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-9f00419e37b4dd8ffc0ba48aeb7726e59b1e131e629de35d74f66a2560d213ee | A ship left port and sailed east at the rate of 20 miles per hour. One hour later, a second ship left the same port at the rate of 25 miles per hour, also traveling east. In how many hours did the second ship overtake the first ship? | [
"4"
] | [
"A ship left port and sailed east at the rate of 20 miles per hour. One hour later, a second ship left the same port at the rate of 25 miles per hour, also traveling east. In how many hours did the second ship overtake the first ship?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-0a91f4cd3bfe904bcb9f960fa1d53330754113a4917d00a675a2a76e5bef26ba | A ship sails 4 km up the river and then another 33 km down the river having spent an hour on the whole trip. Find the speed of the ship in dead water if the speed of the river is 6.5 km/h. | [
"32.5 km/h"
] | [
"A ship sails 4 km up the river and then another 33 km down the river having spent an hour on the whole trip. Find the speed of the ship in dead water if the speed of the river is 6.5 km/h."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-422cc69e1578616612b1a78188595be3d36ec3145b99498b1be1628b649e3085 | A ship sails at 20 kilometres/hour in a straight line which is, at its closest, 1 kilometre from a port. A tug-boat with a maximum speed of 12 kilometres/hour leaves the port and intercepts the ship at the latest possible time for which the interception is still possible. How far does the tug-boat travel? | [
"\\(\\frac{5}{4}\\) km"
] | [
"A ship sails at 20 kilometres/hour in a straight line which is, at its closest, 1 kilometre from a port. A tug-boat with a maximum speed of 12 kilometres/hour leaves the port and intercepts the ship at the latest possible time for which the interception is still possible. How far does the tug-boat travel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-c233474144dd2e8fcf0ce31a6ff8b8228671aac5c3078ce597f7701b470b0929 | A ship sails between two towns, its speed in good weather differs from that in bad weather. On Monday, during its voyage, good weather lasted an hour longer than bad weather. On Tuesday the ship ran in good weather as long as it ran in bad weather the previous day, and in bad weather it ran 1 h 4 min longer. On Wednesd... | [
"117 km, 24 km/h, 22.5 km/h"
] | [
"A ship sails between two towns, its speed in good weather differs from that in bad weather. On Monday, during its voyage, good weather lasted an hour longer than bad weather. On Tuesday the ship ran in good weather as long as it ran in bad weather the previous day, and in bad weather it ran 1 h 4 min longer. On We... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6707d180ac9e2a26678c57f9e512d67a30e9a01d17da0b146cf37f09351a3ed5 | A ship sets sail from port for 25 nautical miles due north. Then, the ship heads at a bearing of S80°W for 15 nautical miles. The ship will now return to port. Determine the bearing and distance the ship must travel to return to port. | [
"Bearing S33.41°W, Distance approximately 26 nautical miles."
] | [
"A ship sets sail from port for 25 nautical miles due north. Then, the ship heads at a bearing of S80°W for 15 nautical miles. The ship will now return to port. Determine the bearing and distance the ship must travel to return to port."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 18
} | 19 | 0 | 0 |
omr-ws-94df638c85586e55e57e80febef8bbd3d901ca19fbfe8451a5c146fdaa2894e0 | A ship spots a lighthouse 1 mile up the coast from the harbor and measures the angle between the line-of-sight observations of the harbor and the lighthouse to be 20 degrees. After traveling directly toward the harbor for 5 minutes at 12 miles per hour, the new angle is measured to be 30 degrees. How far is the ship fr... | [
"1.88 miles"
] | [
"A ship spots a lighthouse 1 mile up the coast from the harbor and measures the angle between the line-of-sight observations of the harbor and the lighthouse to be 20 degrees. After traveling directly toward the harbor for 5 minutes at 12 miles per hour, the new angle is measured to be 30 degrees. How far is the sh... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 29
} | 30 | 0 | 0 |
omr-ws-1bb5af6cc91938ed68f2899a562ff010e8fae03b77c2bb99faf854a9f45f7f9d | A ship starting from port $A$ and going to port $C$ must pass a lighthouse $B$ on the way, the distance between $A$ and $B$ being 140 km and from $B$ to $C$ 100 km. Three hours later a high-speed cutter left port $A$ in the direction of the ship and, having overtaken the ship, gave a command to increase the speed by 5 ... | [
"20 km/h, 60 km/h"
] | [
"A ship starting from port $A$ and going to port $C$ must pass a lighthouse $B$ on the way, the distance between $A$ and $B$ being 140 km and from $B$ to $C$ 100 km. Three hours later a high-speed cutter left port $A$ in the direction of the ship and, having overtaken the ship, gave a command to increase the speed ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-01a06ab1c5bff4e8a6657f6513bfa82b4b06d7dfb23b73f0776537839f675b88 | A ship's officer sees a lighthouse at a 42-degree angle to the path of the ship. After the ship travels 1800 meters, the lighthouse is at a 90-degree angle to the ship's path. What is the distance between the ship and the lighthouse at this second sight? | [
"\\( 1620.72 \\) meters"
] | [
"A ship's officer sees a lighthouse at a 42-degree angle to the path of the ship. After the ship travels 1800 meters, the lighthouse is at a 90-degree angle to the ship's path. What is the distance between the ship and the lighthouse at this second sight?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-378f3b5caebc7c02168fed51b8cbbcc9a6c2c111e98e909f545903e53ad4a9d9 | A shirt regularly priced at $\$30$ is on sale for $20\%$ off. Mary has a coupon that will take another $25\%$ off of the sale price. What single percent discount would give the same final price as the two successive discounts? | [
"40\\%"
] | [
"A shirt regularly priced at $\\$30$ is on sale for $20\\%$ off. Mary has a coupon that will take another $25\\%$ off of the sale price. What single percent discount would give the same final price as the two successive discounts?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1d8e1fd8579452aad97011d09948ea2306fe86e66153286d8e6fea5fe5f0a328 | A shop sells a multi-scoop ice cream called random ice-cream, made by flipping a fair coin to decide whether to include a vanilla scoop, then doing the same for the flavors chocolate, strawberry, and butterscotch. Let the probability that two random ice-creams have at least one flavor in common be $\frac{m}{n}$, where ... | [
"1"
] | [
"A shop sells a multi-scoop ice cream called random ice-cream, made by flipping a fair coin to decide whether to include a vanilla scoop, then doing the same for the flavors chocolate, strawberry, and butterscotch. Let the probability that two random ice-creams have at least one flavor in common be $\\frac{m}{n}$, ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1367aeb4a002ba961bfb7c91dcd192fa22d9759b306d2e877c17feecd8d34837 | A shop sells pencils where each pencil costs $1$ dollar, a bulk package of 5 pencils costs $4$ dollars, and a bulk package of 17 pencils costs $12$ dollars. If Mr. Lee needs 2012 pencils and he pays $n$ dollars to buy pencils, what is the smallest value of $n$? (Note that he may buy more pencils than he needs.) | [
"1421"
] | [
"A shop sells pencils where each pencil costs $1$ dollar, a bulk package of 5 pencils costs $4$ dollars, and a bulk package of 17 pencils costs $12$ dollars. If Mr. Lee needs 2012 pencils and he pays $n$ dollars to buy pencils, what is the smallest value of $n$? (Note that he may buy more pencils than he needs.)"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-cf3593035927c300029238cece5cd911a7aba772c951ecf23a8a57c97035feec | A shop sells three types of shoes, "A", "B", and "C". Each day, the number of customers is the same, and no customer buys multiple shoes on the same day. The transition probabilities are as follows: - 10% of customers who bought shoe A will buy shoe B the next day. - 10% of customers who bought shoe A will buy shoe C t... | [
"55\\frac{5}{9}\\%"
] | [
"A shop sells three types of shoes, \"A\", \"B\", and \"C\". Each day, the number of customers is the same, and no customer buys multiple shoes on the same day. The transition probabilities are as follows:\n- 10% of customers who bought shoe A will buy shoe B the next day.\n- 10% of customers who bought shoe A will... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-4b4be384e5004201fc0a7ee8fee6129ccf9ef82520e24aa6a36bbfff9808b871 | A shopkeeper makes a profit of 20% when giving 3 pencils free with every 5 pens bought and suffers a loss of 25% when giving 6 pencils free with every 2 pens bought. Find the approximate profit percent made by the shopkeeper when he gives 4 pencils free with every 6 pens bought. (Assume that the pencils are identical a... | [
"18.4%"
] | [
"A shopkeeper makes a profit of 20% when giving 3 pencils free with every 5 pens bought and suffers a loss of 25% when giving 6 pencils free with every 2 pens bought. Find the approximate profit percent made by the shopkeeper when he gives 4 pencils free with every 6 pens bought. (Assume that the pencils are identi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-46b4e448441715fafc67807f4bcdc2dc8ed70fe22864424217e9e3f5a261cd60 | A shopping mall has an average of 4100 daily customers. During a big sale last week, the average number was 4838. What percent did the average increase? | [
"18\\%"
] | [
"A shopping mall has an average of 4100 daily customers. During a big sale last week, the average number was 4838. What percent did the average increase?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6fcea2efb1636c7077c729ccc9b106456674c3e1e883274e309d20fe5ef699aa | A side of a rectangle with area $14$ divides a side of a square at the ratio $1:3$. Find the area of the square. | [
"50"
] | [
"A side of a rectangle with area $14$ divides a side of a square at the ratio $1:3$. Find the area of the square."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2eda8d7b6168913f0d0257a5b07e78b0abed18dd74fc3e4c62cb7efff07702be | A side of the base of a regular quadrangular prism is $a$. A plane is drawn through the diagonal of the lower base and a vertex of the upper base which cuts two adjacent lateral faces of the prism along straight lines which make an angle $\alpha $. Find the volume of the prism. | [
"$\\frac{a^3\\sqrt {2\\cos \\alpha }}{2\\sin (\\alpha /2)}$"
] | [
"A side of the base of a regular quadrangular prism is $a$. A plane is drawn through the diagonal of the lower base and a vertex of the upper base which cuts two adjacent lateral faces of the prism along straight lines which make an angle $\\alpha $. Find the volume of the prism."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-52364a6b10f9cb93f07fdd48caf0e343bbfdb642f8ae50e1bb476f156d117f96 | A side of triangle \(ABC\) measures \(a\). The values of the sides form an arithmetic progression and the values of the angles form another. Calculate the value of the area of the triangle. | [
"\\(\\frac{\\sqrt{3}}{4} a^2\\)"
] | [
"A side of triangle \\(ABC\\) measures \\(a\\). The values of the sides form an arithmetic progression and the values of the angles form another. Calculate the value of the area of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-3fce140917c4a1851d69bdfebed4348a9c676227cbd554f46b08b3befb73d6fd | A sidewalk of uniform width surrounds a flowerbed that is 11 ft wide and 13 ft long. The area of the sidewalk is 180 square feet. Find the width of the sidewalk. | [
"3"
] | [
"A sidewalk of uniform width surrounds a flowerbed that is 11 ft wide and 13 ft long. The area of the sidewalk is 180 square feet. Find the width of the sidewalk."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-225b0492e322990a5d35e169e6f87a4c8b2f7c7d9a962966d624c03521369216 | A sign painter paints individual numerals for a row of 50 houses. The houses are numbered with consecutive integers from 1 to 50. How many 9s are painted? | [
"5"
] | [
"A sign painter paints individual numerals for a row of 50 houses. The houses are numbered with consecutive integers from 1 to 50. How many 9s are painted?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01a4c9dd0538e74321f453b73739789a93a6517a681a848cf5d4c5ab7fc70e43 | A silo is created by attaching a hemisphere to the top of a right-circular cylinder with an equal radius. If \( h \) denotes the height and \( r \) the radius of the cylinder, then \( h - \frac{1}{3}r = 5 \). The volume of the silo is \( 80\pi \). Find its surface area (including the bottom). | [
"84\\pi"
] | [
"A silo is created by attaching a hemisphere to the top of a right-circular cylinder with an equal radius. If \\( h \\) denotes the height and \\( r \\) the radius of the cylinder, then \\( h - \\frac{1}{3}r = 5 \\). The volume of the silo is \\( 80\\pi \\). Find its surface area (including the bottom)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-ebc5d261339c7c7e1eb588153b366b92a2b0065ba5c4ab00655275cc83b4bd38 | A silver matrix is an \( n \times n \) matrix with entries from \(\{0, 1\}\) such that the sum of the entries in each row and each column is even. Let \( S(n) \) be the number of distinct silver matrices for a given \( n \). Calculate the last three digits of \( S(1997) \). | [
"536"
] | [
"A silver matrix is an \\( n \\times n \\) matrix with entries from \\(\\{0, 1\\}\\) such that the sum of the entries in each row and each column is even. Let \\( S(n) \\) be the number of distinct silver matrices for a given \\( n \\). Calculate the last three digits of \\( S(1997) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-905ee7e5e3c6926d095dac17350371dd2c0d86da1eb8101a336ed0354d119fbd | A silver matrix is an \( n \times n \) matrix with entries from \(\{0, 1\}\) such that the sum of the entries in each row and each column is even. Let \( f(n) \) be the number of distinct \( n \times n \) silver matrices. Calculate \( f(4) \). | [
"512"
] | [
"A silver matrix is an \\( n \\times n \\) matrix with entries from \\(\\{0, 1\\}\\) such that the sum of the entries in each row and each column is even. Let \\( f(n) \\) be the number of distinct \\( n \\times n \\) silver matrices. Calculate \\( f(4) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2eec11b858350987eb2d9474f276269910ca1527328719b8c8dc5c043bedc37b | A silvered ball of radius \( r = 5 \text{ cm} \) is suspended such that its center is \( 1 \text{ m} \) above the floor and \( 2 \text{ m} \) horizontally away from a person of height \( h = 2 \text{ m} \). There is also a large plane mirror standing vertically at some distance from the person. If the person sees his i... | [
"158 m"
] | [
"A silvered ball of radius \\( r = 5 \\text{ cm} \\) is suspended such that its center is \\( 1 \\text{ m} \\) above the floor and \\( 2 \\text{ m} \\) horizontally away from a person of height \\( h = 2 \\text{ m} \\). There is also a large plane mirror standing vertically at some distance from the person. If the ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ce62ca22e58cd960b96607b8eee8b1fc323e111319cc4a4f63a3d787a4691dc | A similar game is played with ten people, each lined up and wearing a red or blue hat chosen uniformly at random. Each person can see the hats of those in front of them but not their own or those behind them. Each person can bet on the color of their hat. The payout is equal to their final score, which can be negative.... | [
"9"
] | [
"A similar game is played with ten people, each lined up and wearing a red or blue hat chosen uniformly at random. Each person can see the hats of those in front of them but not their own or those behind them. Each person can bet on the color of their hat. The payout is equal to their final score, which can be nega... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-95527c39d56ec99f932ce49f051c1a4e4a706c6fde1bd57b8bbf6478520368fa | A simple connected graph $G$ has $a$ vertices of degree $4$, $b$ vertices of degree $3$, and no other vertices, where $a$ and $b$ are positive integers. Find the smallest real number $c$ so that the following statement is true: “If $a/b > c$, then $G$ must be quasicolorable.” Here, a graph $G$ is called [i]quasicolorab... | [
"\\dfrac{3}{4}"
] | [
"A simple connected graph $G$ has $a$ vertices of degree $4$, $b$ vertices of degree $3$, and no other vertices, where $a$ and $b$ are positive integers. Find the smallest real number $c$ so that the following statement is true: “If $a/b > c$, then $G$ must be quasicolorable.” \n\nHere, a graph $G$ is called [i]qua... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-8472b778dc5a85150f915ef47893bd6f5b2fba9cff7157f0b2cb0f80509d562e | A simple graph $G$ is called permutationary if there exists a permutation $\sigma$ of its vertices such that for every edge $uv$ in $G$, the vertices $\sigma(u)$ and $\sigma(v)$ are not adjacent in $G$. Let $G$ be a permutationary graph with $n$ vertices and $m$ edges. If $G$ and its complement $\overline{G}$ are both ... | [
"11"
] | [
"A simple graph $G$ is called permutationary if there exists a permutation $\\sigma$ of its vertices such that for every edge $uv$ in $G$, the vertices $\\sigma(u)$ and $\\sigma(v)$ are not adjacent in $G$. Let $G$ be a permutationary graph with $n$ vertices and $m$ edges. If $G$ and its complement $\\overline{G}$ ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f6f403c4deacf6dd854915225a470f917c16c877fcc4e2eb61ebe0d8fede75b | A simple graph is beautiful if any two adjacent vertices have different degrees. For any $n \geq 2$, let $f(n)$ be the maximal number of edges in a beautiful graph. Find constants $a$ and $b$ such that $$\lim_{n\to\infty} \frac{\binom{n}{2} - f(n)}{n^a} = b$$ | [
"\\(a = \\frac{3}{2}, b = \\frac{\\sqrt{2}}{3}\\)"
] | [
"A simple graph is beautiful if any two adjacent vertices have different degrees. For any $n \\geq 2$, let $f(n)$ be the maximal number of edges in a beautiful graph. Find constants $a$ and $b$ such that\n\n$$\\lim_{n\\to\\infty} \\frac{\\binom{n}{2} - f(n)}{n^a} = b$$"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ecb34de6f36bd572d566d7cad6a56888a8d9b7354f1f58c1a969aaa5da95afc2 | A single atom of Uranium-238 rests at the origin. Each second, the particle has a \( \frac{1}{4} \) chance of moving one unit in the negative \( x \) direction and a \( \frac{1}{2} \) chance of moving one unit in the positive \( x \) direction. If the particle reaches \( (-3,0) \), it ignites a fission that will consum... | [
"919"
] | [
"A single atom of Uranium-238 rests at the origin. Each second, the particle has a \\( \\frac{1}{4} \\) chance of moving one unit in the negative \\( x \\) direction and a \\( \\frac{1}{2} \\) chance of moving one unit in the positive \\( x \\) direction. If the particle reaches \\( (-3,0) \\), it ignites a fission... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4b4a716055304495fcb236a0e10fa61530fd7045cc8a35ff0525380b0bfe1fb3 | A single atom of Uranium-238 rests at the origin. Each second, the particle has a \(1/4\) chance of moving one unit in the negative x direction and a \(1/2\) chance of moving in the positive x direction. If the particle reaches \((-3, 0)\), it ignites a fission that will consume the earth. If it reaches \((7, 0)\), it ... | [
"919"
] | [
"A single atom of Uranium-238 rests at the origin. Each second, the particle has a \\(1/4\\) chance of moving one unit in the negative x direction and a \\(1/2\\) chance of moving in the positive x direction. If the particle reaches \\((-3, 0)\\), it ignites a fission that will consume the earth. If it reaches \\((... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-d8130c0120a89e80558f1cb193043d7d507bb24ffdf0ade0afe65cce090ac531 | A single bench section at a school event can hold either 7 adults or 11 children. When \( N \) bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of \( N \)? | [
"18"
] | [
"A single bench section at a school event can hold either 7 adults or 11 children. When \\( N \\) bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of \\( N \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-79ea86bd88057cdec23f3f65358c9c31eb2326d1a157236ce43c25d926d3e048 | A single cell can either die with probability $0.1$ or split into two cells with probability $0.9$. Each cell in the new generation dies or splits into two cells independently with the same probabilities. Find the probability distribution for the number of cells in any generation. | [
"G_n(s) = 0.1 + 0.9[G_{n-1}(s)]^2"
] | [
"A single cell can either die with probability $0.1$ or split into two cells with probability $0.9$. Each cell in the new generation dies or splits into two cells independently with the same probabilities. Find the probability distribution for the number of cells in any generation."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-e76fc046c9435f2d000030487455c59d73a295c11e95a08aa3ea4ed09b7518d6 | A single cell is placed in an infinite culture medium. It has a 1/2 chance of dying and a 1/2 chance of splitting into two cells. When a cell splits, both child cells have the same chance of dying as the parent. What is the probability that this cell colony will continue reproducing forever? | [
"0"
] | [
"A single cell is placed in an infinite culture medium. It has a 1/2 chance of dying and a 1/2 chance of splitting into two cells. When a cell splits, both child cells have the same chance of dying as the parent. What is the probability that this cell colony will continue reproducing forever?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2a093bd5a65eb13335f8776b704224a64674ac3524e7ef5d1974e7f544674fff | A single cell is placed in an infinite culture medium. It has a 1/2 chance of dying and a 1/2 chance of splitting into two cells. When a cell splits, each child cell has half the chance of dying as its parent. What is the probability that this cell colony will continue reproducing forever? | [
"2 - \\sqrt{2}"
] | [
"A single cell is placed in an infinite culture medium. It has a 1/2 chance of dying and a 1/2 chance of splitting into two cells. When a cell splits, each child cell has half the chance of dying as its parent. What is the probability that this cell colony will continue reproducing forever?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f61fcd4f6f0466de39b3fd7df919214f59a4d0bd1883ca745e4e40161c87b71e | A single section at a stadium can hold either 7 adults or 11 children. When \( N \) sections are completely filled, an equal number of adults and children will be seated in them. What is the least possible value of \( N \)? | [
"18"
] | [
"A single section at a stadium can hold either 7 adults or 11 children. When \\( N \\) sections are completely filled, an equal number of adults and children will be seated in them. What is the least possible value of \\( N \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7fc2b5ed8608d2459e27e509350cb7e315d65cc495c2ec299d75ff11f9277487 | A single-celled organism contains N particles, some of which are of type A and the others of type B. The cell is said to be in state i, where \(0 \leq i \leq N\), if it contains exactly i particles of type A. Daughter cells are formed by cell division, but first each particle replicates itself; the daughter cell inheri... | [
"\\( T(1) = 5 \\) and \\( T(2) = 5 \\)"
] | [
"A single-celled organism contains N particles, some of which are of type A and the others of type B. The cell is said to be in state i, where \\(0 \\leq i \\leq N\\), if it contains exactly i particles of type A. Daughter cells are formed by cell division, but first each particle replicates itself; the daughter ce... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 25
} | 28 | 0 | 0 |
omr-ws-787d3511a481710e7456faf44e4495572c4adbe4fce409dfc21cf0b7cf941c28 | A sister borrowed a wad of money from her mom and went shopping. She spent $18 on a new CD, then half of her remaining money on a new dress, $11 on lunch, one-third of her remaining money on a book, $12 on gas, and one-fourth of her remaining money on a discounted CD. She also slipped $2 to someone and gave her mom $10... | [
"124"
] | [
"A sister borrowed a wad of money from her mom and went shopping. She spent $18 on a new CD, then half of her remaining money on a new dress, $11 on lunch, one-third of her remaining money on a book, $12 on gas, and one-fourth of her remaining money on a discounted CD. She also slipped $2 to someone and gave her mo... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee1485ef4c7fda6acb369a523ec6290f2b6070d6bc74e5032eedb7bd2b3e6616 | A six-digit number $x$, with no digits other than $0$, $1$, or $2$, is complete if and only if $x$ has each of the digits $0$, $1$, and $2$ at least once. How many complete numbers are there? | [
"360"
] | [
"A six-digit number $x$, with no digits other than $0$, $1$, or $2$, is complete if and only if $x$ has each of the digits $0$, $1$, and $2$ at least once. How many complete numbers are there?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-27a46fd03760e78d47f375522c0189ff10daa6219a11bbcbe6262bf6e6fcfd45 | A six-digit number (base 10) is called squarish if it satisfies the following conditions: - None of its digits are zero. - It is a perfect square. - The first two digits, the middle two digits, and the last two digits of the number are all perfect squares when considered as two-digit numbers. How many squarish numbers ... | [
"2"
] | [
"A six-digit number (base 10) is called squarish if it satisfies the following conditions:\n- None of its digits are zero.\n- It is a perfect square.\n- The first two digits, the middle two digits, and the last two digits of the number are all perfect squares when considered as two-digit numbers.\n\nHow many squari... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-4405037016c870e175be2873bd37371fb70552e0844d4e28149b3c886e3add44 | A six-digit number begins with the digit 2. If this digit is moved from the first to the last place without violating the sequence of the remaining digits, then the newly obtained number will be three times the original number. Find the original number. | [
"285714"
] | [
"A six-digit number begins with the digit 2. If this digit is moved from the first to the last place without violating the sequence of the remaining digits, then the newly obtained number will be three times the original number. Find the original number."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9c0e3c7d90138cdd982c49af7120a2d3d69b9b8f35615bd61852e2633c3562f1 | A six-digit number is called a lucky number if each of its digits is either 4 or 7. Calculate the remainder when the sum of all six-digit lucky numbers is divided by 13. | [
"0"
] | [
"A six-digit number is called a lucky number if each of its digits is either 4 or 7. Calculate the remainder when the sum of all six-digit lucky numbers is divided by 13."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a1a8892180a37b14a343408a80105dae3d7f7c6263c2d19fc4e1c1d7b6734c52 | A six-digit number is defined as "amazing" if it is made up of only using digits 1, 3, and 9 (all 3 must be present to form an "amazing" number) and such that there are an even number of 9's. How many "amazing" numbers are there? | [
"240"
] | [
"A six-digit number is defined as \"amazing\" if it is made up of only using digits 1, 3, and 9 (all 3 must be present to form an \"amazing\" number) and such that there are an even number of 9's. How many \"amazing\" numbers are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6578dc8c5d6cc4cb478a89e6d6f614d2cec190cdf94ebaad7d5f50a69c86e6ab | A six-digit palindrome is a positive integer between $100,000$ and $999,999$ (inclusive) which is the same read forwards and backwards in base ten. How many composite six-digit palindromes are there? | [
"0"
] | [
"A six-digit palindrome is a positive integer between $100,000$ and $999,999$ (inclusive) which is the same read forwards and backwards in base ten. How many composite six-digit palindromes are there?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c461a10306a96481061a59ec38df22737109231526913a51339b8cb248bbcbdb | A six-digit positive integer is in set $F$ if each digit is either $1$, $2$, $3$, or $6$, and every pair of adjacent digits is relatively prime. Let $p(n)$ be the product of the digits of $n$. Compute the total product of $p(k)$ as $k$ ranges over the elements of $F$. | [
"18445"
] | [
"A six-digit positive integer is in set $F$ if each digit is either $1$, $2$, $3$, or $6$, and every pair of adjacent digits is relatively prime. Let $p(n)$ be the product of the digits of $n$. Compute the total product of $p(k)$ as $k$ ranges over the elements of $F$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-183c1cf3127eb2e74db1f976d5f86540a5b0016919b530aa0023c644d34a2589 | A six-dimensional "cube" (a $6$-cube) has vertices at the points $(\pm 3,\pm 3,\pm 3,\pm 3,\pm 3,\pm 3)$. This $6$-cube is divided into $6^6=46656$ smaller congruent "unit" $6$-cubes. How many 2-D squares are faces of one or more of the unit $6$-cubes? | [
"1296540"
] | [
"A six-dimensional \"cube\" (a $6$-cube) has vertices at the points $(\\pm 3,\\pm 3,\\pm 3,\\pm 3,\\pm 3,\\pm 3)$. This $6$-cube is divided into $6^6=46656$ smaller congruent \"unit\" $6$-cubes. How many 2-D squares are faces of one or more of the unit $6$-cubes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 5,
"tir": 18
} | 52 | 0 | 0 |
omr-ws-fb62fb62c597672e0a99cace7f2fbc120b8021be7e06a359fc4d4195b010b0a5 | A six-foot lamp post is directly 20 feet across the street from a 2 feet by 2 feet sign. The light on the top of the lamp post causes the sign to cast a shadow. What is the area of the shadow? | [
"25"
] | [
"A six-foot lamp post is directly 20 feet across the street from a 2 feet by 2 feet sign. The light on the top of the lamp post causes the sign to cast a shadow. What is the area of the shadow?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-92dbc5196789927f05272d7844276251c46931bd1f8e353f8fe93f7789563839 | A six-pound weight is attached to a four-foot nylon cord that is looped over two pegs. The pegs \( A \) and \( C \) are one foot apart and on the same level. When the weight is released, the system obtains an equilibrium position. Determine angle \( ABC \) for this equilibrium position, assuming negligible friction and... | [
"120^\\circ"
] | [
"A six-pound weight is attached to a four-foot nylon cord that is looped over two pegs. The pegs \\( A \\) and \\( C \\) are one foot apart and on the same level. When the weight is released, the system obtains an equilibrium position. Determine angle \\( ABC \\) for this equilibrium position, assuming negligible f... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-1fffd84e62335c12d1877895483d2af2cf859ade88c511cf32574630408a3101 | A six-sided die has faces labeled 1 through 6. It is weighted so that a 3 is 3 times as likely to be rolled as a 1; a 3 and a 6 are equally likely; and a 1, a 2, a 4, and a 5 are equally likely. What is the probability of rolling a 3? | [
"\\(\\frac{3}{10}\\)"
] | [
"A six-sided die has faces labeled 1 through 6. It is weighted so that a 3 is 3 times as likely to be rolled as a 1; a 3 and a 6 are equally likely; and a 1, a 2, a 4, and a 5 are equally likely. What is the probability of rolling a 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-d9bb34f0345dbf1630ac0bc8d2e9f7b951a288d2ba4c509b7d0f9d46d14b6d05 | A six-sided die is cast repeatedly until the same number appears on consecutive throws. What is the expected value of the number of throws required? | [
"7"
] | [
"A six-sided die is cast repeatedly until the same number appears on consecutive throws. What is the expected value of the number of throws required?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cd0f5f946cf6bba7ceba2ab976dcacaa4d704fd43a5320ad01ca947817425b1d | A six-sided polygon with vertices inscribed in a circle has side lengths 3, 3, 3, 5, 5, and 5 in that order. A chord of the circle divides the hexagon into two trapezoids: one with three sides of length 3 and one with three sides of length 5. Express the length of that chord as a reduced fraction. | [
"\\dfrac{360}{49}"
] | [
"A six-sided polygon with vertices inscribed in a circle has side lengths 3, 3, 3, 5, 5, and 5 in that order. A chord of the circle divides the hexagon into two trapezoids: one with three sides of length 3 and one with three sides of length 5. Express the length of that chord as a reduced fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-95e76c9db801849532b6ca2a74ae6af5ff724ed6abe051ed8d10e70fb15e5649 | A six-spoke cart starts rolling. After each twist, one of its spokes breaks with equal probability. If spokes from both sides of a wheel break, the wheel collapses. What is the probability that the wheel collapses after three twists? Express the probability as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are ... | [
"61"
] | [
"A six-spoke cart starts rolling. After each twist, one of its spokes breaks with equal probability. If spokes from both sides of a wheel break, the wheel collapses. What is the probability that the wheel collapses after three twists? Express the probability as a fraction \\( \\frac{a}{b} \\) where \\( a \\) and \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4e8060eb045955418dc3c6c9f48d21afec67322a3b1f1ab12e2b6fa6057224fc | A size-$n$ tromino is defined as an L-shaped figure consisting of $n+2$ unit squares. Determine the number of ways to tile a size-$2013$ tromino using size-$1$ trominos, and provide the last three digits of this number. | [
"096"
] | [
"A size-$n$ tromino is defined as an L-shaped figure consisting of $n+2$ unit squares. Determine the number of ways to tile a size-$2013$ tromino using size-$1$ trominos, and provide the last three digits of this number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-2a18a3cc09e11430c25e3b08e7bc560796c53a212f9977fca89fb1fba5f926b6 | A skill has a $5\%$ chance of activating every $0.1 + 1.01\sqrt{\frac{200}{299}}$ seconds. The activation lasts for $8$ seconds or until the next activation (which "refreshes" the skill), whichever is smaller. Over a "large" amount of time, what is the "uptime" of the skill, defined to be the ratio between the time whe... | [
"0.4"
] | [
"A skill has a $5\\%$ chance of activating every $0.1 + 1.01\\sqrt{\\frac{200}{299}}$ seconds. The activation lasts for $8$ seconds or until the next activation (which \"refreshes\" the skill), whichever is smaller. Over a \"large\" amount of time, what is the \"uptime\" of the skill, defined to be the ratio betwee... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-94c72cf4e69e0375851ff2df4341a9ba4cb1315a7fbbcc8e2b054e97936797af | A skill has a $a\%$ chance of activating every $b$ seconds. The activation lasts for $c$ seconds or until the next activation (which "refreshes" the skill), whichever is smaller. Over a "large" amount of time, what is the "uptime" of the skill, defined to be the ratio between the time where the skill is activated and t... | [
"\\(\\frac{a \\cdot c}{100 \\cdot b}\\)"
] | [
"A skill has a $a\\%$ chance of activating every $b$ seconds. The activation lasts for $c$ seconds or until the next activation (which \"refreshes\" the skill), whichever is smaller. Over a \"large\" amount of time, what is the \"uptime\" of the skill, defined to be the ratio between the time where the skill is act... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef8cdcf368cc3bbe1c828603a6b48664d2342c5772d88f6979c0decb55ec9c35 | A skillet is removed from an oven at 375 degrees F and placed in a room at 70 degrees F. After 5 minutes, the skillet's temperature is 325 degrees F. How long will it take for the skillet's temperature to reach 150 degrees F? | [
"37.37 minutes"
] | [
"A skillet is removed from an oven at 375 degrees F and placed in a room at 70 degrees F. After 5 minutes, the skillet's temperature is 325 degrees F. How long will it take for the skillet's temperature to reach 150 degrees F?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 26
} | 43 | 0 | 0 |
omr-ws-b0165744fe319c2d877b83566f85545a7965acb72ba9bdc216a8aeee2342f96a | A skydiving stunt involves two skydivers, Fred and Harry. Fred steps from a balloon at an altitude of 850 feet, free falls for eight seconds, and then deploys his chute. Harry steps from the balloon \( t_1 \) seconds after Fred, free falls for \( t_2 \) additional seconds, and then deploys his chute. After \( t_3 \) ad... | [
"50"
] | [
"A skydiving stunt involves two skydivers, Fred and Harry. Fred steps from a balloon at an altitude of 850 feet, free falls for eight seconds, and then deploys his chute. Harry steps from the balloon \\( t_1 \\) seconds after Fred, free falls for \\( t_2 \\) additional seconds, and then deploys his chute. After \\(... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-b469f6064827c5db045038c3021f4571406ade149c8c57d5c283e5966b61e6aa | A slant asymptote of the rational expression $y = \frac{2x^2 + 3x - 7}{x-3}$ is the line approached by the equation as $x$ approaches $\infty$ or $-\infty$. If this line is of the form $y = mx + b$, find $m+b$. | [
"11"
] | [
"A slant asymptote of the rational expression $y = \\frac{2x^2 + 3x - 7}{x-3}$ is the line approached by the equation as $x$ approaches $\\infty$ or $-\\infty$. If this line is of the form $y = mx + b$, find $m+b$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bed7d8aea7c2056df9a6a1039ce5cbe899bf1b6fa29360a87ef40d61baff1653 | A sleeping rabbit lies in the interior of a convex $2024$-gon. A hunter picks three vertices of the polygon and lays a trap which covers the interior and the boundary of the triangular region determined by them. Determine the minimum number of times the hunter needs to lay traps to guarantee that the rabbit will be tra... | [
"2022"
] | [
"A sleeping rabbit lies in the interior of a convex $2024$-gon. A hunter picks three vertices of the polygon and lays a trap which covers the interior and the boundary of the triangular region determined by them. Determine the minimum number of times the hunter needs to lay traps to guarantee that the rabbit will b... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-61bf416f7c67db6e8f742e5399cc9f4a5228981c73c33745b1facf2dbbfa35be | A slope field is defined by \( y' = \frac{1}{2} \left( \frac{y}{x} + \frac{x}{y} \right) \). A curve in this field passes through the point \((2,1)\). Find the \(x\)-intercept of this curve. | [
"-3"
] | [
"A slope field is defined by \\( y' = \\frac{1}{2} \\left( \\frac{y}{x} + \\frac{x}{y} \\right) \\). A curve in this field passes through the point \\((2,1)\\). Find the \\(x\\)-intercept of this curve."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cd3ba52d9e143e7495a8d95c9dd4a85784d13ca63d80eae6bdd9fdcdd1b8e44d | A slot machine works on inserting a \$1 coin. If the player wins, the coin is returned with an additional \$1 coin; otherwise, the original coin is lost. The probability of winning is 1/2 unless the previous play resulted in a win, in which case the probability is \( p < 1/2 \). If the cost of maintaining the machine a... | [
"\\( p > \\frac{1 + 3c}{2 + 2c} \\)"
] | [
"A slot machine works on inserting a \\$1 coin. If the player wins, the coin is returned with an additional \\$1 coin; otherwise, the original coin is lost. The probability of winning is 1/2 unless the previous play resulted in a win, in which case the probability is \\( p < 1/2 \\). If the cost of maintaining the ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4365064be7cdf2e5de28e5eca59bec4bf49cfe85335fb076f1710d35eb384ce | A small airplane has 4 rows of seats with 3 seats in each row. Eight passengers have boarded the plane and are distributed randomly among the seats. What is the probability that there will be 2 adjacent seats in the same row for the next boarding married couple? | [
"\\(\\frac{20}{33}\\)"
] | [
"A small airplane has 4 rows of seats with 3 seats in each row. Eight passengers have boarded the plane and are distributed randomly among the seats. What is the probability that there will be 2 adjacent seats in the same row for the next boarding married couple?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 30
} | 78 | 0 | 0 |
omr-ws-866d52c21ddf6fa173e29a290d87ba6453bfdbcd8348d22894f5a508f54cdcd8 | A small army of 120 soldiers has enough food for 6 days. At the start of every day, 4 soldiers are killed. How many days will the food last? | [
"6"
] | [
"A small army of 120 soldiers has enough food for 6 days. At the start of every day, 4 soldiers are killed. How many days will the food last?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4eaaa48bd7657f479b7a6a9c6e55115feeb36a87ee5e2e8d4b480c48351f9416 | A small circle \(A\) of radius \(\frac{1}{3}\) rotates, without slipping, inside and tangent to a unit circle \(B\). Let \(p\) be a fixed point on \(A\), and compute the length of the closed curve traced out by \(p\) as \(A\) rotates inside \(B\). | [
"\\dfrac{16}{3}"
] | [
"A small circle \\(A\\) of radius \\(\\frac{1}{3}\\) rotates, without slipping, inside and tangent to a unit circle \\(B\\). Let \\(p\\) be a fixed point on \\(A\\), and compute the length of the closed curve traced out by \\(p\\) as \\(A\\) rotates inside \\(B\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-8a179aabb9832e68aa5d0511802cf8a7c930b49e208ae795aae25b3f38d33b64 | A small circle is internally tangent to a big circle at point P. AB is a chord of the big circle that intersects the small circle at points C and D. If ∠APB = 120° and the radius of the big circle is 10 units, find the length of segment CD. | [
"\\dfrac{10\\sqrt{3}}{3}"
] | [
"A small circle is internally tangent to a big circle at point P. AB is a chord of the big circle that intersects the small circle at points C and D. If ∠APB = 120° and the radius of the big circle is 10 units, find the length of segment CD."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-b35d0defe1df9bc32f196c894aa4c810f2a7adf9267cfbf0544d644e6652dd9c | A small circle with radius \( r \) rolls around a larger circle with radius \( R \). Given that \( \frac{R}{r} = n \) where \( n \) is a natural number, find the area using line integrals. How to parametrize this line? | [
"\\pi r^2 (n + 1)(n + 2)"
] | [
"A small circle with radius \\( r \\) rolls around a larger circle with radius \\( R \\). Given that \\( \\frac{R}{r} = n \\) where \\( n \\) is a natural number, find the area using line integrals. How to parametrize this line?"
] | [
"aops_c7_college_math"
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"cot"
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"cot": 16
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omr-ws-ab14b129898be8de9c50fff5a8c9a9656212bc1c4cc88bde34096f19cc2c6029 | A small corporation borrowed $900,000: some at 7%, some at 8%, and some at 11%. How much was borrowed at 8% if the annual interest was $72,250 and the amount borrowed at 8% was $50,000 more than the amount borrowed at 11%? | [
"701642.25"
] | [
"A small corporation borrowed $900,000: some at 7%, some at 8%, and some at 11%. How much was borrowed at 8% if the annual interest was $72,250 and the amount borrowed at 8% was $50,000 more than the amount borrowed at 11%?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e925d7c793cfde5d2a808bd3f1f5d57b6f3696a2faad99d9bc21d2726695a9f | A small fish tank requires 60 square inches of wrapping paper and can hold 1/2 liter of water. If another fish tank with similar dimensions requires 240 square inches of wrapping paper, how many liters of water can it hold? | [
"4"
] | [
"A small fish tank requires 60 square inches of wrapping paper and can hold 1/2 liter of water. If another fish tank with similar dimensions requires 240 square inches of wrapping paper, how many liters of water can it hold?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-97cb2d9c8ae56f5f4ac5917c7cb194553fac87ffc57e5974ac99dd85bff62ecb | A small furniture company produces bookcases and sideboards. Producing a bookcase requires \( \frac{4}{3} \) m\(^2\) of wood board, \( \frac{4}{3} \) kg of nails, and \( \frac{2}{3} \) man-hours. Producing a sideboard requires 2 m\(^2\) of wood board, 1.5 kg of nails, and 2 man-hours. The profit for selling a bookcase ... | [
"990000"
] | [
"A small furniture company produces bookcases and sideboards. Producing a bookcase requires \\( \\frac{4}{3} \\) m\\(^2\\) of wood board, \\( \\frac{4}{3} \\) kg of nails, and \\( \\frac{2}{3} \\) man-hours. Producing a sideboard requires 2 m\\(^2\\) of wood board, 1.5 kg of nails, and 2 man-hours. The profit for s... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5229a7f8b531e8a92e4f24e5316647ccfc45e8248445d02c3c998e187b0a9c0f | A small hose fills a swimming pool in 16 hours, and a large hose fills the same pool in 12 hours. The small hose is turned on at 8:00 AM, and the large hose is turned on at 10:00 AM. Both hoses are used until 3:00 PM. Rounded to the nearest percent, how full is the pool at 3:00 PM? | [
"85%"
] | [
"A small hose fills a swimming pool in 16 hours, and a large hose fills the same pool in 12 hours. The small hose is turned on at 8:00 AM, and the large hose is turned on at 10:00 AM. Both hoses are used until 3:00 PM. Rounded to the nearest percent, how full is the pool at 3:00 PM?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
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"tir": 30
} | 45 | 0 | 0 |
omr-ws-ea98c6f3999192f68d114996e7a615d5c62e0d5a99a6ec8d263341300eddcb62 | A small jet plane whose speed is 600 mph in still air travels 1500 miles to another town and, after traveling for the same length of time on the return trip, still has 300 miles to go. How fast was the wind blowing? | [
"\\( \\frac{200}{3} \\) mph"
] | [
"A small jet plane whose speed is 600 mph in still air travels 1500 miles to another town and, after traveling for the same length of time on the return trip, still has 300 miles to go. How fast was the wind blowing?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-a275734672eb2f4097c4f363b7495bf38678788194718ad29481da40e032f9b9 | A small octagon is formed by connecting vertices and midpoints of a regular hexagon. What fraction of the hexagon is shaded by this octagon? | [
"\\(\\frac{1}{3}\\)"
] | [
"A small octagon is formed by connecting vertices and midpoints of a regular hexagon. What fraction of the hexagon is shaded by this octagon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2a8a93cafea2e71b912f6937833da6480b6765c23be7edb7c35ff662579cb614 | A small pack of cards consists of 3 red cards, 3 blue cards, 3 yellow cards, and 3 green cards. The pack is shuffled thoroughly, and the first four cards are turned face up. What is the probability that there is exactly one card of each color? | [
"\\(\\frac{9}{55}\\)"
] | [
"A small pack of cards consists of 3 red cards, 3 blue cards, 3 yellow cards, and 3 green cards. The pack is shuffled thoroughly, and the first four cards are turned face up. What is the probability that there is exactly one card of each color?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-4257c3e1766e66f2e83a3d61bb236eec4c475fad65c68583b1102b2226748d81 | A small pizza costs \$4 and has 6 slices. A large pizza costs \$9 and has 14 slices. If the MMATHS organizers got at least 400 slices of pizza as cheaply as possible, how many large pizzas did they buy? | [
"26"
] | [
"A small pizza costs \\$4 and has 6 slices. A large pizza costs \\$9 and has 14 slices. If the MMATHS organizers got at least 400 slices of pizza as cheaply as possible, how many large pizzas did they buy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-e769b27c1eacc03d3d6b284974672bf9b2f5a09015646bbef1dc6d7f04d81241 | A small rectangle is removed from a larger rectangular sheet of paper. Construct a line through the remaining paper that divides it into two parts with equal areas. | [
"\\text{Draw a line connecting the centers of the original and removed rectangles.}"
] | [
"A small rectangle is removed from a larger rectangular sheet of paper. Construct a line through the remaining paper that divides it into two parts with equal areas."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-79e7bd9bd70f846a0b2bc10d6b635c2025a4097a1b2d8744bc623efaef38e8af | A small robot first jumps 10 cm to the west, then 15 cm to the south, 20 cm to the east, and 25 cm to the north. After every 4 jumps, it repeats these jumps in the same order. After a total of $N$ jumps, the robot is 610 cm east and 585 cm north of its starting point. What is $N$? | [
"239"
] | [
"A small robot first jumps 10 cm to the west, then 15 cm to the south, 20 cm to the east, and 25 cm to the north. After every 4 jumps, it repeats these jumps in the same order. After a total of $N$ jumps, the robot is 610 cm east and 585 cm north of its starting point. What is $N$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f6e4f422291cca67a8188b8b3bd1bd7bffb4ef228a0247e9c41d6f1d4b198192 | A small software company publishes a total of 36 new programs each year, with at least four of these being games. The number of utility programs published is never more than twice the number of educational programs. The company makes an annual profit of 5000 dollars on each computer game, 8000 dollars on each education... | [
"4 computer games, 32 educational programs, 0 utility programs"
] | [
"A small software company publishes a total of 36 new programs each year, with at least four of these being games. The number of utility programs published is never more than twice the number of educational programs. The company makes an annual profit of 5000 dollars on each computer game, 8000 dollars on each educ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 29
} | 37 | 0 | 0 |
omr-ws-3d9f803de4e40d4aebcde7e8cd8d2b2e5f548789d315691811d0723a579914a0 | A small square $PQRS$ is contained within a larger square. Extend $PQ$ to $A$, $QR$ to $B$, $RS$ to $C$, and $SP$ to $D$ such that $A$, $B$, $C$, and $D$ lie on the four sides of the larger square in order. If the side length of the larger square is 10 units and the side length of the smaller square is 4 units, calcula... | [
"59"
] | [
"A small square $PQRS$ is contained within a larger square. Extend $PQ$ to $A$, $QR$ to $B$, $RS$ to $C$, and $SP$ to $D$ such that $A$, $B$, $C$, and $D$ lie on the four sides of the larger square in order. If the side length of the larger square is 10 units and the side length of the smaller square is 4 units, ca... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0ae460e9baa6d4442822c1e6814005a7417baf349d721fa2b58cf3607f0c6db1 | A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the area of the small square is exactly $\frac1{1985}$. | [
"32"
] | [
"A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the area of the small square is exactly $\\frac1{1985}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-666a9d6f087fc3a38c7be871ba3085af65571bab2977698747614e356743588a | A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the area of the small square is exactly $\frac{1}{1985}$. | [
"32"
] | [
"A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the area of the small square is exactly $\\frac{1}{1985}$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-bb7cfc1d9410e96caf6c23c353b9b9f7de670f60904115bd60d05696f595d7e1 | A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly $\frac1{1985}$. | [
"032"
] | [
"A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly $\\frac1{1985}$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 34,
"tir": 1
} | 35 | 0 | 0 |
omr-ws-03f84c96c8906e169810bdcff35cd3f64363dfc2041d5d4ddfc3cd5af1a692e3 | A small square is constructed inside a square of area 1 by dividing each side of the unit square into \( n \) equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of \( n \) if the area of the small square is exactly \( \frac{1}{1985} \). | [
"32"
] | [
"A small square is constructed inside a square of area 1 by dividing each side of the unit square into \\( n \\) equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of \\( n \\) if the area of the small square is exactly \\( \\frac{1}{1985} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 1
} | 12 | 0 | 0 |
omr-ws-56637a155021b279d007c6c69cebb7f715681b89f2ddec8a59bf6c36c810edd3 | A small tent, formed by a square frame with a vertical pole in the center and covered with fabric, requires 4 square meters of fabric. If two such frames are placed side by side with vertical poles in the center of each and covered with fabric to form a large tent, how much fabric is needed for the large tent? | [
"8"
] | [
"A small tent, formed by a square frame with a vertical pole in the center and covered with fabric, requires 4 square meters of fabric. If two such frames are placed side by side with vertical poles in the center of each and covered with fabric to form a large tent, how much fabric is needed for the large tent?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-7af1ac5021dc10058df1a9110347ccbcc4f73ddd0fe1ae801b49782751305e9f | A small tree 5 feet from a lamp post casts a shadow 4 feet long. If the lamp post were 2 feet taller, the shadow would only be 3 feet long. How tall is the tree? | [
"\\(\\frac{24}{5}\\)"
] | [
"A small tree 5 feet from a lamp post casts a shadow 4 feet long. If the lamp post were 2 feet taller, the shadow would only be 3 feet long. How tall is the tree?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-1f471040d0ea7020c2f9fc2719dc65b42a2c49456c7e9e89d2f15879f3ce6eca | A small wheel of 1 inch radius is put inside a circular track of 10 inch radius. If the small wheel is rotated clockwise on the circular track without slipping until it comes back to its starting point $P$, how many turns will the small wheel make? | [
"9"
] | [
"A small wheel of 1 inch radius is put inside a circular track of 10 inch radius. If the small wheel is rotated clockwise on the circular track without slipping until it comes back to its starting point $P$, how many turns will the small wheel make?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f18e89863705eb5611c5e00df8332cdc5846672fb649f0ae642b2e0c3f4b175 | A smaller circle is inscribed in the space between the inscribed circle and two edges of the equilateral triangle with an edge length of 1. What is the radius of the smaller circle? | [
"\\(\\frac{\\sqrt{3}}{18}\\)"
] | [
"A smaller circle is inscribed in the space between the inscribed circle and two edges of the equilateral triangle with an edge length of 1. What is the radius of the smaller circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-3bf311b5eae3f2e2617798499083cedf499ed037b133a2f347c6ea44659c87c2 | A smaller rectangle is inscribed in a larger rectangle such that each side of the larger rectangle contains a vertex of the smaller rectangle. If the ratio of the side lengths of the two rectangles is equal, what are all possible ratios? | [
"1"
] | [
"A smaller rectangle is inscribed in a larger rectangle such that each side of the larger rectangle contains a vertex of the smaller rectangle. If the ratio of the side lengths of the two rectangles is equal, what are all possible ratios?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cd09c32800f6f6d09b25f4120419804ae588711cb7065a4edf24466b14e49b8e | A smaller square was cut out of a larger square such that one of the sides of the smaller square lies on the side of the larger square. The perimeter of the resulting octagon is 40% larger than the perimeter of the original square. By what percentage is the area of the resulting octagon less than the area of the origin... | [
"36%"
] | [
"A smaller square was cut out of a larger square such that one of the sides of the smaller square lies on the side of the larger square. The perimeter of the resulting octagon is 40% larger than the perimeter of the original square. By what percentage is the area of the resulting octagon less than the area of the o... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7df6fa312b4d44baa8bfaa9b1b2fda560f8b856e510c4666c17ac6f708d452e7 | A smooth function \( f: \mathbb{R} \to \mathbb{R} \) satisfies \( f(0) = 0 \), \( f'(0) = 1 \), and for all \( n \ge 0 \) and \( x \in \mathbb{R} \), \( |f^{(n)}(x)| \le 1 \). Calculate the value of \( f\left(\frac{\pi}{2}\right) \). | [
"1"
] | [
"A smooth function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) satisfies \\( f(0) = 0 \\), \\( f'(0) = 1 \\), and for all \\( n \\ge 0 \\) and \\( x \\in \\mathbb{R} \\), \\( |f^{(n)}(x)| \\le 1 \\). Calculate the value of \\( f\\left(\\frac{\\pi}{2}\\right) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a8718b2fb9e2b699e2565fb522d014b517b01bab7c1d4d4347d59c2bc79ac1b9 | A smooth solid consists of a right circular cylinder of height $h$ and base-radius $r$, surmounted by a hemisphere of radius $r$ and center $O$. The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched over the highest point of the solid and held down at th... | [
"\\arcsin\\left(\\dfrac{\\sqrt{3}}{4}\\right)"
] | [
"A smooth solid consists of a right circular cylinder of height $h$ and base-radius $r$, surmounted by a hemisphere of radius $r$ and center $O$. The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched over the highest point of the solid and held down ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3af6b3a34fa3fcb3bc856b89966c70196cc3b490c3f0143fad432de8be5ef1f5 | A snail crawled along a straight line for 6 minutes while several people watched it in progress. Each person watched the snail for 1 minute, and the snail was always watched by at least one person. What is the maximum possible distance the snail could crawl during these 6 minutes? | [
"6 feet"
] | [
"A snail crawled along a straight line for 6 minutes while several people watched it in progress. Each person watched the snail for 1 minute, and the snail was always watched by at least one person. What is the maximum possible distance the snail could crawl during these 6 minutes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-63215677da200922d459b5aba4e91d22262a852c977b095dc16a53742617a890 | A snail is at the bottom of a well, 115 feet deep. On day 1, it starts climbing up the side. That night it rests and slips down one foot. It repeats this process, climbing up 3 feet each day and sliding down one foot each night until it reaches the top of the well. On what day number will the snail reach the top? | [
"57"
] | [
"A snail is at the bottom of a well, 115 feet deep. On day 1, it starts climbing up the side. That night it rests and slips down one foot. It repeats this process, climbing up 3 feet each day and sliding down one foot each night until it reaches the top of the well. On what day number will the snail reach the top?"... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-45dc14a40fa8e2f081076c73f46fc22286f697c3f4ed048de2d2b9e95b993a45 | A snail starts at the origin of an Argand diagram and walks along the real axis for an hour, covering a distance of 8 metres. At the end of each hour, it changes its direction by $\frac{\pi}{4}$ anticlockwise, and in each hour it walks $\frac{1}{\sqrt{2}}$ times as far as it did in the previous hour. Find where it is a... | [
"\\( 10 + 10i \\)"
] | [
"A snail starts at the origin of an Argand diagram and walks along the real axis for an hour, covering a distance of 8 metres. At the end of each hour, it changes its direction by $\\frac{\\pi}{4}$ anticlockwise, and in each hour it walks $\\frac{1}{\\sqrt{2}}$ times as far as it did in the previous hour. Find wher... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-ae82cdf30b233e0d530b51cc63ece2fd3f81203c90deafb0564004e85654cd4f | A snail starts at the origin of an Argand diagram and walks along the real axis for an hour, covering a distance of 8 metres. At the end of each hour, it changes its direction by $\frac{\pi}{4}$ anticlockwise, and in each hour it walks $\frac{1}{\sqrt{2}}$ times as far as it did in the previous hour. Find where it is a... | [
"\\( 6.5 + \\frac{17i}{2} \\)"
] | [
"A snail starts at the origin of an Argand diagram and walks along the real axis for an hour, covering a distance of 8 metres. At the end of each hour, it changes its direction by $\\frac{\\pi}{4}$ anticlockwise, and in each hour it walks $\\frac{1}{\\sqrt{2}}$ times as far as it did in the previous hour. Find wher... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-63c2cdd08c532ee7cd6f1317468bdb9e93af39fb7fa234a130be4af0d353840a | A snail starts at the origin of an Argand diagram and walks along the real axis for an hour, covering a distance of 8 metres. At the end of each hour, it changes its direction by $\frac{\pi}{4}$ anticlockwise, and in each hour it walks $\frac{1}{\sqrt{2}}$ times as far as it did in the previous hour. Find where it is e... | [
"\\( 8 + 8i \\)"
] | [
"A snail starts at the origin of an Argand diagram and walks along the real axis for an hour, covering a distance of 8 metres. At the end of each hour, it changes its direction by $\\frac{\\pi}{4}$ anticlockwise, and in each hour it walks $\\frac{1}{\\sqrt{2}}$ times as far as it did in the previous hour. Find wher... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-4e949bbc15ab77922dc5f0a841452d3d3d806523d329b15b0d1063bfa4a102f6 | A snake forms a circle with a primary length of 1 meter and eats 5 cm of its tail every minute. What is the time required to reduce the radius of the circle to half? | [
"10"
] | [
"A snake forms a circle with a primary length of 1 meter and eats 5 cm of its tail every minute. What is the time required to reduce the radius of the circle to half?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-ca42330a742d6016f5e099f10eb4cd6200cb8e837ee2908ce9bd838adf120e4c | A snake of length \( n \) is a permutation \( x_1, x_2, \ldots, x_n \) of the integers \( 1, 2, \ldots, n \) such that \[ x_1 < x_2, x_2 > x_3, x_3 < x_4, \ldots \] Given that \( s_1 = s_2 = 1 \), \( s_3 = 2 \), and \( s_4 = 5 \), find \( s_7 \), the number of different snakes of length 7. | [
"272"
] | [
"A snake of length \\( n \\) is a permutation \\( x_1, x_2, \\ldots, x_n \\) of the integers \\( 1, 2, \\ldots, n \\) such that\n\\[ x_1 < x_2, x_2 > x_3, x_3 < x_4, \\ldots \\]\nGiven that \\( s_1 = s_2 = 1 \\), \\( s_3 = 2 \\), and \\( s_4 = 5 \\), find \\( s_7 \\), the number of different snakes of length 7."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-367c89a903de94998c729a40ea0c4179275bc7ee8ff1d2eb19fdfb318b5b1c2e | A snakelike arrangement of length \( n \) is defined as an ordering of the numbers from \( 1 \) to \( n \) such that every number at an even index is greater than the two numbers directly on either side of it, and every number at an odd index is less than the two numbers directly on either side of it. If a number is on... | [
"E_n"
] | [
"A snakelike arrangement of length \\( n \\) is defined as an ordering of the numbers from \\( 1 \\) to \\( n \\) such that every number at an even index is greater than the two numbers directly on either side of it, and every number at an odd index is less than the two numbers directly on either side of it. If a n... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-e94a5754f3de155ec0296f006079f160e7ca409a724a14cfd763e145823b44e2 | A sniper has a 0.8 chance to hit the target if he hit his last shot and a 0.7 chance to hit the target if he missed his last shot. It is known that he missed on the 1st shot and hit on the 3rd shot. What is the probability that he hit the second shot? | [
"\\(\\frac{8}{11}\\)"
] | [
"A sniper has a 0.8 chance to hit the target if he hit his last shot and a 0.7 chance to hit the target if he missed his last shot. It is known that he missed on the 1st shot and hit on the 3rd shot. What is the probability that he hit the second shot?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-3d453a39d2a55fd4fc18559dc4adc958c378a5fa69b6d718e66aca211dc929d3 | A snowball is melting such that the radius is shrinking at a rate of 5 in/hr. At what rate will the volume be decreasing at the moment when the radius of the snowball is 8 inches? | [
"\\(-1280 \\pi\\) cubic inches per hour."
] | [
"A snowball is melting such that the radius is shrinking at a rate of 5 in/hr. At what rate will the volume be decreasing at the moment when the radius of the snowball is 8 inches?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 21
} | 39 | 0 | 0 |
omr-ws-7834ea0966c5702d58fb08d5e14ed8c54018de152dc3fdbff1a78e1fc38f65fe | A snowball of spherical shape is melting. How fast is the radius changing when the radius is 2 inches and the snow is melting at a rate of 10 cubic inches per minute? | [
"\\(\\frac{-5}{8\\pi}\\)"
] | [
"A snowball of spherical shape is melting. How fast is the radius changing when the radius is 2 inches and the snow is melting at a rate of 10 cubic inches per minute?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-0e2790249979be14ae485423323bc27f32a8912422922fe69fd2ab47454c0266 | A snowplow (x) leaves the house at 1:00 PM with a speed inversely proportional to the amount of snow it is pushing, where the snow starts falling at a constant rate \( r \) at 12:00 PM. At 2:00 PM, a second snowplow (y) leaves the house with a speed also inversely proportional to the amount of snow it is clearing. When... | [
"2:718 PM"
] | [
"A snowplow (x) leaves the house at 1:00 PM with a speed inversely proportional to the amount of snow it is pushing, where the snow starts falling at a constant rate \\( r \\) at 12:00 PM. At 2:00 PM, a second snowplow (y) leaves the house with a speed also inversely proportional to the amount of snow it is clearin... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 1
} | 4 | 0 | 0 |
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