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-cd4c79f60148b10d65f46426d5fb7be0a6063526ed55ca77f5d86384a3b7382b | A right triangle is located so that its hypotenuse lies in a plane $\pi $ and the legs form with this plane the angles $\alpha $ and $\beta $, respectively. Find the angle between the plane of the triangle and the $\pi $ plane. | [
"$\\arcsin \\sqrt {\\sin ^2 \\alpha +\\sin ^2 \\beta }$"
] | [
"A right triangle is located so that its hypotenuse lies in a plane $\\pi $ and the legs form with this plane the angles $\\alpha $ and $\\beta $, respectively. Find the angle between the plane of the triangle and the $\\pi $ plane."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b66a98d213f95e369be8a26c8092461b3ad87d00c15367cd07aa1c6280dece2b | A right triangle with integer leg lengths is called "cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles? | [
"118"
] | [
"A right triangle with integer leg lengths is called \"cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d14f4a17e6838397a96bc914a6b945613face40d41ca84606c62ada574e30394 | A right triangle with integer sides has a perimeter (measured in zugs) numerically equal to its area (measured in square-zugs). Find all possible values for this triangle’s area. | [
"24, 30"
] | [
"A right triangle with integer sides has a perimeter (measured in zugs) numerically equal to its area (measured in square-zugs). Find all possible values for this triangle’s area."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
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"cot": 1,
"tir": 22
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omr-ws-6d55cc8ecd84d3c492c98b3740c84f61b8dbe47fcb39f522aeabfbfe828da2b3 | A right triangle with integral side lengths and one angle of 15 degrees has a hypotenuse with length numerically equal to the triangle's area. Find this area. | [
"8"
] | [
"A right triangle with integral side lengths and one angle of 15 degrees has a hypotenuse with length numerically equal to the triangle's area. Find this area."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
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omr-ws-d99d18520d652dfbfbc48d0ab5bc386abcaa39d16f45700170e52621b0b18731 | A right triangle with integral side lengths is rotated around the x-axis to form a solid with volume $m$. The same right triangle is rotated around the y-axis to form a solid with volume $n$. Given that the sum of the side lengths is less than 100, minimize $\frac{m}{n} + \frac{n}{m}$. | [
"\\(\\frac{841}{420}\\)"
] | [
"A right triangle with integral side lengths is rotated around the x-axis to form a solid with volume $m$. The same right triangle is rotated around the y-axis to form a solid with volume $n$. Given that the sum of the side lengths is less than 100, minimize $\\frac{m}{n} + \\frac{n}{m}$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
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"cot",
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"cot": 15,
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omr-ws-9fee2feb8aef934c3cc10b6120b5524b228ba71dd059c0f1e387f4db98951695 | A right triangle with legs $10$ and $8$ is similar to another right triangle with corresponding legs $x$ and $5$, respectively. What is the value of $x$? Express your answer as a decimal to the nearest hundredth. | [
"6.25"
] | [
"A right triangle with legs $10$ and $8$ is similar to another right triangle with corresponding legs $x$ and $5$, respectively. What is the value of $x$? Express your answer as a decimal to the nearest hundredth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ad64ccb379e8852707e180170f7131fd176824ccc1c224f7a6af2f56da76c6a0 | A right triangle with sides 3, 4, and 5 is made to move such that the ends of the hypotenuse are always on the circle \(x^2 + y^2 = \frac{25}{2}\). Find the locus of the other vertex. | [
"\\(x^2 + y^2 = \\frac{1}{2}\\) or \\(x^2 + y^2 = \\frac{49}{2}\\)"
] | [
"A right triangle with sides 3, 4, and 5 is made to move such that the ends of the hypotenuse are always on the circle \\(x^2 + y^2 = \\frac{25}{2}\\). Find the locus of the other vertex."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 3
} | 5 | 0 | 0 |
omr-ws-1ef51f6fda2c94c95eee74201532da9c04e7e93194e77c9ba54639386fb7ecf4 | A right triangle's hypotenuse has projections of length 4 and 9. Find the area of the triangle. | [
"18"
] | [
"A right triangle's hypotenuse has projections of length 4 and 9. Find the area of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-6abf90a500f113d83456fc7a6b20334202fe3c6394dbb12dab82e826098c95b3 | A right triangle, when rotated around its longer leg, forms a cone with a volume of $100\pi$. Calculate the length of the path that passes through each vertex of the triangle during a $180^\circ$ rotation around the point of intersection of its angle bisectors, given that the sum of the diameters of the inscribed and c... | [
"\\pi \\left(2\\sqrt{2} + \\sqrt{13} + 2\\sqrt{26}\\right)"
] | [
"A right triangle, when rotated around its longer leg, forms a cone with a volume of $100\\pi$. Calculate the length of the path that passes through each vertex of the triangle during a $180^\\circ$ rotation around the point of intersection of its angle bisectors, given that the sum of the diameters of the inscribe... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c44b113aa588e5d88e92530e98c8e4aa181e19db1744f5a2757dea4fed4b01f3 | A right triangular lot measures 50 m by 120 m. The owner wants to put a right triangular garden inside the lot where each side is \( x \) meters from the nearby sides of the lot. What must be the value of \( x \) such that the area of the garden is half the area of the lot? The three sides of the garden are parallel to... | [
"\\( 10 (2 - \\sqrt{2}) \\)"
] | [
"A right triangular lot measures 50 m by 120 m. The owner wants to put a right triangular garden inside the lot where each side is \\( x \\) meters from the nearby sides of the lot. What must be the value of \\( x \\) such that the area of the garden is half the area of the lot? The three sides of the garden are pa... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 43,
"genselect": 7,
"tir": 21
} | 71 | 0 | 0 |
omr-ws-d532780a788ff911167b80870f7f0ee972844e259e52d9388b6c4fd160acf93b | A right triangular prism is inscribed in a sphere of radius $R$. The base of the prism is a right-angled triangle with an acute angle $\alpha $ and its largest lateral face is a square. Find the volume of the prism. | [
"$V=\\frac{R^3\\sin 2\\alpha }{\\sqrt 2}$"
] | [
"A right triangular prism is inscribed in a sphere of radius $R$. The base of the prism is a right-angled triangle with an acute angle $\\alpha $ and its largest lateral face is a square. Find the volume of the prism."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-30165fb257e0c2ad7c77e628aee36f55c68420981f33eb6ccc6f6baeb1648f72 | A right, rectangular prism has three faces with areas of $6,8$ and $12$ square inches. What is the volume of the prism, in cubic inches? | [
"24"
] | [
"A right, rectangular prism has three faces with areas of $6,8$ and $12$ square inches. What is the volume of the prism, in cubic inches?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8e1b78af9c9c09098098327b2aa21d9316ff77fd34d4b23e53c8e9dbe46a4892 | A right-angle triangle \(ABC\) has shorter sides \(BC = a\) and \(AC = 2a\). Squares \(BDEC\), \(CFGA\), and \(AHIB\) are drawn on the outside of the triangle. The entire figure is surrounded by a rectangle \(MHIJDKFLG\) with \(MHIJ\) parallel to \(AB\). The triangle and the three squares are shaded. What is the ratio ... | [
"\\(\\frac{60}{99}\\)"
] | [
"A right-angle triangle \\(ABC\\) has shorter sides \\(BC = a\\) and \\(AC = 2a\\). Squares \\(BDEC\\), \\(CFGA\\), and \\(AHIB\\) are drawn on the outside of the triangle. The entire figure is surrounded by a rectangle \\(MHIJDKFLG\\) with \\(MHIJ\\) parallel to \\(AB\\). The triangle and the three squares are sha... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-512441122a0c30fde284f398f048b627c26dffed9a7433799ad3ca1458d12d0b | A right-angled triangle \(ABC\) is given. Its cathetus \(AB\) is the base of a regular triangle \(ADB\) lying in the exterior of \(ABC\), and its hypotenuse \(AC\) is the base of a regular triangle \(AEC\) lying in the interior of \(ABC\). Lines \(DE\) and \(AB\) meet at point \(M\). The whole configuration except poin... | [
"\\( M \\) is the reflection of \\( A \\) over \\( B \\)."
] | [
"A right-angled triangle \\(ABC\\) is given. Its cathetus \\(AB\\) is the base of a regular triangle \\(ADB\\) lying in the exterior of \\(ABC\\), and its hypotenuse \\(AC\\) is the base of a regular triangle \\(AEC\\) lying in the interior of \\(ABC\\). Lines \\(DE\\) and \\(AB\\) meet at point \\(M\\). The whole ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-78dc81b22b180ead4b14e99caef7f57e9d5fa2c2c994f95e34a5921068478235 | A right-angled triangle has a perimeter of $60$ and the altitude to the hypotenuse has a length of $12$. Determine the lengths of the sides. | [
"(15, 20, 25) or (20, 15, 25)"
] | [
"A right-angled triangle has a perimeter of $60$ and the altitude to the hypotenuse has a length of $12$. Determine the lengths of the sides."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-fcc0cf8302bcc38772fa976d3d57d46efdf8bc6933483c5f7b7888145ea5c40c | A right-angled triangle has legs \(a\) and \(b\) with \(a > b\). The right angle is bisected, splitting the original triangle into two smaller triangles. Find the distance between the orthocenters of the two smaller triangles using coordinate geometry methods or otherwise. | [
"\\(\\frac{(a-b)\\sqrt{a^{2}+b^{2}}}{a+b}\\)"
] | [
"A right-angled triangle has legs \\(a\\) and \\(b\\) with \\(a > b\\). The right angle is bisected, splitting the original triangle into two smaller triangles. Find the distance between the orthocenters of the two smaller triangles using coordinate geometry methods or otherwise."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 12
} | 34 | 0 | 0 |
omr-ws-d01157bc9cccabe6f88a2578e9a902a4592da4561b90e05b0b5e9afc300e7f68 | A right-angled triangle has the property that when a square is drawn externally on each side of the triangle, the six vertices of the squares that are not vertices of the triangle are concyclic. Given that the area of the triangle is $9$ cm$^2$, determine the lengths of the sides of the triangle. | [
"\\(3\\sqrt{2}, 3\\sqrt{2}, 6\\)"
] | [
"A right-angled triangle has the property that when a square is drawn externally on each side of the triangle, the six vertices of the squares that are not vertices of the triangle are concyclic. Given that the area of the triangle is $9$ cm$^2$, determine the lengths of the sides of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 12
} | 13 | 0 | 0 |
omr-ws-6219aca21bc414a1563c2925c52fb9813d316478270c2278cd758cd60cc71f83 | A right-angled triangle is divided by the median lowered on the hypotenuse into two triangles with circles inscribed in them. The segment of the outer tangent of these circles is 11, and the segment of the inner tangent is 7. Find the perimeter of the original right triangle. | [
"140"
] | [
"A right-angled triangle is divided by the median lowered on the hypotenuse into two triangles with circles inscribed in them. The segment of the outer tangent of these circles is 11, and the segment of the inner tangent is 7. Find the perimeter of the original right triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-dc52451ad754136352d94c9146e09376f80caefb6dc152644531e7236d8e7c1b | A right-angled triangle is inscribed in the parabola $y = x^2$, with its hypotenuse parallel to the $x$-axis. What can be the altitude of the triangle drawn to the hypotenuse? | [
"1"
] | [
"A right-angled triangle is inscribed in the parabola $y = x^2$, with its hypotenuse parallel to the $x$-axis. What can be the altitude of the triangle drawn to the hypotenuse?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9084aaa8e67be1febcf5f17be16ef09dd851f08afae5847129f2bf6b453a9ebe | A right-angled triangle with sides $3, 4, 5$ lies inside the circle $2x^2 + 2y^2 = 25$. The triangle is moved inside the circle such that its hypotenuse always forms a chord of the circle. What is the locus of the vertex opposite to the hypotenuse? | [
"\\( x^2 + y^2 = \\frac{1}{2} \\)"
] | [
"A right-angled triangle with sides $3, 4, 5$ lies inside the circle $2x^2 + 2y^2 = 25$. The triangle is moved inside the circle such that its hypotenuse always forms a chord of the circle. What is the locus of the vertex opposite to the hypotenuse?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 6,
"tir": 5
} | 17 | 0 | 0 |
omr-ws-01e6a978dad58d111a9eff4f2e7e136aada9e799b4cfd51f13b4d188cbbe48af | A rigid square wire loop with a side of $1$ cm was cut into pieces at exactly three points of the contour. What is the largest area that the new contour can cover? | [
"\\(1 \\text{ cm}^2\\)"
] | [
"A rigid square wire loop with a side of $1$ cm was cut into pieces at exactly three points of the contour. What is the largest area that the new contour can cover?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-360f35fe1f766e560b2399dc69db95fa8a19d1e122a77d8b6322ba9a870c000e | A ring is formed by bending a cylinder and gluing the faces together. The top and bottom portions of the cylinder are cut at an angle to form the ring. Given that the inside radius of the ring is 10 and the radius of the cylinder is 4, what is the volume of the ring? | [
"448\\pi^2"
] | [
"A ring is formed by bending a cylinder and gluing the faces together. The top and bottom portions of the cylinder are cut at an angle to form the ring. Given that the inside radius of the ring is 10 and the radius of the cylinder is 4, what is the volume of the ring?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d20283d79330db881eee5aeba379793870e41cf26e7e71bc63429e8467cf5d94 | A ring of six identical spheres, each tangent to the spheres next to it, is placed on the surface of a larger sphere so that each sphere in the ring is tangent to the larger sphere at six evenly spaced points in a circle. If the radius of the larger sphere is \( 5 \), and the circle containing the evenly spaced points ... | [
"\\dfrac{15}{7}"
] | [
"A ring of six identical spheres, each tangent to the spheres next to it, is placed on the surface of a larger sphere so that each sphere in the ring is tangent to the larger sphere at six evenly spaced points in a circle. If the radius of the larger sphere is \\( 5 \\), and the circle containing the evenly spaced ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8db11d71b06ef1ab630cda17f8360b04a8bea797ffb71d1201e7b02e9bf43e07 | A rising number is a positive integer, each digit of which is larger than each of the digits to its left, and no digit of which is 0. For example, 12689 is a rising number. When all five-digit rising numbers are listed from smallest to largest, the 97th number in the list does not contain the digit | [
"5"
] | [
"A rising number is a positive integer, each digit of which is larger than each of the digits to its left, and no digit of which is 0. For example, 12689 is a rising number. When all five-digit rising numbers are listed from smallest to largest, the 97th number in the list does not contain the digit"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-aaf4c03ba5042421d562d1539f1e314f2adf1cbf85d887e940972cfd699ade49 | A river flows at speed \( u \). A boat has speed \( v \) relative to the water. If its velocity is at an angle \( \alpha \) relative to the direction of the river, what angle \( \alpha \) minimizes the time taken to cross the river? | [
"\\(\\alpha = 90^\\circ\\)"
] | [
"A river flows at speed \\( u \\). A boat has speed \\( v \\) relative to the water. If its velocity is at an angle \\( \\alpha \\) relative to the direction of the river, what angle \\( \\alpha \\) minimizes the time taken to cross the river?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-2a48a60835f196f2a40cfa2b6828a82352dc5aaf9372f69098c4c83e970f27a1 | A river flows at speed \( u \). A boat has speed \( v \) relative to the water. If its velocity is at an angle \( \alpha \) relative to the direction of the river, what is its speed relative to the river bank? | [
"\\(\\sqrt{ u^{2}+v^{2}+2 \\cdot u \\cdot v \\cdot \\cos \\alpha }\\)"
] | [
"A river flows at speed \\( u \\). A boat has speed \\( v \\) relative to the water. If its velocity is at an angle \\( \\alpha \\) relative to the direction of the river, what is its speed relative to the river bank?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-f2f89592ecbdec61f68f8d9a8b07dad59b1c5f323acdeeda0d8748e7b28d7d95 | A river flows between two houses \( A \) and \( B \), the houses standing some distances away from the banks. Where should a bridge be built on the river so that a person going from \( A \) to \( B \), using the bridge to cross the river, may do so by the shortest path? Assume that the banks of the river are straight a... | [
"\\frac{a d_B + b d_A}{d_A + d_B}"
] | [
"A river flows between two houses \\( A \\) and \\( B \\), the houses standing some distances away from the banks. Where should a bridge be built on the river so that a person going from \\( A \\) to \\( B \\), using the bridge to cross the river, may do so by the shortest path? Assume that the banks of the river a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-f51e6c981c4de753e550c241dc1a935f29e38c1674bd9058e1d52423278c56e5 | A river is bounded by the lines \(x=0\) and \(x=25\), with a current of 2 units/s in the positive y-direction. At \(t=0\), a mallard is at \((0, 0)\), and a wigeon is at \((25, 0)\). They start swimming with a constant speed such that they meet at \((x,22)\). The mallard has a speed of 4 units/s relative to the water, ... | [
"\\(\\frac{100}{7}\\)"
] | [
"A river is bounded by the lines \\(x=0\\) and \\(x=25\\), with a current of 2 units/s in the positive y-direction. At \\(t=0\\), a mallard is at \\((0, 0)\\), and a wigeon is at \\((25, 0)\\). They start swimming with a constant speed such that they meet at \\((x,22)\\). The mallard has a speed of 4 units/s relati... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3dcaf9067a13f2bc254e25fa6d350f1e1537f3f2b53eb74b1e8d60ef6642f4a2 | A river is flowing at 2 miles per hour. You are rowing against it at 4 miles per hour. After an hour, your hat falls into the river and you immediately turn back to row downstream at the same speed. How long will it take you to reach the hat? | [
"1"
] | [
"A river is flowing at 2 miles per hour. You are rowing against it at 4 miles per hour. After an hour, your hat falls into the river and you immediately turn back to row downstream at the same speed. How long will it take you to reach the hat?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-df0697161ccd64209fcb28968b8c27a99f68bba3daaf576783de8c94d3355020 | A riverboat theater offers bus tours to groups. Hiring the bus costs $360, to be shared equally by the group members. Theater tickets, normally $30 each, are discounted by $0.25 \times \text{number of people in the group}$. How many members must be in the group so that the cost of the theater tour (bus fare plus theate... | [
"23"
] | [
"A riverboat theater offers bus tours to groups. Hiring the bus costs $360, to be shared equally by the group members. Theater tickets, normally $30 each, are discounted by $0.25 \\times \\text{number of people in the group}$. How many members must be in the group so that the cost of the theater tour (bus fare plus... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6cabc3af03e0f8a8517785fb87d9bb25328ddfae4c76aaead9245670f9fb93bb | A road company is trying to build a system of highways in a country with 21 cities. Each highway runs between two cities. A trip is a sequence of distinct cities \(C_1, \cdots, C_n\) for which there is a highway between \(C_i\) and \(C_{i+1}\). The company wants to fulfill the following two constraints: 1. For any orde... | [
"21"
] | [
"A road company is trying to build a system of highways in a country with 21 cities. Each highway runs between two cities. A trip is a sequence of distinct cities \\(C_1, \\cdots, C_n\\) for which there is a highway between \\(C_i\\) and \\(C_{i+1}\\). The company wants to fulfill the following two constraints:\n\n... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-0664ebe0eddb5617b0d5566c016a89516227f78d22656863150e86761a5742d7 | A road has houses numbered from $1$ to $n$, where $n$ is a three-digit number. Exactly $\frac{1}{k}$ of the numbers start with the digit $2$, where $k$ is a positive integer. Find the possible values of $n$. | [
"\\( n = 110, 121, 132, 143, 154, 165, 176, 187, 198, 235, 282, 333, 444, 555, 666, 777, 888, 999 \\)"
] | [
"A road has houses numbered from $1$ to $n$, where $n$ is a three-digit number. Exactly $\\frac{1}{k}$ of the numbers start with the digit $2$, where $k$ is a positive integer. Find the possible values of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 26
} | 30 | 0 | 0 |
omr-ws-8b6b6c31dec72e467a702ca12a99330e4aac00eab15bb5b31001071b4f33a462 | A road has houses numbered from $1$ to $n,$ where $n$ is a three-digit number. Exactly $\frac{1}{k}$ of the numbers start with the digit 2, where $k$ is a positive integer. Find the possible values of $n.$ | [
"999"
] | [
"A road has houses numbered from $1$ to $n,$ where $n$ is a three-digit number. Exactly $\\frac{1}{k}$ of the numbers start with the digit 2, where $k$ is a positive integer. Find the possible values of $n.$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-9ee30ce003927fbe65f80a6a784ff6bec7cadd30ae68f22209a05d87a274f412 | A road has houses numbered from 1 to \( n \), where \( n \) is a three-digit number. Exactly \(\frac{1}{k}\) of the numbers start with the digit 2, where \( k \) is a positive integer. Find the possible values of \( n \). | [
"300, 400, 500, 600, 700, 800, 900"
] | [
"A road has houses numbered from 1 to \\( n \\), where \\( n \\) is a three-digit number. Exactly \\(\\frac{1}{k}\\) of the numbers start with the digit 2, where \\( k \\) is a positive integer. Find the possible values of \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a30ed41820549d7da8a28eb657166203cdf7954a33c05739064ba0f38d1ec78b | A road is $300$ miles long. A car, $A$, starts at noon from one end and drives toward the other end at $50$ mph. At the same time another car, $B$, driving uniformly at $100$ mph, starts from the other end together with a fly traveling $150$ mph. When the fly meets car $A$, it immediately turns and flies toward car $B$... | [
"1:48"
] | [
"A road is $300$ miles long. A car, $A$, starts at noon from one end and drives toward the other end at $50$ mph. At the same time another car, $B$, driving uniformly at $100$ mph, starts from the other end together with a fly traveling $150$ mph. When the fly meets car $A$, it immediately turns and flies toward ca... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c1727414e34a31d02b518ea5e0e921da35955562f1c2c5c5c96e57cd49bf269a | A road is to be constructed to pass through a small mountain. A two-dimensional model for the shape of the mountain is the curve with equation \( y = 24 - x^2 \). A rectangular passageway is to be made through the center of the mountain. The top vertices of the rectangle touch the curve, and the bottom vertices are on ... | [
"50"
] | [
"A road is to be constructed to pass through a small mountain. A two-dimensional model for the shape of the mountain is the curve with equation \\( y = 24 - x^2 \\). A rectangular passageway is to be made through the center of the mountain. The top vertices of the rectangle touch the curve, and the bottom vertices ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-50b8a3493d67e225f9ed355da5db5735e1413f6fa2b2d4df22c3da49a5ca4afd | A road leading from city $A$ to city $B$ first runs uphill for 3 km, then it is level for 5 km and then runs downhill for 6 km. A messenger sets out from $A$ in the direction of $B$ and having covered half the distance, finds out that he must return to pick up some packages he has forgotten. In 3 hours and 36 minutes a... | [
"Uphill 3 km/h, over the level ground 4 km/h, downhill 5 km/h"
] | [
"A road leading from city $A$ to city $B$ first runs uphill for 3 km, then it is level for 5 km and then runs downhill for 6 km. A messenger sets out from $A$ in the direction of $B$ and having covered half the distance, finds out that he must return to pick up some packages he has forgotten. In 3 hours and 36 minu... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a02da3257a1011b4e5398981714e97ec28872e903aaeb7ff986d72ba2f894b1 | A robber starts one mile ahead of a cop. The robber runs one mile per hour faster than the cop. The robber must run until he is 10 miles away from the cop to get to safety. A bloodhound runs back and forth at 10 miles per hour between the robber and the cop. How many miles will the dog run? | [
"90"
] | [
"A robber starts one mile ahead of a cop. The robber runs one mile per hour faster than the cop. The robber must run until he is 10 miles away from the cop to get to safety. A bloodhound runs back and forth at 10 miles per hour between the robber and the cop. How many miles will the dog run?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-607d8e5b14bb556c68e6d27a33f188d23b2a9276d1a85ac51c5523f0ed64c8fa | A robot generates a sequence of non-negative integers starting from $0$ such that each term after the first term is either $0$, or the previous term plus $1$, modulo $1000$. The robot stops generating the sequence when the number $999$ is written exactly three times. Let $N$ denote the number of possible sequences the ... | [
"000"
] | [
"A robot generates a sequence of non-negative integers starting from $0$ such that each term after the first term is either $0$, or the previous term plus $1$, modulo $1000$. The robot stops generating the sequence when the number $999$ is written exactly three times. Let $N$ denote the number of possible sequences... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7cbc0440cbf97e2b28237428b46c1575d810f0dd1cd7d69ec50db11957b8441b | A robot has a row of 10 boxes. A number of rocks are dropped into the rightmost box, and if in any box there are at least 7 rocks, the robot will take out 7 rocks from that box, put 5 rocks in the box immediately to the left of the original box, and discard the other 2 rocks. The robot will continue this process until ... | [
"279"
] | [
"A robot has a row of 10 boxes. A number of rocks are dropped into the rightmost box, and if in any box there are at least 7 rocks, the robot will take out 7 rocks from that box, put 5 rocks in the box immediately to the left of the original box, and discard the other 2 rocks. The robot will continue this process u... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-9ca084c5a8d0ccf6c148358177561465b4e7d091f1d99fb9e854cd435067a9d0 | A robot has a row of 10 boxes. Rocks are dropped into the rightmost box, and if any box contains at least 7 rocks, the robot removes 7 rocks from that box, places 5 rocks in the box immediately to the left, and discards the other 2 rocks. This process continues until no box has more than 6 rocks. Determine the largest ... | [
"343"
] | [
"A robot has a row of 10 boxes. Rocks are dropped into the rightmost box, and if any box contains at least 7 rocks, the robot removes 7 rocks from that box, places 5 rocks in the box immediately to the left, and discards the other 2 rocks. This process continues until no box has more than 6 rocks. Determine the lar... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-77d103fef7f6e273eedd008a1a1658f55294a6fd9dd4766639613eced2312c64 | A robot is moving through the edges of a 4-dimensional unit cube (4-cube) starting from the vertex $(0,0,0,0)$. Each turn consists of moving to an adjacent vertex. In how many ways can the robot return to $(0,0,0,0)$ after $4042$ turns? Note that it is allowed for the robot to pass through $(0,0,0,0)$ before the $4042$... | [
"\\( 2^{4041} + 2^{8081} \\)"
] | [
"A robot is moving through the edges of a 4-dimensional unit cube (4-cube) starting from the vertex $(0,0,0,0)$. Each turn consists of moving to an adjacent vertex. In how many ways can the robot return to $(0,0,0,0)$ after $4042$ turns? Note that it is allowed for the robot to pass through $(0,0,0,0)$ before the $... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 8
} | 53 | 0 | 0 |
omr-ws-476ad051d99e0416c90df21ccc16ea3d3b72c6091ee03eb5ac62571be33afad9 | A robot is placed at point $P$ on the $x$-axis but different from $(0,0)$ and $(1,0)$ and can only move along the axis either to the left or to the right. Two players play the following game. Player $A$ gives a distance and Player $B$ gives a direction, and the robot will move the indicated distance along the indicated... | [
"\\( x = \\frac{n}{2^k} \\) where \\( n \\) and \\( k \\) are positive integers, \\( n \\) is odd, and \\( 0 \\le x \\le 1 \\)"
] | [
"A robot is placed at point $P$ on the $x$-axis but different from $(0,0)$ and $(1,0)$ and can only move along the axis either to the left or to the right. Two players play the following game. Player $A$ gives a distance and Player $B$ gives a direction, and the robot will move the indicated distance along the indi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 10
} | 20 | 0 | 0 |
omr-ws-b0a3efbecbd573c912da7525b46e0d230f0896f971bb8f188ab6b1a252c141cf | A robot is standing at an integer point on a number line. The robot flips two fair coins. If either or both come up heads, it takes one step to the right, to the next larger integer. If both come up tails, it takes one step to the left, to the next smaller integer. If it ever reaches the point 0, the robot stops and sh... | [
"\\dfrac{1}{9}"
] | [
"A robot is standing at an integer point on a number line. The robot flips two fair coins. If either or both come up heads, it takes one step to the right, to the next larger integer. If both come up tails, it takes one step to the left, to the next smaller integer. If it ever reaches the point 0, the robot stops a... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f0c09a3494f891201f224e03fa4f5b4036019f3078d132faeb3ba1298e67abb6 | A robot jumps from point $A$ to point $B$ on a segment $AB$ that is $10$ meters long. The robot can move $3$ meters forward or $2$ meters backward in each step but must stay within segment $AB$ and can only land on $B$ on its final step. How many different ways are there for the robot to jump from $A$ to $B$? | [
"3"
] | [
"A robot jumps from point $A$ to point $B$ on a segment $AB$ that is $10$ meters long. The robot can move $3$ meters forward or $2$ meters backward in each step but must stay within segment $AB$ and can only land on $B$ on its final step. How many different ways are there for the robot to jump from $A$ to $B$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-4fcb5d7bd178a45cfbbc01406b4824ecca7ce5b5797162ef94aacfcf3c33e7c4 | A robot moves in the plane in a straight line, but every one meter it turns $90^{\circ}$ to the right or to the left. At some point it reaches its starting point without having visited any other point more than once, and stops immediately. What are the possible path lengths of the robot? | [
"\\(4n\\) where \\(n \\geq 3\\) and \\(n \\in \\mathbb{Z}^+\\)"
] | [
"A robot moves in the plane in a straight line, but every one meter it turns $90^{\\circ}$ to the right or to the left. At some point it reaches its starting point without having visited any other point more than once, and stops immediately. What are the possible path lengths of the robot?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8680a5e90ab8ae22c29d093c26ceb13d28fd49616437c941da2204e55fc992d8 | A robot moves on a 6x6 grid where each square has an arrow pointing to an adjacent square. When the robot moves to a square, it changes the arrow on that square to point to the previous square it came from (except for the first square). After the robot has moved over all squares twice (except the first square), calcula... | [
"35"
] | [
"A robot moves on a 6x6 grid where each square has an arrow pointing to an adjacent square. When the robot moves to a square, it changes the arrow on that square to point to the previous square it came from (except for the first square). After the robot has moved over all squares twice (except the first square), ca... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a088e6b88c34aec585ffd6ab789b89a18e6670dba9b383a946c591c2fe7e18d0 | A robot moves on a chessboard, starting from an arbitrary vertex of a $1 \times 1$ square (room). The robot can move up, down, left, or right, and it has two memory values $A$ and $B$, initially set to $0$. The robot's movements affect $A$ and $B$ as follows: - Moving up increases $A$ by $1$. - Moving down decreases $A... | [
"36"
] | [
"A robot moves on a chessboard, starting from an arbitrary vertex of a $1 \\times 1$ square (room). The robot can move up, down, left, or right, and it has two memory values $A$ and $B$, initially set to $0$. The robot's movements affect $A$ and $B$ as follows:\n- Moving up increases $A$ by $1$.\n- Moving down decr... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9af262bc9c0e624883c4402f58542860e883de869070ab942661aca044ad989f | A robot moves on the edges of an infinite chessboard, starting at a vertex \( V \). In each step, the robot moves one unit to the north, south, east, or west, and updates two counters \( A \) and \( B \) as follows: - Moving north: \( A \) is replaced by \( A+1 \) - Moving south: \( A \) is replaced by \( A-1 \) - Movi... | [
"12"
] | [
"A robot moves on the edges of an infinite chessboard, starting at a vertex \\( V \\). In each step, the robot moves one unit to the north, south, east, or west, and updates two counters \\( A \\) and \\( B \\) as follows:\n- Moving north: \\( A \\) is replaced by \\( A+1 \\)\n- Moving south: \\( A \\) is replaced ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-84f51c9d7c10eea1c128f3f105edf47c61f978ff65b9dfe868b0b1a3a4715df1 | A robot on the number line starts at $1$. During the first minute, the robot writes down the number $1$. Each minute thereafter, it moves by one, either left or right, with equal probability. It then multiplies the last number it wrote by $n/t$, where $n$ is the number it just moved to, and $t$ is the number of minutes... | [
"no_valid_answer_found"
] | [
"A robot on the number line starts at $1$. During the first minute, the robot writes down the number $1$. Each minute thereafter, it moves by one, either left or right, with equal probability. It then multiplies the last number it wrote by $n/t$, where $n$ is the number it just moved to, and $t$ is the number of mi... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31d836fbc4440ae3d9e3b17aea68a51e99bdd8da5d2ef069d639abec2bdbf80f | A robot shuffles a 9-card deck by splitting it into two piles and interleaving them. The robot repeats this shuffling procedure \( n \) times until the cards are in the same order as they were initially. Find the smallest possible value of \( n \). | [
"6"
] | [
"A robot shuffles a 9-card deck by splitting it into two piles and interleaving them. The robot repeats this shuffling procedure \\( n \\) times until the cards are in the same order as they were initially. Find the smallest possible value of \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-3c86d6689e8555afa06c05b439a5b5e9dcf6f43e9b9e6359e6f76ff41d83630d | A robot starts at the origin and travels to the point $(3,3)$. It can only change direction at a lattice point and can only go north, east, or northeast. How many different routes are possible? | [
"63"
] | [
"A robot starts at the origin and travels to the point $(3,3)$. It can only change direction at a lattice point and can only go north, east, or northeast. How many different routes are possible?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 14
} | 36 | 0 | 0 |
omr-ws-3c9a3c0c76328b2fab10e701c10c86553c62013b4794582ad5414c7a3e636cd5 | A robot starts in the bottom left corner of a $4 \times 4$ grid of squares. How many ways can it travel to each square exactly once and then return to its start if it is only allowed to move to an adjacent (not diagonal) square at each step? | [
"36"
] | [
"A robot starts in the bottom left corner of a $4 \\times 4$ grid of squares. How many ways can it travel to each square exactly once and then return to its start if it is only allowed to move to an adjacent (not diagonal) square at each step?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-1ee5c4d59559b56277946a56fafe2087ece45655f8b71daf072ca07df2968c40 | A robot starts in the bottom left corner of a 4 × 4 grid of squares. How many ways can it travel to each square exactly once and then return to its start if it is only allowed to move to an adjacent (not diagonal) square at each step? | [
"6"
] | [
"A robot starts in the bottom left corner of a 4 × 4 grid of squares. How many ways can it travel to each square exactly once and then return to its start if it is only allowed to move to an adjacent (not diagonal) square at each step?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c9f895ff48866c87ed0bf6b5f06e71d475fa764b5fad71e91dcd06bbaa06963 | A robot walks in a straight line, taking steps of either 1 meter or 2 meters, each with probability 1/2. Given that the robot is $n$ meters from the starting point, what is the probability that its last step has been 2 meters? (The answer must be in terms of only $n$) | [
"\\(\\frac{2(2^{n-1} + (-1)^{n-2})}{2^{n+1} + (-1)^n}\\)"
] | [
"A robot walks in a straight line, taking steps of either 1 meter or 2 meters, each with probability 1/2. Given that the robot is $n$ meters from the starting point, what is the probability that its last step has been 2 meters? (The answer must be in terms of only $n$)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 39,
"genselect": 7,
"tir": 8
} | 54 | 0 | 0 |
omr-ws-984b652fcbd6091d7929797a9e6d8d0504a7dd2aedfb237a439a00a86b95e621 | A robotic arm is programmed to read instructions encoded as a sequence of $0$s and $1$s. A $0$ followed immediately by a $1$ causes the arm to add a ball to the bin, while a $1$ followed by a $0$ causes the arm to remove a ball from the bin. A $1$ followed by a $1$ or a $0$ followed by a $0$ causes no action by the arm... | [
"2008"
] | [
"A robotic arm is programmed to read instructions encoded as a sequence of $0$s and $1$s. A $0$ followed immediately by a $1$ causes the arm to add a ball to the bin, while a $1$ followed by a $0$ causes the arm to remove a ball from the bin. A $1$ followed by a $1$ or a $0$ followed by a $0$ causes no action by th... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-ae1e7c229bfcaf91d398ca674292afb1cca70b838676354a691ee32f3ff47002 | A robotic lawnmower is located in the middle of a large lawn and can only move straight ahead and turn in directions that are multiples of $60^\circ$. Given that it starts facing north, determine the length of the fence required to delimit the entire part of the lawn that the robot can reach by traveling along a curve ... | [
"60"
] | [
"A robotic lawnmower is located in the middle of a large lawn and can only move straight ahead and turn in directions that are multiples of $60^\\circ$. Given that it starts facing north, determine the length of the fence required to delimit the entire part of the lawn that the robot can reach by traveling along a ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8dbe9f245eb7338e22e3574f504c670ede481f9654c0c654b9c5cfea4b0178d6 | A rock is dropped into a deep well. The sound of the impact is heard 7 seconds after the rock is dropped. Ignoring air resistance and assuming the speed of sound is 1100 feet per second, set up and solve a quadratic equation to find the depth of the well in feet. | [
"656.088 feet"
] | [
"A rock is dropped into a deep well. The sound of the impact is heard 7 seconds after the rock is dropped. Ignoring air resistance and assuming the speed of sound is 1100 feet per second, set up and solve a quadratic equation to find the depth of the well in feet."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-1cce670b4aab04b74c7c5b4952959449195fae26af8bc3146c85bb37a363fa29 | A rock travels through an \( n \times n \) board, stepping to a neighboring cell by a side at each turn, such that each cell is visited exactly once. Bob assigns the integers from 1 to \( n^2 \) to the cells in the order the rock visits them. Let \( M \) be the greatest difference between the numbers in neighboring cel... | [
"\\( 2n-1 \\)"
] | [
"A rock travels through an \\( n \\times n \\) board, stepping to a neighboring cell by a side at each turn, such that each cell is visited exactly once. Bob assigns the integers from 1 to \\( n^2 \\) to the cells in the order the rock visits them. Let \\( M \\) be the greatest difference between the numbers in nei... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 3,
"tir": 2
} | 11 | 0 | 0 |
omr-ws-a1059107af040ea40eb87fdde15ea28337ed2e522b097153f150ac3056b8cee8 | A rocket is fired vertically from a point 5 miles from the tracking station. For the first 20 seconds, its angle of elevation changes at $2^{\circ}$ per second. Find the rocket's velocity when the angle of elevation is $30^{\circ}$. | [
"\\(\\frac{2\\pi \\text{ miles}}{27 \\text{ seconds}}\\)"
] | [
"A rocket is fired vertically from a point 5 miles from the tracking station. For the first 20 seconds, its angle of elevation changes at $2^{\\circ}$ per second. Find the rocket's velocity when the angle of elevation is $30^{\\circ}$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-303f7e15a9e7f9e3075a958bed437a9bc7294788baafa3e5761257f4c9541f3f | A rocket is launched on a vertical trajectory and is tracked by a radar station that is 3 km from the launch pad. Find the vertical speed of the rocket at the instant when the distance from the rocket to the radar station is 5 km and that distance is increasing at 5,000 km/h. | [
"6,250"
] | [
"A rocket is launched on a vertical trajectory and is tracked by a radar station that is 3 km from the launch pad. Find the vertical speed of the rocket at the instant when the distance from the rocket to the radar station is 5 km and that distance is increasing at 5,000 km/h."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-933bd5758631d1ea1324e87c643752e6c33a2be4c553d55f384476a9f2c1738a | A rocket is rising vertically from a point on the ground 100 meters from an observer at ground level. The observer notes that the angle of elevation is increasing at a rate of 12 degrees per second when the angle of elevation is 60 degrees. Find the speed of the rocket at that instant. | [
"\\(\\frac{80\\pi}{3}\\)"
] | [
"A rocket is rising vertically from a point on the ground 100 meters from an observer at ground level. The observer notes that the angle of elevation is increasing at a rate of 12 degrees per second when the angle of elevation is 60 degrees. Find the speed of the rocket at that instant."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-6e47802378f311cf48a354c4566929ab825f75992115b956a387fb48861be685 | A rocket of mass 100 kg is fired straight up with a net force of 1,000,000 newtons. It experiences air friction \( f = -100 \left(\frac{\text{Ns}^2}{\text{m}^2}\right)v^2 \). At \( t=0 \), \( v=0 \), and \( y=0 \). Assume the mass of the rocket does not change. a) Write Newton's Second Law for the rocket. b) Solve anal... | [
"\\( V(t) = \\frac{100(e^{200t} - 1)}{1 + e^{200t}} \\)"
] | [
"A rocket of mass 100 kg is fired straight up with a net force of 1,000,000 newtons. It experiences air friction \\( f = -100 \\left(\\frac{\\text{Ns}^2}{\\text{m}^2}\\right)v^2 \\). At \\( t=0 \\), \\( v=0 \\), and \\( y=0 \\). Assume the mass of the rocket does not change.\n\na) Write Newton's Second Law for the ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 7
} | 8 | 0 | 0 |
omr-ws-f672850ce081b56e7cefd233ac4071a3c7f8d9f80c54120b05553bd90061062e | A rocket science class has 69 students. A teacher gives the 69 students a 42-point multiple-choice test. Each question has 4 choices. Every student guesses a random answer on the test. The teacher then curves the test scores using the following method: Let $h$ be the highest score. Let $n_a$ be the raw score of any giv... | [
"35.85"
] | [
"A rocket science class has 69 students. A teacher gives the 69 students a 42-point multiple-choice test. Each question has 4 choices. Every student guesses a random answer on the test. The teacher then curves the test scores using the following method:\nLet $h$ be the highest score.\nLet $n_a$ be the raw score of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-a1f8af99ee51e64305111654b2dbe00330600a1f7d8d0c4aaa7374e487db316a | A rocket's acceleration is modeled by the function \[ a(t) = \frac{u \beta}{m_0 - \beta t} - gt \] where $u$, $\beta$, $m_0$, and $g$ are constants. If a rocket has an initial mass $m_0 = 1000$ kg, exhaust velocity $u = 3000$ m/s, mass flow rate $\beta = 10$ kg/s, and $g = 9.8$ m/s$^2$, find the acceleration of the roc... | [
"50.2"
] | [
"A rocket's acceleration is modeled by the function \n\\[ a(t) = \\frac{u \\beta}{m_0 - \\beta t} - gt \\]\nwhere $u$, $\\beta$, $m_0$, and $g$ are constants. If a rocket has an initial mass $m_0 = 1000$ kg, exhaust velocity $u = 3000$ m/s, mass flow rate $\\beta = 10$ kg/s, and $g = 9.8$ m/s$^2$, find the accelera... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-846c236553fc4311de355aeae0b3e5dbd9bfe3000517cc036ed64e7b6da5f476 | A rod \( R \) is broken into two pieces uniformly at random, and the longer piece is then broken into two pieces uniformly at random. Let \( P(A) \) be the probability that you can form a triangle with the three resulting pieces. Calculate \( e^{P(A)} \). | [
"\\dfrac{4}{e}"
] | [
"A rod \\( R \\) is broken into two pieces uniformly at random, and the longer piece is then broken into two pieces uniformly at random. Let \\( P(A) \\) be the probability that you can form a triangle with the three resulting pieces. Calculate \\( e^{P(A)} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc7c850362dc643d85ee329c3a3e2b486d901a29919e405534b314fcff73e79f | A rod has a length of 8 inches. The linear density of the rod at a point \( x \) inches from the left end is given by \( 2\sqrt{x+1} \) slugs per inch. Find the total mass of the rod and its center of mass. | [
"\\( \\frac{104}{3} \\)"
] | [
"A rod has a length of 8 inches. The linear density of the rod at a point \\( x \\) inches from the left end is given by \\( 2\\sqrt{x+1} \\) slugs per inch. Find the total mass of the rod and its center of mass."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e1731d74fb07ac8a126b06d833669c7840ec0b34a284b89eb93b23594360864 | A rod is broken into three pieces. What is the probability that these three pieces can form a triangle? | [
"\\(\\frac{1}{2}\\ln 2 - \\frac{1}{4}\\)"
] | [
"A rod is broken into three pieces. What is the probability that these three pieces can form a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f8ac9596eb997f5162c5951f210a3a96d1517014d1271b6e2e3943ae91606c75 | A rod is embedded in \( \mathbb{R} \) as the interval \([0, 1]\). The temperature at time \( t \) and point \( x \) is denoted by \( u(t, x) \). The temperature distribution is governed by the partial differential equation \( u_t = k u_{xx} \), where \( k \) is a constant. a.) Describe the temperature distribution at t... | [
"\\( F(t, x) = \\pi |\\cos(\\pi x)| e^{-(\\pi^2 k t)} \\) and as \\( t \\to \\infty \\), \\( F(t, x) \\to 0 \\)."
] | [
"A rod is embedded in \\( \\mathbb{R} \\) as the interval \\([0, 1]\\). The temperature at time \\( t \\) and point \\( x \\) is denoted by \\( u(t, x) \\). The temperature distribution is governed by the partial differential equation \\( u_t = k u_{xx} \\), where \\( k \\) is a constant.\n\na.) Describe the temper... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 20
} | 37 | 0 | 0 |
omr-ws-7923775986ea48d0b4395231585f94e80026897bdbbf867291e82272ceeb86f1 | A rod is first broken into two pieces. One of these pieces is then broken into two more pieces. What is the probability that the resulting three pieces can form a triangle? | [
"\\(\\frac{1}{2}\\ln 2 - \\frac{1}{4}\\)"
] | [
"A rod is first broken into two pieces. One of these pieces is then broken into two more pieces. What is the probability that the resulting three pieces can form a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-943a479143c95b1387d2c0616bc54dd7420a8556dee71aaeecc4be173fe96df8 | A rod of length $2b$ slides on the curve $y = \frac{1}{4}x^2$, $x \geq 0$, such that its midpoint is always tangent to the curve. One end of the rod traces a curve given by the parametric equations $x = 2 \tan \theta - b \cos \theta$ and $y = \tan^2 \theta - b \sin \theta$ for $0 \leq \theta \leq \frac{1}{2} \pi$. If $... | [
"3 - 2\\sqrt{2}"
] | [
"A rod of length $2b$ slides on the curve $y = \\frac{1}{4}x^2$, $x \\geq 0$, such that its midpoint is always tangent to the curve. One end of the rod traces a curve given by the parametric equations $x = 2 \\tan \\theta - b \\cos \\theta$ and $y = \\tan^2 \\theta - b \\sin \\theta$ for $0 \\leq \\theta \\leq \\fr... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cbf9931e130c8905b9ce2d5e7633850efea97e26ce347b2e89b4d4202684cf3b | A rod of length $l$ is given. A spider starts from the left end and crawls rightward at a fixed velocity of $1 \text{ m/s}$. The rod proportionately increases its length by $l$ at the end of every 1 second. Find all values of $l$ for which the spider will reach the other end of the rod. By "proportionately increasing,"... | [
"All \\( l \\)"
] | [
"A rod of length $l$ is given. A spider starts from the left end and crawls rightward at a fixed velocity of $1 \\text{ m/s}$. The rod proportionately increases its length by $l$ at the end of every 1 second. Find all values of $l$ for which the spider will reach the other end of the rod. By \"proportionately incre... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 38,
"tir": 28
} | 66 | 0 | 0 |
omr-ws-3bf0249be4b14cdf754f5b8a38badd42e948ac8fb84cc4ce5fefc7f8d6f97657 | A rod of length 1 is randomly cut into three pieces. What is the probability that the lengths of these three pieces can form a triangle? | [
"\\(\\frac{1}{4}\\)"
] | [
"A rod of length 1 is randomly cut into three pieces. What is the probability that the lengths of these three pieces can form a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-9777f77b26ec9dab879be1ae44b5437497a14ce2e18468cc5a83beeac56608ef | A rod of length 100 meters is inclined at an angle of \(\pi/6\) with the horizontal floor. The horizontal end of the rod slides on the floor at a velocity of 10 m/s. What is the rate of sliding of the middle point of the rod? | [
"10"
] | [
"A rod of length 100 meters is inclined at an angle of \\(\\pi/6\\) with the horizontal floor. The horizontal end of the rod slides on the floor at a velocity of 10 m/s. What is the rate of sliding of the middle point of the rod?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-b2d907de7b94bc98e90fea5bbec6155cacaa96ed9a0dc2512b3c44d5747e1570 | A rod of negligible thickness and length \( L \) is balanced on a circular table of radius \( r \) with \( L \leq 2r \). Assuming that the position of the midpoint of the rod on the table and the direction the rod is pointing are both uniformly distributed, what is the probability that both ends of the rod are on the t... | [
"1 - \\frac{L}{2r}"
] | [
"A rod of negligible thickness and length \\( L \\) is balanced on a circular table of radius \\( r \\) with \\( L \\leq 2r \\). Assuming that the position of the midpoint of the rod on the table and the direction the rod is pointing are both uniformly distributed, what is the probability that both ends of the rod ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5d3b0a430dffbaaa9528ed7f65765f0cbbc9818ddec7c6fb080f7a3d3c5cf63 | A rod of negligible thickness and length \( L \) is balanced on a circular table of radius \( r \) with \( L \leq 2r \). Assuming that the position of the midpoint of the rod on the table and the direction the rod is pointing are both uniformly distributed, what is the probability that one end is on the table and one e... | [
"\\dfrac{L}{\\pi r}"
] | [
"A rod of negligible thickness and length \\( L \\) is balanced on a circular table of radius \\( r \\) with \\( L \\leq 2r \\). Assuming that the position of the midpoint of the rod on the table and the direction the rod is pointing are both uniformly distributed, what is the probability that one end is on the tab... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a1ea81bf7f7b15d07e8954a199d5149e0bf4a986d33dc86177ba04ad6cec06e5 | A roll of quarters contains $40$ quarters and a roll of dimes contains $50$ dimes. James has a jar that contains $83$ quarters and $159$ dimes. Lindsay has a jar that contains $129$ quarters and $266$ dimes. James and Lindsay pool these quarters and dimes and make complete rolls with as many of the coins as possible. I... | [
"\\$5.50"
] | [
"A roll of quarters contains $40$ quarters and a roll of dimes contains $50$ dimes. James has a jar that contains $83$ quarters and $159$ dimes. Lindsay has a jar that contains $129$ quarters and $266$ dimes. James and Lindsay pool these quarters and dimes and make complete rolls with as many of the coins as possi... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-77a3fbb87cd147bcf1fad44455717965c345d63107d5c204a303323e5954e253 | A roll of toilet paper has a diameter of 6 inches. The tissue is rolled around a 2-inch diameter cardboard tube. There are 1000 sheets, each 4 inches by 4 inches. Estimate to the nearest thousandth of an inch, the thickness of each sheet of paper. | [
"0.006"
] | [
"A roll of toilet paper has a diameter of 6 inches. The tissue is rolled around a 2-inch diameter cardboard tube. There are 1000 sheets, each 4 inches by 4 inches. Estimate to the nearest thousandth of an inch, the thickness of each sheet of paper."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ce400ceb5bfb872be4c6756da360393749d1188f9ca0c385d9276ffd978439b0 | A roll of toilet paper with an initial radius of 8 cm is dropped from the roof of a library. The distance fallen after \( t \) seconds is given by \( d = 4.9t^2 \) meters. As the roll unwinds, its cross-sectional area decreases by 5 cm² for each meter it falls. Determine the rate at which the radius is decreasing when ... | [
"\\(-0.9984\\) cm/s"
] | [
"A roll of toilet paper with an initial radius of 8 cm is dropped from the roof of a library. The distance fallen after \\( t \\) seconds is given by \\( d = 4.9t^2 \\) meters. As the roll unwinds, its cross-sectional area decreases by 5 cm² for each meter it falls. Determine the rate at which the radius is decreas... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e3654261027b083766cf0faf6318bf00da08d2c2acd54baa31d46b0134ced918 | A roller coaster travels at a constant speed of 42.0 mph through identical circular horizontal loops, each with a diameter of 152.0 ft. How many complete loops will the roller coaster travel in 46.51 seconds? | [
"5"
] | [
"A roller coaster travels at a constant speed of 42.0 mph through identical circular horizontal loops, each with a diameter of 152.0 ft. How many complete loops will the roller coaster travel in 46.51 seconds?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-bbe4d23de881e63a8b765c3a2e1d28591559439df08888c6ae6132dca4e7c543 | A rook dipped in red paint is initially placed at a corner square on a $12 \times 10$ rectangular board. Two players take turns moving the same rook to a different square in the same row or same column. As it has been dipped in red paint, the rook colors every square it passes over red (This includes both the square it... | [
"First player wins."
] | [
"A rook dipped in red paint is initially placed at a corner square on a $12 \\times 10$ rectangular board. Two players take turns moving the same rook to a different square in the same row or same column. As it has been dipped in red paint, the rook colors every square it passes over red (This includes both the squ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 12
} | 29 | 0 | 0 |
omr-ws-d54601801ee67d6a3363bbcb9d27ac315e495ebdfc96ff770a38e474521cd1ee | A rook is placed on an 8x8 chessboard and can move to any neighboring square (right, left, up, down). After 64 moves, it returns to its starting square and visits each square exactly once. Calculate the absolute difference between the number of vertical moves and the number of horizontal moves. | [
"0"
] | [
"A rook is placed on an 8x8 chessboard and can move to any neighboring square (right, left, up, down). After 64 moves, it returns to its starting square and visits each square exactly once. Calculate the absolute difference between the number of vertical moves and the number of horizontal moves."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-96ee8b4340a3cbb0a11b47e7e67d30f8cae749f822c57a237965ab1a0c81c304 | A rook is placed on any square of an empty $8 \times 8$ chessboard. Additional rooks are then placed, one at a time, each attacking an odd number of the existing rooks. What is the maximum number of rooks that can be placed on the chessboard? | [
"15"
] | [
"A rook is placed on any square of an empty $8 \\times 8$ chessboard. Additional rooks are then placed, one at a time, each attacking an odd number of the existing rooks. What is the maximum number of rooks that can be placed on the chessboard?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-6a6b7aa4dfa67a0df7f99b3f57896c30b8ceb2cac84286545efc6468612adf9e | A rook made a sequence of moves in an $8 \times 8$ chessboard, visiting each cell exactly once and returning to the initial cell. Let $h$ be the number of horizontal moves and $v$ be the number of vertical moves in this sequence. Calculate the absolute difference $|h - v|$. | [
"0"
] | [
"A rook made a sequence of moves in an $8 \\times 8$ chessboard, visiting each cell exactly once and returning to the initial cell. Let $h$ be the number of horizontal moves and $v$ be the number of vertical moves in this sequence. Calculate the absolute difference $|h - v|$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-aa918a059ad19e5743bbd33e43a89064f183f8a1b0140ed6b58ff3279ca167ed | A rook moves alternating horizontally and vertically on an infinite chessboard. The rook moves one square horizontally (in either direction) at the first move, two squares vertically at the second, three horizontally at the third, and so on. Let \( S \) be the set of integers \( n \) with the property that there exists... | [
"502"
] | [
"A rook moves alternating horizontally and vertically on an infinite chessboard. The rook moves one square horizontally (in either direction) at the first move, two squares vertically at the second, three horizontally at the third, and so on. Let \\( S \\) be the set of integers \\( n \\) with the property that the... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-2183cf3012222d75ab538d9850641dea84ef3cc7381460da3754303d13abfbf8 | A rook moves on a $10 \times 10$ chessboard, visiting each square exactly once by moving to adjacent squares. Let $D$ be a main diagonal of the board. Define $f(D)$ as the number of times the rook moves away from $D$ and then immediately returns to $D$ during its path. Calculate the minimum possible value of $f(D)$ for... | [
"8"
] | [
"A rook moves on a $10 \\times 10$ chessboard, visiting each square exactly once by moving to adjacent squares. Let $D$ be a main diagonal of the board. Define $f(D)$ as the number of times the rook moves away from $D$ and then immediately returns to $D$ during its path. Calculate the minimum possible value of $f(D... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-87c76f08aa2616038c0b990cf63c6341f55b2015e699908559d19557410d6be7 | A rook moves on a chessboard without visiting the same square twice and avoids the squares on one of the diagonals (called the fence). What is the maximum number of times the rook can jump over the fence? | [
"6"
] | [
"A rook moves on a chessboard without visiting the same square twice and avoids the squares on one of the diagonals (called the fence). What is the maximum number of times the rook can jump over the fence?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-96b565d90e11e8636b2dc72a9700a45f2317cad2d66ee9e309291adafc09b764 | A rook starts at square a1 on a chessboard with 8 rows and 8 columns. Every minute, the rook randomly moves to a different square either in the same row or the same column. The rook continues moving until it arrives at a square in either row 8 or column h. What is the probability that the rook ends at square a8 after i... | [
"\\(\\frac{1}{6}\\)"
] | [
"A rook starts at square a1 on a chessboard with 8 rows and 8 columns. Every minute, the rook randomly moves to a different square either in the same row or the same column. The rook continues moving until it arrives at a square in either row 8 or column h. What is the probability that the rook ends at square a8 af... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 14
} | 21 | 0 | 0 |
omr-ws-88079f30eae51eddc7bc7d401c9cfd63c4f9104873a598feedda17b73575583e | A room contains a single bulb and $(2^{2^{10}} + 2^{2^9})$ identical switches, out of which exactly one switch operates the bulb. In each attempt, any number of switches can be triggered on/off simultaneously. Find the least number of attempts required to identify the correct switch for the bulb in all possible scenari... | [
"1025"
] | [
"A room contains a single bulb and $(2^{2^{10}} + 2^{2^9})$ identical switches, out of which exactly one switch operates the bulb. In each attempt, any number of switches can be triggered on/off simultaneously. Find the least number of attempts required to identify the correct switch for the bulb in all possible sc... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9f586ada203b91d2d360763768fc47b58701da08a9af5efdd7e9443cfaee48b | A room has $2017$ boxes arranged in a circle. A set of boxes is defined as [i]friendly[/i] if there are at least two boxes in the set, and for each box in the set, if we go clockwise starting from the box, we would pass either $0$ or an odd number of boxes before encountering a new box in the set. Thirty students enter... | [
"435"
] | [
"A room has $2017$ boxes arranged in a circle. A set of boxes is defined as [i]friendly[/i] if there are at least two boxes in the set, and for each box in the set, if we go clockwise starting from the box, we would pass either $0$ or an odd number of boxes before encountering a new box in the set. Thirty students ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-f49448d529fab23d867d3548ff7f51fdb977e81596c37c80a447e78729ddacc4 | A room has eight switches, each controlling a different light. Initially, exactly five of the lights are on. Three people enter the room, one after the other, and each independently flips one switch at random before exiting. What is the probability that after the third person has exited the room, exactly six of the lig... | [
"\\(\\frac{39}{128}\\)"
] | [
"A room has eight switches, each controlling a different light. Initially, exactly five of the lights are on. Three people enter the room, one after the other, and each independently flips one switch at random before exiting. What is the probability that after the third person has exited the room, exactly six of th... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 29
} | 41 | 0 | 0 |
omr-ws-65c568fc94f643f1e46c4ae617ce51e0df6449d3ddafe365bde5ca234f7a59c1 | A room has the following dimensions: a length of 2845 cm, a width of 21000 mm, and a height of 2.86 m. What is the volume of the room in cubic meters? | [
"1708.707 cubic meters"
] | [
"A room has the following dimensions: a length of 2845 cm, a width of 21000 mm, and a height of 2.86 m. What is the volume of the room in cubic meters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d19b7115ac6649e0aa9ef2b6091233e94874864c2995b2456d38fcf73be3b552 | A room has the shape of a triangle $\vartriangle ABC$ with $\angle B = 60^\circ$, $\angle C = 45^\circ$, and $AB = 5$ meters. A bee starts at corner $A$ and flies in a straight line in a random direction, turning $60^\circ$ whenever it hits a wall. Is it possible that after some time the bee will have flown more than a... | [
"Yes"
] | [
"A room has the shape of a triangle $\\vartriangle ABC$ with $\\angle B = 60^\\circ$, $\\angle C = 45^\\circ$, and $AB = 5$ meters. A bee starts at corner $A$ and flies in a straight line in a random direction, turning $60^\\circ$ whenever it hits a wall. Is it possible that after some time the bee will have flown ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-faf120d5f230429ee45dcc284211a320ab01c3e50678c63dbe2c8847e1006425 | A room is 19 feet long and 11 feet wide. Find the ratio of the length of the room to its perimeter. Express your answer in the form $a:b$. | [
"19:60"
] | [
"A room is 19 feet long and 11 feet wide. Find the ratio of the length of the room to its perimeter. Express your answer in the form $a:b$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ddd6b66da1cfe81283edfe193922750db1fd0d29c9c4630f34cd59d002eacde | A room is built in the half-annular region between two concentric semicircles with radii 5 and 10. Jerry is standing on the left endpoint of the diameter of the larger semicircle. The area of the region that Jerry can see is given by $\frac{a\pi - b\sqrt{c}}{d}$, where $c$ is not divisible by the square of any prime in... | [
"181"
] | [
"A room is built in the half-annular region between two concentric semicircles with radii 5 and 10. Jerry is standing on the left endpoint of the diameter of the larger semicircle. The area of the region that Jerry can see is given by $\\frac{a\\pi - b\\sqrt{c}}{d}$, where $c$ is not divisible by the square of any ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-8c727cacf6594f0f4062ea60a844b3fec7c34303d0b739a618da688d3e9f31c2 | A rooster costs $5, a hen costs $3, and chicks are sold at 3 for $1. A farmer bought 100 birds of these three types for $100. How many of each type of bird did he buy? | [
"(12, 4, 84)"
] | [
"A rooster costs $5, a hen costs $3, and chicks are sold at 3 for $1. A farmer bought 100 birds of these three types for $100. How many of each type of bird did he buy?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-818ecceb7f8da42e0d5f4152f496b04bd92293521c201ceaa4d40b3b37445210 | A root of the equation \(5x^3 + Ax^2 + Cx + D = 0\) is 6. When the cubed term is removed, the resulting quadratic equation has a root of 2. When the squared term is also removed, the resulting linear equation has a root of 4. Find the value of \(A + C + D\). | [
"135"
] | [
"A root of the equation \\(5x^3 + Ax^2 + Cx + D = 0\\) is 6. When the cubed term is removed, the resulting quadratic equation has a root of 2. When the squared term is also removed, the resulting linear equation has a root of 4. Find the value of \\(A + C + D\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-1ea38828df96dd2c0a9ad48bdb0616e0aade1bb33cb8a00035ebfca472240536 | A root of unity is a complex number that is a solution to $z^n = 1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^2 + az + b = 0$ for some integers $a$ and $b$. | [
"8"
] | [
"A root of unity is a complex number that is a solution to $z^n = 1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^2 + az + b = 0$ for some integers $a$ and $b$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5590ad38cd77e64c815c1527e49373679e832bf27b564053e6aef6083cba5152 | A rope hangs over a pulley with a sand bag at one end and a monkey at the other. The weight of the monkey is the same as the weight of the sand bag. The rope weighs 4 ounces per foot. The sum of the ages of the monkey and the monkey's mother is 4 years. The monkey's weight in pounds is the same as the mother's age in y... | [
"5"
] | [
"A rope hangs over a pulley with a sand bag at one end and a monkey at the other. The weight of the monkey is the same as the weight of the sand bag. The rope weighs 4 ounces per foot. The sum of the ages of the monkey and the monkey's mother is 4 years. The monkey's weight in pounds is the same as the mother's age... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-225f188f8c5d9f3c1e2f0bfc3f201b6ddc20caa95f5abe433adbca2a4f7f5229 | A rope is wound around a cylinder with a height of 12 meters and a base circle perimeter of 4 meters. The rope makes four full turns around the cylinder and its ends are on the upper and lower circles. What is the shortest length of the rope? | [
"20"
] | [
"A rope is wound around a cylinder with a height of 12 meters and a base circle perimeter of 4 meters. The rope makes four full turns around the cylinder and its ends are on the upper and lower circles. What is the shortest length of the rope?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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