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-b9085a413fc13faaf0470dc55d7cefdb46f2f9815412cfa0e57b4f482f3f4d20 | A finite set $\mathcal{S}$ of points in the plane is defined as [i]balanced[/i] if, for any three different points in $\mathcal{S}$, at least one point has the same distance to the other two. Find the maximum number of elements of a balanced set. | [
"6"
] | [
"A finite set $\\mathcal{S}$ of points in the plane is defined as [i]balanced[/i] if, for any three different points in $\\mathcal{S}$, at least one point has the same distance to the other two. Find the maximum number of elements of a balanced set."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 3
} | 21 | 0 | 0 |
omr-ws-be18abae08f6dab6a4346fd8d9b2b3c807b5bea2f28f0c1abf1ba04e61a224d2 | A finite set $\mathcal{S}$ of positive integers is called cardinal if $\mathcal{S}$ contains the integer $|\mathcal{S}|$, where $|\mathcal{S}|$ denotes the number of distinct elements in $\mathcal{S}$. Let $f$ be a function from the set of positive integers to itself such that for any cardinal set $\mathcal{S}$, the se... | [
"1, 2, 2024"
] | [
"A finite set $\\mathcal{S}$ of positive integers is called cardinal if $\\mathcal{S}$ contains the integer $|\\mathcal{S}|$, where $|\\mathcal{S}|$ denotes the number of distinct elements in $\\mathcal{S}$. Let $f$ be a function from the set of positive integers to itself such that for any cardinal set $\\mathcal{... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c32279e46cf1381d47d49d854a6f2e9481e69be5f2699ccc90f276975e3f887c | A finite set \( M \) of positive integers consists of distinct perfect squares and the number \( 92 \). The average of the numbers in \( M \) is \( 85 \). If we remove \( 92 \) from \( M \), the average drops to \( 84 \). If \( N^2 \) is the largest possible square in \( M \), what is the value of \( N \)? | [
"22"
] | [
"A finite set \\( M \\) of positive integers consists of distinct perfect squares and the number \\( 92 \\). The average of the numbers in \\( M \\) is \\( 85 \\). If we remove \\( 92 \\) from \\( M \\), the average drops to \\( 84 \\). If \\( N^2 \\) is the largest possible square in \\( M \\), what is the value o... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-cf1f8c36413bec7c05a6eccb4d26111a2cef84cff781c9724d9afb611327d10f | A finite set \( M \) of real numbers has at least 4 elements and there exists a bijective function \( f: M \to M \), different from the identity, such that \( ab \leq f(a)f(b) \) for all \( a \neq b \in M \). Let \( S \) be the sum of all elements in \( M \). If \( M \) contains the elements \( -3, -1, 1, \) and \( 3 \... | [
"0"
] | [
"A finite set \\( M \\) of real numbers has at least 4 elements and there exists a bijective function \\( f: M \\to M \\), different from the identity, such that \\( ab \\leq f(a)f(b) \\) for all \\( a \\neq b \\in M \\). Let \\( S \\) be the sum of all elements in \\( M \\). If \\( M \\) contains the elements \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d3d2168c78b837304f70d97f0c376fae1504e032a0b2cbd777e9293e90dfabb2 | A finite set \( S \) of points with \( |S| \ge 3 \) is called [i]good[/i] if for any three distinct elements of \( S \), they are non-collinear and the orthocenter of the triangle formed by these three points is also in \( S \). Find all such good sets. | [
"A rectangle (including a square), a right triangle, and a triangle with its orthocenter included."
] | [
"A finite set \\( S \\) of points with \\( |S| \\ge 3 \\) is called [i]good[/i] if for any three distinct elements of \\( S \\), they are non-collinear and the orthocenter of the triangle formed by these three points is also in \\( S \\). Find all such good sets."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 28,
"genselect": 7,
"tir": 9
} | 44 | 0 | 0 |
omr-ws-d728058786ceec0a65b69960cf31ddf44b89d6a2c9526cafe28665060bf6081b | A finite set \( S \) of positive integers is called cardinal if \( S \) contains the integer \( |S| \), where \( |S| \) denotes the number of distinct elements in \( S \). Let \( f \) be a function from the set of positive integers to itself, such that for any cardinal set \( S \), the set \( f(S) \) is also cardinal. ... | [
"2024"
] | [
"A finite set \\( S \\) of positive integers is called cardinal if \\( S \\) contains the integer \\( |S| \\), where \\( |S| \\) denotes the number of distinct elements in \\( S \\). Let \\( f \\) be a function from the set of positive integers to itself, such that for any cardinal set \\( S \\), the set \\( f(S) \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-007dc26aca0ddad313f6ca4cee7c7b09a4ad51b815312c5ab4ae60bbbb5d6d37 | A finite set of points in the plane, no three of which are collinear, has the property that for every three points \( A, B, C \) from the set, the orthocenter of the triangle \( ABC \) is also in the set. Find all such sets. | [
"A rectangle, a right triangle, or an acute triangle with its orthocenter included."
] | [
"A finite set of points in the plane, no three of which are collinear, has the property that for every three points \\( A, B, C \\) from the set, the orthocenter of the triangle \\( ABC \\) is also in the set. Find all such sets."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-d3531a8a6b2b323391428cd548f14034aa3ccf3131f1c99cd2149074b700f66c | A finite set of segments is given on the plane, the sum of the lengths of which does not exceed $\sqrt{2}$. Find the minimum side length $s$ of a square grid such that the lines of this grid do not intersect any of these segments. | [
"\\sqrt{2}"
] | [
"A finite set of segments is given on the plane, the sum of the lengths of which does not exceed $\\sqrt{2}$. Find the minimum side length $s$ of a square grid such that the lines of this grid do not intersect any of these segments."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-4ed4030ba71494d99759a15fb924442b35c198a64ea0d035f4b680c0a513d3bf | A fire in a building B is reported to two fire stations P and Q, which are 20 km apart on a straight road. P observes the fire at an angle of 60 degrees to the road, and Q observes it at an angle of 45 degrees to the road. Which station should send its team, and how far will the team have to travel? | [
"\\( 20\\sqrt{3} - 20 \\) km or approximately 14.641 km."
] | [
"A fire in a building B is reported to two fire stations P and Q, which are 20 km apart on a straight road. P observes the fire at an angle of 60 degrees to the road, and Q observes it at an angle of 45 degrees to the road. Which station should send its team, and how far will the team have to travel?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-4b88f5bbab1fd1abf446f9db52378e976c3a9a3e71f7367a016cd8dc862158cf | A fire that starts in the steppe spreads in all directions at a speed of $1$ km per hour. A grader with a plow arrived on the fire line at the moment when the fire engulfed a circle with a radius of $1$ km. The grader moves at a speed of $14$ km per hour and cuts a strip with a plow that cuts off the fire. Determine th... | [
"r = e^{\\theta / \\sqrt{195}}"
] | [
"A fire that starts in the steppe spreads in all directions at a speed of $1$ km per hour. A grader with a plow arrived on the fire line at the moment when the fire engulfed a circle with a radius of $1$ km. The grader moves at a speed of $14$ km per hour and cuts a strip with a plow that cuts off the fire. Determi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-6ea36b9c8fcbba8529e4fd5499a8b2bbc71f537e238fd3022b12670cd657f1f1 | A fire that starts in the steppe spreads in all directions at a speed of $1$ km per hour. A grader with a plow arrived on the fire line at the moment when the fire engulfed a circle with a radius of $1$ km. The grader moves at a speed of $14$ km per hour and cuts a strip with a plow that cuts off the fire. Determine th... | [
"\\(\\frac{7}{\\pi}\\)"
] | [
"A fire that starts in the steppe spreads in all directions at a speed of $1$ km per hour. A grader with a plow arrived on the fire line at the moment when the fire engulfed a circle with a radius of $1$ km. The grader moves at a speed of $14$ km per hour and cuts a strip with a plow that cuts off the fire. Determi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-73a0809adf5c074d6073cb0478fccd2593a2624d91ea24c72884755364c60960 | A firetruck is stationed at the intersection of two perpendicular straight highways and can travel at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Determine the area of the region that can be reached by the firetruck within six minutes. The area is expressed as $m/n$ square mile... | [
"731"
] | [
"A firetruck is stationed at the intersection of two perpendicular straight highways and can travel at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Determine the area of the region that can be reached by the firetruck within six minutes. The area is expressed as $m/n$ square... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-d13666538d9f5f23a865853a0e60ce93a9a9624f9c065654597b8f228594a64a | A firetruck is stationed at the intersection of two perpendicular straight highways and can travel at 50 miles per hour along the highways and at 14 miles per hour across the prairie. Determine the area of the region that can be reached by the firetruck within six minutes. The area is expressed as $m/n$ square miles, w... | [
"731"
] | [
"A firetruck is stationed at the intersection of two perpendicular straight highways and can travel at 50 miles per hour along the highways and at 14 miles per hour across the prairie. Determine the area of the region that can be reached by the firetruck within six minutes. The area is expressed as $m/n$ square mil... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 7,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-802cdc77851deefe69c9e41626dadaa2b4c7abaac734927dc6138c30a119c3f0 | A firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of this region is $m/n$ square mile... | [
"183"
] | [
"A firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of this region is $m/n$ square... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-32c16ead5d2bb091447cc81f74c2b27afcfaef91f7f937893a3fc4707924b2ea | A firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at 50 miles per hour along the highways and at 14 miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of this region is \( \frac{m}{n} \) squ... | [
"731"
] | [
"A firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at 50 miles per hour along the highways and at 14 miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of this region is \\( \\frac{m}{n}... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-fc721c9dd3be6b27625e0ba8d99669e205253e0229e35002a0e2ad67b3517420 | A firework travels vertically for 100 meters, then for 75 meters at a 20-degree angle to the vertical, then 50 meters at a 20-degree angle to the horizontal. It then drops vertically to the ground in 1 minute. Find the average speed in meters per second. | [
"3.1263"
] | [
"A firework travels vertically for 100 meters, then for 75 meters at a 20-degree angle to the vertical, then 50 meters at a 20-degree angle to the horizontal. It then drops vertically to the ground in 1 minute. Find the average speed in meters per second."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-22de896da40c7977c8ec0752616d6cfe923abdd9c204224b535fddd8ae736aff | A firing squad is composed of three policemen, Abel, Baker, and Charlie, who have probabilities 0.6, 0.7, and 0.8, respectively, of hitting the victim. Only one of the three bullets is live and is allocated at random. Find the probability that, if the victim was hit by the live bullet, it was Charlie who had the live r... | [
"\\( \\frac{8}{21} \\)"
] | [
"A firing squad is composed of three policemen, Abel, Baker, and Charlie, who have probabilities 0.6, 0.7, and 0.8, respectively, of hitting the victim. Only one of the three bullets is live and is allocated at random. Find the probability that, if the victim was hit by the live bullet, it was Charlie who had the l... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-e19cf65968a257a806543a1a9d43f6b1542419866308ff287711c1e1ed52fbcd | A firm currently produces 375,000 units of output using 5,450 units of capital and 680 units of labor, with a marginal rate of technical substitution (MRTS) of labor for capital of 5. If the manager wants to produce the same amount of output but decrease the use of capital to 5,200 units, how many units of labor will t... | [
"1930"
] | [
"A firm currently produces 375,000 units of output using 5,450 units of capital and 680 units of labor, with a marginal rate of technical substitution (MRTS) of labor for capital of 5. If the manager wants to produce the same amount of output but decrease the use of capital to 5,200 units, how many units of labor w... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-05aafb886aa5ce2ac510e3c31a6bb5ed6732d7bd2cc7b4ab40d23f47daa2e26d | A firm has fixed its expenses in roubles for 100 articles in its budget. Each of the 100 numbers obtained has at most two digits after the decimal point. Each bookkeeper chooses two numbers in his copy of this list, adds them, removes the digits of the sum after the decimal point (if any), and writes the result instead... | [
"51"
] | [
"A firm has fixed its expenses in roubles for 100 articles in its budget. Each of the 100 numbers obtained has at most two digits after the decimal point. Each bookkeeper chooses two numbers in his copy of this list, adds them, removes the digits of the sum after the decimal point (if any), and writes the result in... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-57acdc2710e6ebca7e1eb98f1618c5bf11f9db797a55d6f6a407f93166853e36 | A firm makes two kinds of shoes. Each shoe of the first type requires twice the labor to produce a shoe of the second type. If only shoes of the first type are produced, the firm can make 500 shoes a day. The market limits sales to 150 and 250 shoes of the first and second types, respectively. Profits per shoe are $8 f... | [
"2450"
] | [
"A firm makes two kinds of shoes. Each shoe of the first type requires twice the labor to produce a shoe of the second type. If only shoes of the first type are produced, the firm can make 500 shoes a day. The market limits sales to 150 and 250 shoes of the first and second types, respectively. Profits per shoe are... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-ed51d16c80c782afaed125aa00edf744d9601c4db9b8015483db9527ed7c1577 | A firm manufactures three products P1, P2, and P3 and sells each of them at $100, $250, and $300 respectively. (i) Express this information as a matrix and label it as A. (ii) Customer 1 wishes to buy 3 articles of type P1, 4 of type P2, and 3 of type P3. Customer 2 wishes to buy 1 article of type P1, 2 of type P2, and... | [
"2100"
] | [
"A firm manufactures three products P1, P2, and P3 and sells each of them at $100, $250, and $300 respectively.\n(i) Express this information as a matrix and label it as A.\n(ii) Customer 1 wishes to buy 3 articles of type P1, 4 of type P2, and 3 of type P3. Customer 2 wishes to buy 1 article of type P1, 2 of type ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-0541f7efd2ef0ecd8608bd2aa172351911ab2486702ffd21847f4a04f7d6dbd1 | A first tourist, having cycled for 1.5 h at 16 km/h, makes a stop for 1.5 h and then continues traveling at the same speed. A second tourist starts after him 4 h later, driving a motocycle at a speed of 56 km/h. What distance will they cover before the second tourist overtakes the first one? | [
"56 km"
] | [
"A first tourist, having cycled for 1.5 h at 16 km/h, makes a stop for 1.5 h and then continues traveling at the same speed. A second tourist starts after him 4 h later, driving a motocycle at a speed of 56 km/h. What distance will they cover before the second tourist overtakes the first one?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7a32824046883f872ade9fc96c11e3f2ed60b3f4098b98a8439d3358062985a1 | A first traveler left point $A$ with the speed varying according to the law $v(t)=2t$, and at the same time a second traveler started from point $B$, which is 4 km away from $A$, in pursuit of the first traveler with a constant speed of $2p$ km/h. For what values of $p$ will the second traveler overtake the first one? ... | [
"$p\\ge 2$, the pedestrians will once meet at $p=2$"
] | [
"A first traveler left point $A$ with the speed varying according to the law $v(t)=2t$, and at the same time a second traveler started from point $B$, which is 4 km away from $A$, in pursuit of the first traveler with a constant speed of $2p$ km/h. For what values of $p$ will the second traveler overtake the first ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2d42079d971d4d3a32eab0efa81f875e07cf377b0b88f9f5ee3e6b2b7cca8334 | A first-grade article is more expensive than a second-grade article by as much as a second-grade article is more expensive than a third-grade article, but the difference in price is no more than 40\% of the price of a second-grade article. A factory paid 9600 roubles for several first-grade articles and as much for sev... | [
"14 roubles"
] | [
"A first-grade article is more expensive than a second-grade article by as much as a second-grade article is more expensive than a third-grade article, but the difference in price is no more than 40\\% of the price of a second-grade article. A factory paid 9600 roubles for several first-grade articles and as much f... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f79e9a66d14c1ec138ef32a5d078af7b4ae2676eb7252aa1403002b121edd9c6 | A fish stock in a lake has stopped reproducing due to emissions from a nearby factory. The alteration in the fish stock per time unit is proportionally inverse to the square root of the fish stock. Initially, there were 900 fish, and 657 days later, there were 441 fish. How many days will it take until all of the fish ... | [
"1000 days"
] | [
"A fish stock in a lake has stopped reproducing due to emissions from a nearby factory. The alteration in the fish stock per time unit is proportionally inverse to the square root of the fish stock. Initially, there were 900 fish, and 657 days later, there were 441 fish. How many days will it take until all of the ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 13
} | 32 | 0 | 0 |
omr-ws-4f68f8d4d3a693abb761d36bd9abd2fb4cf725a0367cbe6f44bcc149b603264d | A fish triples in size every hour starting at 2 inches at 12:00 AM. Find its length at 6:30 AM. | [
"1458"
] | [
"A fish triples in size every hour starting at 2 inches at 12:00 AM. Find its length at 6:30 AM."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-287906d7eb634a15efec1c5ec570f9a6c514a2746ef881410c42a45b72f8bc15 | A fisherman has three different favorite places to fish with probabilities of catching a fish per bait drop of 0.6, 0.7, and 0.8 respectively. Given that the fisherman dropped 3 baits and caught only one fish, what is the probability that the fisherman will drop three more baits and catch only one fish? | [
"\\dfrac{47}{250}"
] | [
"A fisherman has three different favorite places to fish with probabilities of catching a fish per bait drop of 0.6, 0.7, and 0.8 respectively. Given that the fisherman dropped 3 baits and caught only one fish, what is the probability that the fisherman will drop three more baits and catch only one fish?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-009506ca9d737c6725c4843959e13e7453605fad6c70a66cb8f66bfadcc4537b | A five-digit integer is made using each of the digits $1, 3, 5, 7, 9$. The integer is greater than $80,000$ and less than $92,000$. The units (ones) digit is $3$. The hundreds and tens digits, in that order, form a two-digit integer that is divisible by $5$. What is the five-digit integer? | [
"91753"
] | [
"A five-digit integer is made using each of the digits $1, 3, 5, 7, 9$. The integer is greater than $80,000$ and less than $92,000$. The units (ones) digit is $3$. The hundreds and tens digits, in that order, form a two-digit integer that is divisible by $5$. What is the five-digit integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-84fa7545a5fd8ee2a3f129e0d9e8ecc6633523f6aed376b7627ff6ee286d685b | A five-digit integer is said to be *balanced* if the sum of any three of its digits is divisible by any of the other two. How many *balanced* numbers are there? | [
"24"
] | [
"A five-digit integer is said to be *balanced* if the sum of any three of its digits is divisible by any of the other two. How many *balanced* numbers are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-b23ebee3afe8b199925bdea282e7c68053da821741e196c39c89e1edc37224e5 | A five-digit integer will be chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit will be less than 5? Express your answer as a common fraction. | [
"\\frac{1}{2}"
] | [
"A five-digit integer will be chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit will be less than 5? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3154a99ed47ce7a6e856ed0ea3f4f1c996be1bae7728d8717ae34f3915f611b0 | A five-digit number \( n = \overline{abcdc} \) is such that when divided by \( 2, 3, 4, 5, 6 \), the remainders are \( a, b, c, d, c \) respectively. What is the remainder when \( n \) is divided by \( 100 \)? | [
"11"
] | [
"A five-digit number \\( n = \\overline{abcdc} \\) is such that when divided by \\( 2, 3, 4, 5, 6 \\), the remainders are \\( a, b, c, d, c \\) respectively. What is the remainder when \\( n \\) is divided by \\( 100 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9523ec1b171ab2606d7a89c4c62a8683b2c184aeeb7b47bfa95840e0c7e8f6ab | A five-digit number \( n = abcde \) is such that when divided respectively by \( 2, 3, 4, 5, 6 \), the remainders are \( a, b, c, d, e \). What is the remainder when \( n \) is divided by \( 100 \)? | [
"11"
] | [
"A five-digit number \\( n = abcde \\) is such that when divided respectively by \\( 2, 3, 4, 5, 6 \\), the remainders are \\( a, b, c, d, e \\). What is the remainder when \\( n \\) is divided by \\( 100 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 10
} | 32 | 0 | 0 |
omr-ws-b67d6fcb963d7c17325feaffe41800ddae76ff312daeefb0b99945602a83fc84 | A five-digit number is written down at random. Find the probability that the number is divisible by 5 and no two consecutive digits are identical. | [
"\\dfrac{11809}{90000}"
] | [
"A five-digit number is written down at random. Find the probability that the number is divisible by 5 and no two consecutive digits are identical."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-9a7ce6ae2f52e823a979e369c4f5f2f06408bc18486c26d72c285001966f45bc | A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$? | [
"18"
] | [
"A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-de506e6112523fc51846faf285619a7f4c5c0d1d32e42e524ba70463f3dfc6be | A five-letter sequence is a word in snake-speak if it contains a letter $R$ that comes directly before a letter $A$. For example, $GRASS$ and $COBRA$ are both words in snake-speak. If I randomly generate a five-letter sequence, what is the probability that it is a word in snake-speak? Assume all letters are upper-case. | [
"\\dfrac{2701}{456976}"
] | [
"A five-letter sequence is a word in snake-speak if it contains a letter $R$ that comes directly before a letter $A$. For example, $GRASS$ and $COBRA$ are both words in snake-speak. If I randomly generate a five-letter sequence, what is the probability that it is a word in snake-speak? Assume all letters are upper-... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-15c8de8d5f8db38baa6bf2281aebe51b1e244eb3ef80f157feed5b0a7fa99ffa | A fixed circle $\omega$ is inscribed into an angle with vertex $C$. An arbitrary circle passing through $C$, touching $\omega$ externally, meets the sides of the angle at points $A$ and $B$. If the radius of $\omega$ is $r$ and the distance from $C$ to the center of $\omega$ is $d$, find the constant value of the perim... | [
"\\dfrac{2(d^2 - r^2)^{3/2}}{d^2}"
] | [
"A fixed circle $\\omega$ is inscribed into an angle with vertex $C$. An arbitrary circle passing through $C$, touching $\\omega$ externally, meets the sides of the angle at points $A$ and $B$. If the radius of $\\omega$ is $r$ and the distance from $C$ to the center of $\\omega$ is $d$, find the constant value of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-a5b2c6ff7300c7d7209477840089d78e18e8b290a6ac2c1dababf90c9b89af96 | A fixed circle $k$ and collinear points $E, F, G$ are given such that points $E$ and $G$ lie outside circle $k$ and $F$ lies inside the circle $k$. Let $ABCD$ be an arbitrary quadrilateral inscribed in circle $k$ such that the points $E, F, G$ lie on lines $AB, AD, DC$ respectively. If $P$ is the fixed point through wh... | [
"3"
] | [
"A fixed circle $k$ and collinear points $E, F, G$ are given such that points $E$ and $G$ lie outside circle $k$ and $F$ lies inside the circle $k$. Let $ABCD$ be an arbitrary quadrilateral inscribed in circle $k$ such that the points $E, F, G$ lie on lines $AB, AD, DC$ respectively. If $P$ is the fixed point throu... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d2ff182c36011fbf20d7e20661024c3addbae819719ad8156079df3578114663 | A fixed circle \( w \) intersects sides \( AB \), \( AC \), and \( BC \) of triangle \( ABC \) at points \( X \), \( Y \), \( Z \), \( T \), \( D \), and \( E \) respectively. Define point \( A' \) such that \( A'D \) and \( A'E \) are tangent to \( w \). Define points \( B' \) and \( C' \) similarly. Let \( P \) be th... | [
"10"
] | [
"A fixed circle \\( w \\) intersects sides \\( AB \\), \\( AC \\), and \\( BC \\) of triangle \\( ABC \\) at points \\( X \\), \\( Y \\), \\( Z \\), \\( T \\), \\( D \\), and \\( E \\) respectively. Define point \\( A' \\) such that \\( A'D \\) and \\( A'E \\) are tangent to \\( w \\). Define points \\( B' \\) and ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-a17cc733c28e3d2bcc338df350aa1329b05c7d8f0b89f13b3e56b339032499b4 | A fixed circle of radius \( a \) is centered at \( (0, a) \). For every angle \( \theta \) (between \( OQ \) and the horizontal axis), the point \( P \) is the midpoint of the segment \( QR \), where \( Q \) is a point on the circle, \( R \) is a point on the line \( y = 2a \), and the points \( O, Q, R \) are collinea... | [
"\\( x = a \\left( \\sin \\theta \\cos \\theta + \\cot \\theta \\right), y = a (1 + \\sin^2 \\theta) \\)"
] | [
"A fixed circle of radius \\( a \\) is centered at \\( (0, a) \\). For every angle \\( \\theta \\) (between \\( OQ \\) and the horizontal axis), the point \\( P \\) is the midpoint of the segment \\( QR \\), where \\( Q \\) is a point on the circle, \\( R \\) is a point on the line \\( y = 2a \\), and the points \\... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 7
} | 12 | 0 | 0 |
omr-ws-aa12151540332d2c58c372b70c4c368c5a40dbe4e633f3bf50e8a493838cc139 | A fixed line $L$ in 3-space and a square of side $s$ in a plane perpendicular to $L$ are given. One vertex of the square is on $L$. As this vertex moves a distance $h$ along $L$, the square turns through a full revolution with $L$ as the axis. Find the volume of the solid generated by this motion. | [
"\\( hs^2 \\)"
] | [
"A fixed line $L$ in 3-space and a square of side $s$ in a plane perpendicular to $L$ are given. One vertex of the square is on $L$. As this vertex moves a distance $h$ along $L$, the square turns through a full revolution with $L$ as the axis. Find the volume of the solid generated by this motion."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-c91650a1510a3c3557c4875e25c9f1f47784a0132426423b3452c908e9f76cbb | A fixed line \( L_1 \) intersects the coordinate axes at \( P(a, 0) \) and \( Q(0, b) \). A variable line \( L_2 \), perpendicular to \( L_1 \), intersects the axes at \( R \) and \( S \). Find the locus of the points of intersection of \( PS \) and \( QR \). | [
"\\( x^2 + y^2 - ax - by = 0 \\)"
] | [
"A fixed line \\( L_1 \\) intersects the coordinate axes at \\( P(a, 0) \\) and \\( Q(0, b) \\). A variable line \\( L_2 \\), perpendicular to \\( L_1 \\), intersects the axes at \\( R \\) and \\( S \\). Find the locus of the points of intersection of \\( PS \\) and \\( QR \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 13
} | 35 | 0 | 0 |
omr-ws-1709f794b1f5ddd12e7c6b0688eb018dde7d0109c78d7e04a85a024ebaf9e94d | A fixed point $L$ lies inside a fixed circle $\omega$, and two points $A_1, B_1$ lie on the circle. Lines $LA_1$ and $LB_1$ are rotated counterclockwise about $A_1$ and $B_1$ by a fixed angle $\alpha$, intersecting $\omega$ again at $A_2$ and $B_2$, respectively. Lines $A_1B_2$ and $A_2B_1$ meet at $P$. If the coordina... | [
"x + y = 4"
] | [
"A fixed point $L$ lies inside a fixed circle $\\omega$, and two points $A_1, B_1$ lie on the circle. Lines $LA_1$ and $LB_1$ are rotated counterclockwise about $A_1$ and $B_1$ by a fixed angle $\\alpha$, intersecting $\\omega$ again at $A_2$ and $B_2$, respectively. Lines $A_1B_2$ and $A_2B_1$ meet at $P$. If the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2545408ee970fabd1eaaaaebb56ed872d79512533edc5b4b51eae164c17b7851 | A flag measures $3 \times 4$ meters and consists of a white cross with a gray background. The white stripes have the same width, and the gray rectangles at the corners are congruent. If the area of the cross is equal to the area of the gray region, what is the width of the cross? | [
"\\( w = 1 \\)"
] | [
"A flag measures $3 \\times 4$ meters and consists of a white cross with a gray background. The white stripes have the same width, and the gray rectangles at the corners are congruent. If the area of the cross is equal to the area of the gray region, what is the width of the cross?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3b659521ca5dea4701b50fae85d1f0eb1da278744b04228be711903af58d64d9 | A flagpole casts a shadow 6 feet high on a wall 10 feet away. At the same time, a 5.5-foot tall person casts a shadow 5 feet long on flat ground. How high is the flagpole? | [
"17"
] | [
"A flagpole casts a shadow 6 feet high on a wall 10 feet away. At the same time, a 5.5-foot tall person casts a shadow 5 feet long on flat ground. How high is the flagpole?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-0e2db3f336739406d5a130ecbfa1d9834e79d6944da5d26388db6c8878fc0ac3 | A flagpole is originally 5 meters tall. A hurricane snaps the flagpole at a point \( x \) meters above the ground so that the upper part, still attached to the stump, touches the ground 1 meter away from the base. What is \( x \)? | [
"2.4"
] | [
"A flagpole is originally 5 meters tall. A hurricane snaps the flagpole at a point \\( x \\) meters above the ground so that the upper part, still attached to the stump, touches the ground 1 meter away from the base. What is \\( x \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-9ea118c56a72c2df389b156d5428a616d73a9f2c00a1873b97422bfad3d7af4c | A flagpole is situated vertically on a hill that makes an angle of 18$^\circ$ with the horizontal. If the shadow of the flagpole is 14 meters when the angle of elevation of the sun is 31$^\circ$, how tall is the flagpole? | [
"\\(3.23\\) meters."
] | [
"A flagpole is situated vertically on a hill that makes an angle of 18$^\\circ$ with the horizontal. If the shadow of the flagpole is 14 meters when the angle of elevation of the sun is 31$^\\circ$, how tall is the flagpole?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 3
} | 5 | 0 | 0 |
omr-ws-62334bf21d215fa4b0493fe3ebf28c91a732ba7f291546a881469785bb825447 | A flask contains a solution of sodium chloride. $\frac{1}{n}$th part of the solution is poured into a test tube and evaporated until the percentage of sodium chloride in the test tube is doubled. The evaporated solution is then poured back into the flask. This increases the percentage of sodium chloride in the flask by... | [
"x=(2n-1)p"
] | [
"A flask contains a solution of sodium chloride. $\\frac{1}{n}$th part of the solution is poured into a test tube and evaporated until the percentage of sodium chloride in the test tube is doubled. The evaporated solution is then poured back into the flask. This increases the percentage of sodium chloride in the fl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d915a773d2c571e83e163f525239e97bc3080b5b321795921b3c0bd7d216902 | A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7.$ Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\tfrac{m}{n},$ where $m$... | [
"173"
] | [
"A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7.$ Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\\tfrac{m}{n},$ whe... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 3
} | 49 | 0 | 0 |
omr-ws-9fb7e0c72ea8ebb2458bfe5c06e73a34ae78b7bd468219a179338a501de6ed10 | A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is 7. Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\frac{m}{n}$, where $m$ an... | [
"173"
] | [
"A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is 7. Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\\frac{m}{n}$, where ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 6,
"tir": 4
} | 54 | 0 | 0 |
omr-ws-b026ad63a6819bb5cd8453cc8ce08ceb50e947a816861ccf35151a2f7c8e6573 | A flat circular disc of radius $\sqrt{5}$ lies in the xy-plane in the region $x^2 + y^2 \leq 5$, $z = 0$. The disc has a varying density given by $\rho(r) = \frac{1}{3 + r^2}$, where $r$ is the distance from the center. A small object of mass $m$ lies above the disc at the point $(0, 0, 2)$. Find the gravitational forc... | [
" -2\\pi G m \\left( \\ln\\frac{3}{2} - \\frac{1}{3} \\right) "
] | [
"A flat circular disc of radius $\\sqrt{5}$ lies in the xy-plane in the region $x^2 + y^2 \\leq 5$, $z = 0$. The disc has a varying density given by $\\rho(r) = \\frac{1}{3 + r^2}$, where $r$ is the distance from the center. A small object of mass $m$ lies above the disc at the point $(0, 0, 2)$. Find the gravitati... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-9988cba63b1371e7ee49a16191492a528f00aad75b433bd8eefffa11dce39e98 | A flat egg-shaped figure is delimited by four arcs of circles: $AB$, $BF$, $FE$, and $EA$. Given that the radius $AO$ has length $1$, determine the area of the figure. | [
"\\((3-\\sqrt{2})\\pi-1\\)"
] | [
"A flat egg-shaped figure is delimited by four arcs of circles: $AB$, $BF$, $FE$, and $EA$. Given that the radius $AO$ has length $1$, determine the area of the figure."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f0d7009503c27d4ff7863f78fb289c425ab6bb59e8ddfc7faeb8e8e5fffdfbf1 | A flea is jumping on a coordinate system from one point with whole-number coordinates to one of the neighboring such points, always jumping either horizontally or vertically. How many different locations can the flea be after 100 jumps, if it started from the origin? | [
"20201"
] | [
"A flea is jumping on a coordinate system from one point with whole-number coordinates to one of the neighboring such points, always jumping either horizontally or vertically. How many different locations can the flea be after 100 jumps, if it started from the origin?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e79c1d2f4e65def262c35204402f0a9b167a22c6d7d68345238a4d76b9695be2 | A flea is jumping on the vertices of a square $ABCD$. It starts at vertex $A$, and in each jump, it moves to an adjacent vertex with a probability of $\frac{1}{2}$ for each. The flea will stop when it has reached all vertices at least once. Determine the probability of each vertex being the last vertex the flea reaches... | [
"\\( \\frac{1}{3} \\)"
] | [
"A flea is jumping on the vertices of a square $ABCD$. It starts at vertex $A$, and in each jump, it moves to an adjacent vertex with a probability of $\\frac{1}{2}$ for each. The flea will stop when it has reached all vertices at least once. Determine the probability of each vertex being the last vertex the flea r... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 5
} | 18 | 0 | 0 |
omr-ws-5eb30c91709d764e59c6ad2c2bcb90bb35944b01d8fe0b1e191e3642d2a2d223 | A flea jumps 1/2 ft right, then 1/4 ft left, 1/8 ft right, and so on. To what point is the flea converging? | [
"\\(\\frac{1}{3}\\)"
] | [
"A flea jumps 1/2 ft right, then 1/4 ft left, 1/8 ft right, and so on. To what point is the flea converging?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-73cb1235f0ba793c79f8a82c4f8365884e5f20e380a25992ddbdb5f444c48de7 | A flea jumps in a straight numbered line. It jumps first from point $0$ to point $1$. Afterwards, if its last jump was from $A$ to $B$, then the next jump is from $B$ to one of the points $B + (B - A) - 1$, $B + (B - A)$, $B + (B-A) + 1$. Let $f(n)$ be the minimum number of jumps the flea needs to make to arrive at poi... | [
"20"
] | [
"A flea jumps in a straight numbered line. It jumps first from point $0$ to point $1$. Afterwards, if its last jump was from $A$ to $B$, then the next jump is from $B$ to one of the points $B + (B - A) - 1$, $B + (B - A)$, $B + (B-A) + 1$. Let $f(n)$ be the minimum number of jumps the flea needs to make to arrive a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-5d25eac5615794eb4f9fbfc06b76b3aec8ec16f88e5a02cdfd875938866af151 | A flea jumps in a straight numbered line. It jumps first from point $0$ to point $1$. Afterwards, if its last jump was from $A$ to $B$, then the next jump is from $B$ to one of the points $B + (B - A) - 1$, $B + (B - A)$, $B + (B-A) + 1$. Suppose the flea arrives at point $n$ exactly twice, where $n$ is a positive inte... | [
"n + 1"
] | [
"A flea jumps in a straight numbered line. It jumps first from point $0$ to point $1$. Afterwards, if its last jump was from $A$ to $B$, then the next jump is from $B$ to one of the points $B + (B - A) - 1$, $B + (B - A)$, $B + (B-A) + 1$. Suppose the flea arrives at point $n$ exactly twice, where $n$ is a positive... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-da3aec0dfe751cfa33a22c9c8da8e81b26ff35fcfda1bd3c11a0233fe039f54f | A flea jumps on a plane, making jumps of the same length, but each time turning $1$ degree to the left or right of the previous direction. After making exactly $40$ jumps, the flea decides to return to its starting point in the minimum number of additional jumps. Determine the minimum number of additional jumps needed ... | [
"320"
] | [
"A flea jumps on a plane, making jumps of the same length, but each time turning $1$ degree to the left or right of the previous direction. After making exactly $40$ jumps, the flea decides to return to its starting point in the minimum number of additional jumps. Determine the minimum number of additional jumps ne... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-0f678a48d6917ff39fbb5a83d9fb6a8cafa28ef5ae2b0f8e0a97d4b27b63bc33 | A flea jumps on the perimeter of a circle with perimeter 120. It jumps $n$ times, the lengths of the jumps are $1, 2, \ldots, n$ in some order. What is the length of the longest arc that can be avoided by the flea if $n=120$? | [
"60"
] | [
"A flea jumps on the perimeter of a circle with perimeter 120. It jumps $n$ times, the lengths of the jumps are $1, 2, \\ldots, n$ in some order. What is the length of the longest arc that can be avoided by the flea if $n=120$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-74881cfeba683ace14908c23472d78dca8780928ecf1eb7e980a7941c050578a | A flea jumps on the perimeter of a circle with perimeter 120. It jumps $n$ times, the lengths of the jumps are $1, 2, \ldots, n$ in some order. What is the length of the longest arc that can be avoided by the flea if $n=60$? | [
"60"
] | [
"A flea jumps on the perimeter of a circle with perimeter 120. It jumps $n$ times, the lengths of the jumps are $1, 2, \\ldots, n$ in some order. What is the length of the longest arc that can be avoided by the flea if $n=60$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-fc9d11b7ae7c425658faff3ba2e0dd8869a467f19a2272fc23f5fb30ebd3b0a4 | A flea jumps on the perimeter of a circle with perimeter 120. It jumps $n$ times, the lengths of the jumps are $1, 2, \ldots, n$ in some order. What is the length of the longest arc that can be avoided by the flea if $n=k$? | [
"\\max\\left(k, 120 - \\frac{k(k + 1)}{2}\\right)"
] | [
"A flea jumps on the perimeter of a circle with perimeter 120. It jumps $n$ times, the lengths of the jumps are $1, 2, \\ldots, n$ in some order. What is the length of the longest arc that can be avoided by the flea if $n=k$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-f278b359eed116f5734cb8a89e235b5fe3cf04d78e37cf48ffd4f0e60fc771d6 | A flea moves in the positive direction on the real Ox axis, starting from the origin. It can only jump distances of $\sqrt{2}$ or $\sqrt{2005}$. Let $n_0$ be the smallest integer such that the flea can reach any interval $[n, n+1]$ for $n \geq n_0$. Find the value of $n_0$. | [
"44"
] | [
"A flea moves in the positive direction on the real Ox axis, starting from the origin. It can only jump distances of $\\sqrt{2}$ or $\\sqrt{2005}$. Let $n_0$ be the smallest integer such that the flea can reach any interval $[n, n+1]$ for $n \\geq n_0$. Find the value of $n_0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-43705333dda1456023f66434d49ea550a1749b3736abffd3d938368838106bce | A flea starts at point 1 on the real number axis and jumps to other marked points $2, 3, \ldots, 2n$ without visiting any point twice. After the $(2n-1)$-th jump, it cannot jump anymore since all other points are visited. It then breaks the rule and jumps back to point 1 with its $2n$-th jump. If the sum of the (non-di... | [
"n"
] | [
"A flea starts at point 1 on the real number axis and jumps to other marked points $2, 3, \\ldots, 2n$ without visiting any point twice. After the $(2n-1)$-th jump, it cannot jump anymore since all other points are visited. It then breaks the rule and jumps back to point 1 with its $2n$-th jump. If the sum of the (... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4a6a51df9d49b45165ad6e10e5cda79cbf4fe2353cd7d868e694fd3bdb2e351a | A flea starts at the coordinate $1$ on the real line. At each second, the flea can jump to the coordinate $a+2$ or $\frac{a}{2}$ from its current coordinate $a$. Determine the number of distinct positions (including the initial position) the flea can be in after $n$ seconds. For example, if $n=1$, the flea can be at co... | [
"2^{n+1} - 1"
] | [
"A flea starts at the coordinate $1$ on the real line. At each second, the flea can jump to the coordinate $a+2$ or $\\frac{a}{2}$ from its current coordinate $a$. Determine the number of distinct positions (including the initial position) the flea can be in after $n$ seconds. For example, if $n=1$, the flea can be... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-1c7567c19dd8214ecbe8af37f611b519161a340c81adda765abbb9806b9e1df7 | A flea starts at the coordinate point $1$ on the real line. Every second, the flea can jump to either $a + 2$ or $\dfrac{a}{2}$ from its current position $a$. Determine the number of distinct positions (including the initial position) that the flea can be in after $n$ seconds. | [
"2^n"
] | [
"A flea starts at the coordinate point $1$ on the real line. Every second, the flea can jump to either $a + 2$ or $\\dfrac{a}{2}$ from its current position $a$. Determine the number of distinct positions (including the initial position) that the flea can be in after $n$ seconds."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-ee5b3c95a96ba2d4a39142d2e4a1b6b26363dfef04a318bdffdd80fd247006b5 | A flea starts at the point $(0, 0)$ in the Cartesian plane and makes $n$ jumps. Each jump is in one of the four cardinal directions, and the length of the $k$-th jump is $2^{k-1}$. If the final position of the flea is $(x, y)$, find the sum of all possible values of $x$ after the $n$-th jump, given that $n=5$ and the f... | [
"31"
] | [
"A flea starts at the point $(0, 0)$ in the Cartesian plane and makes $n$ jumps. Each jump is in one of the four cardinal directions, and the length of the $k$-th jump is $2^{k-1}$. If the final position of the flea is $(x, y)$, find the sum of all possible values of $x$ after the $n$-th jump, given that $n=5$ and ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0f281d3795beeecc605de9f19708b46c2feb0ae4075bc702bd79047fa70af0f7 | A flock of 3k crows holds a speed-flying competition where each group of 3 crows races, and the crow that finishes second wins. This process is repeated until one crow remains. How many of the crows have a chance of being declared the most medium? | [
"\\(3^k - 2 \\cdot 2^k + 2\\)"
] | [
"A flock of 3k crows holds a speed-flying competition where each group of 3 crows races, and the crow that finishes second wins. This process is repeated until one crow remains. How many of the crows have a chance of being declared the most medium?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1194cb9b48d0cdc4ab0159a23af9f7291e0cc98948a412714613b98ab20afdc3 | A flood light is on the ground 45 meters from a building. A thief 2 meters tall runs from the floodlight directly towards the building at 6 meters/sec. How rapidly is the length of his shadow on the building changing when he is 15 meters from the building? | [
"0.6 meters/sec"
] | [
"A flood light is on the ground 45 meters from a building. A thief 2 meters tall runs from the floodlight directly towards the building at 6 meters/sec. How rapidly is the length of his shadow on the building changing when he is 15 meters from the building?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 3
} | 9 | 0 | 0 |
omr-ws-1cb9682b58632a77a44f8a65cf3cf09fbd5ad56397fafd1036e6c3dde85c357a | A floor 9 feet by 12 feet, is to be tiled with 4-inch-by-6-inch tiles. How many tiles are needed to cover the floor? | [
"648"
] | [
"A floor 9 feet by 12 feet, is to be tiled with 4-inch-by-6-inch tiles. How many tiles are needed to cover the floor?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7577265b2d173f28f320fe630478b569c92eff5efcfa2c14b529467f2758c87a | A floor has an area of $3 \ m^2$. It is covered by 5 rugs, each with an area of $1 \ m^2$. Let $A$ be the maximum possible area of overlap between any two rugs. Calculate the minimum possible value of $A$. | [
"\\dfrac{1}{5}"
] | [
"A floor has an area of $3 \\ m^2$. It is covered by 5 rugs, each with an area of $1 \\ m^2$. Let $A$ be the maximum possible area of overlap between any two rugs. Calculate the minimum possible value of $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-8bfe6d890cd44e872ee6b4bb1c22c2a2e83e256283137823132ad4c7a7c501f7 | A floor is made of rectangles of dimensions $l \times 2l$ cm. A coin with a diameter of $l$ cm is dropped on the floor. What is the probability that the coin will fall on the meeting point of three rectangles? | [
"\\(\\frac{\\pi}{4}\\)"
] | [
"A floor is made of rectangles of dimensions $l \\times 2l$ cm. A coin with a diameter of $l$ cm is dropped on the floor. What is the probability that the coin will fall on the meeting point of three rectangles?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ccd957b6f68d362fca88b1235c2aa1d9396509e681684683e994877abd651683 | A floor measuring $5$ by $8$ feet needs to be tiled using thirteen $3$ feet by $1$ foot rectangles and one $1$ foot by $1$ foot square. If no overlapping or cutting is permitted, which positions can the $1$ foot by $1$ foot square be in? | [
"(1,2), (1,5), (1,8), (2,1), (2,4), (2,7), (3,3), (3,6), (4,2), (4,5), (4,8), (5,1), (5,4), (5,7)"
] | [
"A floor measuring $5$ by $8$ feet needs to be tiled using thirteen $3$ feet by $1$ foot rectangles and one $1$ foot by $1$ foot square. If no overlapping or cutting is permitted, which positions can the $1$ foot by $1$ foot square be in?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-6cf58e7fadbe8eda57b73a386e8b376502f27c3fd47daa34be3ca7da83e6f12d | A floor tile has the shape of a regular polygon. If the tile is removed from the floor and rotated through $50^\circ$, it will fit back exactly into its original place in the floor. What is the least number of sides that the polygon can have? | [
"36"
] | [
"A floor tile has the shape of a regular polygon. If the tile is removed from the floor and rotated through $50^\\circ$, it will fit back exactly into its original place in the floor. What is the least number of sides that the polygon can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1d04fb8ee44699ee11abea9695e30728f9d6d2ab23ca9523d2e12ffc5eb2b168 | A flower shape is formed by drawing a circle with radius \( n \), and then drawing 5 petals of equal size. Each petal is an arc with its center on the circumference of the larger circle, and the petals touch adjacent petals. Find the area of the flower in terms of \( n \). | [
"\\dfrac{(41 - 5\\sqrt{5})\\pi n^2}{16}"
] | [
"A flower shape is formed by drawing a circle with radius \\( n \\), and then drawing 5 petals of equal size. Each petal is an arc with its center on the circumference of the larger circle, and the petals touch adjacent petals. Find the area of the flower in terms of \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-920f7b29c7f225f6b504df2103fa90ece4bbc93852073fb33dbe44f1407faf80 | A flower-like shape is formed by attaching semi-circles onto each side of a regular hexagon. If the hexagon has an area of $54\sqrt{3}$, what is the largest possible distance between two points on the flower? | [
"\\(6 + 6\\sqrt{3}\\)"
] | [
"A flower-like shape is formed by attaching semi-circles onto each side of a regular hexagon. If the hexagon has an area of $54\\sqrt{3}$, what is the largest possible distance between two points on the flower?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-883d5eba97ad3e36f07d7c4a74bfc0ee9c097e463cfb0ef5cfc3162a682333c2 | A fly and \(k\) spiders are placed in some vertices of a \(2012 \times 2012\) lattice. One move consists of the fly moving to an adjacent vertex or staying where it is, followed by each spider moving to an adjacent vertex or staying where it is. Spiders and the fly know each other's positions at all times. a) Find the ... | [
"3"
] | [
"A fly and \\(k\\) spiders are placed in some vertices of a \\(2012 \\times 2012\\) lattice. One move consists of the fly moving to an adjacent vertex or staying where it is, followed by each spider moving to an adjacent vertex or staying where it is. Spiders and the fly know each other's positions at all times.\na... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6d3bb55a1e11ae128bd5b340a7446a03e632d471a1e3010a772d74afe41ec538 | A fly and a blind spider are on opposite corners of a unit cube. The fly never moves, and the spider moves at random from its initial corner to another corner connected by an edge. What is the expected number of moves before the spider reaches the fly? | [
"10"
] | [
"A fly and a blind spider are on opposite corners of a unit cube. The fly never moves, and the spider moves at random from its initial corner to another corner connected by an edge. What is the expected number of moves before the spider reaches the fly?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-a6259f21b611da3b0aa03fdf16813611bc5183985df883be08546bc5530decc4 | A fly flies inside a regular tetrahedron with edge length $a$. What is the shortest length of the flight the fly should take to visit every face and return to the initial spot? | [
"\\dfrac{4a}{3}"
] | [
"A fly flies inside a regular tetrahedron with edge length $a$. What is the shortest length of the flight the fly should take to visit every face and return to the initial spot?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-c59a314eb55188c91f6cbe03c9e52add6727b37e0d636cdc88d63860dc1dab33 | A fly flies inside a right tetrahedron \(ABCD\). The fly visits each face of the tetrahedron \((ABC), (ABD), (ACD), (BCD)\) and returns to the starting point. Find the minimum distance \(S\) that the fly can travel. | [
"2\\sqrt{2}"
] | [
"A fly flies inside a right tetrahedron \\(ABCD\\). The fly visits each face of the tetrahedron \\((ABC), (ABD), (ACD), (BCD)\\) and returns to the starting point. Find the minimum distance \\(S\\) that the fly can travel."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-e45d7a7e9147da540504f97eb495b9042552c36e6946993f6dc264c0af3e2f47 | A fly is moving on a 3x2 grid, starting at the point labeled \( F \) (1, 2). The fly moves one square at a time. If the fly reaches the bottom row, it flies out. If it reaches the top-right point (3, 2), it is safe. What is the probability that the fly flies out safely? | [
"\\dfrac{1}{5}"
] | [
"A fly is moving on a 3x2 grid, starting at the point labeled \\( F \\) (1, 2). The fly moves one square at a time. If the fly reaches the bottom row, it flies out. If it reaches the top-right point (3, 2), it is safe. What is the probability that the fly flies out safely?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1a425ae67c106b3bfc2c54369c2da0d809ed4c4e011490cec00002d2fcc19afe | A fly is on the edge of a ceiling of a circular room with a radius of 58 feet. The fly walks straight across the ceiling to the opposite edge, passing through the center of the circle. It then walks straight to another point on the edge of the circle but not back through the center. The third part of the journey is str... | [
"280"
] | [
"A fly is on the edge of a ceiling of a circular room with a radius of 58 feet. The fly walks straight across the ceiling to the opposite edge, passing through the center of the circle. It then walks straight to another point on the edge of the circle but not back through the center. The third part of the journey i... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-726ac0add7622b7a01cae41e71b943d59a2928be438ceaac0b354ba83068ed08 | A fly is sitting on a vertex of a cube. Each time the fly moves to one of the three adjacent vertices with equal probability. In how many ways can the fly travel through each vertex exactly once in 7 moves? | [
"18"
] | [
"A fly is sitting on a vertex of a cube. Each time the fly moves to one of the three adjacent vertices with equal probability. In how many ways can the fly travel through each vertex exactly once in 7 moves?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 5,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-00ff0a1b8e93cdf8872050b6d22716deb72b6c9429b6965e6baeef3914995fde | A fly is tied to a rope of length $2$, and the other end of the rope is tied to a vertex of a unit cube. The fly can fly anywhere outside the cube, as long as the rope does not pass through the interior of the cube. What volume of space can the fly cover? | [
"\\dfrac{127}{12} \\pi"
] | [
"A fly is tied to a rope of length $2$, and the other end of the rope is tied to a vertex of a unit cube. The fly can fly anywhere outside the cube, as long as the rope does not pass through the interior of the cube. What volume of space can the fly cover?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-e27ce943ea7c5a51fea1e12fbde0b1c788c28623a71142c2bae335b2a856f3a7 | A fly starts at one corner of a cube and must travel to the diagonally opposite corner, moving only along the faces of the cube. The cube's faces are divided into 3x3 squares, each with a side length of 1 cm, making each side of the cube 3 cm. The fly can only move on three faces that touch each other. How many differe... | [
"148"
] | [
"A fly starts at one corner of a cube and must travel to the diagonally opposite corner, moving only along the faces of the cube. The cube's faces are divided into 3x3 squares, each with a side length of 1 cm, making each side of the cube 3 cm. The fly can only move on three faces that touch each other. How many di... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-83f46d3e14b003c4632f22ff06ee5a74ad4cc1851bf77771b08e064ee2ec5647 | A fly starts from $(0, 16)$, flies straight to $(8, 0)$, then to $(0, -4)$, then to $(-2, 0)$, and so on, spiraling to the origin, each time intersecting the coordinate axes at a point half as far from the origin as its previous intercept. If the fly flies at a constant speed of 2 units per second, how many seconds wil... | [
"\\(\\frac{160}{3}\\)"
] | [
"A fly starts from $(0, 16)$, flies straight to $(8, 0)$, then to $(0, -4)$, then to $(-2, 0)$, and so on, spiraling to the origin, each time intersecting the coordinate axes at a point half as far from the origin as its previous intercept. If the fly flies at a constant speed of 2 units per second, how many second... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1619d84cbdd3968973a58c2f44de381c87f3cf787f841c3d4ea308273a0955fe | A fly starts on the second hand of a clock at noon and switches to the other hand each time the hand it is on crosses another hand. How many revolutions has the fly taken by midnight, given that no three hands coincide between noon and midnight? | [
"245"
] | [
"A fly starts on the second hand of a clock at noon and switches to the other hand each time the hand it is on crosses another hand. How many revolutions has the fly taken by midnight, given that no three hands coincide between noon and midnight?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-51cf3bdc43f66c360907dc85bd12b4264d3cbd607d370b8ab9ab7eeb26361107 | A fly trapped inside a cubical box with side length $1$ meter decides to visit each corner of the box exactly once, starting and ending at the same corner. It can either fly or crawl in a straight line between corners. What is the maximum possible length, in meters, of its path? | [
"\\( 4\\sqrt{2} + 4\\sqrt{3} \\)"
] | [
"A fly trapped inside a cubical box with side length $1$ meter decides to visit each corner of the box exactly once, starting and ending at the same corner. It can either fly or crawl in a straight line between corners. What is the maximum possible length, in meters, of its path?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 43,
"genselect": 4,
"tir": 17
} | 64 | 0 | 0 |
omr-ws-2c5b0c80b4d0e9e41f3b71db9ea3f7ba1b54cacdc40ade6f900ecae8fb9c3b86 | A fly's path is given by the vector \( \mathbf{r}(t) = (1+t^2) \mathbf{i} + (-4t+3) \mathbf{j} + (1-t^2) \mathbf{k} \). The fly is moving above a tilted table defined by the plane \( x + y + 10z = 5 \). Determine the speed of the fly's shadow along the table at any time \( t \) before it lands. | [
"\\sqrt{\\dfrac{246t^2 - 72t + 808}{51}}"
] | [
"A fly's path is given by the vector \\( \\mathbf{r}(t) = (1+t^2) \\mathbf{i} + (-4t+3) \\mathbf{j} + (1-t^2) \\mathbf{k} \\). The fly is moving above a tilted table defined by the plane \\( x + y + 10z = 5 \\). Determine the speed of the fly's shadow along the table at any time \\( t \\) before it lands."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-f3471987b18b3833a4a0f2fb622bb9a847ef67c83de74936e316123e026fb480 | A flying monkey is leashed to an upper corner of a barn in the shape of a cube with side length 20. Given that the leash has a length of 12, determine the total volume of space where the monkey can fly (Assume the monkey has negligible volume). | [
"288\\pi"
] | [
"A flying monkey is leashed to an upper corner of a barn in the shape of a cube with side length 20. Given that the leash has a length of 12, determine the total volume of space where the monkey can fly (Assume the monkey has negligible volume)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-e34a8155f8c4f5e72b71dcb0d68e2dc8d0f8604b56b16935932ba6076ae008d1 | A flying saucer can jump from any point $A$ to any other point $A'$ inside the lunar orbit, with the Earth or the Moon placed at the midpoint of segment $AA'$. Given that the flying saucer can make an unlimited number of jumps, determine the minimum number of jumps required for the flying saucer to travel from a point ... | [
"1"
] | [
"A flying saucer can jump from any point $A$ to any other point $A'$ inside the lunar orbit, with the Earth or the Moon placed at the midpoint of segment $AA'$. Given that the flying saucer can make an unlimited number of jumps, determine the minimum number of jumps required for the flying saucer to travel from a p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-96480cc507814b6cfdf5b7125699cfe7264ae4e206eefbe37b6817a3348be06d | A football is made by sewing together regular pentagons and regular hexagons. Each pentagon is surrounded by 5 hexagons, and each hexagon is surrounded by 3 pentagons. Determine the number of pentagons and hexagons used. | [
"12 pentagons, 20 hexagons"
] | [
"A football is made by sewing together regular pentagons and regular hexagons. Each pentagon is surrounded by 5 hexagons, and each hexagon is surrounded by 3 pentagons. Determine the number of pentagons and hexagons used."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 25
} | 27 | 0 | 0 |
omr-ws-2ba3b4af7317617515d3538d736c2cfe75e8a5d359520651cb7728bf65efce4e | A football is made by sewing together some black and white leather patches. The black patches are regular pentagons of the same size. The white patches are regular hexagons of the same size. Each pentagon is bordered by $5$ hexagons. Each hexagon is bordered by $3$ pentagons and $3$ hexagons. We need $12$ pentagons to ... | [
"36"
] | [
"A football is made by sewing together some black and white leather patches. The black patches are regular pentagons of the same size. The white patches are regular hexagons of the same size. Each pentagon is bordered by $5$ hexagons. Each hexagon is bordered by $3$ pentagons and $3$ hexagons. We need $12$ pentagon... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7aedef6eaad4566cf467467a89a441fbcd1ce0df3933113de5099a9c4bf5faf9 | A football player shoots $n$ times at the goal. The probability of getting the ball in the net at shot $S_k$ is $\frac{1}{2^k}$. What is the probability of getting an even number of shots in the net after the $n$ shots? | [
"$\\frac{1}{2}$"
] | [
"A football player shoots $n$ times at the goal. The probability of getting the ball in the net at shot $S_k$ is $\\frac{1}{2^k}$. What is the probability of getting an even number of shots in the net after the $n$ shots?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-805acdc76bccd784dfb5a814f98df0f13202c797cb8c7ce26135527aafa38681 | A football team has 23 players and uses 11 players in each game. Last year, they played 40 games. What is the minimum number of games that the player who played the most games could have participated in? | [
"20"
] | [
"A football team has 23 players and uses 11 players in each game. Last year, they played 40 games. What is the minimum number of games that the player who played the most games could have participated in?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-d2aa56e841aace18e8a53783fa33f282988ee8465afe670fca410bbd56079965 | A football team runs back and forth on a 100-yard field. They start by running the full 100 yards, then half the previous distance, and continue halving the distance each time. What is the coordinate of the point that they approach? | [
"\\(\\frac{200}{3}\\)"
] | [
"A football team runs back and forth on a 100-yard field. They start by running the full 100 yards, then half the previous distance, and continue halving the distance each time. What is the coordinate of the point that they approach?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-8d6be3a2b451808cca9ec7570ddde2c40231b3a4c1ac6b1e7768a811a4d89b90 | A footbridge is in the shape of an arc of a circle. The bridge is 7 ft tall and 28 ft wide. What is the radius of the circle that contains the bridge? Round your answer to the nearest tenth. | [
"17.5"
] | [
"A footbridge is in the shape of an arc of a circle. The bridge is 7 ft tall and 28 ft wide. What is the radius of the circle that contains the bridge? Round your answer to the nearest tenth."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-703990698bb4b02f8c758b9abd86b4e37fb347143f75aa57f25cea152b0d88f0 | A foreman noticed an inspector checking a $3"$-hole with a $2"$-plug and a $1"$-plug and suggested that two more gauges be inserted to be sure that the fit was snug. If the new gauges are alike, then the diameter, $d$, of each, to the nearest hundredth of an inch, is: | [
"0.86"
] | [
"A foreman noticed an inspector checking a $3\"$-hole with a $2\"$-plug and a $1\"$-plug and suggested that two more gauges be inserted to be sure that the fit was snug. If the new gauges are alike, then the diameter, $d$, of each, to the nearest hundredth of an inch, is:"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-48bd4796a8aff3ef41453d30f2b1d6b7d65348f19b036de44875bff64be0808f | A forest grows up $p$ percent during a summer, but gets reduced by $x$ units between two summers. At the beginning of this summer, the size of the forest has been $a$ units. How large should $x$ be if we want the forest to increase $q$ times in $n$ years? | [
"x = \\frac{a p \\left( \\left(1 + \\frac{p}{100}\\right)^n - q \\right)}{100 \\left( \\left(1 + \\frac{p}{100}\\right)^n - 1 \\right)}"
] | [
"A forest grows up $p$ percent during a summer, but gets reduced by $x$ units between two summers. At the beginning of this summer, the size of the forest has been $a$ units. How large should $x$ be if we want the forest to increase $q$ times in $n$ years?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6131a065d932f04d1504e6e6704db004d85394d6ea0029f6ebed7dc99a3261ce | A forest is in the shape of a rectangle 5 miles long and 3 miles wide. The density of trees at a distance of \( x \) miles from the 5-mile side is given by \( p(x) \) trees per square mile. Which expression gives the number of trees in the forest? | [
"\\( 5\\int_0^3 p(x) \\: dx \\)"
] | [
"A forest is in the shape of a rectangle 5 miles long and 3 miles wide. The density of trees at a distance of \\( x \\) miles from the 5-mile side is given by \\( p(x) \\) trees per square mile. Which expression gives the number of trees in the forest?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-f08f84b6f0310ebf0eff663087cdd3766b2a7f13f17c113720f2e91547b2af2b | A forester mixes gasoline and oil to make 2 gallons of mixture for his two-cycle chain-saw engine. The mixture is 32 parts gasoline and 1 part two-cycle oil. How much gasoline must be added to bring the mixture to 40 parts gasoline and 1 part oil? | [
"\\(\\frac{16}{33}\\)"
] | [
"A forester mixes gasoline and oil to make 2 gallons of mixture for his two-cycle chain-saw engine. The mixture is 32 parts gasoline and 1 part two-cycle oil. How much gasoline must be added to bring the mixture to 40 parts gasoline and 1 part oil?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-4d958556ee2835cb0d60d09a956125514f8a8abd34481c2a3253901c391f9c97 | A forester plants 2000 trees along a fence. On each tree, he writes the number of oaks among itself and its neighboring trees (left and right). How many distinct sequences of these numbers can be created if an oak is a specific type of tree? | [
"2^{2000}"
] | [
"A forester plants 2000 trees along a fence. On each tree, he writes the number of oaks among itself and its neighboring trees (left and right). How many distinct sequences of these numbers can be created if an oak is a specific type of tree?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.