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-e90eca6e23d3b5e05ec3ed54d08f538d595d39ddb9b81dae3ef39baf27593ebd | A rectangle has an area of 400 $\text{cm}^2$. If the rectangle is a square, what is its perimeter? | [
"80"
] | [
"A rectangle has an area of 400 $\\text{cm}^2$. If the rectangle is a square, what is its perimeter?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a5d48c0d8c2fe18b8a6d47bac46fa0a227e13d2722d42f99fd76c9d3c01e3f0c | A rectangle has an area of 9 square inches. The diagonals of the rectangle intersect at a 60-degree angle. What is the length of the longer side of the rectangle? | [
"\\(3^{5/4}\\)"
] | [
"A rectangle has an area of 9 square inches. The diagonals of the rectangle intersect at a 60-degree angle. What is the length of the longer side of the rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-b4399c5bfe3133fe88c4621871137cc3acfc7a8233c18edcc73f0b260de249ed | A rectangle has area $k^2$ for some $k$. If there are 3 possible distinct values for the perimeter, find the smallest possible value for $k$. Note: $k$ is a positive integer. | [
"4"
] | [
"A rectangle has area $k^2$ for some $k$. If there are 3 possible distinct values for the perimeter, find the smallest possible value for $k$. Note: $k$ is a positive integer."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-317ff0ed4398b1778a5fbcac67c896aa960842aac2917dc2a652f72fd0de614c | A rectangle has its longest side tangent to a circle and two of its vertices on the circle. The longest side of the rectangle is 16, and the radius of the circle is 10. Find the area of the rectangle. | [
"64"
] | [
"A rectangle has its longest side tangent to a circle and two of its vertices on the circle. The longest side of the rectangle is 16, and the radius of the circle is 10. Find the area of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-0d53e9a87b9c3155460e211273975f5452727bc85e882b4458cc58ee7159e47d | A rectangle has its vertices at lattice points and its sides parallel to the axes. Its smaller side has length $k$. The rectangle is divided into triangles whose vertices are all lattice points, such that each triangle has area $\frac{1}{2}$. Let $N$ be the number of right-angled triangles in this division. Find the mi... | [
"2k"
] | [
"A rectangle has its vertices at lattice points and its sides parallel to the axes. Its smaller side has length $k$. The rectangle is divided into triangles whose vertices are all lattice points, such that each triangle has area $\\frac{1}{2}$. Let $N$ be the number of right-angled triangles in this division. Find ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-9cc1587a2cf19a8530129f8e55585a3d3cff0e2953d5b36d47cc13badcfe1315 | A rectangle has side lengths 6 and 8. There are relatively prime positive integers \( m \) and \( n \) such that \( \frac{m}{n} \) is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find \( m + n \). | [
"203"
] | [
"A rectangle has side lengths 6 and 8. There are relatively prime positive integers \\( m \\) and \\( n \\) such that \\( \\frac{m}{n} \\) is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find \\( m + n ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 7,
"tir": 2
} | 13 | 0 | 0 |
omr-ws-97500bee044a0d7a570faa75ddc06417de230bdd056bcbc05584f4d00861747c | A rectangle has side lengths of 4 and 8, and a square with side length 5 is positioned such that three of its vertices lie on three different sides of the rectangle. What is the area of the region inside both the square and the rectangle? | [
"\\( 15 \\frac{5}{8} \\)"
] | [
"A rectangle has side lengths of 4 and 8, and a square with side length 5 is positioned such that three of its vertices lie on three different sides of the rectangle. What is the area of the region inside both the square and the rectangle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 20,
"genselect": 8,
"tir": 4
} | 32 | 0 | 0 |
omr-ws-65ec103ffde6b898236f04d25315cd93fdcd020fa4cf47ff3d11ac1b510aa766 | A rectangle has sides of length $a$ and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with... | [
"720"
] | [
"A rectangle has sides of length $a$ and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 8,
"tir": 6
} | 39 | 0 | 0 |
omr-ws-b902fd9282c12b0f0da128397a6c326e53502931bfd13130abf3e5593852b77f | A rectangle has sides of length \(a\) and 36. A hinge is installed at each vertex of the rectangle and at the midpoint of each side of length 36. The sides of length \(a\) can be pressed toward each other, keeping those two sides parallel, so the rectangle becomes a convex hexagon. When the figure is a hexagon with the... | [
"720"
] | [
"A rectangle has sides of length \\(a\\) and 36. A hinge is installed at each vertex of the rectangle and at the midpoint of each side of length 36. The sides of length \\(a\\) can be pressed toward each other, keeping those two sides parallel, so the rectangle becomes a convex hexagon. When the figure is a hexagon... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 20,
"genselect": 7,
"tir": 4
} | 31 | 0 | 0 |
omr-ws-43ea12a58514d9aba296f16b8c7290655a66a975ae32921af37ea6c7256f75fb | A rectangle has vertices at the points $(0, 0)$, $(x, 0)$, $(0, x^n)$, and $(x, x^n)$ for positive real $x$ and positive real $n$. What is the probability that a point randomly selected within this rectangle lies beneath the curve $y = x^n$? Express the probability in terms of $n$. | [
"\\(\\frac{1}{n+1}\\)"
] | [
"A rectangle has vertices at the points $(0, 0)$, $(x, 0)$, $(0, x^n)$, and $(x, x^n)$ for positive real $x$ and positive real $n$. What is the probability that a point randomly selected within this rectangle lies beneath the curve $y = x^n$? Express the probability in terms of $n$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-54a73c1b9c5acfa3d82f9eae43bf04f0825637e6d37a08579b975e7373f6a2c3 | A rectangle having area $1$ is inscribed in a circle having area $2$. How long is the diagonal of the rectangle? | [
"2\\sqrt{\\dfrac{2}{\\pi}}"
] | [
"A rectangle having area $1$ is inscribed in a circle having area $2$. How long is the diagonal of the rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-47365a3b64c53a456fe0ea000aad1758162dc3ae69a4197a3310204f2c2d30ef | A rectangle having integer length and width has a perimeter of 100 units. What is the number of square units in the least possible area? | [
"49"
] | [
"A rectangle having integer length and width has a perimeter of 100 units. What is the number of square units in the least possible area?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0490acf7e4f65acfb26d2ed749801d5860646946b3b9a73ab5d6173ad594b853 | A rectangle is bounded by the x and y axis and the line \( y = \frac{6 - x}{2} \). Find the dimensions that maximize the area of the rectangle. | [
"\\(3\\) and \\(\\frac{3}{2}\\)"
] | [
"A rectangle is bounded by the x and y axis and the line \\( y = \\frac{6 - x}{2} \\). Find the dimensions that maximize the area of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 21
} | 34 | 0 | 0 |
omr-ws-0c23f0717a4c7ce555ca14db6c11a729ae13926e5dff51ad9043d37ab1d7f517 | A rectangle is built inside a rhombus such that the distances from the vertices of the rhombus to the sides of the rectangle are equal to 2 and 3. What is the area of the rectangle? | [
"24"
] | [
"A rectangle is built inside a rhombus such that the distances from the vertices of the rhombus to the sides of the rectangle are equal to 2 and 3. What is the area of the rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-bc0758955c862ed58325f588f75896fb9d8a67248c4e52cfaabce5e3b94dbeba | A rectangle is called cool if the number of square units in its area is equal to twice the number of units in its perimeter. A cool rectangle also must have integer side lengths. What is the sum of all the different possible areas of cool rectangles? | [
"236"
] | [
"A rectangle is called cool if the number of square units in its area is equal to twice the number of units in its perimeter. A cool rectangle also must have integer side lengths. What is the sum of all the different possible areas of cool rectangles?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-382eab706cf1c4dd09eb283658b5a6ca57893c0c7de261f5a04796eec3b680bc | A rectangle is cut by lines parallel to its sides into $m \times n$ rectangular pieces. Each part is covered with a card, on the back of which the area of this part is written. What is the least number of cards that must be turned over to determine the area of the original rectangle? | [
"\\( m + n - 1 \\)"
] | [
"A rectangle is cut by lines parallel to its sides into $m \\times n$ rectangular pieces. Each part is covered with a card, on the back of which the area of this part is written. What is the least number of cards that must be turned over to determine the area of the original rectangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-e8de8843a6bd8e87d20ea7eb50a8344ca9b190053a32b93bb306c5a6eb7bf3ee | A rectangle is cut into 10 squares, with the side of the smallest square being 3. Find the dimensions of the rectangle. | [
"\\(29 \\times 57\\)"
] | [
"A rectangle is cut into 10 squares, with the side of the smallest square being 3. Find the dimensions of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9ae5e38663e4b25c8598feec14e1d560456bfbf57d1d243c076362ee75735500 | A rectangle is cut into 6 squares, with the gray square in the middle having a side length of 1. What is the area of the rectangle? | [
"143"
] | [
"A rectangle is cut into 6 squares, with the gray square in the middle having a side length of 1. What is the area of the rectangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-604630cf5d2cb395cd2093f7d74cacbc71a4e37fafda02b1b0b60600d70e331f | A rectangle is cut into four smaller rectangles such that three of these smaller rectangles have perimeters of 11, 16, and 19, respectively. What is the largest possible perimeter of the fourth smaller rectangle? | [
"19"
] | [
"A rectangle is cut into four smaller rectangles such that three of these smaller rectangles have perimeters of 11, 16, and 19, respectively. What is the largest possible perimeter of the fourth smaller rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-61156fbb950dcd9454b0bf5f42826d0c7c2f012e17b9a265f3d40cbe25b55836 | A rectangle is cut out of grid paper with cell side length 1. Carlson then cuts out a square adjacent to the boundary of the rectangle. After this, the perimeter of the remaining figure is the same as the original area of the rectangle, and the area of the remaining figure is the same as the original perimeter of the r... | [
"\\( x = 2 \\)"
] | [
"A rectangle is cut out of grid paper with cell side length 1. Carlson then cuts out a square adjacent to the boundary of the rectangle. After this, the perimeter of the remaining figure is the same as the original area of the rectangle, and the area of the remaining figure is the same as the original perimeter of ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-a910fc4b2514436e4b6b1d12f88ef470edb3fb5c33bf3f40aa7c8bc7c79c127a | A rectangle is cut using only vertical or horizontal cuts. What is the smallest possible number of cuts needed to cut the rectangle into 36 pieces? | [
"10"
] | [
"A rectangle is cut using only vertical or horizontal cuts. What is the smallest possible number of cuts needed to cut the rectangle into 36 pieces?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3a1f6c8f759939f2bace8b3b77c886753c1fde3e9c440cd50fcafc15234491f2 | A rectangle is decomposed into 9 squares with side lengths of 1, 4, 7, 8, 9, 10, 14, 15, and 18. Find the dimensions of the rectangle. | [
"32 \\times 33"
] | [
"A rectangle is decomposed into 9 squares with side lengths of 1, 4, 7, 8, 9, 10, 14, 15, and 18. Find the dimensions of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3333b8b98e02f9d2cc55d612abf550595ca03c2fb132da53ff468fb3a558be0f | A rectangle is decomposed into nine squares whose sides measure 1, 4, 7, 8, 9, 10, 14, 15, and 18 units respectively. What are the dimensions of the rectangle? | [
"32 and 33"
] | [
"A rectangle is decomposed into nine squares whose sides measure 1, 4, 7, 8, 9, 10, 14, 15, and 18 units respectively. What are the dimensions of the rectangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 22
} | 35 | 0 | 0 |
omr-ws-d5db23ba39fd2beb9ceae7b500b6043e7a3a4ec64ac5cefeb2ebd5c761922540 | A rectangle is dissected into eleven congruent pieces. If it is possible for these pieces to not all be rectangles, find the minimum number of non-rectangular pieces in such a dissection. If all pieces must be rectangles, enter 0. | [
"11"
] | [
"A rectangle is dissected into eleven congruent pieces. If it is possible for these pieces to not all be rectangles, find the minimum number of non-rectangular pieces in such a dissection. If all pieces must be rectangles, enter 0."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-249f3aa40ca3e56d3d43a23dc96e9b4e6d08325dee1af5cef5d89fdd39ac1484 | A rectangle is divided into $2 \times 1$ and $1 \times 2$ dominoes, and in each domino, a diagonal is drawn such that no two diagonals share common endpoints. If the rectangle has dimensions $m \times n$, find the sum of all possible values of $m+n$ for which exactly two corners of the rectangle are endpoints of these ... | [
"no_valid_answer_found"
] | [
"A rectangle is divided into $2 \\times 1$ and $1 \\times 2$ dominoes, and in each domino, a diagonal is drawn such that no two diagonals share common endpoints. If the rectangle has dimensions $m \\times n$, find the sum of all possible values of $m+n$ for which exactly two corners of the rectangle are endpoints o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6286b529ba276a7ff9111fbbec65b5c7e7bc917aed95eacbf1c1006e93e11792 | A rectangle is divided into $9$ smaller rectangles by parallel lines to its sides. Four of these smaller rectangles have areas given as $10$, $15$, $20$, and $25$. If $A$ is the total area of the rectangle, find the minimum possible value of $A$. | [
"70 + 30\\sqrt{5}"
] | [
"A rectangle is divided into $9$ smaller rectangles by parallel lines to its sides. Four of these smaller rectangles have areas given as $10$, $15$, $20$, and $25$. If $A$ is the total area of the rectangle, find the minimum possible value of $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-016db59f6a39e60e1b356d5cc235a0a93e9c8fda23cedc48f45fb36879b8aef5 | A rectangle is divided into $n^2$ smaller rectangles by $n - 1$ horizontal lines and $n - 1$ vertical lines. If there are exactly 5660 smaller rectangles that are not congruent, what is the minimum value of $n$ for which this is possible? | [
"13"
] | [
"A rectangle is divided into $n^2$ smaller rectangles by $n - 1$ horizontal lines and $n - 1$ vertical lines. If there are exactly 5660 smaller rectangles that are not congruent, what is the minimum value of $n$ for which this is possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-03dbe8584874b13fa9daca1f203a94da3a56c1ce3edf6608e35222638694c80b | A rectangle is divided into 5 rectangles with integer sides and areas 3, 4, 7, 10, and 12. Determine all possible perimeters of the original rectangle. | [
"24"
] | [
"A rectangle is divided into 5 rectangles with integer sides and areas 3, 4, 7, 10, and 12. Determine all possible perimeters of the original rectangle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b81b84c479698d375d6136b380c7e732e75109d2a850bfaff3f5379bf09cb09e | A rectangle is divided into 54 unit squares. The sides of the rectangle have length greater than 3. How many squares with area 5 and vertices among the midpoints of the vertices of the unit squares are there? | [
"45"
] | [
"A rectangle is divided into 54 unit squares. The sides of the rectangle have length greater than 3. How many squares with area 5 and vertices among the midpoints of the vertices of the unit squares are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-978185f135a39349b0fc771c29888e2cbc60ba573f16b6516d5fd0f85ddb2291 | A rectangle is divided into 8 squares with whole sides. The two squares on the bottom side (on the left) are the same, and the three shaded squares have side 1. Calculate the sides of all subdivision squares. | [
"5"
] | [
"A rectangle is divided into 8 squares with whole sides. The two squares on the bottom side (on the left) are the same, and the three shaded squares have side 1. Calculate the sides of all subdivision squares."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-c2dedecc0c48d8c1dc764ce626bef4e548c4991cfd15bf129680402551086a61 | A rectangle is divided into 9 smaller rectangles by four segments. The perimeters of four of these smaller rectangles are given. Determine the perimeter of the central rectangle, given that it is an integer and does not match any of the perimeters of the rectangles whose perimeters are provided. | [
"22"
] | [
"A rectangle is divided into 9 smaller rectangles by four segments. The perimeters of four of these smaller rectangles are given. Determine the perimeter of the central rectangle, given that it is an integer and does not match any of the perimeters of the rectangles whose perimeters are provided."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-a52c603e2e3f530f1a5e894c5e1866f1828bb52b381af037416da8a3fec7c6bc | A rectangle is divided into 9 smaller rectangular parts using four straight cuts. Perimeters are written in each part as follows: \[ \begin{array}{ | l | c | r | } \hline 14 & 16 & 12 \\ \hline 18 & 14 & 10 \\ \hline 16 & 18 & 14 \\ \hline \end{array} \] It is known that exactly one of these perimeters is incorrect. Id... | [
"18"
] | [
"A rectangle is divided into 9 smaller rectangular parts using four straight cuts. Perimeters are written in each part as follows:\n\\[\n\\begin{array}{ | l | c | r | }\n\\hline\n14 & 16 & 12 \\\\ \\hline\n18 & 14 & 10 \\\\ \\hline\n16 & 18 & 14 \\\\ \\hline\n\\end{array}\n\\]\nIt is known that exactly one of these... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-69e3a7a5fe5768b154dde05954293fa3e7ee99dc0ddddc382fa846884557e563 | A rectangle is divided into 9 squares with integral side lengths. The adjacent sides of the rectangle are coprime. Find the perimeter of the rectangle. | [
"260"
] | [
"A rectangle is divided into 9 squares with integral side lengths. The adjacent sides of the rectangle are coprime. Find the perimeter of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-54178310e1cf160235e6f0d38a1c0f33c8f5c93635296e4e50a87fdc14e972d8 | A rectangle is divided into four sections using three segments. The areas of three of the sections are 3, 4, and 5. Find the area of the gray section. | [
"\\dfrac{20}{3}"
] | [
"A rectangle is divided into four sections using three segments. The areas of three of the sections are 3, 4, and 5. Find the area of the gray section."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-080fb37b44bf372c707bda60f673b039fd71891964993855a39bda4ed5df486b | A rectangle is divided into four smaller rectangles by cutting it into two segments equal to its sides. Three of these smaller rectangles have areas of 8, 12, and 18, respectively. What is the largest possible area of the original large rectangle? | [
"30"
] | [
"A rectangle is divided into four smaller rectangles by cutting it into two segments equal to its sides. Three of these smaller rectangles have areas of 8, 12, and 18, respectively. What is the largest possible area of the original large rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17960e7659db0714f2df2d2d791a138a4bd5ca8dcc81bb15dfff958f52bc7eec | A rectangle is divided into four smaller rectangles by two straight lines parallel to its sides. One of the smaller rectangles is a square, and the perimeters of the rectangles adjacent to it on the side are 20 cm and 22 cm. Find the area of the original rectangle. | [
"110 cm^2"
] | [
"A rectangle is divided into four smaller rectangles by two straight lines parallel to its sides. One of the smaller rectangles is a square, and the perimeters of the rectangles adjacent to it on the side are 20 cm and 22 cm. Find the area of the original rectangle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-65238916d149fd0361184f003812e646b4b6fd4123dbfe14441b6c15224b10b6 | A rectangle is divided into right triangles with legs of lengths $1$ and $2$. Let $N$ be the total number of these triangles. If the rectangle has an area of $2023$, find the remainder when $N$ is divided by $4$. | [
"3"
] | [
"A rectangle is divided into right triangles with legs of lengths $1$ and $2$. Let $N$ be the total number of these triangles. If the rectangle has an area of $2023$, find the remainder when $N$ is divided by $4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe5f6c0f4f0d9fbaa81f08ab09953a629dd9eca2356ab51716c3d42443983bed | A rectangle is divided into smaller rectangles by horizontal and vertical lines such that there is at least one horizontal and one vertical section. The areas of all the smaller rectangles form an arithmetic progression. If the smallest rectangle has an area of $A$ and the largest rectangle has an area of $B$, find the... | [
"\\dfrac{\\sqrt{A^2 + 8AB} - 3A}{4}"
] | [
"A rectangle is divided into smaller rectangles by horizontal and vertical lines such that there is at least one horizontal and one vertical section. The areas of all the smaller rectangles form an arithmetic progression. If the smallest rectangle has an area of $A$ and the largest rectangle has an area of $B$, fin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-1d792af228aaa7cb580393075f2f1bd757080ed0ad6272a5acc123aaafbae1eb | A rectangle is divided into smaller rectangles, each of which has at least one integer side length. Let the dimensions of the original rectangle be $a \times b$. If $a$ and $b$ are not integers, find the minimum possible value of $a + b$ under the given conditions. | [
"3"
] | [
"A rectangle is divided into smaller rectangles, each of which has at least one integer side length. Let the dimensions of the original rectangle be $a \\times b$. If $a$ and $b$ are not integers, find the minimum possible value of $a + b$ under the given conditions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-1bd9f10306b7379b5ef6307cc0321fa36a23350e7d12e1fbcc02df97b1d88313 | A rectangle is divided into smaller rectangles, each of which has at least one side with a natural number length. Let $L$ be the length and $W$ be the width of the original rectangle. If $L$ is not a natural number, find the smallest possible value of $W$ such that the area of the original rectangle is $2023$. | [
"2"
] | [
"A rectangle is divided into smaller rectangles, each of which has at least one side with a natural number length. Let $L$ be the length and $W$ be the width of the original rectangle. If $L$ is not a natural number, find the smallest possible value of $W$ such that the area of the original rectangle is $2023$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-088040ab337ad21217fd99ec10026f4b450e138feba14bfe78693070259b6a04 | A rectangle is divided into squares with a side of $1$ cm. The total length of the drawn lines (including the perimeter of the rectangle) is $1$ meter. Find the dimensions of the rectangle. | [
"\\(22\\) by \\(1\\)"
] | [
"A rectangle is divided into squares with a side of $1$ cm. The total length of the drawn lines (including the perimeter of the rectangle) is $1$ meter. Find the dimensions of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4df3d57deca903e283e40b2c869ae977969947a1f578bedb6b71e2d06b34708f | A rectangle is divided into squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18. Determine the dimensions of the rectangle. | [
"32 × 33"
] | [
"A rectangle is divided into squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18. Determine the dimensions of the rectangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-4f3db63ab59f1c50f1e13c8a107f1d709a9031c228241ada66423284a00dd291 | A rectangle is drawn such that the vertices of a uniform hexagon lie on the sides of the rectangle. If the area of the hexagon is 24, find the area of the rectangle. | [
"48"
] | [
"A rectangle is drawn such that the vertices of a uniform hexagon lie on the sides of the rectangle. If the area of the hexagon is 24, find the area of the rectangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-cd9eefa290cac8c5e21602f127a18262be663763e9bd5a0994a4195026ed64bd | A rectangle is formed using the lines of a grid of unit squares and is completely covered using three types of tiles such that no two tiles overlap: - Type $a$ tiles are isosceles right triangles with a hypotenuse of two units. - Type $b$ tiles are squares with a side length of 1 unit. - Type $c$ tiles are parallelogra... | [
"2"
] | [
"A rectangle is formed using the lines of a grid of unit squares and is completely covered using three types of tiles such that no two tiles overlap:\n- Type $a$ tiles are isosceles right triangles with a hypotenuse of two units.\n- Type $b$ tiles are squares with a side length of 1 unit.\n- Type $c$ tiles are para... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-edb4dfd4f27da7743fa1151a1c869ec71541a9f007906960f2c7d24fe00c97ea | A rectangle is inscribed in a circle of radius $1$, with all its vertices on the circumference. Find the maximum possible perimeter of the rectangle, and express your answer as a simplified expression involving $\sqrt{2}$. | [
"4\\sqrt{2}"
] | [
"A rectangle is inscribed in a circle of radius $1$, with all its vertices on the circumference. Find the maximum possible perimeter of the rectangle, and express your answer as a simplified expression involving $\\sqrt{2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fa6378b0b51dcd98d5a3e1b4e4908869c00323a4ad7124e268e1c6e7552bc8ef | A rectangle is inscribed in a circle of unit radius. The perimeter of the rectangle is equal to three-fourths of the circumference of the circle. Find the area of the rectangle. | [
"\\(\\frac{9\\pi^2}{32} - 2\\)."
] | [
"A rectangle is inscribed in a circle of unit radius. The perimeter of the rectangle is equal to three-fourths of the circumference of the circle. Find the area of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-68b2cec3714c4505814446b88cc37b52ced3f4a67ba7d360a7a15475677b4f21 | A rectangle is inscribed in a circle of unit radius. The rectangle has an area equal to half the area of the circle. What is the perimeter of the rectangle? | [
"\\(2\\sqrt{4 + \\pi}\\)"
] | [
"A rectangle is inscribed in a circle of unit radius. The rectangle has an area equal to half the area of the circle. What is the perimeter of the rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f22646164113944c087923fd931f8aeb9fa69655c98f4abe3c6a4f4a2adfdef3 | A rectangle is inscribed in a circle with an area of $32\pi$. The area of the rectangle is $34$. Find the perimeter of the rectangle. | [
"28"
] | [
"A rectangle is inscribed in a circle with an area of $32\\pi$. The area of the rectangle is $34$. Find the perimeter of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-662ddcfe30057abae39932fa676165629fc1be4b254fe52833889e9bafe63b88 | A rectangle is inscribed in a circle with radius 10, with one of its sides along the diameter of the circle. If the rectangle's area is maximized, what is the ratio of the rectangle's length to its width? Express your answer as a simplified fraction. | [
"2"
] | [
"A rectangle is inscribed in a circle with radius 10, with one of its sides along the diameter of the circle. If the rectangle's area is maximized, what is the ratio of the rectangle's length to its width? Express your answer as a simplified fraction."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-e800f0ad7edf55ed2af757b35e8fb78eed86a8e5be0dee0a30479ed2ff43a60c | A rectangle is inscribed in a circle with radius 10, with one of its sides along the diameter of the circle. What must be the ratio of the rectangle's length and width for its area to be the greatest? | [
"1:2",
"\\(\\frac{1}{2}\\)"
] | [
"A rectangle is inscribed in a circle with radius 10, with one of its sides along the diameter of the circle. What must be the ratio of the rectangle's length and width for its area to be the greatest?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 2
} | 47 | 0 | 0 |
omr-ws-dd28952fe59c0130bcd881aa0f32e5776fb7701e8e68299df7512b0f971c5ad6 | A rectangle is inscribed in a sector of a circle of radius $1$ and angle $\theta = \frac{\pi}{3}$. Find the maximum area of such a rectangle. | [
"$\\frac{1}{2\\sqrt{3}}$"
] | [
"A rectangle is inscribed in a sector of a circle of radius $1$ and angle $\\theta = \\frac{\\pi}{3}$. Find the maximum area of such a rectangle."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-51911f8cfae43fe5accb13b4767cc05ecd621a831d7bf6aa8a0b5c09d6ea004b | A rectangle is inscribed in a sector of a circle of radius $1$ with a central angle of $\theta = \pi / 3$. What is the maximum possible area for the rectangle? | [
"2 - \\sqrt{3}"
] | [
"A rectangle is inscribed in a sector of a circle of radius $1$ with a central angle of $\\theta = \\pi / 3$. What is the maximum possible area for the rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-73d92045db592719f0904e07729d7ce2c2b1d8a614c77faede5981d2349bfdb1 | A rectangle is inscribed in a semicircle of radius 2. Let \( P = (x, y) \) be the point in quadrant 1 that is a vertex of the rectangle and is on the circle. Express the area \( A \) of the rectangle as a function of \( x \). | [
"\\( 2x \\sqrt{4 - x^2} \\)"
] | [
"A rectangle is inscribed in a semicircle of radius 2. Let \\( P = (x, y) \\) be the point in quadrant 1 that is a vertex of the rectangle and is on the circle. Express the area \\( A \\) of the rectangle as a function of \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-b2c6ecaf2394b5b49f03cfdcc2ee8a3262528ec9372e4906a0cb6ceb60195c31 | A rectangle is inscribed in a triangle such that its base coincides with the base $b$ of the triangle. The altitude of the triangle is $h$, and the altitude $x$ of the rectangle is half the base of the rectangle. Determine $x$. | [
"\\( x = \\frac{bh}{2h + b} \\)"
] | [
"A rectangle is inscribed in a triangle such that its base coincides with the base $b$ of the triangle. The altitude of the triangle is $h$, and the altitude $x$ of the rectangle is half the base of the rectangle. Determine $x$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-cdaa56f85f738a01e5964247a4a3e7acb82d03c6d15e01e4d09aadd36462f0eb | A rectangle is inscribed in a triangle such that its base coincides with the base \( b \) of the triangle. The altitude of the triangle is \( h \), and the altitude \( x \) of the rectangle is half the base of the rectangle. Determine \( x \). | [
"\\( x = \\frac{bh}{2h + b} \\)"
] | [
"A rectangle is inscribed in a triangle such that its base coincides with the base \\( b \\) of the triangle. The altitude of the triangle is \\( h \\), and the altitude \\( x \\) of the rectangle is half the base of the rectangle. Determine \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-9db5d0c723b9892d77d822618287d397c6f7206275f10b8cee4350340c9e80d1 | A rectangle is inscribed in a triangle with sides $5$, $\frac{5}{3}\sqrt{3}$, and $\frac{10}{3}\sqrt{3}$. Find the maximum area of the rectangle. | [
"\\(\\frac{15 - 5\\sqrt{3}}{2}\\)"
] | [
"A rectangle is inscribed in a triangle with sides $5$, $\\frac{5}{3}\\sqrt{3}$, and $\\frac{10}{3}\\sqrt{3}$. Find the maximum area of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-33a5ba271ca27fd12d76049c5ebe01d2693a147a5034f83eca9dc8a30f2e0c7c | A rectangle is inscribed in an isosceles right triangle so that two of its vertices are on the hypotenuse, and the other two are on the legs. Find the sides of the rectangle if they are in a ratio $5:2$ and the triangle's hypotenuse is $45$. | [
"\\(\\frac{225\\sqrt{2}}{14}\\) and \\(\\frac{45\\sqrt{2}}{7}\\)"
] | [
"A rectangle is inscribed in an isosceles right triangle so that two of its vertices are on the hypotenuse, and the other two are on the legs. Find the sides of the rectangle if they are in a ratio $5:2$ and the triangle's hypotenuse is $45$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 10
} | 15 | 0 | 0 |
omr-ws-b801cd98d8b6ec571a5bc635e9dded5783512a126c3075e2ff63d0981449ba5b | A rectangle is inscribed in the parabola \( y = 1 - x^2 \) with its base on the x-axis and its upper two vertices on the parabola. Find the maximum area of such a rectangle and give its dimensions. | [
"\\(\\frac{4\\sqrt{3}}{9}\\)"
] | [
"A rectangle is inscribed in the parabola \\( y = 1 - x^2 \\) with its base on the x-axis and its upper two vertices on the parabola. Find the maximum area of such a rectangle and give its dimensions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 15
} | 18 | 0 | 0 |
omr-ws-6ee2b6e91f3c847d3a2d0a033d83dc4a9898aea244c4d2894d4966a12d516780 | A rectangle is inscribed inside a triangle with base \( b \) and height \( h \). Find the ratio of the area of the triangle to the area of the rectangle when the area of the rectangle is at a maximum. | [
"\\(\\frac{1}{2}\\)"
] | [
"A rectangle is inscribed inside a triangle with base \\( b \\) and height \\( h \\). Find the ratio of the area of the triangle to the area of the rectangle when the area of the rectangle is at a maximum."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b31b3792798a58ab8a53152928ba7e36f4f16d5f62bd58dc3c30ff641b73b3d | A rectangle is partitioned into 100 L-trominos and some $1 \times 3$ tiles. Someone chose 96 of these L-trominoes and merged them into pairs into 48 $2 \times 3$ rectangles. Given that the remaining 4 L-trominoes can be translated and merged into two $2 \times 3$ rectangles, calculate the total number of distinct ways ... | [
"12"
] | [
"A rectangle is partitioned into 100 L-trominos and some $1 \\times 3$ tiles. Someone chose 96 of these L-trominoes and merged them into pairs into 48 $2 \\times 3$ rectangles. Given that the remaining 4 L-trominoes can be translated and merged into two $2 \\times 3$ rectangles, calculate the total number of distin... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-86deb21460714d620ffaa48e5e48ddd0d4bb07a1fa4cbb2735f9af4c5da1bfeb | A rectangle is partitioned into 5 regions. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible? | [
"540"
] | [
"A rectangle is partitioned into 5 regions. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88786013818483d08d0db8356b3ee255cae4f612ea8aacbb7c8b4eba2d641e9d | A rectangle is partitioned into finitely many small rectangles. A point is called a cross point if it belongs to four different small rectangles. A segment is called maximal if there is no other segment containing it. Let \( R \) be the number of small rectangles, \( C \) be the number of cross points, and \( M \) be t... | [
"3"
] | [
"A rectangle is partitioned into finitely many small rectangles. A point is called a cross point if it belongs to four different small rectangles. A segment is called maximal if there is no other segment containing it. Let \\( R \\) be the number of small rectangles, \\( C \\) be the number of cross points, and \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-6587b1847cb230667224cd812352b531b6de6f94d7da94654c1e4f7c6740dafd | A rectangle is placed inside a quarter circle of radius $1$, such that its vertices all lie on the perimeter of the quarter circle and one vertex coincides with the centre of the (whole) circle. Let the perimeter of such a rectangle be $P$. (a) Calculate the value of $P$ when the rectangle is a square. (b) Determine th... | [
"2\\sqrt{2}"
] | [
"A rectangle is placed inside a quarter circle of radius $1$, such that its vertices all lie on the perimeter of the quarter circle and one vertex coincides with the centre of the (whole) circle. Let the perimeter of such a rectangle be $P$.\n(a) Calculate the value of $P$ when the rectangle is a square.\n(b) Deter... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de756a589df1acf665854193cf01c98b02bd7d887c0951050dd21636aa0fba3d | A rectangle is split into 11 smaller rectangles, all of which are similar to the initial rectangle. The smallest rectangles have a lower side length of 1. Determine the perimeter of the large rectangle. | [
"30"
] | [
"A rectangle is split into 11 smaller rectangles, all of which are similar to the initial rectangle. The smallest rectangles have a lower side length of 1. Determine the perimeter of the large rectangle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8cd74392cf7ae3f7274207397b7d18916b26227de7ee81623b35e5fde8523a94 | A rectangle is split into 9 squares. If the side lengths of the rectangle are relatively prime, what is the perimeter of the rectangle? | [
"130"
] | [
"A rectangle is split into 9 squares. If the side lengths of the rectangle are relatively prime, what is the perimeter of the rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-209bf65cae5a17bbf98afd7ec014ea750efe60e391617f8a88b4821c19ec0753 | A rectangle measures 3233 inches by 3232 inches. The total area of a huge tile board made of these rectangles is \(3233^{3231}\) square inches. Find the difference between the area of a single rectangle and the sum of all the digits of \(3233^{3231}\). | [
"3"
] | [
"A rectangle measures 3233 inches by 3232 inches. The total area of a huge tile board made of these rectangles is \\(3233^{3231}\\) square inches. Find the difference between the area of a single rectangle and the sum of all the digits of \\(3233^{3231}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-9605b0887f54fdd40e4365c952245b4e894d941b0629cc730d33edb3d93ed2bd | A rectangle measures 6 meters by 10 meters. Drawn on each side of the rectangle is a semicircle that has the endpoints of its diameter on the vertices of the rectangle. What percent larger is the area of the large semicircles than the area of the small semicircles? Express your answer to the nearest whole number. | [
"178\\%"
] | [
"A rectangle measures 6 meters by 10 meters. Drawn on each side of the rectangle is a semicircle that has the endpoints of its diameter on the vertices of the rectangle. What percent larger is the area of the large semicircles than the area of the small semicircles? Express your answer to the nearest whole number."... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-539d2b71403f64c89778998f80b972d470671e943d68e9fb5b5ee32af86e0cab | A rectangle of dimensions $A \times B$ is divided into smaller rectangles of dimensions $1 \times N$. Given that $A$ and $B$ are positive integers and $N$ is a fixed positive integer, find the sum of all possible values of $N$ if $A = 120$ and $B = 168$. | [
"780"
] | [
"A rectangle of dimensions $A \\times B$ is divided into smaller rectangles of dimensions $1 \\times N$. Given that $A$ and $B$ are positive integers and $N$ is a fixed positive integer, find the sum of all possible values of $N$ if $A = 120$ and $B = 168$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ffc598e15c59cf71ca4b418af2a7ba4f6b5674e3a1f6b2f8b288fda9cefca68 | A rectangle of dimensions $a \times b$ ($a > b$) is cut into right triangles such that any common side of two triangles is a leg for one and a hypotenuse for the other. Let $r$ be the smallest possible value of $\frac{a}{b}$ for which such a configuration exists. Find the value of $r^2$. | [
"\\dfrac{3 + \\sqrt{5}}{2}"
] | [
"A rectangle of dimensions $a \\times b$ ($a > b$) is cut into right triangles such that any common side of two triangles is a leg for one and a hypotenuse for the other. Let $r$ be the smallest possible value of $\\frac{a}{b}$ for which such a configuration exists. Find the value of $r^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-f2ec1466d51154d550917cc324ffd98f310769ce8068f9427aafd5fd23ba8bb0 | A rectangle of dimensions \( m \times n \) is divided into figures. The figures are of two types: 1. A \( 3 \times 4 \) block: ``` *** *** *** *** ``` 2. A \( 2 \times 3 \) block with a specific pattern: ``` o** o** **o **o ``` (with their rotations and reflections). If the rectangle can be completely covered by these ... | [
"24"
] | [
"A rectangle of dimensions \\( m \\times n \\) is divided into figures. The figures are of two types:\n1. A \\( 3 \\times 4 \\) block:\n```\n***\n***\n***\n***\n```\n2. A \\( 2 \\times 3 \\) block with a specific pattern:\n```\no**\no**\n**o\n**o\n```\n(with their rotations and reflections). If the rectangle can be... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-60882b288dbdcaebe39bfab188b1d956133d595c694722bde6a7d013912943b1 | A rectangle of paper 20 cm wide and 30 cm long is folded to form a figure with two gray rectangles $A$ and $B$, and four white triangles. If the perimeter of rectangle $A$ is 34 cm, determine the area of rectangle $B$ in cm². | [
"180"
] | [
"A rectangle of paper 20 cm wide and 30 cm long is folded to form a figure with two gray rectangles $A$ and $B$, and four white triangles. If the perimeter of rectangle $A$ is 34 cm, determine the area of rectangle $B$ in cm²."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-a8c1528dc8b8b18ebe6ded4d61880169caa7349b292419d979be346e2873de6e | A rectangle of paper with dimensions 3 cm by 9 cm is folded along a straight line such that two opposite vertices coincide, forming a pentagon. Calculate the area of the pentagon. | [
"\\dfrac{39}{2}"
] | [
"A rectangle of paper with dimensions 3 cm by 9 cm is folded along a straight line such that two opposite vertices coincide, forming a pentagon. Calculate the area of the pentagon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-0045803d379e27ffc3509846996e497a76f0ae97ea7b8a2f91c8994599d66aee | A rectangle of perimeter 30 is inscribed in a circle. What is the area of the rectangle if the radius is 7? | [
"\\(\\frac{29}{2}\\)"
] | [
"A rectangle of perimeter 30 is inscribed in a circle. What is the area of the rectangle if the radius is 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-ef3712c18164b769e6bdada4a771fe734e68acb4274f642468967b9d1724e9d9 | A rectangle of size $2021 \times 4300$ is given. A billiard ball is launched from a point A inside it along a straight line forming an angle of $45^\circ$ with the sides of the rectangle. After reaching a side, the ball is reflected at an angle of $45^\circ$; if the ball enters a corner, it leaves it along the same lin... | [
"145"
] | [
"A rectangle of size $2021 \\times 4300$ is given. A billiard ball is launched from a point A inside it along a straight line forming an angle of $45^\\circ$ with the sides of the rectangle. After reaching a side, the ball is reflected at an angle of $45^\\circ$; if the ball enters a corner, it leaves it along the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-b115bc0576cd1425cc64f8141b600c5026e8d96991881c0ff3a9d020dffdddcc | A rectangle of size $5 \times 11$ was cut along the cell lines into two identical shapes, from which you can make a square, and a number of individual cells. What is the smallest number of individual cells that could be formed? | [
"19"
] | [
"A rectangle of size $5 \\times 11$ was cut along the cell lines into two identical shapes, from which you can make a square, and a number of individual cells. What is the smallest number of individual cells that could be formed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-39b6637941b6820d8abfcc9b6539451b0e9425f7d5fcc3c0bc1d46ed5b91e69d | A rectangle swimming pool with dimensions 'l' and 'b' is situated in a grass field. A goat is tied to a corner of the pool using a rope. Determine the maximum possible area in which the goat can graze without the grazing areas in clockwise and counterclockwise directions overlapping. | [
"\\dfrac{\\pi}{2}(l^2 + b^2)"
] | [
"A rectangle swimming pool with dimensions 'l' and 'b' is situated in a grass field. A goat is tied to a corner of the pool using a rope. Determine the maximum possible area in which the goat can graze without the grazing areas in clockwise and counterclockwise directions overlapping."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-03e8c575dccae3d9f6f3df22e8ffb3767accc042d836f3b7f603ad993dc0b97d | A rectangle that is inscribed in a larger 6 by 8 rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in the 6 by 8 rectangle, the smallest p... | [
"194"
] | [
"A rectangle that is inscribed in a larger 6 by 8 rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in the 6 by 8 rectangle, the small... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-20bd8cc47e631c28c44c9e79af96a38d671050f7ba14da985514aaffcde91d06 | A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter ... | [
"448"
] | [
"A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perim... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-82de086f1da55beeda8ae85a94ced1dc140ca1062dfea113b5913124a7b8a94b | A rectangle with a diagonal of length \( x \) is twice as long as it is wide. What is the area of the rectangle? | [
"\\dfrac{2x^2}{5}"
] | [
"A rectangle with a diagonal of length \\( x \\) is twice as long as it is wide. What is the area of the rectangle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-af511a8f111d1a017ca44cfdc863195270648442d809c7151af2bc9062a77800 | A rectangle with a perimeter of 176 is divided into five congruent rectangles. What is the perimeter of one of the five congruent rectangles? | [
"80"
] | [
"A rectangle with a perimeter of 176 is divided into five congruent rectangles. What is the perimeter of one of the five congruent rectangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-067f1fbed97f063a295b164543a287e3fe2f8487d9b3baadaa1dc31a20846c8d | A rectangle with area $n$, where $n$ is a positive integer, can be divided into $n$ equal squares and also into $n + 98$ equal squares. Find the dimensions of this rectangle. | [
"\\( 3 \\times 42, 6 \\times 21, 24 \\times 24 \\)"
] | [
"A rectangle with area $n$, where $n$ is a positive integer, can be divided into $n$ equal squares and also into $n + 98$ equal squares. Find the dimensions of this rectangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 5
} | 10 | 0 | 0 |
omr-ws-480cba1737d41ad189384ea7c09b77402accf312aeb9367e40ecd9f39b6f7975 | A rectangle with base 12 cm and height 6 cm meets a right triangle with base 3 cm and height 6 cm to form a quadrilateral. This quadrilateral is divided into two rectangles $A$ and $B$, and a trapezoid $C$, such that these three new quadrilaterals have the same area. Calculate the perimeter of rectangle $A$ in cm. | [
"21cm"
] | [
"A rectangle with base 12 cm and height 6 cm meets a right triangle with base 3 cm and height 6 cm to form a quadrilateral. This quadrilateral is divided into two rectangles $A$ and $B$, and a trapezoid $C$, such that these three new quadrilaterals have the same area. Calculate the perimeter of rectangle $A$ in cm.... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-aed24ca4c7ec8a33e3fff16946d1b8bf6f867ca31d6b965ac5921af832c2fa74 | A rectangle with dimensions $16 \times 20$ is divided into $10$ smaller equal rectangles. What is the perimeter of each of the $10$ smaller rectangles? | [
"24"
] | [
"A rectangle with dimensions $16 \\times 20$ is divided into $10$ smaller equal rectangles. What is the perimeter of each of the $10$ smaller rectangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-77c91f9581d4bac4b210ee029fd635e7fa2cae58e04f8bd5400a71252b8a96b2 | A rectangle with dimensions $p \times q$, where $p$ and $q$ are relatively coprime positive integers with $p < q$, is divided into $1 \times 1$ squares. The diagonal from the bottom-left vertex to the top-right vertex cuts triangles from some of these squares. Find the sum of the perimeters of all such triangles. | [
"\\((p+1)\\left( \\sqrt{\\left( \\frac{p}{q}\\right)^2+1}+1+\\frac{p}{q}\\right)\\)"
] | [
"A rectangle with dimensions $p \\times q$, where $p$ and $q$ are relatively coprime positive integers with $p < q$, is divided into $1 \\times 1$ squares. The diagonal from the bottom-left vertex to the top-right vertex cuts triangles from some of these squares. Find the sum of the perimeters of all such triangles... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-59628dd7625f647590f9310eb7f7c75f10c601892a2ef8742fdbb1794c5456be | A rectangle with dimensions 1524 (width) by 1200 (height) is rotated 60° clockwise about its center. What are the dimensions of the smallest possible rectangle, also resting flat, required to completely contain the rotated rectangle? | [
"1802 width, 1920 height"
] | [
"A rectangle with dimensions 1524 (width) by 1200 (height) is rotated 60° clockwise about its center. What are the dimensions of the smallest possible rectangle, also resting flat, required to completely contain the rotated rectangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 18
} | 36 | 0 | 0 |
omr-ws-7ab32c13e41a50cdea7104de659a0d5a90a324553ae1d4a2f645317b7bc93cc8 | A rectangle with dimensions 5m by 12m is surrounded by a path of uniform width. How wide will this path be if the area of the rectangle is the same as the area of the path? | [
"3 meters"
] | [
"A rectangle with dimensions 5m by 12m is surrounded by a path of uniform width. How wide will this path be if the area of the rectangle is the same as the area of the path?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-229b2d0bddcdc9e28ccbc1e369cf81c9b7282fa19a2dd70643500a20df476fc7 | A rectangle with dimensions AxB lies in the 2D plane. You are viewing the plane from an angle that makes the rectangle appear as a line of minimum length. What is the minimum length of the line in terms of A and B? | [
"\\min(A, B)"
] | [
"A rectangle with dimensions AxB lies in the 2D plane. You are viewing the plane from an angle that makes the rectangle appear as a line of minimum length. What is the minimum length of the line in terms of A and B?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-417ef303894c761dcba40d4afa47790abaa69098be513e4ad9b02fde64f7f051 | A rectangle with integer dimensions has a perimeter of 122 cm. It is divided into \( n \) equal smaller rectangles, each with a perimeter of 64 cm and with sides of integer length. What are the possible dimensions of the large rectangle? | [
"\\( (31, 30) \\)"
] | [
"A rectangle with integer dimensions has a perimeter of 122 cm. It is divided into \\( n \\) equal smaller rectangles, each with a perimeter of 64 cm and with sides of integer length. What are the possible dimensions of the large rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 3
} | 6 | 0 | 0 |
omr-ws-c7e51acd10b271ecec3a35558a22082dfae8899b76cbf462612fd7a7e89da564 | A rectangle with integer dimensions has an area numerically twice its perimeter. What is the maximum numerical value of its area? | [
"64"
] | [
"A rectangle with integer dimensions has an area numerically twice its perimeter. What is the maximum numerical value of its area?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c424efea23bb48a786a83116cf6272937bcdfbced1715786ec077633fe293315 | A rectangle with integer side lengths is cut into squares with side lengths $2$, $2$, $3$, $3$, $5$, $5$, $7$, $7$, $8$, $8$, $9$, $9$. Determine the perimeter of the original rectangle. | [
"90"
] | [
"A rectangle with integer side lengths is cut into squares with side lengths $2$, $2$, $3$, $3$, $5$, $5$, $7$, $7$, $8$, $8$, $9$, $9$. Determine the perimeter of the original rectangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-98be45d18793283911566de77a7b91fd3e77beac810c693b6fe6b29f229609cf | A rectangle with integer sides \( m \) and \( n \) is divided into unit squares. A ray of light enters the rectangle through one of the vertices and travels along the angle bisector of the right angle. The ray is reflected successively off the sides of the rectangle. How many unit squares does the ray of light cross? | [
"\\( m + n - \\gcd(m, n) \\)"
] | [
"A rectangle with integer sides \\( m \\) and \\( n \\) is divided into unit squares. A ray of light enters the rectangle through one of the vertices and travels along the angle bisector of the right angle. The ray is reflected successively off the sides of the rectangle. How many unit squares does the ray of light... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-4f61e600a9027d34e0cd611e9e38638b86b5c2f8e6fcbb74cc4904f070790273 | A rectangle with integer sides \(ABCD\) and area \(756\) is divided into three rectangles with integer sides: \(AFHG\), \(FBIH\), and \(GICD\). The area of \(FBIH\) is three times that of \(AFHG\), and the area of \(GICD\) is five times the area of \(AFHG\). Determine all possibilities for the length of \(AD\). | [
"756"
] | [
"A rectangle with integer sides \\(ABCD\\) and area \\(756\\) is divided into three rectangles with integer sides: \\(AFHG\\), \\(FBIH\\), and \\(GICD\\). The area of \\(FBIH\\) is three times that of \\(AFHG\\), and the area of \\(GICD\\) is five times the area of \\(AFHG\\). Determine all possibilities for the le... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-8d6d07643254365d324986e25260749ff6c28d394e4e046a5de15fa514adb46c | A rectangle with integer-length sides and area $36$ is cut from a square with sides of length $20$, such that one of the sides of the rectangle forms part of one of the sides of the square. What is the largest possible perimeter of the remaining shape? | [
"116"
] | [
"A rectangle with integer-length sides and area $36$ is cut from a square with sides of length $20$, such that one of the sides of the rectangle forms part of one of the sides of the square. What is the largest possible perimeter of the remaining shape?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 40,
"genselect": 4,
"tir": 2
} | 46 | 0 | 0 |
omr-ws-c5d96bac52085f76caedafd388eb0b72fc99402d9e63fe82a14bb696e721dfac | A rectangle with length $\frac{\pi}{4}$ and height $4$ is bisected by the $x$-axis and is positioned in the first and fourth quadrants, with its leftmost edge on the $y$-axis. The graph of $y = \sin{x} + C$ divides the area of the rectangle in half. Compute the value of $C$. | [
"\\(\\frac{4}{\\pi} \\left(\\frac{\\sqrt{2}}{2} - 1\\right)\\)"
] | [
"A rectangle with length $\\frac{\\pi}{4}$ and height $4$ is bisected by the $x$-axis and is positioned in the first and fourth quadrants, with its leftmost edge on the $y$-axis. The graph of $y = \\sin{x} + C$ divides the area of the rectangle in half. Compute the value of $C$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-1a83823179f8a63fc3160b2cd4b5952c2ecce25c385bb6fa72b9613b17f399f8 | A rectangle with length \( x \) has an area of 3, and \( x^3 d = 20 \), where \( d \) is the length of the diagonal of the rectangle. Find \( d \). | [
"2.5"
] | [
"A rectangle with length \\( x \\) has an area of 3, and \\( x^3 d = 20 \\), where \\( d \\) is the length of the diagonal of the rectangle. Find \\( d \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-779ff80c93172df6853b8ba18f735ec14dc873174870fd75489e78b3297c7f6d | A rectangle with one side of length 1 is divided into 4 smaller rectangles by perpendicular lines. The areas of three of these smaller rectangles are at least 1, and the area of the fourth rectangle is at least 2. Let \( L \) be the length of the other side of the rectangle. Find the minimum possible value of \( L \). | [
"3 + 2\\sqrt{2}"
] | [
"A rectangle with one side of length 1 is divided into 4 smaller rectangles by perpendicular lines. The areas of three of these smaller rectangles are at least 1, and the area of the fourth rectangle is at least 2. Let \\( L \\) be the length of the other side of the rectangle. Find the minimum possible value of \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-4a7fcf04836764a7e8db1db2c428de101523a6cab136bf59de4532263af1c95a | A rectangle with side lengths $1$ and $3$, a square with side length $1$, and a rectangle $R$ are inscribed inside a larger square. The sum of all possible values for the area of $R$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | [
"67"
] | [
"A rectangle with side lengths $1$ and $3$, a square with side length $1$, and a rectangle $R$ are inscribed inside a larger square. The sum of all possible values for the area of $R$ can be written in the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a8738763ca2a302d84053d4b1f0dbb931d283b699ce2e09a6b46aed360f93b7 | A rectangle with side lengths $3$ and $4$ is rotated around its diagonal. The volume of the resulting figure can be expressed in the form $\frac{a\pi}{b}$, where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a+b$ is divided by $1000$. | [
"589"
] | [
"A rectangle with side lengths $3$ and $4$ is rotated around its diagonal. The volume of the resulting figure can be expressed in the form $\\frac{a\\pi}{b}$, where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a+b$ is divided by $1000$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-346f74b0d928a62b7269b046908fe20ca7c0ffa4805f9b5f6635fb0422ac7840 | A rectangle with side lengths $6$ and $8$ is divided into $6 \times 8$ unit squares, and each unit square is further divided into two isosceles right triangles by its diagonal. Circular disks with a diameter of $\frac{1}{2}$ are to be placed in the plane of the rectangle so that each of the $96$ triangular surfaces has... | [
"24"
] | [
"A rectangle with side lengths $6$ and $8$ is divided into $6 \\times 8$ unit squares, and each unit square is further divided into two isosceles right triangles by its diagonal. Circular disks with a diameter of $\\frac{1}{2}$ are to be placed in the plane of the rectangle so that each of the $96$ triangular surfa... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-10e6ce62f1bec06fc286b681da57d1356d25c7fa4b0a2c5637ed67a6008ceb42 | A rectangle with side lengths $a_0, b_0$ is dissected into smaller rectangles with side lengths $a_k, b_k, 1 \le k \le n$. The sides of smaller rectangles are parallel to the corresponding sides of the big rectangle. Calculate the minimum value of $\sum_{k=1}^{n} |\sin a_k \sin b_k| - |\sin a_0 \sin b_0|$ for a given d... | [
"0"
] | [
"A rectangle with side lengths $a_0, b_0$ is dissected into smaller rectangles with side lengths $a_k, b_k, 1 \\le k \\le n$. The sides of smaller rectangles are parallel to the corresponding sides of the big rectangle. Calculate the minimum value of $\\sum_{k=1}^{n} |\\sin a_k \\sin b_k| - |\\sin a_0 \\sin b_0|$ f... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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