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-03b6afe9163558d9b8d13391a5de64ec80e6696af22dc5fc9c4b7cfdbff6cfe7 | A cruise ship has 100 compartments with capacities of 101, 102, ..., 200 people respectively. Suppose there are $n$ people on the ship. The captain wants to transfer all guests from one compartment $A$ to another compartment $B$ without exceeding the capacity of compartment $B$, in order to free up a compartment for a ... | [
"8824"
] | [
"A cruise ship has 100 compartments with capacities of 101, 102, ..., 200 people respectively. Suppose there are $n$ people on the ship. The captain wants to transfer all guests from one compartment $A$ to another compartment $B$ without exceeding the capacity of compartment $B$, in order to free up a compartment f... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-ef281c30924e5d270ce31bfafceb6c6d16a4e369e31b757f2a22efb1c69f7ce8 | A cruise ship has 100 compartments with capacities of 101, 102, ..., 200 people respectively. Suppose there are \( n \) people on the ship. The captain wants to give a VIP passenger a personal compartment by transferring all guests from one compartment \( A \) to another compartment \( B \) without exceeding the capaci... | [
"8824"
] | [
"A cruise ship has 100 compartments with capacities of 101, 102, ..., 200 people respectively. Suppose there are \\( n \\) people on the ship. The captain wants to give a VIP passenger a personal compartment by transferring all guests from one compartment \\( A \\) to another compartment \\( B \\) without exceeding... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-de484eaf8a08bdc383671bde5754636bc58d1a79d083f7d718ea18f5bd679585 | A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$. Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set $\{V, W, X, Y, Z\}$. Using this correspondence, the cryptogra... | [
"108"
] | [
"A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$. Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set $\\{V, W, X, Y, Z\\}$. Using this correspondence, the cr... | [
"c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-1fdf36147017b493971ea98d6b3b4a4a38aa8b2a73cb37379bf173f2b81af9e3 | A cryptographic method encodes positive integers by first expressing them in base 5 and then establishing a 1-to-1 correspondence between the digits and the elements of the set {$ V, W, X, Y, Z$}. Given that three consecutive integers in increasing order are coded as $VYZ$, $VYX$, and $VVW$, respectively, determine the... | [
"108"
] | [
"A cryptographic method encodes positive integers by first expressing them in base 5 and then establishing a 1-to-1 correspondence between the digits and the elements of the set {$ V, W, X, Y, Z$}. Given that three consecutive integers in increasing order are coded as $VYZ$, $VYX$, and $VVW$, respectively, determin... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 2
} | 48 | 0 | 0 |
omr-ws-1a9b400082e816ff33da7ae7ba8464c697235f9bef1117297e9eef4338ea85ee | A crystal is inscribed in a cube of unit volume such that the crystal's vertices lie on the midpoints of the cube's edges. Find the volume of the crystal. | [
"\\dfrac{5}{6}"
] | [
"A crystal is inscribed in a cube of unit volume such that the crystal's vertices lie on the midpoints of the cube's edges. Find the volume of the crystal."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8eb1d497a88a448d23698438cea0db239b698ed356a57bd66b160ede6b81149d | A cube $3\times3\times3$ is built from $26$ white and $1$ black cubes $1\times1\times1$ such that the black cube is in the center. A larger cube $3n\times 3n\times 3n$ is formed by $n^3$ such $3\times3\times3$-cubes. What is the smallest number of white cubes which should be colored in red in such a way that every whit... | [
"\\( n^2(n+1) \\)"
] | [
"A cube $3\\times3\\times3$ is built from $26$ white and $1$ black cubes $1\\times1\\times1$ such that the black cube is in the center. A larger cube $3n\\times 3n\\times 3n$ is formed by $n^3$ such $3\\times3\\times3$-cubes. What is the smallest number of white cubes which should be colored in red in such a way th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e94b28409499a3b76bebc9cd412392cf61d5354f1290df23bc619e20a091044 | A cube $ABCDA'B'C'D'$ has an edge length of 2. The point $K$ is the center of the edge $AB$. The plane containing the points $B', D', K$ intersects the edge $AD$ at point $L$. Calculate the volume of the pyramid with apex $A$ and base the quadrilateral $D'B'KL$. | [
"1"
] | [
"A cube $ABCDA'B'C'D'$ has an edge length of 2. The point $K$ is the center of the edge $AB$. The plane containing the points $B', D', K$ intersects the edge $AD$ at point $L$. Calculate the volume of the pyramid with apex $A$ and base the quadrilateral $D'B'KL$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0b1ac1128e9d4edea80263b8bf083faa6598d7dac8d22e761dc6c1ac36071327 | A cube $ABCDEFGH$ with an edge length of $6$ cm is divided by plane cuts through the points $A,F,H$; $B,E,G$; $C,F,H$; $D,E,G$; $E,B,D$; $F,A,C$; $G,B,D$; and $H,A,C$. Calculate the volume of each of these partial bodies. | [
"18"
] | [
"A cube $ABCDEFGH$ with an edge length of $6$ cm is divided by plane cuts through the points $A,F,H$; $B,E,G$; $C,F,H$; $D,E,G$; $E,B,D$; $F,A,C$; $G,B,D$; and $H,A,C$. Calculate the volume of each of these partial bodies."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-7ed68e58b4cbe559b457ebc326c98146e11a564d53e0fd6b1bf6e7ea7b77a156 | A cube $ABCDEFGH$ with an edge length of $6$ cm is divided by plane cuts through the points $A,F,H$; $B,E,G$; $C,F,H$; $D,E,G$; $E,B,D$; $F,A,C$; $G,B,D$; and $H,A,C$. Determine the number of these partial bodies. | [
"13"
] | [
"A cube $ABCDEFGH$ with an edge length of $6$ cm is divided by plane cuts through the points $A,F,H$; $B,E,G$; $C,F,H$; $D,E,G$; $E,B,D$; $F,A,C$; $G,B,D$; and $H,A,C$. Determine the number of these partial bodies."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-6a03e5bf9a45dbbfe5418b8e52d03e71cc22d2fb3879d65455352465ffdbd392 | A cube $ABCDEFGH$ with edge length $1$ is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$, respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\frac{p}{q}$, where $p$ and $q$ are r... | [
"089",
"89"
] | [
"A cube $ABCDEFGH$ with edge length $1$ is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\\overline{AB}$ and $\\overline{CG}$, respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\\frac{p}{q}$, where $p$ and $... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 60,
"genselect": 6,
"tir": 16
} | 82 | 0 | 0 |
omr-ws-b6c9f6e9d5332ce64fc87871223c88d789fd0b72e951f5cddb2ee1ad06803e85 | A cube $ABCDEFGH$ with mirrored interior has a laser beam entering at vertex $A$ in a direction not parallel to any of the cube's sides. The beam is reflected off the sides, absorbed at the edges, and leaves the cube at a vertex if it hits one. Determine the set of vertices where the laser beam can leave the cube after... | [
"For even $n$, the set is $\\{G\\}$. For odd $n$, the set is $\\{B, C, D, E, F, H\\}$."
] | [
"A cube $ABCDEFGH$ with mirrored interior has a laser beam entering at vertex $A$ in a direction not parallel to any of the cube's sides. The beam is reflected off the sides, absorbed at the edges, and leaves the cube at a vertex if it hits one. Determine the set of vertices where the laser beam can leave the cube ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-ebc81affcf3de5e2d13625f8cb2df3792743eab69f967bfd97ed3f19d9a73f43 | A cube $Q$ of edge length 2011 is divided into $2011^3$ unit cubes. Some of these unit cubes must be chosen such that each face of each unit cube is contained in the surface of one of the chosen cubes. What is the smallest number of cubes that should be chosen? | [
"24240802"
] | [
"A cube $Q$ of edge length 2011 is divided into $2011^3$ unit cubes. Some of these unit cubes must be chosen such that each face of each unit cube is contained in the surface of one of the chosen cubes. What is the smallest number of cubes that should be chosen?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-ce9d2ccc3f24ac8aa873c1c5229776140e1140241e9253944dd63e09e6314e0b | A cube $T$ with edge length $2n$ consists of $(2n)^3$ unit cubes. If one unit cube is removed from $T$, the remaining solid can be constructed using "pieces," where a piece is defined as a cube $S$ with edge length 2, consisting of 8 unit cubes, with one unit cube removed. Calculate the minimum number of pieces require... | [
"31"
] | [
"A cube $T$ with edge length $2n$ consists of $(2n)^3$ unit cubes. If one unit cube is removed from $T$, the remaining solid can be constructed using \"pieces,\" where a piece is defined as a cube $S$ with edge length 2, consisting of 8 unit cubes, with one unit cube removed. Calculate the minimum number of pieces ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2df8731d389a4af2dbee78d6a419ab8af19bf26626e2ae5dd6391363394ee8dc | A cube $W$ with edge length 2 is composed of 8 smaller cubes $W_1, \ldots, W_8$ each with edge length 1. Each of the six faces of a small cube is either colored white or black. Given that there are 24 white faces and 24 black faces among the 48 faces of the smaller cubes, find the number of distinct ways to arrange the... | [
"30"
] | [
"A cube $W$ with edge length 2 is composed of 8 smaller cubes $W_1, \\ldots, W_8$ each with edge length 1. Each of the six faces of a small cube is either colored white or black. Given that there are 24 white faces and 24 black faces among the 48 faces of the smaller cubes, find the number of distinct ways to arran... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de09f48139f2bdb507e839a246e36f4c58f591c7ad7731757cb032e11a888d7c | A cube $n \times n \times n$ is composed of unit cubes. Consider a polygonal line that consists of segments connecting the centers of two adjacent cubes (i.e., sharing a face). A face of a unit cube is marked if the polygonal line passes through it. Paint the edges of the unit cubes in two colors, red and blue, such th... | [
"2^{27}"
] | [
"A cube $n \\times n \\times n$ is composed of unit cubes. Consider a polygonal line that consists of segments connecting the centers of two adjacent cubes (i.e., sharing a face). A face of a unit cube is marked if the polygonal line passes through it. Paint the edges of the unit cubes in two colors, red and blue, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-b982dcfaf9ce576a492ede1dd16afffac9f5df7f8a409477a09389759fcfd3a0 | A cube 10x10x10 is constructed from 1000 white unit cubes. Velly chooses a certain amount of parallelepipeds 1x1x10, no two of which have a common vertex or an edge, and repaints them in black. Polly can choose an arbitrary number of unit cubes and ask Velly for their color. What is the least amount of unit cubes Polly... | [
"10"
] | [
"A cube 10x10x10 is constructed from 1000 white unit cubes. Velly chooses a certain amount of parallelepipeds 1x1x10, no two of which have a common vertex or an edge, and repaints them in black. Polly can choose an arbitrary number of unit cubes and ask Velly for their color. What is the least amount of unit cubes ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-820929c545e2cd0d6f79e9879c9edc19a3b9cc319f74c18c401bf3d7fba2f03a | A cube 4 units on each side is composed of 64 unit cubes. Two faces of the larger cube that share an edge are painted blue, and the cube is disassembled into 64 unit cubes. Two of the unit cubes are selected uniformly at random. What is the probability that one of two selected unit cubes will have exactly two painted f... | [
"\\frac{1}{14}"
] | [
"A cube 4 units on each side is composed of 64 unit cubes. Two faces of the larger cube that share an edge are painted blue, and the cube is disassembled into 64 unit cubes. Two of the unit cubes are selected uniformly at random. What is the probability that one of two selected unit cubes will have exactly two pain... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-df89c9330972271b96b45e6592e9eb9d32f17190da6eaa17496701d7e5cdd621 | A cube XYZTABCD is cut into four pieces by cutting along two planes, BCTX and BDTY. What fraction of the volume of the cube is occupied by the piece containing corner A? | [
"\\(\\frac{1}{3}\\)"
] | [
"A cube XYZTABCD is cut into four pieces by cutting along two planes, BCTX and BDTY. What fraction of the volume of the cube is occupied by the piece containing corner A?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 41,
"genselect": 3,
"tir": 27
} | 71 | 0 | 0 |
omr-ws-44fa91df882a64a63ee0c6a5f6786e70435e81b88f420719de0d28875afc08c4 | A cube \( 2005 \times 2005 \times 2005 \) consists of unit cubes. Some of the unit cubes are colored in black such that for each black cube, there is at least one row (horizontal, vertical, or perpendicular) that contains this cube and does not contain other black cubes. What is the maximal number of black cubes? | [
"2005^2"
] | [
"A cube \\( 2005 \\times 2005 \\times 2005 \\) consists of unit cubes. Some of the unit cubes are colored in black such that for each black cube, there is at least one row (horizontal, vertical, or perpendicular) that contains this cube and does not contain other black cubes. What is the maximal number of black cub... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-444580f909bcc63e855d5d64eb5f3b4083c65ab47ab44d2ecb7afcc25f2032bb | A cube \(ABCD.EFGH\) is cut into four parts by two planes. The first plane is parallel to side \(ABCD\) and passes through the midpoint of edge \(BF\). The second plane passes through the midpoints of edges \(AB\), \(AD\), \(GH\), and \(FG\). Determine the ratio of the volumes of the smallest part to the largest part. | [
"\\dfrac{7}{17}"
] | [
"A cube \\(ABCD.EFGH\\) is cut into four parts by two planes. The first plane is parallel to side \\(ABCD\\) and passes through the midpoint of edge \\(BF\\). The second plane passes through the midpoints of edges \\(AB\\), \\(AD\\), \\(GH\\), and \\(FG\\). Determine the ratio of the volumes of the smallest part to... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-55482e2604afd8f0084726cf537a615d19373cd38b7e374c97e81bf500003a42 | A cube and a sphere have the same surface area, and their centers are on the same point in the 3D plane. Let \( x \) denote the total volume of the space inside the sphere but outside the cube, and \( r \) be the radius of the sphere. Find the value of \( x \) in terms of \( r \). | [
"\\( 2\\pi r^3 \\left(1-\\sqrt{\\frac{6}{\\pi}}\\right) \\left(2-\\sqrt{\\frac{6}{\\pi}}-\\frac{6}{\\pi}\\right) \\)"
] | [
"A cube and a sphere have the same surface area, and their centers are on the same point in the 3D plane. Let \\( x \\) denote the total volume of the space inside the sphere but outside the cube, and \\( r \\) be the radius of the sphere. Find the value of \\( x \\) in terms of \\( r \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c548823f3f3782272d880ccfae89f715d015c4b348eef40565c7966ee3c6573e | A cube and a sphere have the same surface area. What is the ratio of the volume of the sphere to the volume of the cube? | [
"\\(\\frac{\\sqrt{6\\pi}}{\\pi}\\)"
] | [
"A cube and a sphere have the same surface area. What is the ratio of the volume of the sphere to the volume of the cube?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-55ef5b214390b66d78171beb02710f30a96339f831a8fd8cfe92d9bdd44b5b30 | A cube consisting of $(2N)^3$ unit cubes is pierced by several needles parallel to the edges of the cube (each needle pierces exactly $2N$ unit cubes). Each unit cube is pierced by at least one needle. Let us call any subset of these needles “regular” if there are no two needles in this subset that pierce the same unit... | [
"36"
] | [
"A cube consisting of $(2N)^3$ unit cubes is pierced by several needles parallel to the edges of the cube (each needle pierces exactly $2N$ unit cubes). Each unit cube is pierced by at least one needle. Let us call any subset of these needles “regular” if there are no two needles in this subset that pierce the same... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c4afd2fd1f914c6723a45f61c3b8789c63229472cdb0829e322d43dae534194b | A cube consisting of $(2N)^3$ unit cubes is pierced by several needles parallel to the edges of the cube (each needle pierces exactly $2N$ unit cubes). Each unit cube is pierced by at least one needle. Let us call any subset of these needles “regular” if there are no two needles in this subset that pierce the same unit... | [
"\\(2N^2\\)"
] | [
"A cube consisting of $(2N)^3$ unit cubes is pierced by several needles parallel to the edges of the cube (each needle pierces exactly $2N$ unit cubes). Each unit cube is pierced by at least one needle. Let us call any subset of these needles “regular” if there are no two needles in this subset that pierce the same... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-acedf94ff27ef850a0353b8e71f57df9e6319679c00a52b50954bcce804c865c | A cube frame is assembled from sticks $1$ m long. On the frame sit $9$ spiders, while the distance between any two (measured the shortest path along the edges of the cube) is not less than $R$. At what maximum $R$ is that possible? | [
"\\dfrac{2}{3}"
] | [
"A cube frame is assembled from sticks $1$ m long. On the frame sit $9$ spiders, while the distance between any two (measured the shortest path along the edges of the cube) is not less than $R$. At what maximum $R$ is that possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-8d29302c30798fa69f469e5fc769d19d512a8fcb30043f519c88845f62c140b6 | A cube has a surface area of 150 $\text{ft}^2$. Let $V$ be the volume, in $\text{ft}^3$, of the smallest sphere that can contain the cube. $V$ can be written in the form $\frac{a\pi\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers with $\gcd(a, c) = 1$ and $b$ indivisible by the square of a prime. Find $a + ... | [
"130"
] | [
"A cube has a surface area of 150 $\\text{ft}^2$. Let $V$ be the volume, in $\\text{ft}^3$, of the smallest sphere that can contain the cube. $V$ can be written in the form $\\frac{a\\pi\\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers with $\\gcd(a, c) = 1$ and $b$ indivisible by the square of a prime.... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-1b2b0fd766f683482053e7183618d5c019c7eda7035b9b5c669fb7421ef3caca | A cube has a surface area of 216 square centimeters. What is the volume of the cube, in cubic centimeters? | [
"216"
] | [
"A cube has a surface area of 216 square centimeters. What is the volume of the cube, in cubic centimeters?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3b237e7237da0fb4bb80fdf0caf496e2b8a54b04eab503c7840c07ca73521fb3 | A cube has a surface area of 96 square inches. If the surface area of the cube is increased by 300%, what is the new volume in cubic inches? | [
"512"
] | [
"A cube has a surface area of 96 square inches. If the surface area of the cube is increased by 300%, what is the new volume in cubic inches?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-2abbf92011fabef3475512ce4ae2bc78733f4359f96efccf718b6ec0df565806 | A cube has an edge length of 2. A smaller cube with an edge length of 1 is glued on top of the larger cube such that one of its faces rests entirely on the top face of the larger cube. What is the percent increase in the surface area from the original cube to the new solid formed? | [
"17"
] | [
"A cube has an edge length of 2. A smaller cube with an edge length of 1 is glued on top of the larger cube such that one of its faces rests entirely on the top face of the larger cube. What is the percent increase in the surface area from the original cube to the new solid formed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 4,
"tir": 18
} | 37 | 0 | 0 |
omr-ws-1fccce9c60a436f45625e4cefe338ab147f03dde6777d753b9d73f0535c0eb8d | A cube has an internal point \( P \) such that the perpendicular distances from \( P \) to the six faces of the cube are 1 cm, 2 cm, 3 cm, 4 cm, 5 cm, and 6 cm. How many other internal points of the cube have this property? | [
"47"
] | [
"A cube has an internal point \\( P \\) such that the perpendicular distances from \\( P \\) to the six faces of the cube are 1 cm, 2 cm, 3 cm, 4 cm, 5 cm, and 6 cm. How many other internal points of the cube have this property?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 1
} | 24 | 0 | 0 |
omr-ws-da9d92ec569aaefacc54452edc86d058795e290f02be247ca6a8f643051f4e7a | A cube has edges of length 1 cm and has a dot marked in the centre of the top face. The cube is sitting on a flat table. The cube is rolled, without lifting or slipping, in one direction so that at least two of its vertices are always touching the table. The cube is rolled until the dot is again on the top face. The le... | [
"\\dfrac{1+\\sqrt{5}}{2}"
] | [
"A cube has edges of length 1 cm and has a dot marked in the centre of the top face. The cube is sitting on a flat table. The cube is rolled, without lifting or slipping, in one direction so that at least two of its vertices are always touching the table. The cube is rolled until the dot is again on the top face... | [
"MATH_training_set"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6f6ab5f3900bc4698ef7a323ff06ba1acd1ddacb157ce325d283a468539d584c | A cube has its faces painted in 6 different colors, with each face divided into 4 equal squares. The numbers from 1 to 24 are arranged in these squares such that the sum of every 3 numbers whose squares share a vertex of the cube is a multiple of 3, and the sum of every two numbers whose squares share a side on the sam... | [
"6 \\times (8!)^3"
] | [
"A cube has its faces painted in 6 different colors, with each face divided into 4 equal squares. The numbers from 1 to 24 are arranged in these squares such that the sum of every 3 numbers whose squares share a vertex of the cube is a multiple of 3, and the sum of every two numbers whose squares share a side on th... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-95af9e23941d939e533048d2f3b391c445668bbf98e9f3028494783c2645b172 | A cube has its vertices on the Riemann sphere \( S \) and its edges parallel to the coordinate axes. Write the coordinates of the vertices and find the stereographic projections of the vertices. | [
" \\left( \\pm \\frac{\\sqrt{3} - 1}{2}, \\pm \\frac{\\sqrt{3} - 1}{2} \\right) "
] | [
"A cube has its vertices on the Riemann sphere \\( S \\) and its edges parallel to the coordinate axes. Write the coordinates of the vertices and find the stereographic projections of the vertices."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-0bc3ab46a769393a16089c202b86524b9ca0c2af31b7a23bcb8bf3cfbb4341d5 | A cube has opposite vertices at the origin \((0, 0, 0)\) and \((2, 2, 2)\). State the equations of the four space diagonals. Are the diagonals perpendicular? | [
"The equations of the four space diagonals are \\( AG: \\vec{r} = (2t, 2t, 2t) \\), \\( BH: \\vec{r} = (2-2t, 2t, 2t) \\), \\( CE: \\vec{r} = (2-2t, 2-2t, 2t) \\), \\( DF: \\vec{r} = (2t, 2-2t, 2t) \\). The diagonals are not perpendicular."
] | [
"A cube has opposite vertices at the origin \\((0, 0, 0)\\) and \\((2, 2, 2)\\). State the equations of the four space diagonals. Are the diagonals perpendicular?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 29
} | 39 | 0 | 0 |
omr-ws-080a7fc883d9ba7dfc8829836ce1d433b2ff792865b27adbdf8aaa46f2a060f9 | A cube is assembled with 27 white smaller cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the reassembled cube is completely black on the outside? | [
"\\(\\frac{8! \\cdot 12! \\cdot 6! \\cdot 1!}{27!} \\cdot \\frac{1}{8^8 \\cdot 12^{12} \\cdot 6^6}\\) or approximately \\(1.83 \\times 10^{-37}\\)."
] | [
"A cube is assembled with 27 white smaller cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the reassembled cube is completely black on the outside?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 4
} | 13 | 0 | 0 |
omr-ws-a3112d60ee27e02f10b76f94ec34b89dd4078062d4d50a8d0ecf83ab9e116002 | A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) | [
"7"
] | [
"A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \\times 2 \\times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 30
} | 30 | 0 | 0 |
omr-ws-4451b336df70aa1d5630d020f2cdee2ede24d0b312025b26e49e4b8cfd6e6227 | A cube is cut by a plane such that the cross section is a pentagon. Let $\ell$ be the length of the longest side of the pentagon. Calculate the minimum possible value of $|\ell-1|$. | [
"\\dfrac{\\sqrt{6} - 2}{2}"
] | [
"A cube is cut by a plane such that the cross section is a pentagon. Let $\\ell$ be the length of the longest side of the pentagon. Calculate the minimum possible value of $|\\ell-1|$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-8f8121bc7299a6993c1bc9fd4a8cff9e081dab10d0c2fab02eb827b4effde674 | A cube is cut from side to side by a plane that passes through the midpoints of three edges that are on different faces. Determine the plane figure that appears in the cut (the section) and calculate its area. | [
"$\\frac{s^2\\sqrt{3}}{8}$",
"\\dfrac{\\sqrt{3}}{8}a^2"
] | [
"A cube is cut from side to side by a plane that passes through the midpoints of three edges that are on different faces. Determine the plane figure that appears in the cut (the section) and calculate its area."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-56d284cc26d8782733320fd092d22fdaa802b6eed56f8a69bf567d3a8fca2658 | A cube is cut into 36 smaller cubes such that 32 of them are of the same size and 4 of them are bigger in size. How have the cuts been made? | [
"The cube was cut into 8 smaller cubes, then 4 of them were each cut into 8 smaller cubes."
] | [
"A cube is cut into 36 smaller cubes such that 32 of them are of the same size and 4 of them are bigger in size. How have the cuts been made?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9eeac7b2bb23e8f74820070a64bb0f1c0af865a748579ef53a23d71cff886a37 | A cube is cut into two parts by a cutting plane such that one of the parts is a tetrahedron with maximum volume. Find the ratio of the volume of the two parts. | [
"5:1"
] | [
"A cube is cut into two parts by a cutting plane such that one of the parts is a tetrahedron with maximum volume. Find the ratio of the volume of the two parts."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-5f144cc84e764b09d98e82855d90072351b7e4bbf60eb28816d6aa56b3136a5a | A cube is decomposed into a finite number of rectangular parallelepipeds. Let $V_c$ be the volume of the cube's circumsphere and $V_{p_i}$ be the volume of the circumsphere of the $i$-th parallelepiped. If the sum of the volumes of all the circumspheres of the parallelepipeds equals $V_c$, find the ratio of the side le... | [
"2"
] | [
"A cube is decomposed into a finite number of rectangular parallelepipeds. Let $V_c$ be the volume of the cube's circumsphere and $V_{p_i}$ be the volume of the circumsphere of the $i$-th parallelepiped. If the sum of the volumes of all the circumspheres of the parallelepipeds equals $V_c$, find the ratio of the si... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-380870fde38350b7a60cc8adf567a61c9a9d10d07475afd4bab782d5772abf3d | A cube is divided into $8$ smaller cubes of the same size. Each of these small cubes is then divided again into $8$ smaller cubes of the same size, and this process is repeated $4$ more times. What is the ratio between the sum of the total surface areas of all the small cubes resulting from the last division and the to... | [
"32"
] | [
"A cube is divided into $8$ smaller cubes of the same size. Each of these small cubes is then divided again into $8$ smaller cubes of the same size, and this process is repeated $4$ more times. What is the ratio between the sum of the total surface areas of all the small cubes resulting from the last division and t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-b8359d2e06e3b65e881c7e9bfb744cd4cddffc114918fc9bed68dea644de4bd3 | A cube is divided into 27 equal smaller cubes. What is the maximum possible number of smaller cubes that a plane can intersect? | [
"19"
] | [
"A cube is divided into 27 equal smaller cubes. What is the maximum possible number of smaller cubes that a plane can intersect?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 13
} | 45 | 0 | 0 |
omr-ws-c866742e085abedb90e20d4ddd81e0f7a73c94bda8f520910f1c65a9fb618a33 | A cube is divided into 27 unit cubes. A sphere is inscribed in each of the corner unit cubes, and another sphere is placed tangent to these 8 spheres. What is the smallest possible value for the radius of the last sphere? | [
"\\(\\sqrt{3} - \\frac{1}{2}\\)"
] | [
"A cube is divided into 27 unit cubes. A sphere is inscribed in each of the corner unit cubes, and another sphere is placed tangent to these 8 spheres. What is the smallest possible value for the radius of the last sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-46c4a6435c64dd438897576c6312b0660fa3f4f2fc51ed67078d3873cf81beea | A cube is divided into a finite number of smaller cubes. If the side lengths of these smaller cubes are represented as distinct positive integers, calculate the minimum possible number of smaller cubes that must be present in such a division. | [
"54"
] | [
"A cube is divided into a finite number of smaller cubes. If the side lengths of these smaller cubes are represented as distinct positive integers, calculate the minimum possible number of smaller cubes that must be present in such a division."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-7e0082ed1400668b2fee062359ad248015986224cb6bea5754e623e4a9747cc4 | A cube is formed by soldering twelve 3-inch lengths of wire at the vertices. If a fly starts at one vertex and travels along the edges without retracing any distance, what is the total distance it would travel before coming to any vertex a second time? | [
"24"
] | [
"A cube is formed by soldering twelve 3-inch lengths of wire at the vertices. If a fly starts at one vertex and travels along the edges without retracing any distance, what is the total distance it would travel before coming to any vertex a second time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-df632109584c2b1d9b5298b012245538dc6f6fbd78b8e4a3088da73187cc504e | A cube is inscribed in a regular quadrangular pyramid so that its four vertices are found on the lateral edges, and the remaining four, in the plane of its base. Determine the edge of the cube, if the altitude of the pyramid is equal to $H$, and the lateral edge, to $l$. | [
"$x=\\frac{H\\sqrt {2(l^2-H^2)}}{H+\\sqrt {2(l^2-H^2)}}$"
] | [
"A cube is inscribed in a regular quadrangular pyramid so that its four vertices are found on the lateral edges, and the remaining four, in the plane of its base. Determine the edge of the cube, if the altitude of the pyramid is equal to $H$, and the lateral edge, to $l$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2d37640646d9b89a1bfeef1f4e64f1bcbc53614528540f39e8566aa7c84d8ae8 | A cube is inscribed in a regular quadrangular pyramid so that its vertices lie on the slant heights of the pyramid. Find the ratio of the volume of the pyramid to the volume of the cube, if the angle between the altitude of the pyramid and its lateral face is equal to $\alpha $. | [
"$\\frac{V}{V_1}=\\frac{\\left(\\sqrt 2+2\\tan\\alpha \\right)^3\\cot\\alpha }{6}$"
] | [
"A cube is inscribed in a regular quadrangular pyramid so that its vertices lie on the slant heights of the pyramid. Find the ratio of the volume of the pyramid to the volume of the cube, if the angle between the altitude of the pyramid and its lateral face is equal to $\\alpha $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2bf9d310c29f26ff24c1ff2bb830a9f0f6ba90cf34896e6243e35bb0f56651a8 | A cube is inscribed in a right circular cone such that one face of the cube lies on the base of the cone. If the ratio of the height of the cone to the radius of the cone is $2 : 1$, what fraction of the cone's volume does the cube take up? Express your answer in simplest radical form. | [
"\\(\\frac{60\\sqrt{2} - 84}{\\pi}\\)"
] | [
"A cube is inscribed in a right circular cone such that one face of the cube lies on the base of the cone. If the ratio of the height of the cone to the radius of the cone is $2 : 1$, what fraction of the cone's volume does the cube take up? Express your answer in simplest radical form."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-f1bc740ae401a00d2038517f5ff11ca481162779934d776a82a507f3e788dd1d | A cube is inscribed into a hemisphere of radius $R$ so that its four vertices lie on the base of the hemisphere and the other four belong to the spherical surface of the half-ball. Calculate the volume of the cube. | [
"$\\frac{2}{3}R^3\\sqrt {\\frac{2}{3}}$"
] | [
"A cube is inscribed into a hemisphere of radius $R$ so that its four vertices lie on the base of the hemisphere and the other four belong to the spherical surface of the half-ball. Calculate the volume of the cube."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0f7bde765b1173fd76c05f00e462b4db0895e2f4c17438003ea801e9f5f19cfe | A cube is intersect ed by a plane passing through its diagonal. How must this plane be drawn to obtain the section of the least area? | [
"\\(\\frac{a^2\\sqrt{6}}{2}\\)"
] | [
"A cube is intersect ed by a plane passing through its diagonal. How must this plane be drawn to obtain the section of the least area?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5714b2ffc63bd6faa0af7d37449a37e5473e5da7478b6c642e8d46989f193f19 | A cube is made by soldering twelve 3-inch lengths of wire at the vertices. If a fly alights at one of the vertices and then walks along the edges, what is the greatest distance it could travel before coming to any vertex a second time, without retracing any distance? | [
"24"
] | [
"A cube is made by soldering twelve 3-inch lengths of wire at the vertices. If a fly alights at one of the vertices and then walks along the edges, what is the greatest distance it could travel before coming to any vertex a second time, without retracing any distance?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-4b9f43b149e6d0e7dcfd9610867978323670922e4a7c61ca6a0823d87b23ccab | A cube is made up of unit cubes, with 36 unit cubes on each face. If each face of the cube is painted, what fraction of the smaller unit cubes are painted? | [
"\\(\\frac{19}{27}\\)"
] | [
"A cube is made up of unit cubes, with 36 unit cubes on each face. If each face of the cube is painted, what fraction of the smaller unit cubes are painted?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-72032b892b01c4e0b006825cf2504ce76bb582a9c20787ea231ddf2325bb141c | A cube is painted on all 6 sides and is then cut into 8 smaller cubes. Each of these smaller cubes is rotated at random and then they are put back together. What is the probability that both at least half of the cube's surface is painted and at least two faces are completely painted? | [
"\\(\\frac{15}{256}\\)"
] | [
"A cube is painted on all 6 sides and is then cut into 8 smaller cubes. Each of these smaller cubes is rotated at random and then they are put back together. What is the probability that both at least half of the cube's surface is painted and at least two faces are completely painted?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e8dcbb998b0c43d708e35aa60a388ae0439bed7d3532c15e49c2dcf02ad2d4d | A cube is painted red on all faces, cut into 27 smaller cubes, and reassembled into another cube. What is the probability that the reassembled cube will still have all-red faces? | [
"\\(\\frac{1}{8!12!6! \\cdot 8^8 \\cdot 6^6 \\cdot 12^{12}}\\)"
] | [
"A cube is painted red on all faces, cut into 27 smaller cubes, and reassembled into another cube. What is the probability that the reassembled cube will still have all-red faces?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-afbe6ca5f877466b492c01f98facf519245313554dca541d21915f586adb3ef4 | A cube is painted so that one side is blue, two sides are red, and three sides are green. How many different such cubes can be painted? Two cubes are considered the same if one cube can be rotated in any way to match the second cube. | [
"3"
] | [
"A cube is painted so that one side is blue, two sides are red, and three sides are green. How many different such cubes can be painted? Two cubes are considered the same if one cube can be rotated in any way to match the second cube."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-226fb95e9c639298f0298b32a2cf996e011ca858d5acd13043b1cbbfe1c54044 | A cube is placed above ground where 5 of its sides (faces) are colored white and the other one side (face) is colored black. In the beginning, the black-colored face is one of the bases of the cube. Then, the cube is rotated on one of its edges which are on the ground such that the bases change, and this process is rep... | [
"\\dfrac{43}{128}"
] | [
"A cube is placed above ground where 5 of its sides (faces) are colored white and the other one side (face) is colored black. In the beginning, the black-colored face is one of the bases of the cube. Then, the cube is rotated on one of its edges which are on the ground such that the bases change, and this process i... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-0b8020254353273c59313534bbad03b8d595e8042915744b5fb227d0b0ddad7e | A cube is rotated $90$ degrees around the horizontal axis, then $90$ degrees around the vertical axis, then again $90$ degrees around the horizontal axis, then vertical, and so on. Will the cube return to its original position, and if so, after how many rotations? | [
"12"
] | [
"A cube is rotated $90$ degrees around the horizontal axis, then $90$ degrees around the vertical axis, then again $90$ degrees around the horizontal axis, then vertical, and so on. Will the cube return to its original position, and if so, after how many rotations?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8eb9d3f129f8307641eeb11b9100dfebf8151e1957822f3b47af29949d79a48c | A cube is sectioned with 9 planes parallel to an edge. Each plane divides the cube into two quadrilateral prisms with a volume ratio of $\frac{1}{n}$, where $n$ is a fixed positive integer. Determine the number of distinct sets of 3 planes that intersect at a common line. | [
"0"
] | [
"A cube is sectioned with 9 planes parallel to an edge. Each plane divides the cube into two quadrilateral prisms with a volume ratio of $\\frac{1}{n}$, where $n$ is a fixed positive integer. Determine the number of distinct sets of 3 planes that intersect at a common line."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-7d0df18adaa2fb3f83b2e262e0ab76b709287d5ededa74df520ffa80873b263b | A cube is sliced by a plane so that the cross section is a regular hexagon. What is the length of a side of such a regular hexagon (assuming the length of an edge of the cube is $l$)? How many distinct slices are there of this type? | [
"6"
] | [
"A cube is sliced by a plane so that the cross section is a regular hexagon. What is the length of a side of such a regular hexagon (assuming the length of an edge of the cube is $l$)? How many distinct slices are there of this type?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-a04626b1cfb955795aa79912e46fdf2766ba11cb6a4c27ef6477060dcd1fe8e5 | A cube is sliced by a plane so that the cross section is an equilateral triangle. What is the maximum possible length of a side of such an equilateral triangle (assuming the length of an edge of the cube is $l$)? How many distinct slices are there of this type? | [
"8"
] | [
"A cube is sliced by a plane so that the cross section is an equilateral triangle. What is the maximum possible length of a side of such an equilateral triangle (assuming the length of an edge of the cube is $l$)? How many distinct slices are there of this type?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-208ed6c0909ba153a453ae42256fbbe155fdfa7553436d0ac9ec6dfce7ffc3c8 | A cube is split into 8 parallelepipeds by three planes parallel to its faces. The resulting parts are painted in a chessboard pattern (black and white, no two black parts sharing a face, no two white parts sharing a face). The volumes of the black parallelepipeds are 1, 6, 8, and 12. Find the volumes of the white paral... | [
"24, 4, 3, 2"
] | [
"A cube is split into 8 parallelepipeds by three planes parallel to its faces. The resulting parts are painted in a chessboard pattern (black and white, no two black parts sharing a face, no two white parts sharing a face). The volumes of the black parallelepipeds are 1, 6, 8, and 12. Find the volumes of the white ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 12
} | 26 | 0 | 0 |
omr-ws-f7a2e14090d228291947b8d81b2bde680e0fe1c437534d0375d9a62aa33edf07 | A cube lies on a plane. After being rolled over its edges, it returns to its initial location with the same face up. If the top face has been rotated by a certain angle, what is the sum of all possible angles (in degrees) through which the top face could have been rotated? | [
"450"
] | [
"A cube lies on a plane. After being rolled over its edges, it returns to its initial location with the same face up. If the top face has been rotated by a certain angle, what is the sum of all possible angles (in degrees) through which the top face could have been rotated?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-b0ac47b88ce9a8103b4a5ad7941d00dbff1c47101d8fc1630c2313dd00427468 | A cube measuring 100 units on each side is painted only on the outside and cut into unit cubes. How many unit cubes have paint only on two sides? | [
"1176"
] | [
"A cube measuring 100 units on each side is painted only on the outside and cut into unit cubes. How many unit cubes have paint only on two sides?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-3fb4773a1a659b4c58775878e36b4caee9d499d68e624040fc3899c3c16d45bc | A cube numbered from 1 to 20 is dropped and breaks into 7 pieces. Find the probability that no two consecutive numbers are on the same piece. The probability is of the form $\frac{m}{n}$. Find $m+n$. | [
"6^{19} + 7^{19}"
] | [
"A cube numbered from 1 to 20 is dropped and breaks into 7 pieces. Find the probability that no two consecutive numbers are on the same piece. The probability is of the form $\\frac{m}{n}$. Find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-680abc1e3af33f691ae2a1a8ed668fbd29c00f534e0d2d81abb9217544335486 | A cube of 8cm x 8cm x 8cm is divided into smaller cubes of 1cm x 1cm x 1cm. Each smaller cube is numbered based on the smallest volume enclosed by extending its sides to the outer surface of the largest cube. Find the sum of the numbers on the cubes along the two body diagonals of the largest cube. | [
"400"
] | [
"A cube of 8cm x 8cm x 8cm is divided into smaller cubes of 1cm x 1cm x 1cm. Each smaller cube is numbered based on the smallest volume enclosed by extending its sides to the outer surface of the largest cube. Find the sum of the numbers on the cubes along the two body diagonals of the largest cube."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-6c627e73d680bd845a053383e091d732402483b480e6abe156dd98084199613e | A cube of cheese $C = \{(x, y, z) | 0 \le x, y, z \le 1\}$ is cut along the planes $x = y$, $y = z$, and $z = x$. How many pieces are there? | [
"6"
] | [
"A cube of cheese $C = \\{(x, y, z) | 0 \\le x, y, z \\le 1\\}$ is cut along the planes $x = y$, $y = z$, and $z = x$. How many pieces are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4772f8c97d2042cc3e7c610971d3f28e54fb51c9a6cebecdbd8dbb764f0208f0 | A cube of dimensions 20 * 20 * 20 is divided into 8000 cubic units, each containing a number. The sum of the numbers in each line and each column of 20 small cubes, parallel to one of the edges of the cube, is 1. One of the small cubes contains the number 10. Three layers of 1 * 20 * 20, parallel to the faces of the cu... | [
"333"
] | [
"A cube of dimensions 20 * 20 * 20 is divided into 8000 cubic units, each containing a number. The sum of the numbers in each line and each column of 20 small cubes, parallel to one of the edges of the cube, is 1. One of the small cubes contains the number 10. Three layers of 1 * 20 * 20, parallel to the faces of t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-eab87768050d43473ec211e01faffda5da18d638dc4053217602d93292b3375d | A cube of dimensions 6x6x6 is made up of smaller 1x1x1 cubes. Some of these smaller cubes are completely red, and the others are completely blue. It is known that any 2x2x2 sub-cube within the larger cube consists of 3 red cubes and 5 blue cubes. Calculate the total number of red cubes in the 8 corner positions of the ... | [
"3"
] | [
"A cube of dimensions 6x6x6 is made up of smaller 1x1x1 cubes. Some of these smaller cubes are completely red, and the others are completely blue. It is known that any 2x2x2 sub-cube within the larger cube consists of 3 red cubes and 5 blue cubes. Calculate the total number of red cubes in the 8 corner positions of... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-04430ad0fd2ad6fc3fb54659d81c35a842253715d20720bf744a84d8f4d26d1b | A cube of side $3$ metres has three holes, each with a side of $1$ metre, running from the centre of one face to the centre of the opposite face. What is the total surface area of the solid in square metres? | [
"72"
] | [
"A cube of side $3$ metres has three holes, each with a side of $1$ metre, running from the centre of one face to the centre of the opposite face. What is the total surface area of the solid in square metres?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 6,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-703fd21fc22ea5928b040355dddd66b0f0dce099669616253124b9fd13326a31 | A cube of side $n$ is cut into $n^3$ unit cubes. Some of these cubes are marked such that the centers of any three marked cubes do not form a right-angled triangle with legs parallel to the sides of the cube. Find the maximum possible number of marked cubes, denoted as $m$. | [
"\\( n \\)"
] | [
"A cube of side $n$ is cut into $n^3$ unit cubes. Some of these cubes are marked such that the centers of any three marked cubes do not form a right-angled triangle with legs parallel to the sides of the cube. Find the maximum possible number of marked cubes, denoted as $m$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-afce7dc4d40d62427c0e4cecb3d25fc490a4823c4be30765caca89e6c326db46 | A cube of side 1 has vertices P(0,0,1), Q(1,0,1), R(1,1,1), S(0,1,1), T(0,0,0), U(1,0,0), V(1,1,0), and W(0,1,0). The plane through P and the centers of faces TUVW and UQRV intersects UV at X. What is the length of UX? | [
"\\(\\frac{2}{3}\\)"
] | [
"A cube of side 1 has vertices P(0,0,1), Q(1,0,1), R(1,1,1), S(0,1,1), T(0,0,0), U(1,0,0), V(1,1,0), and W(0,1,0). The plane through P and the centers of faces TUVW and UQRV intersects UV at X. What is the length of UX?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5dae393f88be07a887b477413fdd95e151afba41ecbb56bfe27f590b37e917f9 | A cube of side \( n \) consists of \( n^3 \) unit cubes, where \( n \) is even. One arbitrarily selects \( \frac{3n^2}{2} \) unit cubes. Let \( T(n) \) be the number of right-angled triangles whose vertices lie in the centers of three of the selected unit cubes and whose legs are parallel to the edges of the cube. Find... | [
"72"
] | [
"A cube of side \\( n \\) consists of \\( n^3 \\) unit cubes, where \\( n \\) is even. One arbitrarily selects \\( \\frac{3n^2}{2} \\) unit cubes. Let \\( T(n) \\) be the number of right-angled triangles whose vertices lie in the centers of three of the selected unit cubes and whose legs are parallel to the edges o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b51e95e31d28f4357367f4734925d9c964d48fcde4b0ec286d18ce0ef34a084 | A cube of side length $2$ is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one of the faces of the first cube. What is the length of a side of the second cube? | [
"\\(\\frac{2}{3}\\)"
] | [
"A cube of side length $2$ is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one of the faces of the first cube. What is the length of a side of the second cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-483c56f9bc4e43fd09c8d488a34cef11dc1f2c7147cca943f7a5f257e3f0cb2e | A cube of side length $n$ is divided into unit cubes by partitions (each partition separates a pair of adjacent unit cubes). What is the smallest number of partitions that can be removed so that from each unit cube, one can reach the surface of the cube without passing through a partition? | [
"\\( 3(n-2)^2(n-3) \\)"
] | [
"A cube of side length $n$ is divided into unit cubes by partitions (each partition separates a pair of adjacent unit cubes). What is the smallest number of partitions that can be removed so that from each unit cube, one can reach the surface of the cube without passing through a partition?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-609a8a47db778fa37013ecf4ee1cd4f4274a01904c466a097635ec07230e2f7f | A cube of side length $n$ is divided into unit cubes by partitions. What is the smallest number of partitions that can be removed so that from each unit cube, one can reach the surface of the cube without passing through a partition? | [
"\\( n-2 \\)"
] | [
"A cube of side length $n$ is divided into unit cubes by partitions. What is the smallest number of partitions that can be removed so that from each unit cube, one can reach the surface of the cube without passing through a partition?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-385284e78bcac5fdd588763f1d9fefc7105df24cb4c9deaea4c048aaadc8de40 | A cube of side length 1 is cut into three pieces of equal volume by two planes passing through the diagonal of the top face. One plane cuts the edge UV at the point P. What is the length PV? | [
"\\dfrac{2}{3}"
] | [
"A cube of side length 1 is cut into three pieces of equal volume by two planes passing through the diagonal of the top face. One plane cuts the edge UV at the point P. What is the length PV?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-81f76123801d7cb5e3611018fcd2d96ce4f7c0cb73af9769d29fa021c0dd602f | A cube of side length 1 unit is placed on a horizontal plane such that one of its major diagonals is perpendicular to the plane. The sun, assumed to be at infinity, illuminates all three faces of the cube facing upwards. Calculate the area of the shadow cast by the cube on the plane. | [
"\\sqrt{3}"
] | [
"A cube of side length 1 unit is placed on a horizontal plane such that one of its major diagonals is perpendicular to the plane. The sun, assumed to be at infinity, illuminates all three faces of the cube facing upwards. Calculate the area of the shadow cast by the cube on the plane."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-22d9020d9783287f2b0b352672ae2e5804824a81805e0747684878cb4ed086d9 | A cube of side length \( 2 \) gets dropped into a hollow cone-shaped container and gets stuck inside such that the cone’s axis is perfectly aligned with one of the main diagonals of the cube. Additionally, the three vertices touching the sides of the cone are a distance \( 6 \) away from the cone’s tip. If the shortest... | [
"6"
] | [
"A cube of side length \\( 2 \\) gets dropped into a hollow cone-shaped container and gets stuck inside such that the cone’s axis is perfectly aligned with one of the main diagonals of the cube. Additionally, the three vertices touching the sides of the cone are a distance \\( 6 \\) away from the cone’s tip. If the... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-94ba9480671abf3e2c70ca8c2fc948215698db93507fda35f2ee489eb8f0561d | A cube of wood sits in a fire. It burns at the same rate on five sides but remains untouched on the bottom side. Determine what proportion of the cube has burnt away when the height of the remaining block is twice the width. | [
"\\(\\frac{25}{27}\\)"
] | [
"A cube of wood sits in a fire. It burns at the same rate on five sides but remains untouched on the bottom side. Determine what proportion of the cube has burnt away when the height of the remaining block is twice the width."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 29
} | 43 | 0 | 0 |
omr-ws-b5d61c135636eb8e55d052fb83c2dba558b9ac0910a130502dc0c17199f175e1 | A cube stands on one of the squares of an infinite chessboard. On each face of the cube, there is an arrow pointing in one of the four directions parallel to the sides of the face. Anton looks at the cube from above and rolls it over an edge in the direction pointed by the arrow on the top face. After a series of such ... | [
"24"
] | [
"A cube stands on one of the squares of an infinite chessboard. On each face of the cube, there is an arrow pointing in one of the four directions parallel to the sides of the face. Anton looks at the cube from above and rolls it over an edge in the direction pointed by the arrow on the top face. After a series of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-50e9f77e87807bbd1a99403ce4fe22a5a64b7d52baa4b5c7543ca1ccae5ad58f | A cube was split into 8 parallelepipeds by three planes parallel to its faces. The resulting parts were painted in a chessboard pattern. The volumes of the black parallelepipeds are 1, 6, 8, 12. Find the volumes of the white parallelepipeds. | [
"27"
] | [
"A cube was split into 8 parallelepipeds by three planes parallel to its faces. The resulting parts were painted in a chessboard pattern. The volumes of the black parallelepipeds are 1, 6, 8, 12. Find the volumes of the white parallelepipeds."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a114d5ec4f150618e38e7f9d802d8c8bba032c3078f36e9208fd159b5344f65f | A cube with 3-inch edges is to be constructed from 27 smaller cubes with 1-inch edges. Twenty-one of the cubes are colored red and 6 are colored white. If the 3-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white? | [
"\\(\\frac{5}{54}\\)"
] | [
"A cube with 3-inch edges is to be constructed from 27 smaller cubes with 1-inch edges. Twenty-one of the cubes are colored red and 6 are colored white. If the 3-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 25
} | 47 | 0 | 0 |
omr-ws-fbe1d4ecec617d12d6cd5fe30f42c74ff782c509fe5f9134988d14f828de3287 | A cube with a diagonal of (4, 4, 4) rotates about its diagonal at 3 radians per second. What is the velocity vector of the point (1, 3, 2)? | [
"\\((- \\sqrt{3}, - \\sqrt{3}, 2 \\sqrt{3})\\)"
] | [
"A cube with a diagonal of (4, 4, 4) rotates about its diagonal at 3 radians per second. What is the velocity vector of the point (1, 3, 2)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-f3422d6fa5470a6f0b67ca33f13638c64d3293e8ec739e4bd61d2f5184451142 | A cube with a side length of 12 cm has a part cut out in the form of a cuboid with rib lengths of 5 cm, 2 cm, and 3 cm. Find the total surface area of the remaining part of the cube. | [
"864 cm²"
] | [
"A cube with a side length of 12 cm has a part cut out in the form of a cuboid with rib lengths of 5 cm, 2 cm, and 3 cm. Find the total surface area of the remaining part of the cube."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 2
} | 23 | 0 | 0 |
omr-ws-9817d5796381e0e4a585082173194fb38d23d3d1a62b3a21b050b38d48aeea5a | A cube with a side length of 5 inches is painted and divided into unit cubes. What is the ratio, expressed as a fraction in lowest terms, between the number of cubes with 2 or 3 faces painted to the number of cubes with no faces painted? | [
"\\(\\frac{46}{27}\\)"
] | [
"A cube with a side length of 5 inches is painted and divided into unit cubes. What is the ratio, expressed as a fraction in lowest terms, between the number of cubes with 2 or 3 faces painted to the number of cubes with no faces painted?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5300d5ef608b89520d9180f3735b0b918f0f8bc2d260ad0727421a199716fb61 | A cube with a volume of eight is combined with 7 others to form a larger cube. What is the surface area of the new larger cube? | [
"96"
] | [
"A cube with a volume of eight is combined with 7 others to form a larger cube. What is the surface area of the new larger cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-ee9c21fe963244905baacf7a1ccabcdfade0b9505281d67c55e19da255749dc9 | A cube with an edge length of 1 is intersected by a plane that does not pass through any of its vertices and intersects its edges only at the midpoints. Find the area of the formed section. Consider all possible cases. | [
"\\(\\frac{3\\sqrt{3}}{4}\\)"
] | [
"A cube with an edge length of 1 is intersected by a plane that does not pass through any of its vertices and intersects its edges only at the midpoints. Find the area of the formed section. Consider all possible cases."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-bb9efa89fc3572ea8fe889fb0c010ca48a8b1d97518ebed33860b704fe8f8912 | A cube with an edge length of 4 units has the same volume as a square-based pyramid with base edge lengths of 8 units and a height of $h$ units. What is the value of $h$? | [
"3"
] | [
"A cube with an edge length of 4 units has the same volume as a square-based pyramid with base edge lengths of 8 units and a height of $h$ units. What is the value of $h$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d8100433f4ebb18abbd3f4b77361e617e65943ec482d70a90c2420b62cbd095a | A cube with an edge of $n$ cm is divided into $n^3$ mini-cubes with edges of length $1$ cm. If we draw with blue all the diagonals of all the faces of the cube, upon how many mini-cubes do we find blue segments? | [
"12n - 16 - 6(n \\bmod 2)"
] | [
"A cube with an edge of $n$ cm is divided into $n^3$ mini-cubes with edges of length $1$ cm. If we draw with blue all the diagonals of all the faces of the cube, upon how many mini-cubes do we find blue segments?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-aa0d58b960b36b2fec8a3d0af41eed224e678faceda4d6906b829876e0e6bd5a | A cube with an edge of $n$ cm is divided into $n^3$ mini-cubes with edges of length $1$ cm. Only the exterior of the cube is colored. How many of the mini-cubes have at least two colored faces? | [
"\\(12(n-2) + 8\\)"
] | [
"A cube with an edge of $n$ cm is divided into $n^3$ mini-cubes with edges of length $1$ cm. Only the exterior of the cube is colored. How many of the mini-cubes have at least two colored faces?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-b22b2a773ac2fb99e7504db6979c8d45a5e2ae407fb438327a2225fa9836d927 | A cube with an edge of $n$ cm is divided into $n^3$ mini-cubes with edges of length $1$ cm. Only the exterior of the cube is colored. How many of the mini-cubes have only one colored face? | [
"\\(6(n-2)^2\\)"
] | [
"A cube with an edge of $n$ cm is divided into $n^3$ mini-cubes with edges of length $1$ cm. Only the exterior of the cube is colored. How many of the mini-cubes have only one colored face?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-4e9325635d69d771495b1e94bc915b0f447ccfeb17dbddadb94b4dfdda982657 | A cube with edge length $10$ is cut into $27$ parallelepipeds by three pairs of planes parallel to the faces. The interior parallelepiped has edge lengths $1$, $2$, and $3$. Find the sum of the volumes of the $8$ parallelepipeds at the corners of the cube. | [
"504"
] | [
"A cube with edge length $10$ is cut into $27$ parallelepipeds by three pairs of planes parallel to the faces. The interior parallelepiped has edge lengths $1$, $2$, and $3$. Find the sum of the volumes of the $8$ parallelepipeds at the corners of the cube."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 2
} | 34 | 0 | 0 |
omr-ws-85710788afef55163109ab1945d6290f8e0e197703133f18177281e51d662864 | A cube with edge length $2n + 1$ is dissected into small cubes of size $1 \times 1 \times 1$ and bars of size $2 \times 2 \times 1$. Find the least possible number of $1 \times 1 \times 1$ cubes in such a dissection. | [
"\\(2n + 1\\)"
] | [
"A cube with edge length $2n + 1$ is dissected into small cubes of size $1 \\times 1 \\times 1$ and bars of size $2 \\times 2 \\times 1$. Find the least possible number of $1 \\times 1 \\times 1$ cubes in such a dissection."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-7c5575a824ce37cc6eea453195bcef42780206b79d329035e2b987c94895be5b | A cube with edge length $3$ is divided into $27$ unit cubes. The numbers $1, 2, \ldots, 27$ are distributed arbitrarily over the unit cubes, with one number in each cube. We form the $27$ possible row sums (there are nine such sums of three integers for each of the three directions parallel with the edges of the cube).... | [
"24"
] | [
"A cube with edge length $3$ is divided into $27$ unit cubes. The numbers $1, 2, \\ldots, 27$ are distributed arbitrarily over the unit cubes, with one number in each cube. We form the $27$ possible row sums (there are nine such sums of three integers for each of the three directions parallel with the edges of the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-1e97fbf593a2a325812bf41a7989cb2e7ee9e12d2109890b71f9aa2325cd61b0 | A cube with edge length 4 units is sliced by a plane that passes through two diagonally opposite vertices of the cube. What is the area of the resulting cross section? | [
"\\(8\\sqrt{6}\\)"
] | [
"A cube with edge length 4 units is sliced by a plane that passes through two diagonally opposite vertices of the cube. What is the area of the resulting cross section?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 8
} | 10 | 0 | 0 |
omr-ws-3c83fa5fe5c1711d0d485dafb71535905bf1b13055e475ebce46bf6358989c25 | A cube with edge length \(a\) is placed on an infinite chessboard such that its vertices are aligned with the vertices of the chessboard squares. Let \(N(a)\) be the maximum number of chessboard vertices that can be covered by the cube. Find \(N(3)\). | [
"16"
] | [
"A cube with edge length \\(a\\) is placed on an infinite chessboard such that its vertices are aligned with the vertices of the chessboard squares. Let \\(N(a)\\) be the maximum number of chessboard vertices that can be covered by the cube. Find \\(N(3)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48797d4ec4b3a1704eb616b6a9abbec8abe92cdaf53de704f29af0cf86db5585 | A cube with side length $1$ contains $9$ spheres of equal radius $r$. Eight of the spheres are tangent to three of the faces of the cube, one in each corner. The last sphere is centered at the cube's center and is tangent to the other eight spheres. Find the radius $r$ of the spheres. | [
"\\( \\frac{2\\sqrt{3} - 3}{2} \\)"
] | [
"A cube with side length $1$ contains $9$ spheres of equal radius $r$. Eight of the spheres are tangent to three of the faces of the cube, one in each corner. The last sphere is centered at the cube's center and is tangent to the other eight spheres. Find the radius $r$ of the spheres."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-77f69f937219abf92ee7e0b24f73628b1535b917aa71e0ae2b8b437d00d505d8 | A cube with side length $1$ is sliced by a plane that passes through two diagonally opposite vertices $A$ and $C$ and the midpoints $B$ and $D$ of two opposite edges not containing $A$ and $C$. What is the area of quadrilateral $ABCD$? | [
"\\(\\frac{\\sqrt{6}}{2}\\)"
] | [
"A cube with side length $1$ is sliced by a plane that passes through two diagonally opposite vertices $A$ and $C$ and the midpoints $B$ and $D$ of two opposite edges not containing $A$ and $C$. What is the area of quadrilateral $ABCD$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-894b3657a35cb4fb268340e9018b101de4ccfd4a5ca3b5f9edea40dca4aa810a | A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids? | [
"350"
] | [
"A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 37,
"tir": 16
} | 53 | 0 | 0 |
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