What is the minimum distance between any point on the circle $x^2 + y^2 = 25$ and the line $y = -\\frac{3}{4}x + \\frac{75}{4}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-cfdfa3a704eaf024d2a98b5d2d898aa99df79a48198e434b58ff0d2224175126 | What is the minimum distance between the graphs of \( y = e^x \) and \( y = \ln(x) \)? | [
"\\sqrt{2}"
] | [
"What is the minimum distance between the graphs of \\( y = e^x \\) and \\( y = \\ln(x) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-40c4b4dc7bb9e81e70d13d05ffa50093082b07a39a1626ce819b00118ad6f000 | What is the minimum distance between the graphs of \( y = e^x \) and \( y = x \)? | [
"\\(\\frac{\\sqrt{2}}{2}\\)"
] | [
"What is the minimum distance between the graphs of \\( y = e^x \\) and \\( y = x \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-50b6cae73bf97139c195a2587ad3a68351ec127c492c6c620179468c87e59a76 | What is the minimum distance between the graphs of \( y = x^2 + 2x + 2 \) and \( y = -x^2 - 6x - 10 \)? | [
"2\\sqrt{2}"
] | [
"What is the minimum distance between the graphs of \\( y = x^2 + 2x + 2 \\) and \\( y = -x^2 - 6x - 10 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-6fd41a3ce4ef5271848df78d00b9518564d5dc752e06ba6b6c901421f61226f0 | What is the minimum distance between the line \( y = \frac{4x}{7} + \frac{1}{5} \) and any lattice point in the plane? (Lattice points are points with integer coordinates) | [
"\\(\\frac{2\\sqrt{65}}{325}\\)"
] | [
"What is the minimum distance between the line \\( y = \\frac{4x}{7} + \\frac{1}{5} \\) and any lattice point in the plane? (Lattice points are points with integer coordinates)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-bc00ad6328a667f18515c88190bf1fb3558e29a80bdddc000ed56d59fd55d99d | What is the minimum distance between the points $(2019, 470)$ and $(21a - 19b, 19b + 21a)$ for $a, b \in \mathbb{Z}$? | [
"1.2"
] | [
"What is the minimum distance between the points $(2019, 470)$ and $(21a - 19b, 19b + 21a)$ for $a, b \\in \\mathbb{Z}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66298eb274f51ac9124517cb8c68085c0b6443568c697a2407549fa7d84165f9 | What is the minimum distance between two parabolas given by the equations $ax^2 + bx + c = y$ and $nx^2 + px + q = y$? | [
"0"
] | [
"What is the minimum distance between two parabolas given by the equations $ax^2 + bx + c = y$ and $nx^2 + px + q = y$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-04c217bff5bde59723d18a39f793d333229218d894b1dcd66377524846f741ca | What is the minimum distance from the point $(0,8)$ to the x-axis and then to the point $(7,3)$? | [
"$\\sqrt{170}$"
] | [
"What is the minimum distance from the point $(0,8)$ to the x-axis and then to the point $(7,3)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-48ba3d19ce50ecaa320c4d8a962f4eeda6ee475e1ce1fc26ff30d6116afa5b84 | What is the minimum element, if any, of the set \( B = \left\{ b \in \mathbb{N} \mid \frac{\left| \bigcap_{k=b}^{b+138} S_k \right|}{\left| \bigcup_{k=b}^{b+138} S_k \right|} \geq \frac{1}{2} \right\} \), where \( S_m = \{ n \in \mathbb{N} \mid 138m \leq n \leq 139m \} \)? | [
"19044"
] | [
"What is the minimum element, if any, of the set \\( B = \\left\\{ b \\in \\mathbb{N} \\mid \\frac{\\left| \\bigcap_{k=b}^{b+138} S_k \\right|}{\\left| \\bigcup_{k=b}^{b+138} S_k \\right|} \\geq \\frac{1}{2} \\right\\} \\), where \\( S_m = \\{ n \\in \\mathbb{N} \\mid 138m \\leq n \\leq 139m \\} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-29e42d7aa2c27f8e5fd2e1988de0dfab395bebb2468976174d2dfed2d9b99df4 | What is the minimum length \( n \) of a string of digits such that the string contains all possible permutations of 2 digits, 3 digits, and \( k \) digits? | [
"10003"
] | [
"What is the minimum length \\( n \\) of a string of digits such that the string contains all possible permutations of 2 digits, 3 digits, and \\( k \\) digits?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-8e31d5bb2523b8e64ab84e1c245434bd6b19f2ab95d6c29aa44badcab07b53f6 | What is the minimum number $k$ of weights with integer values needed for a mechanical balance to measure any object of integer mass between 1 and 2009 (inclusive)? | [
"8"
] | [
"What is the minimum number $k$ of weights with integer values needed for a mechanical balance to measure any object of integer mass between 1 and 2009 (inclusive)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-199a8563b8f8c3a9b62e5b893f4ceadd13d19dbe7558830d0549d27cc2e9a03b | What is the minimum number $n$ of lattice points $(x, y, z)$ in 3-dimensional space such that the midpoint between some two of these points is also a lattice point? | [
"9"
] | [
"What is the minimum number $n$ of lattice points $(x, y, z)$ in 3-dimensional space such that the midpoint between some two of these points is also a lattice point?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b46824ab582339e73c2d2c1e5b2d02fe181c79d0e71a08ea8202865f3d07271a | What is the minimum number $n$ such that if we choose any $n$ points inside a $3 \times 4$ rectangle, at least two of them are at a mutual distance of at most $\sqrt{5}$? | [
"6"
] | [
"What is the minimum number $n$ such that if we choose any $n$ points inside a $3 \\times 4$ rectangle, at least two of them are at a mutual distance of at most $\\sqrt{5}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-c9c44a888c83e8a0e749d6a21010d59bbb2d849b7b2379a077bbf0f2d73c4e32 | What is the minimum number $p_n$ of coins needed to buy an item priced at $n$ dollars, given that the available coins are 1, 5, and 10 dollars? | [
"\\( p_n = n - \\left\\lfloor\\frac{n}{10}\\right\\rfloor - 4\\left\\lfloor\\frac{n}{5}\\right\\rfloor \\)"
] | [
"What is the minimum number $p_n$ of coins needed to buy an item priced at $n$ dollars, given that the available coins are 1, 5, and 10 dollars?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 6
} | 19 | 0 | 0 |
omr-ws-13fa2c457ba78b7933cbad702b70980d4b8fadf1d9ab51377f3f910f68275af9 | What is the minimum number \( M \) such that after deleting \( M \) \( 1 \times 1 \) squares from an \( m \times n \) rectangle, it is no longer possible to place a disc of radius 1.1 in the remaining space? | [
"\\( M = \\begin{cases} \n\\frac{mn}{2} & \\text{if both } m \\text{ and } n \\text{ are even} \\\\\n\\frac{m(n-1)}{2} & \\text{if exactly one of } m \\text{ or } n \\text{ is even} \\\\\n\\frac{(m-1)n}{2} & \\text{if both } m \\text{ and } n \\text{ are odd}\n\\end{cases} \\)"
] | [
"What is the minimum number \\( M \\) such that after deleting \\( M \\) \\( 1 \\times 1 \\) squares from an \\( m \\times n \\) rectangle, it is no longer possible to place a disc of radius 1.1 in the remaining space?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-359dfaaed312d9b64a6017f0493548c7357bfbbcfb27689242f36b24f56fe954 | What is the minimum number \( N \) such that by choosing \( N \) cells on a \( 1000 \times 1000 \) chessboard covered with \( 1 \times 10 \) dominoes, we can determine the exact placement of all dominoes on the board? | [
"100000"
] | [
"What is the minimum number \\( N \\) such that by choosing \\( N \\) cells on a \\( 1000 \\times 1000 \\) chessboard covered with \\( 1 \\times 10 \\) dominoes, we can determine the exact placement of all dominoes on the board?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-b4b28b0fe20e64a488589ba6c8463f653464a3bc7860c800c761544988e002ab | What is the minimum number \( X \) such that, when the 3-sets of \( X \) objects are colored red or green, there must exist a 5-set such that each of the 4-sets of this 5-set contain an even number of 3-sets of each color? | [
"8"
] | [
"What is the minimum number \\( X \\) such that, when the 3-sets of \\( X \\) objects are colored red or green, there must exist a 5-set such that each of the 4-sets of this 5-set contain an even number of 3-sets of each color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-71d407b33efe49f05a8b1bb4f6d831d65ce3270f50a3496ac90ba09cc574596a | What is the minimum number \( X \) such that, when the 3-sets of a set of \( X \) objects are colored red or green, there must exist a 4-set such that it contains an even number of 3-sets of each color (i.e., 2 red and 2 green, or all red, or all green)? | [
"5"
] | [
"What is the minimum number \\( X \\) such that, when the 3-sets of a set of \\( X \\) objects are colored red or green, there must exist a 4-set such that it contains an even number of 3-sets of each color (i.e., 2 red and 2 green, or all red, or all green)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-ef5d981bde062d23474d0aae5bc2eb87047661650ed6e54cf901dbea117774b8 | What is the minimum number \( n \) such that we can place \( n \) points in four-dimensional space, connect all the points, and ensure that no lattice points lie on the edges of the four-dimensional figure formed, except at the vertices? | [
"5"
] | [
"What is the minimum number \\( n \\) such that we can place \\( n \\) points in four-dimensional space, connect all the points, and ensure that no lattice points lie on the edges of the four-dimensional figure formed, except at the vertices?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-02a1b38b3f341471f3c2aac69a86c6114a494481e80c6b9f33395ae82fa7980f | What is the minimum number of $1 \times k$ dominoes that can be placed on an $n \times n$ square such that no dominoes overlap and each domino is aligned with the rows and columns? | [
"\\dfrac{n^2}{k}"
] | [
"What is the minimum number of $1 \\times k$ dominoes that can be placed on an $n \\times n$ square such that no dominoes overlap and each domino is aligned with the rows and columns?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-cae3d39ca022a3bb766ee7dc519f3cddec64bd2d9792648b486248a92c7c31b9 | What is the minimum number of 3-digit numbers that need to be chosen such that there exist seven of them with the same sum of digits? | [
"147"
] | [
"What is the minimum number of 3-digit numbers that need to be chosen such that there exist seven of them with the same sum of digits?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 33,
"genselect": 8,
"tir": 16
} | 57 | 0 | 0 |
omr-ws-a060f2ae6ee052ba4a795d57077e79fb5814f0cf62b29c36c541557c1d940611 | What is the minimum number of acute-angle triangles into which an obtuse-angle triangle can be divided? | [
"7"
] | [
"What is the minimum number of acute-angle triangles into which an obtuse-angle triangle can be divided?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 4
} | 27 | 0 | 0 |
omr-ws-e8b66564a180414c833278bb455aa17fce03d2cdf5e60edae5f94a4af4f026eb | What is the minimum number of arithmetic progressions that can be formed from a subset of size 50 of the set $\{1, 2, 3, \ldots, 500\}$? | [
"0"
] | [
"What is the minimum number of arithmetic progressions that can be formed from a subset of size 50 of the set $\\{1, 2, 3, \\ldots, 500\\}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-e1db5ccc96ada9ac6e2e5dca9537ab1f2aa312a4625a48bb213c138b1af5c38a | What is the minimum number of balls lost for there to exist a valid bingo board for which it is impossible to win? | [
"5"
] | [
"What is the minimum number of balls lost for there to exist a valid bingo board for which it is impossible to win?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-886b4741fdd4566b79243ba43c244fff3526e98fc0058863b5eb6dd5290c1516 | What is the minimum number of balls that must be drawn from a box containing 28 red balls, 20 green balls, 19 yellow balls, 13 blue balls, 11 white balls, and 9 black balls, without replacement, to guarantee that at least 15 balls of a single color will be drawn? | [
"76"
] | [
"What is the minimum number of balls that must be drawn from a box containing 28 red balls, 20 green balls, 19 yellow balls, 13 blue balls, 11 white balls, and 9 black balls, without replacement, to guarantee that at least 15 balls of a single color will be drawn?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-bc8512f30c0959e2bd758b43b58d47b938fef7cb875a8ff317709cbf6955a3c4 | What is the minimum number of balls you need to draw from a set of 100 balls numbered from 1 to 100, without replacement, to have a probability greater than 50% of drawing at least two consecutive numbers? | [
"9"
] | [
"What is the minimum number of balls you need to draw from a set of 100 balls numbered from 1 to 100, without replacement, to have a probability greater than 50% of drawing at least two consecutive numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-3a1d0baafb92a895b3b51ae903461473d5ec4a660b4f3a626e9e59e02d579ee7 | What is the minimum number of blows needed to split a 3x3x3 Rubik's cube into 27 smaller cubes, given that pieces can be rearranged between blows? | [
"6"
] | [
"What is the minimum number of blows needed to split a 3x3x3 Rubik's cube into 27 smaller cubes, given that pieces can be rearranged between blows?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-bdb3cdf677f1d080f4b43b3c058970956cbf6d84b1a46fa893771d918998cd05 | What is the minimum number of cards that you should take from a bag containing 900 numbered cards (from 100 to 999) to ensure that at least 6 cards have digits that sum up to the same value? | [
"124"
] | [
"What is the minimum number of cards that you should take from a bag containing 900 numbered cards (from 100 to 999) to ensure that at least 6 cards have digits that sum up to the same value?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-f8a0f47242e6127aada5d79c234431570a7b0d2eba70c14a49ebe818cf3bff83 | What is the minimum number of cells that can be colored black in a $300 \times 300$ white square so that no three black cells form a corner, and after painting any white cell, this condition is violated? | [
"45000"
] | [
"What is the minimum number of cells that can be colored black in a $300 \\times 300$ white square so that no three black cells form a corner, and after painting any white cell, this condition is violated?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 3,
"tir": 5
} | 35 | 0 | 0 |
omr-ws-ff0ede036fa7c82541d411b4889d819f3b5db4a2f19a08c1699d3bdff555c4fd | What is the minimum number of checkers starting on an 8x8 checkerboard required in order for the checkerboard to end up filled after some amount of time, given that every second a checker is placed on each unoccupied space that is adjacently bordered by at least two occupied spaces? | [
"8"
] | [
"What is the minimum number of checkers starting on an 8x8 checkerboard required in order for the checkerboard to end up filled after some amount of time, given that every second a checker is placed on each unoccupied space that is adjacently bordered by at least two occupied spaces?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-d5ea1ef2c410c8dbc8ad74eae4171079b161602aed690960c2e689d30be4d84f | What is the minimum number of checkers that can be placed on the remaining 60 squares of an $8 \times 8$ board (with the middle $2 \times 2$ removed) so that every square is threatened by at least one checker? (A checker threatens the square it occupies and any square it can move to in one move without crossing the rem... | [
"12"
] | [
"What is the minimum number of checkers that can be placed on the remaining 60 squares of an $8 \\times 8$ board (with the middle $2 \\times 2$ removed) so that every square is threatened by at least one checker? (A checker threatens the square it occupies and any square it can move to in one move without crossing ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-5ac6e504771dc023476a1d0d7ddf72b750f3c86a763d3cb0935def0a8b1285ec | What is the minimum number of circles of radius $1$ needed to cover a circle whose radius is the golden number $\phi = \frac{1+\sqrt{5}}{2}$? | [
"5"
] | [
"What is the minimum number of circles of radius $1$ needed to cover a circle whose radius is the golden number $\\phi = \\frac{1+\\sqrt{5}}{2}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-1d352128d3b3f69932cb55976f2e5bae781aa676266539c73f7e47e228089b7f | What is the minimum number of classes of a partition from the set \(\{1, 2, \ldots, 2012\}\) such that for any three numbers in the same class, their greatest common divisor is 1? | [
"503"
] | [
"What is the minimum number of classes of a partition from the set \\(\\{1, 2, \\ldots, 2012\\}\\) such that for any three numbers in the same class, their greatest common divisor is 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 10
} | 58 | 0 | 0 |
omr-ws-fdafecdfa127857ceb4849846a012756e3d9d575ee172c15a067f81652f4d017 | What is the minimum number of colors \( k \) needed to color the subsets of \(\{1, 2, 3, \ldots, n\}\) such that no two different subsets \( A \) and \( B \) have \( A \), \( B \), and \( A \cup B \) all the same color? | [
"\\( n+1 \\)"
] | [
"What is the minimum number of colors \\( k \\) needed to color the subsets of \\(\\{1, 2, 3, \\ldots, n\\}\\) such that no two different subsets \\( A \\) and \\( B \\) have \\( A \\), \\( B \\), and \\( A \\cup B \\) all the same color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 17
} | 17 | 0 | 0 |
omr-ws-469d8ba3eba11a89c09522993e13ed040d5451ba8a2a1178ae31a91ae263cd0a | What is the minimum number of colors required for a legal coloring of the regions in a map formed by 2001 lines in the plane, where no two lines are parallel and no three lines are concurrent? | [
"2"
] | [
"What is the minimum number of colors required for a legal coloring of the regions in a map formed by 2001 lines in the plane, where no two lines are parallel and no three lines are concurrent?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-09ea8c3c3c8b1e95915d99826e4bae39e5694ce45f74dfe8a5d62cb5612804a5 | What is the minimum number of colors required for a legal coloring of the vertices in a map formed by 2001 lines in the plane, where no two lines are parallel and no three lines are concurrent? | [
"3"
] | [
"What is the minimum number of colors required for a legal coloring of the vertices in a map formed by 2001 lines in the plane, where no two lines are parallel and no three lines are concurrent?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 1
} | 23 | 0 | 0 |
omr-ws-7ba73a922883bda4fd448563f0dd55cd08b5f4e0a86d0a73efd40be41f85e9aa | What is the minimum number of colors required to color the integers $1, 2, \ldots, 2013$ such that for any two integers $a$ and $b$ of the same color, the product $ab$ is not divisible by $2014$? Provide your answer as an integer. | [
"3"
] | [
"What is the minimum number of colors required to color the integers $1, 2, \\ldots, 2013$ such that for any two integers $a$ and $b$ of the same color, the product $ab$ is not divisible by $2014$? Provide your answer as an integer."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-12fd3eb850cd99c0900f1e138647956314de8bd380b80b0f20270076b9a14ddb | What is the minimum number of colors required to color the set of natural numbers such that any two numbers that differ by a factorial get a different color? | [
"4"
] | [
"What is the minimum number of colors required to color the set of natural numbers such that any two numbers that differ by a factorial get a different color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 21,
"genselect": 1
} | 22 | 0 | 0 |
omr-ws-43f2b4c2b1e5be908198c33ceab90d34991b0f0ad382f1b8ab2c6dcc3580f858 | What is the minimum number of colours $k$ required to colour the dominoes on a $2020 \times 2020$ board such that, for every configuration of dominoes, it is possible to colour the dominoes in $k$ colours with the condition that each domino of colour $x$ has at most 2 adjacent dominoes of the same colour $x$? Dominoes ... | [
"3"
] | [
"What is the minimum number of colours $k$ required to colour the dominoes on a $2020 \\times 2020$ board such that, for every configuration of dominoes, it is possible to colour the dominoes in $k$ colours with the condition that each domino of colour $x$ has at most 2 adjacent dominoes of the same colour $x$? Dom... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-afcfd9aac42f83018249fe66e872d6c3904deb45ab2f83822375da46ff187f43 | What is the minimum number of colours required to colour all positive real numbers such that any two numbers whose ratio is 4 or 8 have different colours? | [
"3"
] | [
"What is the minimum number of colours required to colour all positive real numbers such that any two numbers whose ratio is 4 or 8 have different colours?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-91db497b8d14677b03d225c8c6974bc1140697a7dc6dd58487ef2709ac460076 | What is the minimum number of comparisons necessary between pairs of objects to arrange five objects in order of their size? | [
"6"
] | [
"What is the minimum number of comparisons necessary between pairs of objects to arrange five objects in order of their size?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dfeeeb79df6bbc5400115faaaed140ccb6c3a240960424b849643ed368a446b8 | What is the minimum number of comparisons needed to arrange the sequence \(a_1, a_2, a_3, \ldots, a_n\) in order of size? | [
"\\(n - 1\\)"
] | [
"What is the minimum number of comparisons needed to arrange the sequence \\(a_1, a_2, a_3, \\ldots, a_n\\) in order of size?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca68767cfd18727352adc57a4ebcb454f98ab08cfa874d634267962a58debac2 | What is the minimum number of coordinates needed to uniquely locate a regular hexagon? | [
"6"
] | [
"What is the minimum number of coordinates needed to uniquely locate a regular hexagon?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8d6aa259838b2adcc9c962fe2254c6d3505c80c5828e7543ae0ba9e18ce31b7a | What is the minimum number of custodians required to watch all the rooms in a gallery consisting of $n$ rooms ($n > 1$), where each custodian can watch all rooms that can be reached by one move of a chess rook (horizontally or vertically adjacent rooms)? | [
"$\\lfloor \\frac{n}{2} \\rfloor$"
] | [
"What is the minimum number of custodians required to watch all the rooms in a gallery consisting of $n$ rooms ($n > 1$), where each custodian can watch all rooms that can be reached by one move of a chess rook (horizontally or vertically adjacent rooms)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b38acb3d17b89a7964428931b3a3a27949e5714152fe711b3f9bc62c8ccb534 | What is the minimum number of custodians required to watch all the rooms in a gallery of $n$ rooms ($n > 2$), where each custodian can watch all rooms reachable by one move of a chess queen from their room? | [
"\\(\\left \\lfloor{ \\frac{n}{3}}\\right \\rfloor\\)"
] | [
"What is the minimum number of custodians required to watch all the rooms in a gallery of $n$ rooms ($n > 2$), where each custodian can watch all rooms reachable by one move of a chess queen from their room?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eb04bf9adae4506f858202ceaff60aece3b82c2c21d61d864867b0114c089e94 | What is the minimum number of cuts needed to cut a 4x8 grid of squares into their individual squares, given that Bob can fold and stack the paper? | [
"5"
] | [
"What is the minimum number of cuts needed to cut a 4x8 grid of squares into their individual squares, given that Bob can fold and stack the paper?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2c7dc5bc2fe803b532f0bb5739a0969ed695228c2168dec64e177d556d6b281e | What is the minimum number of cuts required to cut a $4 \times 4 \times 4$ cube into $64$ unit cubes ($1 \times 1 \times 1$), given that cuts must be straight and parallel to a face of the cube, and multiple pieces can be cut in a single cut if they are aligned in a straight line? | [
"6"
] | [
"What is the minimum number of cuts required to cut a $4 \\times 4 \\times 4$ cube into $64$ unit cubes ($1 \\times 1 \\times 1$), given that cuts must be straight and parallel to a face of the cube, and multiple pieces can be cut in a single cut if they are aligned in a straight line?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-2419104ecff33c9c5393fa12b7166b2a8e82513ff394731a7a22d1617d4a8ef3 | What is the minimum number of detectors $Q$ required to guarantee the determination of the location of the treasure in a $2 \times 4$ grid, where each detector can scan a specific subgrid $[a,b] \times [c,d]$ with $1 \le a \le b \le 2$ and $1 \le c \le d \le 4$? | [
"3"
] | [
"What is the minimum number of detectors $Q$ required to guarantee the determination of the location of the treasure in a $2 \\times 4$ grid, where each detector can scan a specific subgrid $[a,b] \\times [c,d]$ with $1 \\le a \\le b \\le 2$ and $1 \\le c \\le d \\le 4$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 21,
"genselect": 8
} | 29 | 0 | 0 |
omr-ws-9328b6bf8a03014878357c7e87206989652f7fa7d2e463144afb7068be8177f8 | What is the minimum number of different distances between \( n \) points in 2-dimensional space? | [
"\\Omega\\left( \\frac{n}{\\log n} \\right)"
] | [
"What is the minimum number of different distances between \\( n \\) points in 2-dimensional space?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-2d2b3cb8a91ed14ce8f3a224650855b51efb40bdb640198151a71addc59b5064 | What is the minimum number of different distances between \( n \) points in 3-dimensional space? | [
"O(n^{2/3})"
] | [
"What is the minimum number of different distances between \\( n \\) points in 3-dimensional space?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-b918e091ac1d695ec4b32126fcda7c8384d1d59d9f54d960bd45ca442250f1aa | What is the minimum number of digits to the right of the decimal point needed to express the fraction $\dfrac{123\,456\,789}{2^{26} \cdot 5^4}$ as a decimal? | [
"26"
] | [
"What is the minimum number of digits to the right of the decimal point needed to express the fraction $\\dfrac{123\\,456\\,789}{2^{26} \\cdot 5^4}$ as a decimal?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3c17eaa0fbc6d6f85dee4596c250f214163c804f671706fce31acff181411dab | What is the minimum number of dimensions a hypercube must be in such that it is impossible to color each of the 2d edges of the hypercube in two colors without forming a set of four coplanar points that form a square and its two diagonals, all being the same color? | [
"4"
] | [
"What is the minimum number of dimensions a hypercube must be in such that it is impossible to color each of the 2d edges of the hypercube in two colors without forming a set of four coplanar points that form a square and its two diagonals, all being the same color?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-5bc8483933c4d70bd6be62bfbcbf6678b636ffe623aa964239cc9a9ad7529c3c | What is the minimum number of discs that must be placed on a chessboard so that it is impossible to fit an L-shaped domino (2x1) on any empty fields? (The domino cannot be rotated.) | [
"20"
] | [
"What is the minimum number of discs that must be placed on a chessboard so that it is impossible to fit an L-shaped domino (2x1) on any empty fields? (The domino cannot be rotated.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b2cd5ac485d4450d74a826de4be27c7ddc47bbaf8b278c505be2fc0c9ac9242 | What is the minimum number of disk-shaped napkins of radius $1$ required to cover a $100 \times \sqrt{3}$ rectangular table completely? The napkins are allowed to overlap and protrude the table's edges. | [
"100"
] | [
"What is the minimum number of disk-shaped napkins of radius $1$ required to cover a $100 \\times \\sqrt{3}$ rectangular table completely? The napkins are allowed to overlap and protrude the table's edges."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-776d86340e5aa9532d86e16923e70e6f185f54f13f4f83705d5769434b2478e3 | What is the minimum number of distances between $n$ points that need to be specified so that a unique non-degenerate $n$-gon is determined? | [
"2n - 3"
] | [
"What is the minimum number of distances between $n$ points that need to be specified so that a unique non-degenerate $n$-gon is determined?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-0d8916c3624afb5248db99a6a5dad6847075e033eb87a874d450d9a245dae9c0 | What is the minimum number of distinct Set cards such that a set is guaranteed? | [
"21"
] | [
"What is the minimum number of distinct Set cards such that a set is guaranteed?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e2790b64fc76d5e4535acfe83e6c3e5f19fa06c4097902abe552bf9281778271 | What is the minimum number of distinct complex zeros that the polynomial $P(x) = ax^n + x^{2014} + 1$ can have, where $a$ is a real number and $n$ is a positive integer not equal to 2014? | [
"1007"
] | [
"What is the minimum number of distinct complex zeros that the polynomial $P(x) = ax^n + x^{2014} + 1$ can have, where $a$ is a real number and $n$ is a positive integer not equal to 2014?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 38
} | 38 | 0 | 0 |
omr-ws-ea3fd20f65ef43f8ee93d105720ae8eb93aadf493a86dcc7e6ffff0270fc4710 | What is the minimum number of dominoes required to cover a chessboard such that no additional domino can be added? | [
"22"
] | [
"What is the minimum number of dominoes required to cover a chessboard such that no additional domino can be added?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 17,
"genselect": 3,
"tir": 3
} | 23 | 0 | 0 |
omr-ws-fc3b484571697844c80cf1a25504ac7ca4176ef59eb17e642d039d8f09e405e9 | What is the minimum number of dominos (each a $1 \times 2$ or $2 \times 1$ rectangle) needed to be placed on an $8 \times 8$ chessboard such that no additional domino can be placed on the board? | [
"22"
] | [
"What is the minimum number of dominos (each a $1 \\times 2$ or $2 \\times 1$ rectangle) needed to be placed on an $8 \\times 8$ chessboard such that no additional domino can be placed on the board?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 3,
"tir": 1
} | 14 | 0 | 0 |
omr-ws-705cbb996dc99f34cc2db4861a6beb6278d23178bac2f394c82d0ae0ec7f6679 | What is the minimum number of edges in a bipartite graph with $a$ and $b$ vertices in each set such that the graph does not contain a cycle? | [
"\\(a + b - 1\\)"
] | [
"What is the minimum number of edges in a bipartite graph with $a$ and $b$ vertices in each set such that the graph does not contain a cycle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-38a6d081ba596342e88d650e4b6bab22528b9c206668f660b9e2f471166aef25 | What is the minimum number of edges of a regular polygon that is not constructible by straightedge and compass? | [
"7"
] | [
"What is the minimum number of edges of a regular polygon that is not constructible by straightedge and compass?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01b7156d8f4e922202d03c5484a23ccf3f7733a46c561297b831cea021529704 | What is the minimum number of edges required to guarantee a four-cycle in a graph with $n$ vertices? | [
"\\left\\lfloor \\frac{n^{3/2}}{2} \\right\\rfloor + 1"
] | [
"What is the minimum number of edges required to guarantee a four-cycle in a graph with $n$ vertices?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-28832db19fc526d7855b6b74816997bbe808e1abbc8e2a2d677577bf7a233316 | What is the minimum number of edges required to guarantee a three-cycle in a graph with $n$ vertices? | [
"\\( 2n - 3 \\)"
] | [
"What is the minimum number of edges required to guarantee a three-cycle in a graph with $n$ vertices?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a7702b1d4f336b8701e2a3b7d6612a703acaa49e2030e3530622d5a4e2f31af7 | What is the minimum number of elements in an integer array such that a subset of 5 elements whose sum is divisible by 5 can be found? | [
"9"
] | [
"What is the minimum number of elements in an integer array such that a subset of 5 elements whose sum is divisible by 5 can be found?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-407984a7f91b1c5dc0741fa1fcf21518d31f56ead28e28eeab9a4c4b98698c7c | What is the minimum number of elements that must be removed from the set $\{1, 2, 3, \ldots, 100\}$ so that no three remaining elements $x, y, z$ satisfy $xy = z$? | [
"10"
] | [
"What is the minimum number of elements that must be removed from the set $\\{1, 2, 3, \\ldots, 100\\}$ so that no three remaining elements $x, y, z$ satisfy $xy = z$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-7ab943eafdcd3303fe70481572d696bb5c51ca8ebf775cb21b5f53bf9648f811 | What is the minimum number of equilateral triangle-shaped foods (each with a side length of 5 units) needed to completely fill an equilateral triangle-shaped tray with a side length of 50 units, such that no additional food can be placed inside the tray without moving the already placed foods? | [
"25 to 31"
] | [
"What is the minimum number of equilateral triangle-shaped foods (each with a side length of 5 units) needed to completely fill an equilateral triangle-shaped tray with a side length of 50 units, such that no additional food can be placed inside the tray without moving the already placed foods?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cdc4f71ef33f9c50fa9e68de0806bf54a02100fd2f3895dd07482be3ab0823e9 | What is the minimum number of equilateral triangles, of side length 1 unit, needed to cover an equilateral triangle of side length 10 units? | [
"100"
] | [
"What is the minimum number of equilateral triangles, of side length 1 unit, needed to cover an equilateral triangle of side length 10 units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-05469e383d5a82ff8ef36aefdcabafad3c8920c7628018315d71ddc0cfdeeb86 | What is the minimum number of games needed to determine the winner in a double elimination tournament with 10 teams? | [
"18"
] | [
"What is the minimum number of games needed to determine the winner in a double elimination tournament with 10 teams?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-0076f7dd6a0e35447e7b921853db2846646a3831d0864c605b2c8ba5760f8f35 | What is the minimum number of incongruent squares needed to tile a non-square rectangle? | [
"9"
] | [
"What is the minimum number of incongruent squares needed to tile a non-square rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-451864663ee704e4eabd769bb3e81cd03470242e4743af1e2d07db9a2cc9453e | What is the minimum number of incongruent squares that tile a (non-square) rectangle? | [
"2"
] | [
"What is the minimum number of incongruent squares that tile a (non-square) rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-47209595a62412c59fb379e72441b2fe1f60fb40331cf8779142cf35085c3950 | What is the minimum number of jellybeans Grace must take out of her pocket to ensure that she has one of each color, given that she has 8 red, 4 green, and 4 blue jellybeans? | [
"13"
] | [
"What is the minimum number of jellybeans Grace must take out of her pocket to ensure that she has one of each color, given that she has 8 red, 4 green, and 4 blue jellybeans?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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