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-023ab784a6e5dacd7d97335e83e66579e280538db3df1e35e223e9f62f13c8d9 | An acute triangle \(ABC\) is inscribed in a circle of radius 1 with center \(O\); all the angles of \(ABC\) are greater than \(45^\circ\). \(B_{1}\) is the foot of the perpendicular from \(B\) to \(CO\), \(B_{2}\) is the foot of the perpendicular from \(B_{1}\) to \(AC\). Similarly, \(C_{1}\) is the foot of the perpend... | [
"\\(\\frac{1}{2}\\)"
] | [
"An acute triangle \\(ABC\\) is inscribed in a circle of radius 1 with center \\(O\\); all the angles of \\(ABC\\) are greater than \\(45^\\circ\\). \\(B_{1}\\) is the foot of the perpendicular from \\(B\\) to \\(CO\\), \\(B_{2}\\) is the foot of the perpendicular from \\(B_{1}\\) to \\(AC\\). Similarly, \\(C_{1}\\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-7cc328cf67a60a24102c99a17e62f923465d926ef0c1c7a73df65e5e9e6faef2 | An acute triangle \(ABC\) is inscribed in a circle. Two altitudes \(AM\) and \(CN\) are extended to meet the circle at \(P\) and \(Q\) respectively. Given that \(PQ:AC = 3:2\), find \(\cos \angle ABC\). | [
"\\(\\frac{3}{4}\\)"
] | [
"An acute triangle \\(ABC\\) is inscribed in a circle. Two altitudes \\(AM\\) and \\(CN\\) are extended to meet the circle at \\(P\\) and \\(Q\\) respectively. Given that \\(PQ:AC = 3:2\\), find \\(\\cos \\angle ABC\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 24,
"genselect": 7,
"tir": 6
} | 37 | 0 | 0 |
omr-ws-ae6cd3fdb58e2b83f032486dea975d31c8242b286fdec7c0605cc33393d6e930 | An acute triangle \(PNK\) is located in a circle with diameter \(MN\). Let \(A\) be the intersection of \(MN\) with \(PK\) and \(H\) be a point on the arc \(PN\) that does not contain \(K\). The circumcircle of triangle \(PAH\) intersects \(MN\) and \(PN\) at points \(B\) and \(D\), respectively. Circles with diameter ... | [
"8"
] | [
"An acute triangle \\(PNK\\) is located in a circle with diameter \\(MN\\). Let \\(A\\) be the intersection of \\(MN\\) with \\(PK\\) and \\(H\\) be a point on the arc \\(PN\\) that does not contain \\(K\\). The circumcircle of triangle \\(PAH\\) intersects \\(MN\\) and \\(PN\\) at points \\(B\\) and \\(D\\), respe... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-070e71b24b636864190d57e6bbe06dbea4c0b863f00f01d52dbdb547345a0da2 | An acute triangle and a circle centered at the triangle's incenter both have an area of \(4\pi\). Given that \(\frac{3}{4}\) of the circle's circumference lies outside the triangle, find the total area of the regions that are inside the triangle and outside the circle. | [
"3\\pi + 3"
] | [
"An acute triangle and a circle centered at the triangle's incenter both have an area of \\(4\\pi\\). Given that \\(\\frac{3}{4}\\) of the circle's circumference lies outside the triangle, find the total area of the regions that are inside the triangle and outside the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-d18f2ab4449b8db1170a06356aed90e2dbd3376533fcfce08303512e5ce6c5c4 | An acute triangle is inscribed in a circle. The resulting three minor arcs of the circle are reflected about the corresponding sides of the triangle. Let $P$ be the point of concurrency of the reflected arcs. If the triangle has side lengths 5, 12, and 13, calculate the distance from $P$ to the circumcenter of the tria... | [
"\\dfrac{13}{2}"
] | [
"An acute triangle is inscribed in a circle. The resulting three minor arcs of the circle are reflected about the corresponding sides of the triangle. Let $P$ be the point of concurrency of the reflected arcs. If the triangle has side lengths 5, 12, and 13, calculate the distance from $P$ to the circumcenter of the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-a86ad4130409ca4ddb6c7c4f17f83b87bd905f59531c971846d73ebdd223eddd | An acute-angled triangle $ABC$ is such that $\angle A = 60^\circ$. A billiard ball starts from vertex $A$ along the bisector of angle $A$, reflects off side $BC$ according to the law "the angle of reflection equals the angle of incidence," and continues along a straight line without any further reflections. Let $O$ be ... | [
"1"
] | [
"An acute-angled triangle $ABC$ is such that $\\angle A = 60^\\circ$. A billiard ball starts from vertex $A$ along the bisector of angle $A$, reflects off side $BC$ according to the law \"the angle of reflection equals the angle of incidence,\" and continues along a straight line without any further reflections. Le... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-354fea2a66c238c364b865f59d8900366cb25e571a2e305b61e56749c9b0c8bf | An acute-angled triangle $\mathcal{T}$ is inscribed in a circle $\Omega$ with center $O$ and radius $R$. Denote by $\Gamma$ the nine-point circle of $\mathcal{T}$. A circle $\omega$ passes through vertices $A$ and $B$ of $\mathcal{T}$ and the center $O$ of $\Omega$. Let $P$ be the intersection point of the common exter... | [
"1"
] | [
"An acute-angled triangle $\\mathcal{T}$ is inscribed in a circle $\\Omega$ with center $O$ and radius $R$. Denote by $\\Gamma$ the nine-point circle of $\\mathcal{T}$. A circle $\\omega$ passes through vertices $A$ and $B$ of $\\mathcal{T}$ and the center $O$ of $\\Omega$. Let $P$ be the intersection point of the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-3fbeca581fb3fd69918cfc683495cc22153d9bf47b8555c0f8aeb3711f034d38 | An acute-angled triangle \(ABC\) with \(AB < AC\) is inscribed in a circle \(\omega\). Let \(M\) be the centroid of \(\triangle ABC\), and let \(AH\) be an altitude of the triangle. A ray \(MH\) meets \(\omega\) at \(A'\). Let \(O\) be the circumcenter of \(\triangle A'HB\). Calculate the length of \(AO\) if \(AB = 13\... | [
"14"
] | [
"An acute-angled triangle \\(ABC\\) with \\(AB < AC\\) is inscribed in a circle \\(\\omega\\). Let \\(M\\) be the centroid of \\(\\triangle ABC\\), and let \\(AH\\) be an altitude of the triangle. A ray \\(MH\\) meets \\(\\omega\\) at \\(A'\\). Let \\(O\\) be the circumcenter of \\(\\triangle A'HB\\). Calculate the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e757678fe94b5a41af038909b076580f6a9b465eee03d098ad3968bdde5fe4c | An acute-angled triangle \(ABC\) with orthocenter \(H\) is inscribed in a circle \(\omega\) with center \(O\). The line \(d\) passes through \(H\) and intersects arcs \(AB\) and \(AC\) at points \(P\) and \(Q\), respectively. Let \(AA'\) be the diameter of the circle \(\omega\). Lines \(A'P\) and \(A'Q\) intersect \(BC... | [
"169"
] | [
"An acute-angled triangle \\(ABC\\) with orthocenter \\(H\\) is inscribed in a circle \\(\\omega\\) with center \\(O\\). The line \\(d\\) passes through \\(H\\) and intersects arcs \\(AB\\) and \\(AC\\) at points \\(P\\) and \\(Q\\), respectively. Let \\(AA'\\) be the diameter of the circle \\(\\omega\\). Lines \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-71ffa9ac08f257ae2133ec112e4cb8dec957777eaed208007c4a04683bdb483e | An acute-angled triangle \(ABC\) with orthocenter \(H\) is inscribed in a circle \(\omega\) with center \(O\). The line \(d\) passes through \(H\) and intersects the smaller arcs \(AB\) and \(AC\) of the circle \(\omega\) at points \(P\) and \(Q\), respectively. Let \(AA'\) be the diameter of the circle \(\omega\), and... | [
"90"
] | [
"An acute-angled triangle \\(ABC\\) with orthocenter \\(H\\) is inscribed in a circle \\(\\omega\\) with center \\(O\\). The line \\(d\\) passes through \\(H\\) and intersects the smaller arcs \\(AB\\) and \\(AC\\) of the circle \\(\\omega\\) at points \\(P\\) and \\(Q\\), respectively. Let \\(AA'\\) be the diamete... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-dd0e51986e9c26429c2682fc3f189b18b4aa8fe257d4c189e1b2a539c2b212e5 | An acute-angled triangle is inscribed in a circle of radius $R$. The distance between the center of the circle and the point of intersection of the medians of the triangle is $d$. Find the radius of a circle inscribed in a triangle whose vertices are the feet of the altitudes of this triangle. | [
"\\dfrac{R^2 - 9d^2}{4R}"
] | [
"An acute-angled triangle is inscribed in a circle of radius $R$. The distance between the center of the circle and the point of intersection of the medians of the triangle is $d$. Find the radius of a circle inscribed in a triangle whose vertices are the feet of the altitudes of this triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-908509d72cc1b65d2d45abcc0f6bce348ef687762a4025a18bd0ea6647fb53ef | An acute-angled triangle with integer coordinates and no sides parallel to the coordinate axes has an area of 348 and one side of length 29. What is the product of the lengths of the other two sides? | [
"870"
] | [
"An acute-angled triangle with integer coordinates and no sides parallel to the coordinate axes has an area of 348 and one side of length 29. What is the product of the lengths of the other two sides?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-934d30bbafa569c25b87f7e048297107ca353c1ad3711a6f64a41fc0c409f670 | An acute-angled triangle with integer coordinates for its vertices and no sides parallel to the coordinate axes has an area of $348$ and one side of length $29$. What is the product of the lengths of the other two sides? | [
"\\(58\\sqrt{265}\\)"
] | [
"An acute-angled triangle with integer coordinates for its vertices and no sides parallel to the coordinate axes has an area of $348$ and one side of length $29$. What is the product of the lengths of the other two sides?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b70d1a38e87263334aaa53f1b8f2adcf16ed0da3793538b54aea8770d2003232 | An acute-angled triangle with vertices at different integer coordinates and no sides parallel to the coordinate axes has an area of 348 and one side of length 29. What is the product of the lengths of the other two sides? | [
"870"
] | [
"An acute-angled triangle with vertices at different integer coordinates and no sides parallel to the coordinate axes has an area of 348 and one side of length 29. What is the product of the lengths of the other two sides?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5aaecc60662706fe09259f044e7d728e888473ed572b46270362c33259ca50fe | An advertiser is analyzing the growth of likes for their new ad on Youtube. The increase in likes in a given second is equal to 4T_av, where T_av is the midpoint of the time interval. For example, the increase in likes from 3 seconds to 4 seconds is equal to 4 x 3.5. i) If the total likes follow the path as \( l(t) = a... | [
"0",
"\\( b = 0 \\)"
] | [
"An advertiser is analyzing the growth of likes for their new ad on Youtube. The increase in likes in a given second is equal to 4T_av, where T_av is the midpoint of the time interval. For example, the increase in likes from 3 seconds to 4 seconds is equal to 4 x 3.5.\ni) If the total likes follow the path as \\( l... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 6
} | 54 | 0 | 0 |
omr-ws-d26da7b03b1102856d3c7591f19e0bf1278cc9de4cd97298f4574b524961c0d6 | An aerosol can is designed as a cylinder with radius $r$ and height $h$, topped and bottomed by hemispheres of radius $r$. The volume of the cylinder is $\pi r^2h$, and the surface area of the sides of the cylinder is $2\pi rh$. The volume of a sphere is $\frac{4}{3}\pi r^3$, and the surface area is $4\pi r^2$. What ra... | [
"\\( h:r = 4:1 \\)"
] | [
"An aerosol can is designed as a cylinder with radius $r$ and height $h$, topped and bottomed by hemispheres of radius $r$. The volume of the cylinder is $\\pi r^2h$, and the surface area of the sides of the cylinder is $2\\pi rh$. The volume of a sphere is $\\frac{4}{3}\\pi r^3$, and the surface area is $4\\pi r^2... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b415b45815a501d8536bb586868e04516f83885af3644d596422cd4c01820591 | An agricultural sprinkler distributes water in a circular pattern of radius 100 ft. It supplies water to a depth of \( e^{-r} \) feet per hour at a distance of \( r \) feet from the sprinkler. What is the total amount of water supplied per hour to the region inside the circle of radius \( R \) centered at the sprinkler... | [
"\\( 2\\pi \\left( 1 - e^{-R} \\right) \\)"
] | [
"An agricultural sprinkler distributes water in a circular pattern of radius 100 ft. It supplies water to a depth of \\( e^{-r} \\) feet per hour at a distance of \\( r \\) feet from the sprinkler. What is the total amount of water supplied per hour to the region inside the circle of radius \\( R \\) centered at th... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9045fa1789a1749ee0f57b8ea7acdd0db018f928c3c68e54fc0a501342134a9e | An air rescue plane averages 300 mph in still air and carries enough fuel for 5 hours of flying time. If it encounters a constant wind of 30 mph, how far can it fly and return safely? | [
"1485"
] | [
"An air rescue plane averages 300 mph in still air and carries enough fuel for 5 hours of flying time. If it encounters a constant wind of 30 mph, how far can it fly and return safely?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-902a35fd5187a25c12d4b71e4d7a06316b543516641f2163bfca1f98296a0077 | An air-fire station is located \(a\) miles due North of a point \(P\) on a river that runs due East. A popular campground is located at a point \(c\) miles due North of a point \(Q\) on the river. Point \(Q\) is \(b\) miles East of point \(P\). To respond to a fire at the campground, a helicopter has to fly to the rive... | [
"\\(9a^2 - 25c^2 = 4b^2\\)"
] | [
"An air-fire station is located \\(a\\) miles due North of a point \\(P\\) on a river that runs due East. A popular campground is located at a point \\(c\\) miles due North of a point \\(Q\\) on the river. Point \\(Q\\) is \\(b\\) miles East of point \\(P\\). To respond to a fire at the campground, a helicopter has... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-73abe851696e550a540bdb55215717ae8553939196f50818a4ded5349a1a2147 | An airbus leaves airport A at latitude \( x \) and longitude \( y \) for airport B at latitude \( x \) and longitude \( y+z \), where \( 0 < x < 90 \) and \( 0 < z < 180 \). The airbus takes the shortest path, which is the arc AB on a great circle. Find the greatest latitude reached by the airbus in terms of \( x \) an... | [
"\\arctan\\left( \\frac{\\tan x}{\\cos \\frac{z}{2}} \\right)"
] | [
"An airbus leaves airport A at latitude \\( x \\) and longitude \\( y \\) for airport B at latitude \\( x \\) and longitude \\( y+z \\), where \\( 0 < x < 90 \\) and \\( 0 < z < 180 \\). The airbus takes the shortest path, which is the arc AB on a great circle. Find the greatest latitude reached by the airbus in te... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-5c60a481562636f4a15ab5b7e8e6ecb71f12c0ec77b7add828a418a81f32543f | An aircraft can fly at a maximum speed of 30 km/h in still air and intends to fly from point A to point B, which is 850 km to the north-west of A. The wind is blowing at a speed of 20 km/h in the direction due north. Determine the course that the pilot must steer to fly from A to B in the shortest time possible. | [
"16.9 degrees north of west"
] | [
"An aircraft can fly at a maximum speed of 30 km/h in still air and intends to fly from point A to point B, which is 850 km to the north-west of A. The wind is blowing at a speed of 20 km/h in the direction due north. Determine the course that the pilot must steer to fly from A to B in the shortest time possible."
... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 3,
"tir": 14
} | 22 | 0 | 0 |
omr-ws-ef57dd920315a448bb3f2212efbdaee2c085522fd470a74bc274da03fff7b0ca | An airline company is planning to introduce a network of connections between ten different airports of Sawubonia, ranked by priority from first to last. A network is called feasible if it satisfies the following conditions: - All connections operate in both directions. - If there is a direct connection between two airp... | [
"512"
] | [
"An airline company is planning to introduce a network of connections between ten different airports of Sawubonia, ranked by priority from first to last. A network is called feasible if it satisfies the following conditions:\n- All connections operate in both directions.\n- If there is a direct connection between t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-2bf4d822b0ea8271abbdfa724dfb23c840dbf862c19cba8ef7242237704be0c4 | An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network feasible if it satisfies the following conditions: - All connections operate in both directions. - If there ... | [
"512"
] | [
"An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network feasible if it satisfies the following conditions:\n- All connections operate in both directions.\n- If... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-1751daab88aeccbe36b7b17cf6af2ebf1ec7a96f338ae7400972eb0ab3b13701 | An airplane accelerates at $10$ mph per second and decelerates at $15$ mph/sec. Given that its takeoff speed is $180$ mph, and the pilots want enough runway length to safely decelerate to a stop from any speed below takeoff speed, what’s the shortest length that the runway can be allowed to be? Assume the pilots always... | [
"$\\frac{3}{4}$ miles"
] | [
"An airplane accelerates at $10$ mph per second and decelerates at $15$ mph/sec. Given that its takeoff speed is $180$ mph, and the pilots want enough runway length to safely decelerate to a stop from any speed below takeoff speed, what’s the shortest length that the runway can be allowed to be? Assume the pilots a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 2,
"tir": 17
} | 34 | 0 | 0 |
omr-ws-2cfe3ae75b8d6a709a2a9737671f944d2b2efeefcc67502d8d55ea76818ca2b5 | An airplane can fly with the wind a distance of 600 miles in 5 hours. However, the return trip against the wind takes 6 hours. Find the speed of the plane in still air and the speed of the wind. | [
"P = 110, W = 10"
] | [
"An airplane can fly with the wind a distance of 600 miles in 5 hours. However, the return trip against the wind takes 6 hours. Find the speed of the plane in still air and the speed of the wind."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 21
} | 21 | 0 | 0 |
omr-ws-c71c4bcb349ab5a67261baf7e62b930b1decc31cccf554391a78e064b374a3d9 | An airplane flies west at 150 m/s at an altitude of 1000 m. A searchlight on the ground in the same vertical plane must be kept on the airplane. What is the rate of revolution of the searchlight when the airplane is 500 m due east of the searchlight? | [
"\\(\\frac{3}{25}\\) radians/second."
] | [
"An airplane flies west at 150 m/s at an altitude of 1000 m. A searchlight on the ground in the same vertical plane must be kept on the airplane. What is the rate of revolution of the searchlight when the airplane is 500 m due east of the searchlight?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-5865b5055916d1b390fc59a6ed7aaa78bb0564c0618b2a9fd819e59011fc6edc | An airplane flying 3500 feet above the ground sees an airport at an angle of depression of 77°. How far is the airplane from the airport? | [
"3592.06"
] | [
"An airplane flying 3500 feet above the ground sees an airport at an angle of depression of 77°. How far is the airplane from the airport?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 18
} | 36 | 0 | 0 |
omr-ws-7af299525ec0521fa28b4eafd7155fe81fd05d513800fe073579de248a4bb2c6 | An airplane has a bearing of 40 degrees and an airspeed of 240 mph. The wind is blowing from the northwest (bearing 135 degrees) at 30 mph. Determine the bearing of the course and the ground speed of the airplane. | [
"Ground speed = 239.23 mph, Bearing = 47.21 degrees"
] | [
"An airplane has a bearing of 40 degrees and an airspeed of 240 mph. The wind is blowing from the northwest (bearing 135 degrees) at 30 mph. Determine the bearing of the course and the ground speed of the airplane."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 18
} | 18 | 0 | 0 |
omr-ws-6d1ded3765b5ffd087cd64cdf99230441d2f65fb09d5dfec60b54b61738ba4eb | An airplane has three sections: First Class (24 seats), Business Class ($25\%$ of the total number of seats) and Economy ($\frac{2}{3}$ of the total number of seats). How many seats does the plane have? | [
"288"
] | [
"An airplane has three sections: First Class (24 seats), Business Class ($25\\%$ of the total number of seats) and Economy ($\\frac{2}{3}$ of the total number of seats). How many seats does the plane have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0cc5f82576bfe891179a6bf9852688af70a95cba9cbb2c8db3f6834bae9554c4 | An airplane is flying on a bearing of 335 degrees at 430 mph. Find the component form of the velocity of the airplane. (Note: Component form refers to the horizontal and vertical components of the velocity vector.) | [
"\\langle -181.73, 389.71 \\rangle"
] | [
"An airplane is flying on a bearing of 335 degrees at 430 mph. Find the component form of the velocity of the airplane. (Note: Component form refers to the horizontal and vertical components of the velocity vector.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-af2f40b7d437c0e171aaf0527016b00a61165c4607ed3628460f37e5c17898fd | An airplane leaves an aircraft carrier and flies due south at 400 km/h. The carrier proceeds 60° east of north at 32 km/h. If the plane has enough fuel for 5 hours of flying, what is the maximum distance south the pilot can travel, so that the fuel remaining will allow a safe return to the carrier? (You may assume Eart... | [
"1200 km"
] | [
"An airplane leaves an aircraft carrier and flies due south at 400 km/h. The carrier proceeds 60° east of north at 32 km/h. If the plane has enough fuel for 5 hours of flying, what is the maximum distance south the pilot can travel, so that the fuel remaining will allow a safe return to the carrier? (You may assume... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d1088d5eac5716f6ed5cc84ac3beadf59bed610e4f1af33e33260da9596dd692 | An airplane leaves the runway with an angle of climb of 18 degrees and a speed of 275 feet per second. Find the plane's altitude after two minutes. | [
"10198"
] | [
"An airplane leaves the runway with an angle of climb of 18 degrees and a speed of 275 feet per second. Find the plane's altitude after two minutes."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 28
} | 49 | 0 | 0 |
omr-ws-36d9e5e5463fa4d15253029fecdfddd1e50d71fa194efeeb6504f629683fc57c | An airplane loses altitude at the rate of 400 miles per hour. What is the rate of decrease of the visible portion of the surface of the earth when the plane is one mile high? (Assume the earth is a sphere) | [
"-9950000"
] | [
"An airplane loses altitude at the rate of 400 miles per hour. What is the rate of decrease of the visible portion of the surface of the earth when the plane is one mile high? (Assume the earth is a sphere)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-7ce588bd2492a60edfa72a775153d3c0e4e73e83cd52f292906c9b5db23c4287 | An airplane starts from a point on the equator and moves northeast with speed \( v \). Given the radius of the Earth is \( R \), will the airplane reach the North Pole? If so, how long will it take to reach the North Pole? | [
"\\( \\frac{\\sqrt{2}R}{v} \\)"
] | [
"An airplane starts from a point on the equator and moves northeast with speed \\( v \\). Given the radius of the Earth is \\( R \\), will the airplane reach the North Pole? If so, how long will it take to reach the North Pole?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-7da9187e6bd2a5fdb0a542957cf592329356e9f582e80ac4c8074ec0dd63698d | An airplane takes off with a ground velocity of 375 km/h on a heading of N45°W, climbing at an angle of 15 degrees. A 20 km/h wind is blowing from the south. Determine the magnitude of the resultant air and ground velocities. | [
"362"
] | [
"An airplane takes off with a ground velocity of 375 km/h on a heading of N45°W, climbing at an angle of 15 degrees. A 20 km/h wind is blowing from the south. Determine the magnitude of the resultant air and ground velocities."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-d70c2d5ad43c1ae4a73f334b8e58a51541049ff1905716f86042f9156e73a63a | An airplane travels between two cities. The first half of the distance between the cities is traveled at a constant speed of 600 mi/hour, and the second half of the distance is traveled at a constant speed of 900 mi/hour. Find the average speed of the plane. | [
"720"
] | [
"An airplane travels between two cities. The first half of the distance between the cities is traveled at a constant speed of 600 mi/hour, and the second half of the distance is traveled at a constant speed of 900 mi/hour. Find the average speed of the plane."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-770eb8514e2bd952cedda5c95f543599ddb27ad741d848cc7a7fab831be54907 | An airplane was in flight from $A$ to $B$ in a straight line. Due to a head wind, after a certain time, it reduced its speed to $v$ km/h and therefore was $t_1$ minutes late. During a second flight from $A$ to $B$ the airplane for the same reason reduced its speed to the same level but this time $d$ km farther from $A$... | [
"w=\\frac{60vd}{60d+v(t_2-t_1)}\\mbox{ km/h}"
] | [
"An airplane was in flight from $A$ to $B$ in a straight line. Due to a head wind, after a certain time, it reduced its speed to $v$ km/h and therefore was $t_1$ minutes late. During a second flight from $A$ to $B$ the airplane for the same reason reduced its speed to the same level but this time $d$ km farther fro... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95bf4b1b353455a926415ca5ef2c077f69749971e7a5c23061c87ae7479190cd | An airport contains 25 terminals, each pair connected by tunnels. There are exactly 50 bidirectional tunnels and the rest are unidirectional. A group of four terminals is called "connected" if from each terminal in the group, one can reach the other three using only the tunnels connecting them. Find the maximum number ... | [
"210"
] | [
"An airport contains 25 terminals, each pair connected by tunnels. There are exactly 50 bidirectional tunnels and the rest are unidirectional. A group of four terminals is called \"connected\" if from each terminal in the group, one can reach the other three using only the tunnels connecting them. Find the maximum ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-87ce6b0d9160c74445c64c6f4755c0a3a5ec40b869dfc47bac5b610a8fa7e244 | An airport contains 25 terminals, with each pair of terminals connected by tunnels. There are exactly 50 main tunnels that can be traversed in both directions, and the others are one-way tunnels. A group of four terminals is called "good" if from each terminal in the group, it is possible to reach the other three termi... | [
"210"
] | [
"An airport contains 25 terminals, with each pair of terminals connected by tunnels. There are exactly 50 main tunnels that can be traversed in both directions, and the others are one-way tunnels. A group of four terminals is called \"good\" if from each terminal in the group, it is possible to reach the other thre... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-4d7670708fbf32e52e3ce187bca38bdaf5ae94e9bb80781adac58e510f5a05a6 | An airport has a collection of gates arranged in a \(10^{2022} \times 10^{2022}\) grid, labeled 1, 2, 3, \ldots, \(10^{4044}\) in some order. A traveler's gate is the unique gate with the property that its label is less than all neighboring gates (horizontally or vertically). The traveler can ask for the label of a par... | [
"2 \\times 10^{2022} - 1"
] | [
"An airport has a collection of gates arranged in a \\(10^{2022} \\times 10^{2022}\\) grid, labeled 1, 2, 3, \\ldots, \\(10^{4044}\\) in some order. A traveler's gate is the unique gate with the property that its label is less than all neighboring gates (horizontally or vertically). The traveler can ask for the lab... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-4b5b9117a1a78cb198cd39ce0e0299254215989d9953fe502511ea0655a6ff9d | An alarm clock runs 4 minutes slow every hour. It was set right $3 \frac{1}{2}$ hours ago. Now another clock which is correct shows noon. In how many minutes, to the nearest minute, will the alarm clock show noon? | [
"15"
] | [
"An alarm clock runs 4 minutes slow every hour. It was set right $3 \\frac{1}{2}$ hours ago. Now another clock which is correct shows noon. In how many minutes, to the nearest minute, will the alarm clock show noon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-074ebcafaf1fa8a47ef65bcc547613d5c372025a2ef46aa8939feb15c445b21f | An alien has collected 30 fruits with different degrees of sweetness. He has a device that can sort exactly 5 fruits from least to most sugary. How many times must the alien use the device to find the second least sugary fruit among the 30 fruits? | [
"9"
] | [
"An alien has collected 30 fruits with different degrees of sweetness. He has a device that can sort exactly 5 fruits from least to most sugary. How many times must the alien use the device to find the second least sugary fruit among the 30 fruits?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-173a4e3edafa0fa82f196a7fb1748c846745186f15242bbf21ce36ff9ed379d2 | An alien on the surface of a spherical planet can run at a top speed of \( u \). An alien hunter in a spaceship can fly at a top speed of \( v \). The hunter does not know the position of the alien initially but can spot and kill the alien as soon as it comes into the field of vision of the spacecraft. Determine the mi... | [
"1"
] | [
"An alien on the surface of a spherical planet can run at a top speed of \\( u \\). An alien hunter in a spaceship can fly at a top speed of \\( v \\). The hunter does not know the position of the alien initially but can spot and kill the alien as soon as it comes into the field of vision of the spacecraft. Determi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-3a273c89a5eb737fcee23fcdb4fbda08614504af49af3f09e688b4a8b15db1a5 | An alien population starts with one member. Each day, any given member has a $\frac{1}{5}$ probability to produce two offspring, a $\frac{1}{3}$ probability to produce one offspring, and a $\frac{7}{15}$ probability of dying. The probability that the alien population eventually does not die out can be expressed in simp... | [
"4"
] | [
"An alien population starts with one member. Each day, any given member has a $\\frac{1}{5}$ probability to produce two offspring, a $\\frac{1}{3}$ probability to produce one offspring, and a $\\frac{7}{15}$ probability of dying. The probability that the alien population eventually does not die out can be expressed... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-835f162a940931e49bce790bf728f61aac88b26db584c4cf1a0516486589ad28 | An alien race has a unique language consisting of only two letters $X$ and $Z$. In this language, each word consists of at least one letter and no more than 11 letters. For every two words, if the first and second words are written side by side, the result is not a word. Find the maximum number of words in this languag... | [
"4094"
] | [
"An alien race has a unique language consisting of only two letters $X$ and $Z$. In this language, each word consists of at least one letter and no more than 11 letters. For every two words, if the first and second words are written side by side, the result is not a word. Find the maximum number of words in this la... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5d924734eea2ece07b2e83767c74e124727d123b2679e6710e7d3a9cc55c4306 | An alien sitting on a table undergoes one of four actions every hour with equal probability: disappear, do nothing, clone itself (resulting in 2 aliens), or clone itself twice (resulting in 3 aliens). Each new clone behaves independently with the same probabilities. What is the probability that there will be no more al... | [
"\\(\\sqrt{2} - 1\\)"
] | [
"An alien sitting on a table undergoes one of four actions every hour with equal probability: disappear, do nothing, clone itself (resulting in 2 aliens), or clone itself twice (resulting in 3 aliens). Each new clone behaves independently with the same probabilities. What is the probability that there will be no mo... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-633486920094499401822009caa3e1981d5b223b9d89aae96442169314d6237d | An alien species starts with one alien. Each day, each alien independently does one of the following with equal probability: (i) kills itself, (ii) does nothing, (iii) splits into two aliens (killing itself in the process), or (iv) splits into three aliens (killing itself in the process). What is the probability that t... | [
"\\(\\sqrt{2} - 1\\)"
] | [
"An alien species starts with one alien. Each day, each alien independently does one of the following with equal probability: (i) kills itself, (ii) does nothing, (iii) splits into two aliens (killing itself in the process), or (iv) splits into three aliens (killing itself in the process). What is the probability t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-6daf0cf50c83bfb6e0c396a1e698aa65d5cb2575b2d6fa888ff69e188e7ab107 | An alloy consists of two metals (copper and tin) in proportion 1:2; another alloy consists of the same metals in proportion 2:3. How many pounds of each alloy should be taken to get 1,320 pounds of an alloy containing the metals in proportion 17:27? | [
"alloy 1: 270, alloy 2: 1050"
] | [
"An alloy consists of two metals (copper and tin) in proportion 1:2; another alloy consists of the same metals in proportion 2:3. How many pounds of each alloy should be taken to get 1,320 pounds of an alloy containing the metals in proportion 17:27?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 31
} | 35 | 0 | 0 |
omr-ws-3ab78daed8a300c14e8b30ab6b9a0309ddf632101df1884ef540eba6248f98a6 | An alloy of copper and silver contains 2 kg more copper than silver. If a further $\frac{9}{16}$ of the silver in the alloy is added, then the percentage of silver in the new alloy will be equal to that of the copper in the original alloy. Find the mass of the original alloy. | [
"18 kg"
] | [
"An alloy of copper and silver contains 2 kg more copper than silver. If a further $\\frac{9}{16}$ of the silver in the alloy is added, then the percentage of silver in the new alloy will be equal to that of the copper in the original alloy. Find the mass of the original alloy."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de9d1ac6f52e338e6f58a0671430c6685a236c0eec7a61823713cdc123bcf559 | An alphabet consists of six letters which are coded in Morse code as $$\cdot \ ;\ -\ ;\ ..\ ;\ - \, -\ ;\ \cdot \, -\ ;\ -\, .$$ A word was transmitted without spaces between the letters so that the resultant continuous line of dots and dashes contained 12 characters. In how many ways can that word be read? | [
"233"
] | [
"An alphabet consists of six letters which are coded in Morse code as $$\\cdot \\ ;\\ -\\ ;\\ ..\\ ;\\ - \\, -\\ ;\\ \\cdot \\, -\\ ;\\ -\\, .$$ A word was transmitted without spaces between the letters so that the resultant continuous line of dots and dashes contained 12 characters. In how many ways can that word ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2a4a23bdb44c4bb734934d46c8e8c08b1d5767934069064255deb311fccee761 | An alphabet consists of three letters. Some sequences of letters of length two or more are prohibited, and all of the prohibited sequences have different lengths. A word is a sequence of letters of any length. A correct word does not contain any prohibited sequences. Let $N$ be the smallest positive integer such that t... | [
"5"
] | [
"An alphabet consists of three letters. Some sequences of letters of length two or more are prohibited, and all of the prohibited sequences have different lengths. A word is a sequence of letters of any length. A correct word does not contain any prohibited sequences. Let $N$ be the smallest positive integer such t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-5e00f0463550da6c2717a0b9be22d5d6400013b8a5f953beab46cdd6a9feea39 | An alphabet consists of three letters. Some sequences of letters of length two or more are prohibited, and all of the prohibited sequences have different lengths. A word is a sequence of letters of any length. A correct word does not contain any prohibited sequences. Let $f(n)$ be the number of correct words of length ... | [
"7"
] | [
"An alphabet consists of three letters. Some sequences of letters of length two or more are prohibited, and all of the prohibited sequences have different lengths. A word is a sequence of letters of any length. A correct word does not contain any prohibited sequences. Let $f(n)$ be the number of correct words of le... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-84c165358ce5612ebb1564ed6134fa96a3f5bb042009e3381efbdbb96e01f502 | An altitude $BD$ and a bisector $BE$ are drawn in the triangle $A BC$ from the vertex $B$. It is known that the length of the side $| AC | = 1$, and the magnitudes of the angles $BEC$, $ABD$, $ABE$, $BAC$ form an arithmetic progression. Find the length of the side $BC$. | [
"$\\frac{1}{2}$"
] | [
"An altitude $BD$ and a bisector $BE$ are drawn in the triangle $A BC$ from the vertex $B$. It is known that the length of the side $| AC | = 1$, and the magnitudes of the angles $BEC$, $ABD$, $ABE$, $BAC$ form an arithmetic progression. Find the length of the side $BC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b0f2b0e0e9d8c09a5a657fd253e838d212cd7b0e0c90c6ec3e6b656a32d88a47 | An altitude $BD$ is drawn from the vertex $B$ of the isosceles triangle $ABC$ to its base $AC$. The length of each of the lateral sides $A B$ and $BC$ of $\triangle ABC$ is 8 cm. A median $DE$ is drawn in $\triangle BCD$. A circle is inscribed in $\triangle BDE$ which touches the side $BE$ at a point $K$ and the side $... | [
"$\\frac{\\pi }{6}$"
] | [
"An altitude $BD$ is drawn from the vertex $B$ of the isosceles triangle $ABC$ to its base $AC$. The length of each of the lateral sides $A B$ and $BC$ of $\\triangle ABC$ is 8 cm. A median $DE$ is drawn in $\\triangle BCD$. A circle is inscribed in $\\triangle BDE$ which touches the side $BE$ at a point $K$ and th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e96fd33ab723f2d92d0665f4f84ac7692ffae1f1b222a99ac4370b3c64f3869 | An aluminum sphere with an irrational radius can be melted down to form two new spheres with integer radii in two different ways. What is the smallest possible volume of the original sphere? | [
"\\(1729\\)"
] | [
"An aluminum sphere with an irrational radius can be melted down to form two new spheres with integer radii in two different ways. What is the smallest possible volume of the original sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 2
} | 15 | 0 | 0 |
omr-ws-e7e5d25c4a3135ae8807544c11d624a0c2fdd0913e29fce7fc6bf4096b5bf2bf | An amoeba has a 25% chance of dying and a 75% chance of splitting into two amoebas with these same probabilities. Starting with one amoeba, what is the probability that it will produce an infinite chain of amoebas that will never all die out? | [
"\\(\\frac{2}{3}\\)"
] | [
"An amoeba has a 25% chance of dying and a 75% chance of splitting into two amoebas with these same probabilities. Starting with one amoeba, what is the probability that it will produce an infinite chain of amoebas that will never all die out?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-b321d618bccb5078d07f76bcc47cb71946fce2063e23c8e2b6730086ac683cf6 | An amoeba is placed in a puddle one day, and on that same day it splits into two amoebas. The next day, each new amoeba splits into two new amoebas, and so on, so that each day every living amoeba splits into two new amoebas. After one week, how many amoebas are in the puddle? (Assume the puddle has no amoebas before t... | [
"128"
] | [
"An amoeba is placed in a puddle one day, and on that same day it splits into two amoebas. The next day, each new amoeba splits into two new amoebas, and so on, so that each day every living amoeba splits into two new amoebas. After one week, how many amoebas are in the puddle? (Assume the puddle has no amoebas be... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a047c2a927634db7075111037ba196fc1fd7fdfd8621e907f64a3a2c1042259 | An amoeba population starts with a single amoeba. After each minute, each amoeba will either die, split into two amoebas, or stay the same, each with equal probability. What is the probability that the amoeba population will eventually die out? | [
"1"
] | [
"An amoeba population starts with a single amoeba. After each minute, each amoeba will either die, split into two amoebas, or stay the same, each with equal probability. What is the probability that the amoeba population will eventually die out?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d444aabd01355c77770a9c306bd3064234b4ba8ab213d90458e607405018cee3 | An amusement park has a giant double ferris wheel. The double ferris wheel has a 30-meter rotating arm attached at its center to a 25-meter main support (10 meters off the ground). At each end of the rotating arm is attached a ferris wheel measuring 20 meters in diameter rotating counterclockwise. The rotating arm take... | [
"\\( f(t) = 7.5 \\sin\\left(\\frac{\\pi t}{3}\\right) + 25 + 10 \\sin\\left(\\frac{\\pi t}{2}\\right) \\), Period: 12 minutes, Maximum height: 42.5 meters, Least \\( t \\) for maximum height: 3 minutes."
] | [
"An amusement park has a giant double ferris wheel. The double ferris wheel has a 30-meter rotating arm attached at its center to a 25-meter main support (10 meters off the ground). At each end of the rotating arm is attached a ferris wheel measuring 20 meters in diameter rotating counterclockwise. The rotating arm... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-442ba6b295036c615ff1c37636ce474a708d2a6922e34bb223acd03cd74f54b4 | An amusement park purchased equipment from a manufacturer for a list price of $42,000, minus trade discounts of 30/20/15. Calculate the amount of the trade discount on the purchase. | [
"22008"
] | [
"An amusement park purchased equipment from a manufacturer for a list price of $42,000, minus trade discounts of 30/20/15. Calculate the amount of the trade discount on the purchase."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90b20470c42e748c523c5324901cf1cc999f158e7566e62948b34316fc2bb009 | An analog clock has two hands that can move independently. Initially, both hands point to the number 12. The clock performs a sequence of 144 hand movements such that on each movement, one of the two hands moves clockwise to the next number on the clock while the other hand does not move. Let $N$ be the number of seque... | [
"608"
] | [
"An analog clock has two hands that can move independently. Initially, both hands point to the number 12. The clock performs a sequence of 144 hand movements such that on each movement, one of the two hands moves clockwise to the next number on the clock while the other hand does not move. Let $N$ be the number of ... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b688d2faf4b316a7e94960bba410a8de21000ace613a35bcb5f1ee4ce9f8f5ea | An ancient Roman clock is divided into 18 sections marked with Roman numerals. The clock is broken into 5 pieces such that the sum of the numbers on each piece is the same. Show how the clock could be broken into these 5 pieces. | [
"No solution exists as the total sum (171) is not divisible by 5."
] | [
"An ancient Roman clock is divided into 18 sections marked with Roman numerals. The clock is broken into 5 pieces such that the sum of the numbers on each piece is the same. Show how the clock could be broken into these 5 pieces."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-24736f59339dd31e9252a36b32a41fd0a3b8cef7493fd3d3ac0306187892b998 | An ancient human tribe had a hierarchical system where there existed one chief with 2 supporting chiefs (supporting chief A and supporting chief B), each of whom had 2 equal, inferior officers. If the tribe at one point had 10 members, what is the number of different ways to choose the leadership of the tribe? That is,... | [
"151200"
] | [
"An ancient human tribe had a hierarchical system where there existed one chief with 2 supporting chiefs (supporting chief A and supporting chief B), each of whom had 2 equal, inferior officers. If the tribe at one point had 10 members, what is the number of different ways to choose the leadership of the tribe? Th... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f5a148bfe0f4041eb0fbcaa3da4c41f50bca9bc209dec4a4c1aff92e2558d83f | An and Bo play a card game with a set of 2n cards. Each card has a positive integer written on it. The cards are shuffled and arranged in a row, facing up. An starts first and can take one card from either end of the row. They continue to take cards from either end, alternating turns, until Bo takes the last card. An d... | [
"n^2"
] | [
"An and Bo play a card game with a set of 2n cards. Each card has a positive integer written on it. The cards are shuffled and arranged in a row, facing up. An starts first and can take one card from either end of the row. They continue to take cards from either end, alternating turns, until Bo takes the last card.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-36d1b1f9fb5a742978dde8e4f06a6e3ddeedfd0ce1f2039cf155060c01ef0927 | An angle $BAC$ of $60^\circ$ is taken in a plane $P$. The distances from a point $S$ to the vertex $A$, the side $AB$ and the side $AC$ are respectively 25 cm, 7 cm and 20 cm. Find the distance between the point $S$ and the plane $P$. | [
"\\(\\sqrt{37}\\) cm"
] | [
"An angle $BAC$ of $60^\\circ$ is taken in a plane $P$. The distances from a point $S$ to the vertex $A$, the side $AB$ and the side $AC$ are respectively 25 cm, 7 cm and 20 cm. Find the distance between the point $S$ and the plane $P$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e9831995c3f7b8a50d2c629b202b640460e54aa4a8993c038d641bb912639ed0 | An angle $x$ is chosen at random from the interval $0^\circ < x < 90^\circ.$ Let $p$ be the probability that the numbers $\sin^2 x, \cos^2 x,$ and $\sin x \cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n,$ where $d$ is the number of degrees in $\text{arctan}$ $m$ and $m$ and $n$ are positive ... | [
"092"
] | [
"An angle $x$ is chosen at random from the interval $0^\\circ < x < 90^\\circ.$ Let $p$ be the probability that the numbers $\\sin^2 x, \\cos^2 x,$ and $\\sin x \\cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n,$ where $d$ is the number of degrees in $\\text{arctan}$ $m$ and $m$ and ... | [
"MATH_training_set"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-a13de47d963f7cb01b4c6bc52d2d686826db10c9046de3cae59933482845ee3f | An angle \( \alpha \) satisfies \( 0 < \alpha < \frac{\pi}{4} \) and the equation \( \tan \left(\frac{\pi}{2} - \alpha \right) - \tan \alpha = 1 \). Find the values of \( \tan \alpha \) and \( \sin 2\alpha \). | [
"\\( \\tan \\alpha = \\frac{-1 + \\sqrt{5}}{2} \\), \\( \\sin 2\\alpha = \\frac{2\\sqrt{5}}{5} \\)"
] | [
"An angle \\( \\alpha \\) satisfies \\( 0 < \\alpha < \\frac{\\pi}{4} \\) and the equation \\( \\tan \\left(\\frac{\\pi}{2} - \\alpha \\right) - \\tan \\alpha = 1 \\). Find the values of \\( \\tan \\alpha \\) and \\( \\sin 2\\alpha \\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-b9e2a7c79dffdce5499bef80fba932b6d5558bed84b60b32c3522d345cd90222 | An angle \(\angle BAC\) is said to be "laceable" if distinct points \(X_1, X_2, \ldots, X_{2n}\) can be found such that: - \(X_{2k-1}\) is on \(AC\) for each value of \(k\), - \(X_{2k}\) is on \(AB\) for each value of \(k\), - \(AX_1 = X_1X_2 = X_2X_3 = \ldots = X_{2n-1}X_{2n} = X_{2n}A\). Determine the number of lacea... | [
"1"
] | [
"An angle \\(\\angle BAC\\) is said to be \"laceable\" if distinct points \\(X_1, X_2, \\ldots, X_{2n}\\) can be found such that:\n- \\(X_{2k-1}\\) is on \\(AC\\) for each value of \\(k\\),\n- \\(X_{2k}\\) is on \\(AB\\) for each value of \\(k\\),\n- \\(AX_1 = X_1X_2 = X_2X_3 = \\ldots = X_{2n-1}X_{2n} = X_{2n}A\\)... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-87dd38da698b132e264d1e12f0c85961642e30d8aac095fd50a6d51006e0ced4 | An angle bisector $AD$ is drawn in triangle $ABC$. On the side $AB$, point $M$ is marked so that $MD \parallel AC$ and $\angle MDA = \angle MDB$. Find the length of segment $DC$ if $AD = 5$. | [
"5"
] | [
"An angle bisector $AD$ is drawn in triangle $ABC$. On the side $AB$, point $M$ is marked so that $MD \\parallel AC$ and $\\angle MDA = \\angle MDB$. Find the length of segment $DC$ if $AD = 5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-1434baf101e075c95d27e443a315b1417b57ec7425f81f2a751f7b9ad1ffb2d3 | An angle bisector AC divides a trapezoid ABCD into two similar triangles ABC and ACD. Find the perimeter of this trapezoid if the leg AB = 9 cm and the leg CD = 12 cm. | [
"\\( 46 \\) cm or \\( 33 + 6\\sqrt{3} \\) cm"
] | [
"An angle bisector AC divides a trapezoid ABCD into two similar triangles ABC and ACD. Find the perimeter of this trapezoid if the leg AB = 9 cm and the leg CD = 12 cm."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-3809fba322d0e08bf8bf743ec1d7c8fe22aff1f6f348ec950de2ab74669dfe17 | An angle is given in a plane. Using only a compass, one must determine: (a) if this angle is acute. Find the minimal number of circles one must draw to be sure. (b) if this angle equals \(31^\circ\). (One may draw as many circles as one needs). | [
"\\text{No}"
] | [
"An angle is given in a plane. Using only a compass, one must determine:\n(a) if this angle is acute. Find the minimal number of circles one must draw to be sure.\n(b) if this angle equals \\(31^\\circ\\). (One may draw as many circles as one needs)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ae6ee5fda79fab12208362637432be42a381a63a9947f8dfa851e67b7aa87c6f | An angle of depression from the top of a 75-foot tall lighthouse to a ship in the water is 31 degrees. What is the length of the rope attached directly from the ship to the top of the lighthouse? | [
"146"
] | [
"An angle of depression from the top of a 75-foot tall lighthouse to a ship in the water is 31 degrees. What is the length of the rope attached directly from the ship to the top of the lighthouse?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 23
} | 46 | 0 | 0 |
omr-ws-271a030ef2fba57a955cd6d9e3910417f2336c80f3c7e9d6799d3d902ffaae80 | An angle referred to different units has a ratio of 8:5. If the smaller unit is 2 degrees, find the greater unit. | [
"3.2 degrees"
] | [
"An angle referred to different units has a ratio of 8:5. If the smaller unit is 2 degrees, find the greater unit."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 17
} | 31 | 0 | 0 |
omr-ws-d602514f2d866ce79020faab4cfa8fe22c192852ba31ecd4990c1e11af6a43e5 | An angle with a vertex at point \( O \) and a circle inside it. Using a compass and straightedge, construct points \( A \) and \( B \) on the sides of the angle so that the triangle \( OAB \) contains a circle inside it and has the smallest perimeter among all such triangles. | [
"OAB"
] | [
"An angle with a vertex at point \\( O \\) and a circle inside it. Using a compass and straightedge, construct points \\( A \\) and \\( B \\) on the sides of the angle so that the triangle \\( OAB \\) contains a circle inside it and has the smallest perimeter among all such triangles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-f81ec19ea4508372838da227a1461c7fd5069780ab5666985f3c36224bf3c926 | An angler casts a fishing line so that the sinker is projected with a speed of \(V\) m/s from a point 5 metres above a flat sea. The angle of projection is \(\theta\). Assume the equations of motion are \(x'' = 0\) and \(y'' = -g\). i) Let \((x, y)\) be the position of the sinker at time \(t\) seconds after the cast, a... | [
"\\dfrac{121}{8}"
] | [
"An angler casts a fishing line so that the sinker is projected with a speed of \\(V\\) m/s from a point 5 metres above a flat sea. The angle of projection is \\(\\theta\\). Assume the equations of motion are \\(x'' = 0\\) and \\(y'' = -g\\).\n\ni) Let \\((x, y)\\) be the position of the sinker at time \\(t\\) seco... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e82e86d2620d0d7b585200d05698aaab0f94b73aeaf89c4bbe0e0a7b3aaf043 | An annulus is the region between two concentric circles. Inside this annulus, twelve equal white semicircles are placed such that their diameters lie along the diameters of the outer circle. The rest of the annulus is black. What fraction of the annulus is black? | [
"\\(\\frac{1}{4}\\)"
] | [
"An annulus is the region between two concentric circles. Inside this annulus, twelve equal white semicircles are placed such that their diameters lie along the diameters of the outer circle. The rest of the annulus is black. What fraction of the annulus is black?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d347004540082546105e1ea33b48be0a7a0dedc662bcc0bb1d1a7f46bc435fbe | An ant can crawl from any cell to any of the four neighboring cells that have a common side with it (on the same face or across an edge). Help the ant go around all the cells, visiting each of them exactly once, on a Rubik's cube where each face is divided into $16$ cells ($4 \times 4$). | [
"\\text{Yes}"
] | [
"An ant can crawl from any cell to any of the four neighboring cells that have a common side with it (on the same face or across an edge). Help the ant go around all the cells, visiting each of them exactly once, on a Rubik's cube where each face is divided into $16$ cells ($4 \\times 4$)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-08089142b150936eb4731a393baa7c24f51561bb344e8ebab2e7d63f5fa4116c | An ant can crawl from any cell to any of the four neighboring cells that have a common side with it (on the same face or across an edge). Help the ant go around all the cells, visiting each of them exactly once, on a Rubik's cube where each face is divided into $25$ cells ($5 \times 5$). | [
"\\text{Yes}"
] | [
"An ant can crawl from any cell to any of the four neighboring cells that have a common side with it (on the same face or across an edge). Help the ant go around all the cells, visiting each of them exactly once, on a Rubik's cube where each face is divided into $25$ cells ($5 \\times 5$)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-6d9825942489dc11c1facd0ce41ae0a43985a3c311a37c20df20d9e60b3aace2 | An ant can crawl from any cell to any of the four neighboring cells that have a common side with it (on the same face or across an edge). Help the ant go around all the cells, visiting each of them exactly once, on a Rubik's cube where each face is divided into $9$ cells ($3 \times 3$). | [
"\\text{Yes}"
] | [
"An ant can crawl from any cell to any of the four neighboring cells that have a common side with it (on the same face or across an edge). Help the ant go around all the cells, visiting each of them exactly once, on a Rubik's cube where each face is divided into $9$ cells ($3 \\times 3$)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-e1ca75021c30a38f623ce4ccee1712837a8744e1ce7ae5b94208212515b6de68 | An ant crawls along the edges of an octahedron. It will not turn around on an edge until it reaches a vertex and can move to another edge after reaching a vertex. If the ant reaches one vertex 2014 times and each of the other 5 vertices \( n \) times, what is the minimum possible value of \( n \)? | [
"671"
] | [
"An ant crawls along the edges of an octahedron. It will not turn around on an edge until it reaches a vertex and can move to another edge after reaching a vertex. If the ant reaches one vertex 2014 times and each of the other 5 vertices \\( n \\) times, what is the minimum possible value of \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-122e9fc38a2b04243e54ee1008a663001caadb9cf3616325138b9303e727c4e2 | An ant crawls along the surface of a cube with side length $\sqrt{20}$, from vertex $A$ to vertex $B$. $A$ and $B$ are vertices that form an interior diagonal of the cube, and they are not shared by any one face. What is the shortest possible length of the ant’s path? | [
"10"
] | [
"An ant crawls along the surface of a cube with side length $\\sqrt{20}$, from vertex $A$ to vertex $B$. $A$ and $B$ are vertices that form an interior diagonal of the cube, and they are not shared by any one face. What is the shortest possible length of the ant’s path?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-35c0ddffefa8527a48dbb46670b27e63a9741c3a042aeee0f5d8a7b8bdb1449f | An ant crawls from the bottom left corner A to the top right corner B of a 4x5 square grid. Each step the ant takes moves it along one edge of a unit square to an adjacent vertex. How many ways can the ant travel from A to B in exactly 11 steps? | [
"5082"
] | [
"An ant crawls from the bottom left corner A to the top right corner B of a 4x5 square grid. Each step the ant takes moves it along one edge of a unit square to an adjacent vertex. How many ways can the ant travel from A to B in exactly 11 steps?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-d62a3221749d4c437d59b073c00dd6183f007dcd4e1679cdcd7c67962cabbe20 | An ant crawls on a table with constant speed in one direction, then every 15 minutes changes direction by turning 90 degrees. Let $T$ be the time in hours after which the ant returns to its original position for the first time. Find the value of $T$. | [
"1"
] | [
"An ant crawls on a table with constant speed in one direction, then every 15 minutes changes direction by turning 90 degrees. Let $T$ be the time in hours after which the ant returns to its original position for the first time. Find the value of $T$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c57d262ae39fc6b1b5afb1570c9660f13e18c1e3e81852bd6c1084ae3aeda501 | An ant crawls on the surface of a 3x3x3 large cube, made up of smaller cubes, from one corner to the opposite corner. The ant walks only along the edges of the small cubes and covers exactly nine of these edges. How many different paths can the ant take to reach its goal? | [
"384"
] | [
"An ant crawls on the surface of a 3x3x3 large cube, made up of smaller cubes, from one corner to the opposite corner. The ant walks only along the edges of the small cubes and covers exactly nine of these edges. How many different paths can the ant take to reach its goal?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 19
} | 40 | 0 | 0 |
omr-ws-29c500493f98d8694b7cc055d64b9af2752e7df6086ffac21f42d5bbd8219941 | An ant crawls on the surface of a cube with an edge length of $1$ cm. It wants to visit all the edges of the cube and return to the starting point. Find the smallest possible path length. | [
"13"
] | [
"An ant crawls on the surface of a cube with an edge length of $1$ cm. It wants to visit all the edges of the cube and return to the starting point. Find the smallest possible path length."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-baff751472eb570fea770bbc8887510324a31941ed07cea822283d61d3107b13 | An ant has one sock and one shoe for each of its six legs. In how many different orders can the ant put on its socks and shoes, assuming that, on each leg, the sock must be put on before a shoe? | [
"132"
] | [
"An ant has one sock and one shoe for each of its six legs. In how many different orders can the ant put on its socks and shoes, assuming that, on each leg, the sock must be put on before a shoe?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-811fd5003916e10f7b633b2e47991b945c8658d2733c8e409a6527c65806eb24 | An ant is about to step on food that is 10 steps in front of it. The ant can only step forward a multiple of three steps and the rest must step backwards. Determine how many steps the ant takes to reach the food, if it has to take no more than twenty steps. | [
"6"
] | [
"An ant is about to step on food that is 10 steps in front of it. The ant can only step forward a multiple of three steps and the rest must step backwards. Determine how many steps the ant takes to reach the food, if it has to take no more than twenty steps."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-995daa195f298123bba45a1e4c3f202c90ae73157571f3ed178094acf4787b07 | An ant is allowed to crawl either along the edges of a single cube, or along the diagonals of its faces. He must crawl from one vertex to the opposite without visiting any vertex twice. What is the largest number of diagonals such a path can include? | [
"2"
] | [
"An ant is allowed to crawl either along the edges of a single cube, or along the diagonals of its faces. He must crawl from one vertex to the opposite without visiting any vertex twice. What is the largest number of diagonals such a path can include?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6129bb03cc9a0efd67a7d848ffa159622ca43174e2c3436982376684f3d93981 | An ant is at $(2020, 2019)$ in the coordinate plane. Every minute, it can perform one of the following 9 actions from $(x,y)$: - Stay at $(x,y)$ - Move to $(x, y + 1)$ - Move to $(x + 1, y + 1)$ - Move to $(x + 1, y)$ - Move to $(x + 1, y - 1)$ - Move to $(x, y - 1)$ - Move to $(x - 1, y - 1)$ - Move to $(x - 1, y)$ - ... | [
"19"
] | [
"An ant is at $(2020, 2019)$ in the coordinate plane. Every minute, it can perform one of the following 9 actions from $(x,y)$:\n- Stay at $(x,y)$\n- Move to $(x, y + 1)$\n- Move to $(x + 1, y + 1)$\n- Move to $(x + 1, y)$\n- Move to $(x + 1, y - 1)$\n- Move to $(x, y - 1)$\n- Move to $(x - 1, y - 1)$\n- Move to $(... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-33c0e2708584f86e074b7caef2e3d372871ff0053aa74aa52dd0537d46e14a02 | An ant is at $(2020, 2019)$ in the coordinate plane. Every minute, starting with the first minute, she will do one of the following 9 actions from $(x, y)$: - Stay at $(x, y)$ - Move to $(x, y+1)$ - Move to $(x+1, y+1)$ - Move to $(x+1, y)$ - Move to $(x+1, y-1)$ - Move to $(x, y-1)$ - Move to $(x-1, y-1)$ - Move to $(... | [
"19"
] | [
"An ant is at $(2020, 2019)$ in the coordinate plane. Every minute, starting with the first minute, she will do one of the following 9 actions from $(x, y)$:\n- Stay at $(x, y)$\n- Move to $(x, y+1)$\n- Move to $(x+1, y+1)$\n- Move to $(x+1, y)$\n- Move to $(x+1, y-1)$\n- Move to $(x, y-1)$\n- Move to $(x-1, y-1)$\... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6f07a8685b11500c5c789c5696ef2828ae9937b3313541e8862af78c4f3228a0 | An ant is at a vertex of a cube. Every 10 minutes it moves to an adjacent vertex along an edge. If $N$ is the number of one-hour journeys that end at the starting vertex, find the sum of the squares of the digits of $N$. | [
"74"
] | [
"An ant is at a vertex of a cube. Every 10 minutes it moves to an adjacent vertex along an edge. If $N$ is the number of one-hour journeys that end at the starting vertex, find the sum of the squares of the digits of $N$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-2244cfd15907a575231fa8156479e0a2b5b8e4782439e7d5d1ab2b8193369269 | An ant is at a vertex of a cube. Every 10 minutes it moves to an adjacent vertex along an edge. If \( N \) is the number of one-hour journeys that end at the starting vertex, find the sum of the squares of the digits of \( N \). | [
"74"
] | [
"An ant is at a vertex of a cube. Every 10 minutes it moves to an adjacent vertex along an edge. If \\( N \\) is the number of one-hour journeys that end at the starting vertex, find the sum of the squares of the digits of \\( N \\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 28
} | 50 | 0 | 0 |
omr-ws-1c466a10538986944a8aeea9c807cfa24d8d79f59aa7de1f8e56d5a9522bf55d | An ant is at one corner of a unit cube. If the ant must travel on the box’s surface, what is the shortest distance the ant must crawl to reach the opposite corner of the cube? The distance can be written in the form $\sqrt{a}$, where $a$ is a positive integer. Compute $a$. | [
"5"
] | [
"An ant is at one corner of a unit cube. If the ant must travel on the box’s surface, what is the shortest distance the ant must crawl to reach the opposite corner of the cube? The distance can be written in the form $\\sqrt{a}$, where $a$ is a positive integer. Compute $a$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-da2e8591d171e255456c9b71f3b163bf83957ec20fa1708fdc4c2b3d168d2b5e | An ant is at point A and wants to reach point B by only moving in the directions $\to$, $\uparrow$, or $\nearrow$. How many different paths are possible given the grid below? [asy]usepackage("tikz");label("\begin{tikzpicture} \coordinate[label=below:$A$] (A) at (0,0); \coordinate[label=above:$B$] (B) at (3,5); \draw (A... | [
"231"
] | [
"An ant is at point A and wants to reach point B by only moving in the directions $\\to$, $\\uparrow$, or $\\nearrow$. How many different paths are possible given the grid below?\n[asy]usepackage(\"tikz\");label(\"\\begin{tikzpicture}\n\\coordinate[label=below:$A$] (A) at (0,0);\n\\coordinate[label=above:$B$] (B) a... | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-60c5826893e4689af4313389f6ae08f186e7c93dbf37d46f44c18300e13c06b7 | An ant is at the midpoint of a slant line on the outside wall of a conical glass with a slant height of 21 and a radius of 14 at the top rim. The ant sees a honey drop diametrically opposite to it on the inside wall of the glass. What is the integer part of the shortest distance the ant should crawl to reach the honey ... | [
"27"
] | [
"An ant is at the midpoint of a slant line on the outside wall of a conical glass with a slant height of 21 and a radius of 14 at the top rim. The ant sees a honey drop diametrically opposite to it on the inside wall of the glass. What is the integer part of the shortest distance the ant should crawl to reach the h... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-f60d4f9b7e2144c36dd8e19082a46c3cd4d2a794a254e1ce0b951f85fac594b7 | An ant is crawling along the coordinate plane. Each move, it moves one unit up, down, left, or right with equal probability. If it starts at $(0,0)$, what is the probability that it will be at either $(2,1)$ or $(1,2)$ after 6 moves? | [
"0"
] | [
"An ant is crawling along the coordinate plane. Each move, it moves one unit up, down, left, or right with equal probability. If it starts at $(0,0)$, what is the probability that it will be at either $(2,1)$ or $(1,2)$ after 6 moves?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-73ff682122f3b1c096d6970f85f46eecc0ed3d735744a4e5c2842afe0de02f61 | An ant is crawling at a rate of 48 cm per minute along a strip of rubber which is initially one meter long and one cm wide. At the end of each minute, the strip is stretched an additional one meter uniformly. If the ant starts at one end of the strip, find the number of minutes until it reaches the other end. Assume th... | [
"4"
] | [
"An ant is crawling at a rate of 48 cm per minute along a strip of rubber which is initially one meter long and one cm wide. At the end of each minute, the strip is stretched an additional one meter uniformly. If the ant starts at one end of the strip, find the number of minutes until it reaches the other end. Assu... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4e0e83f53c4f427cd0b9664a8492e96dae369189b5a5bb17ac96b1ba21b81240 | An ant is crawling on the edges of a cube, starting from one vertex. What is the maximum number of edges the ant can traverse if it can go through each edge only once? | [
"9"
] | [
"An ant is crawling on the edges of a cube, starting from one vertex. What is the maximum number of edges the ant can traverse if it can go through each edge only once?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-fcf51948638fdf838f14bae0765bc5bfc7db3dc59e0aa6341b1b9eb6801c8066 | An ant is going from $(0,0)$ to $(50,50)$. At each step, the ant can move from $(a,b)$ to $(a+1,b)$, $(a,b+1)$, or $(a+1,b+1)$. The path must not contain any right angles. Determine the number of such paths. | [
"3^{49}"
] | [
"An ant is going from $(0,0)$ to $(50,50)$. At each step, the ant can move from $(a,b)$ to $(a+1,b)$, $(a,b+1)$, or $(a+1,b+1)$. The path must not contain any right angles. Determine the number of such paths."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-7d39361510118d85ae5348fdac1ff6362753a7ccaf9fbc4d46427dcfdee6a98e | An ant is located at the center of one face of a regular octahedron with side length 6. The ant walks to the center of the face directly opposite to its starting face. Find the square of the shortest distance the ant travels, denoted as $m$. | [
"84"
] | [
"An ant is located at the center of one face of a regular octahedron with side length 6. The ant walks to the center of the face directly opposite to its starting face. Find the square of the shortest distance the ant travels, denoted as $m$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.