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-b885f31343b9621e77f839140343362a55d53b980b6879e9e58ba5230c8732d8 | Topsoil costs $\$6$ per cubic foot. What is the cost, in dollars, of 5 cubic yards of topsoil? | [
"810"
] | [
"Topsoil costs $\\$6$ per cubic foot. What is the cost, in dollars, of 5 cubic yards of topsoil?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b5891f741567403314779c75e9a0c9ab0a39eb42e25b343d6ad1d7d01cfb1b6 | Tori's mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered $70\%$ of the arithmetic, $40\%$ of the algebra, and $60\%$ of the geometry problems correctly, she did not pass the test because she got less than $60\%$ of the problems right. How many more questions wo... | [
"5"
] | [
"Tori's mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered $70\\%$ of the arithmetic, $40\\%$ of the algebra, and $60\\%$ of the geometry problems correctly, she did not pass the test because she got less than $60\\%$ of the problems right. How many more que... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-964182a45c9f6e48d829bac8b18e7abff01618a57b59cac5b5011ba3db82366e | Tortle and Tally play a game using a \( 100 \times 100 \) chocolate bar, where they take turns choosing a piece of chocolate and breaking it up (into two pieces) along a gridline. They keep taking turns until they are left with 10,000 small \( 1 \times 1 \) pieces of chocolate, at which point whoever made the last move... | [
"first"
] | [
"Tortle and Tally play a game using a \\( 100 \\times 100 \\) chocolate bar, where they take turns choosing a piece of chocolate and breaking it up (into two pieces) along a gridline. They keep taking turns until they are left with 10,000 small \\( 1 \\times 1 \\) pieces of chocolate, at which point whoever made th... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-17490ba7d088582cab4278d97bb5965b03c4683855bb3df9c9e53e0da8c035e2 | Torus $T$ is the surface produced by revolving a circle with radius $3$ around an axis in the plane of the circle that is a distance $6$ from the center of the circle (so like a donut). Let $S$ be a sphere with a radius $11$. When $T$ rests on the inside of $S$, it is internally tangent to $S$ along a circle with radiu... | [
"127"
] | [
"Torus $T$ is the surface produced by revolving a circle with radius $3$ around an axis in the plane of the circle that is a distance $6$ from the center of the circle (so like a donut). Let $S$ be a sphere with a radius $11$. When $T$ rests on the inside of $S$, it is internally tangent to $S$ along a circle with ... | [
"MATH_training_set"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-6a1f4c1403df91b77a3a3491fd41d08c31799aeb4bd2d281af2f4707bb720827 | Toss a fair coin repeatedly. Each time you get a head, give yourself 1 point. Each time you get a tail, give yourself 2 points. What is the probability that at some time in the game you will have exactly 5 points? | [
"\\(\\frac{21}{32}\\)"
] | [
"Toss a fair coin repeatedly. Each time you get a head, give yourself 1 point. Each time you get a tail, give yourself 2 points. What is the probability that at some time in the game you will have exactly 5 points?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-1c7aeef9eef7ed0b27630e1e3f4b9b4bbb858ac1212f8d6cd6efba05ba960f9a | Toto has 10 boxes each containing 5 beans and 5 boxes each containing 10 beans. Each box contains one pinto bean. If Toto selects a box at random, and then selects a bean from that box at random, what is the probability that he selects a pinto bean? | [
"\\(\\frac{1}{6}\\)\n\nExplanation:\nThe problem asks for the probability that Toto selects a pinto bean. According to the discussion in Post 4, the probability is calculated as follows:\n\\[ P(FP) + P(TP) = \\frac{2}{3} \\cdot \\frac{1}{5} + \\frac{1}{3} \\cdot \\frac{1}{10} = \\frac{1}{6} \\]"
] | [
"Toto has 10 boxes each containing 5 beans and 5 boxes each containing 10 beans. Each box contains one pinto bean. If Toto selects a box at random, and then selects a bean from that box at random, what is the probability that he selects a pinto bean?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-5599b1fd037f0665b26e814c1f7cd103d7639e79d87a028b70155a40d8f9b6d7 | Totoro wants to cut a 3-meter long bar of mixed metals into two parts with equal monetary value. The left meter is bronze, worth 10 zoty per meter, the middle meter is silver, worth 25 zoty per meter, and the right meter is gold, worth 40 zoty per meter. How far, in meters, from the left should Totoro make the cut? | [
"2.0625"
] | [
"Totoro wants to cut a 3-meter long bar of mixed metals into two parts with equal monetary value. The left meter is bronze, worth 10 zoty per meter, the middle meter is silver, worth 25 zoty per meter, and the right meter is gold, worth 40 zoty per meter. How far, in meters, from the left should Totoro make the cut... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a0a257741a2d4d8efa902929d59d178c98c3ea4af2c840b623cbb4c69def96b2 | Tower \(\overline{BC}\) is on the west of point \(A\). The elevation from \(A\) to the top point \(C\) is \(45^{\circ}\). After walking southeast to point \(D\) for \(100\) meters, the elevation from \(D\) to the top point \(C\) is \(30^{\circ}\). Find the height of the tower \(\overline{BC}\) and the azimuth to point ... | [
"\\( \\overline{BC} = 25\\left(\\sqrt{2}+\\sqrt{10}\\right) \\) and the azimuth to point \\( D \\) is \\( 111^{\\circ} \\)."
] | [
"Tower \\(\\overline{BC}\\) is on the west of point \\(A\\). The elevation from \\(A\\) to the top point \\(C\\) is \\(45^{\\circ}\\). After walking southeast to point \\(D\\) for \\(100\\) meters, the elevation from \\(D\\) to the top point \\(C\\) is \\(30^{\\circ}\\). Find the height of the tower \\(\\overline{B... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-ba72d326a05ce548a0cbaf563cb543a4153bb835fea217066985254b77fd99bf | Towers grow at points along a line. All towers start with height 0 and grow at the rate of one meter per second. As soon as any two adjacent towers are each at least 1 meter tall, a new tower begins to grow at a point along the line exactly halfway between those two adjacent towers. Before time 0, there are no towers, ... | [
"\\( n - 1 + 2^{n-1} \\)"
] | [
"Towers grow at points along a line. All towers start with height 0 and grow at the rate of one meter per second. As soon as any two adjacent towers are each at least 1 meter tall, a new tower begins to grow at a point along the line exactly halfway between those two adjacent towers. Before time 0, there are no tow... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f065a4f5017d62c1e02e43e04d1000ee34dba08f3563e0df0d6a771d75faa9b | Towns $A$ and $B$ are located on a river, $B$ downstream from $A$. At 9 a.m. a raft starts floating downstream from $A$ in the direction of $B$ (the rate of the raft relative to the bank of the river is the same as the rate of the current). At that time, a boat starts out from $B$ for $A$ and meets the raft in 5 hours.... | [
"No"
] | [
"Towns $A$ and $B$ are located on a river, $B$ downstream from $A$. At 9 a.m. a raft starts floating downstream from $A$ in the direction of $B$ (the rate of the raft relative to the bank of the river is the same as the rate of the current). At that time, a boat starts out from $B$ for $A$ and meets the raft in 5 h... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-646c30df3c44a7a6bb042d45e773ca2400b2014ea15cdcdeb10432f0b11acd7d | Towns $A$ and $B$ are situated on two straight roads intersecting at the angle $\angle ACB = 60^{\circ}.$ The bus travels from $A$ to $C$ to $B$ in $11$ minutes. Walking directly from $A$ to $B$ takes $1$ hour and $10$ minutes. Walking at $3$ km/h and the bus driving at $30$ km/h, find the distance from $A$ to the inte... | [
"4"
] | [
"Towns $A$ and $B$ are situated on two straight roads intersecting at the angle $\\angle ACB = 60^{\\circ}.$ The bus travels from $A$ to $C$ to $B$ in $11$ minutes. Walking directly from $A$ to $B$ takes $1$ hour and $10$ minutes. Walking at $3$ km/h and the bus driving at $30$ km/h, find the distance from $A$ to t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-8c545a68f1ebd57ef00593bf7ac7352ba4f820e5d2e2e1920170f34338cd7461 | Towns A, B, and C are connected by roads, with at least one road between each pair of towns. There are 33 paths from A to B (including paths through C) and 23 paths from B to C (including paths through A). How many paths are there from A to C (including paths through B)? | [
"21"
] | [
"Towns A, B, and C are connected by roads, with at least one road between each pair of towns. There are 33 paths from A to B (including paths through C) and 23 paths from B to C (including paths through A). How many paths are there from A to C (including paths through B)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-b8e1799afd2c1d1637c1858ae4377b3396a0a516e4510376bcb366bd5c1262dc | Towns \( T_0, T_1, \ldots, T_{n-1} \) are situated clockwise on a circular route of length 1. The town \( T_i \) can supply fuel sufficient to drive a car for distance \( p_i \), and the total amount of fuel is exactly enough to drive once around the circle. Calculate the minimum number of towns \( k \) from which you ... | [
"1"
] | [
"Towns \\( T_0, T_1, \\ldots, T_{n-1} \\) are situated clockwise on a circular route of length 1. The town \\( T_i \\) can supply fuel sufficient to drive a car for distance \\( p_i \\), and the total amount of fuel is exactly enough to drive once around the circle. Calculate the minimum number of towns \\( k \\) f... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-83a5c1359040837e1d1168cdce2fd4f5024094a95e6f5e4690605468a45bfd2c | Trace the curve given by the equation \( y = \frac{x}{1 + x^2} \). | [
"y = \\frac{x}{1 + x^2}"
] | [
"Trace the curve given by the equation \\( y = \\frac{x}{1 + x^2} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0b0be1c9702849f6f1300704d6ce12570a891cf1770e8bc8ee5b0a5da52e18ba | Trace the curve given by the equation \[ y^4 - x^4 - 96y^2 + 100x^2 = 0. \] | [
"y^4 - x^4 - 96y^2 + 100x^2 = 0"
] | [
"Trace the curve given by the equation \\[ y^4 - x^4 - 96y^2 + 100x^2 = 0. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-0db157efcf2e3fc085d4abfd946377b0ae6a01ff567a635075566d57b905c337 | Tracey baked a square cake whose surface is dissected into a $10 \times 10$ grid. She wants to place strawberries in some of the fields such that every $2 \times 2$ subgrid contains at least one strawberry. What is the minimum number of strawberries required? | [
"29"
] | [
"Tracey baked a square cake whose surface is dissected into a $10 \\times 10$ grid. She wants to place strawberries in some of the fields such that every $2 \\times 2$ subgrid contains at least one strawberry. What is the minimum number of strawberries required?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f975f45a43cdd8bce355489d763ff40c0a65d6c522a3fd1ba6101d5f99b2297 | Track stars Noah and Jonah run around a circular track. It takes Noah 3 minutes 20 seconds to run around the track, while it takes Jonah 3 minutes 45 seconds to run around the track. They start running around the track from the same spot going in the same direction. In how many minutes will the two of them be in the sa... | [
"30"
] | [
"Track stars Noah and Jonah run around a circular track. It takes Noah 3 minutes 20 seconds to run around the track, while it takes Jonah 3 minutes 45 seconds to run around the track. They start running around the track from the same spot going in the same direction. In how many minutes will the two of them be in t... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-31bdafe8e9962edde555eff9ec3ce44226aa36373fc81b925c1ebac3ae16df6e | Tracy draws two triangles: one with vertices at $(0, 0)$, $(2, 0)$, and $(1, 8)$ and another with vertices at $(1, 0)$, $(3, 0)$, and $(2, 8)$. What is the area of overlap of the two triangles? | [
"2"
] | [
"Tracy draws two triangles: one with vertices at $(0, 0)$, $(2, 0)$, and $(1, 8)$ and another with vertices at $(1, 0)$, $(3, 0)$, and $(2, 8)$. What is the area of overlap of the two triangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-09440d40a0405dc11092b4cda761b02a375b43ed2794b6f0cd94b2970d045e56 | Tracy had a bag of candies, and none of the candies could be broken into pieces. She ate $\frac{1}{3}$ of them and then gave $\frac{1}{4}$ of what remained to her friend Rachel. Tracy and her mom then each ate 15 candies from what Tracy had left. Finally, Tracy's brother took somewhere from one to five candies, leaving... | [
"72"
] | [
"Tracy had a bag of candies, and none of the candies could be broken into pieces. She ate $\\frac{1}{3}$ of them and then gave $\\frac{1}{4}$ of what remained to her friend Rachel. Tracy and her mom then each ate 15 candies from what Tracy had left. Finally, Tracy's brother took somewhere from one to five candies, ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dae7684caa94b2f9d443a947c8db54fdf6dd85034b16b109f5aef914a8c487b1 | Tracy has been baking a rectangular cake whose surface is dissected by grid lines into square fields. The number of rows is $2^n$ and the number of columns is $2^{n+1}$ where $n \geq 1$ and $n \in \mathbb{N}$. She covers the fields with strawberries such that each row has at least $2n + 2$ strawberries. Let $S(n)$ be t... | [
"672"
] | [
"Tracy has been baking a rectangular cake whose surface is dissected by grid lines into square fields. The number of rows is $2^n$ and the number of columns is $2^{n+1}$ where $n \\geq 1$ and $n \\in \\mathbb{N}$. She covers the fields with strawberries such that each row has at least $2n + 2$ strawberries. Let $S(... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-8f131b8aa023ae3fc6b0a0c65814fadccefc1b4e4ebb1aa3281c393bf721b3ae | Train $A$, whose speed is $v$ km/h, departs after train $B$, whose speed is $v_1$ km/h. The difference between the departure times (the lag of train $A$) is calculated so that both trains simultaneously arrive at the destination. Train $B$ covers $2/ 3$ of the distance and then has to reduce its speed to half. As a res... | [
"The distance to the terminal station is equal to $\\displaystyle\\frac{3a(2v-v_1)}{v}$ km. The problem is solvable only if $v_1<2v$"
] | [
"Train $A$, whose speed is $v$ km/h, departs after train $B$, whose speed is $v_1$ km/h. The difference between the departure times (the lag of train $A$) is calculated so that both trains simultaneously arrive at the destination. Train $B$ covers $2/ 3$ of the distance and then has to reduce its speed to half. As ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-adc593cd7e5d60bd53de78bb20e7aafed4d84ee7860439f6ce52c03a1b81a9db | Train A passes a milestone in 8 seconds before meeting train B. The two trains pass each other in 9 seconds. Then train A passes the same milestone in 12 seconds. What is the ratio of the length of train A to the length of train B? | [
"\\dfrac{2}{1}"
] | [
"Train A passes a milestone in 8 seconds before meeting train B. The two trains pass each other in 9 seconds. Then train A passes the same milestone in 12 seconds. What is the ratio of the length of train A to the length of train B?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-2211044af61b6cb8f39f6a8ba6d35c436320b096b12683e4d9aa11dc00451db8 | Train A will enter Intersection X from the west at a random time between 9:00 am and 2:30 pm; each moment in that interval is equally likely. Train B will enter the same intersection from the north at a random time between 9:30 am and 12:30 pm, independent of Train A; again, each moment in the interval is equally likel... | [
"\\(\\frac{13}{48}\\)"
] | [
"Train A will enter Intersection X from the west at a random time between 9:00 am and 2:30 pm; each moment in that interval is equally likely. Train B will enter the same intersection from the north at a random time between 9:30 am and 12:30 pm, independent of Train A; again, each moment in the interval is equally ... | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-7cfbb1625e18551bc79720d0e6af49dfce91916c8e1fb0d94255fa4106aed9ac | Trains A and B start at the west end of two parallel tracks, and Train C starts at the east end of another parallel track. Train A travels at 60 miles per hour, Train B travels at 30 miles per hour, and Train C travels at 50 miles per hour. How many minutes will it take for Train C to pass the midpoint between Train A ... | [
"\\(\\frac{720}{19}\\) minutes"
] | [
"Trains A and B start at the west end of two parallel tracks, and Train C starts at the east end of another parallel track. Train A travels at 60 miles per hour, Train B travels at 30 miles per hour, and Train C travels at 50 miles per hour. How many minutes will it take for Train C to pass the midpoint between Tra... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-2d1036d07201fedec787077818709c0f78a061859c21a1df888baddef4bb0ec6 | Trains coming from the north arrive every 3 minutes starting at noon and finishing at midnight. Trains coming from the south arrive every 5 minutes starting at noon and finishing at midnight. If you walk to the train station at a random time, on average, how many seconds do you have to wait until either train arrives? | [
"74"
] | [
"Trains coming from the north arrive every 3 minutes starting at noon and finishing at midnight. Trains coming from the south arrive every 5 minutes starting at noon and finishing at midnight. If you walk to the train station at a random time, on average, how many seconds do you have to wait until either train arri... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-1572acbf2acd94713717728dc0af100a6c7013356625f40a850a78dd7f2a6cd0 | Tram ticket costs $1$ Tug ($=100$ tugriks). $20$ passengers have only coins in denominations of $2$ and $5$ tugriks, while the conductor has nothing at all. It turned out that all passengers were able to pay the fare and get change. What is the smallest total number of passengers that the tram could have? | [
"20"
] | [
"Tram ticket costs $1$ Tug ($=100$ tugriks). $20$ passengers have only coins in denominations of $2$ and $5$ tugriks, while the conductor has nothing at all. It turned out that all passengers were able to pay the fare and get change. What is the smallest total number of passengers that the tram could have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-9510516681a3fa78e20e43b91d25aa4bc8ac9748b430b584bff90eca6729d804 | Tram tickets have six-digit numbers (from $000000$ to $999999$). A ticket is called lucky if the sum of its first three digits is equal to that of its last three digits. A ticket is called medium if the sum of all of its digits is $27$. Let $A$ and $B$ denote the numbers of lucky tickets and medium tickets, respectivel... | [
"0"
] | [
"Tram tickets have six-digit numbers (from $000000$ to $999999$). A ticket is called lucky if the sum of its first three digits is equal to that of its last three digits. A ticket is called medium if the sum of all of its digits is $27$. Let $A$ and $B$ denote the numbers of lucky tickets and medium tickets, respec... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5f25360367eb49d91f6cc65d5d485aeb9ce2347a5010d8bfbc03e5aba527e531 | Tram tickets have six-digit numbers (from 000000 to 999999). A ticket is called lucky if the sum of its first three digits is equal to that of its last three digits. A ticket is called medium if the sum of all of its digits is 27. Let A and B denote the numbers of lucky tickets and medium tickets, respectively. Find A-... | [
"0"
] | [
"Tram tickets have six-digit numbers (from 000000 to 999999). A ticket is called lucky if the sum of its first three digits is equal to that of its last three digits. A ticket is called medium if the sum of all of its digits is 27. Let A and B denote the numbers of lucky tickets and medium tickets, respectively. Fi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8c37887d1ff184fbf030cbe3cfd63c4fe0dd8bbb8074e4c53ac7e146ae18fe6b | Tram tickets have six-digit numbers (from 000000 to 999999). A ticket is called lucky if the sum of its first three digits is equal to that of its last three digits. A ticket is called medium if the sum of all of its digits is 27. Let \( A \) and \( B \) denote the numbers of lucky tickets and medium tickets, respectiv... | [
"0"
] | [
"Tram tickets have six-digit numbers (from 000000 to 999999). A ticket is called lucky if the sum of its first three digits is equal to that of its last three digits. A ticket is called medium if the sum of all of its digits is 27. Let \\( A \\) and \\( B \\) denote the numbers of lucky tickets and medium tickets, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-470505a2ac20fabfd164bbae13ec825b65e109ed00e26914d38d9c40513239ea | Transform into a product the expression $$\sin^2 \alpha +\sin^2 \beta +\sin^2 \gamma +2\cos \alpha \cos \beta \cos \gamma -2.$$ | [
"4\\sin\\frac{\\alpha -\\beta +\\gamma }{2}\\sin\\frac{\\gamma -\\alpha +\\beta }{2} \\sin\\frac{\\alpha +\\beta +\\gamma }{2}\\sin\\frac{\\alpha +\\beta -\\gamma }{2}"
] | [
"Transform into a product the expression $$\\sin^2 \\alpha +\\sin^2 \\beta +\\sin^2 \\gamma +2\\cos \\alpha \\cos \\beta \\cos \\gamma -2.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-80e461eb92728b36f98307220e8dfea87523db9ef40444fa05c32cd8d11157ff | Transform the ODE \( y'' + (\tan x)y' + (\cos^2 x) y = 0 \) into an ODE with constant coefficients using the variable \( t = \sin x \), and then solve it. | [
"\\( y(x) = A \\cos(\\sin x) + B \\sin(\\sin x) \\)"
] | [
"Transform the ODE \\( y'' + (\\tan x)y' + (\\cos^2 x) y = 0 \\) into an ODE with constant coefficients using the variable \\( t = \\sin x \\), and then solve it."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-f17e5622ff9579884087c1f278f04bebbd0379fb1076e55ef1b5a6611d074479 | Transform the PDE \( u_x + u_y = u \) into canonical form and solve the resulting equation. | [
"\\( u(x, y) = g(y - x)e^x \\)"
] | [
"Transform the PDE \\( u_x + u_y = u \\) into canonical form and solve the resulting equation."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-862a283f4f39d9a40398ee8c8c9f38c9341175d5a7b839a591272d8841f45f89 | Transform the boundary conditions to be homogeneous for the following PDE: \[ \begin{align*} & {{u}_{t}}=K{{u}_{xx}},\text{ }0<x<L,\text{ }t>0, \\ & u(0,t)={{h}_{0}}(t)\text{ for }t>0, \\ & {{u}_{x}}(L,t)+\alpha u(L,t)={{h}_{1}}(t),\text{ for }t>0, \\ & u(x,0)=f(x),\text{ for }0<x<L. \end{align*} \] | [
"u(x,t) = v(x,t) + \\left( \\frac{h_1(t) - \\alpha h_0(t)}{1 + \\alpha L} \\right)x + h_0(t)"
] | [
"Transform the boundary conditions to be homogeneous for the following PDE:\n\\[\n\\begin{align*}\n& {{u}_{t}}=K{{u}_{xx}},\\text{ }0<x<L,\\text{ }t>0, \\\\ \n& u(0,t)={{h}_{0}}(t)\\text{ for }t>0, \\\\ \n& {{u}_{x}}(L,t)+\\alpha u(L,t)={{h}_{1}}(t),\\text{ for }t>0, \\\\ \n& u(x,0)=f(x),\\text{ for }0<x<L. \n\\end... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-fb5d199e232fea69385b40414d27340f5d4e8db445c89187df7f894f71024f92 | Transform the boundary conditions to be homogeneous for the following PDE: \[ \begin{align*} &{{u}_{t}}=K{{u}_{xx}},\text{ }0<x<L,\text{ }t>0, \\ &{{u}_{x}}(0,t)={{g}_{0}}(t),\text{ }{{u}_{x}}(L,t)={{g}_{1}}(t),\text{ for }t>0, \\ &u(x,0)=f(x),\text{ for }0<x<L. \end{align*} \] | [
"v(x,t) = u(x,t) - \\left( \\frac{g_1(t) - g_0(t)}{2L} x^2 + g_0(t) x \\right)"
] | [
"Transform the boundary conditions to be homogeneous for the following PDE:\n\\[\n\\begin{align*}\n&{{u}_{t}}=K{{u}_{xx}},\\text{ }0<x<L,\\text{ }t>0, \\\\ \n&{{u}_{x}}(0,t)={{g}_{0}}(t),\\text{ }{{u}_{x}}(L,t)={{g}_{1}}(t),\\text{ for }t>0, \\\\ \n&u(x,0)=f(x),\\text{ for }0<x<L.\n\\end{align*}\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-4f6a57631d458da611168e5fc420d1640688be8784a2f2028e7b77323c0b6abb | Transform the cone given by the equation \(x^2 + y^2 = 3z^2\) such that its axis of symmetry is parallel to the vector \(\langle -5, 0, 2 \rangle\). Identify the correct linear transformation and resulting equation of the cone. | [
"\\(\\left(\\frac{2}{\\sqrt{29}}x - \\frac{5}{\\sqrt{29}}z\\right)^2 + y^2 = 3\\left(\\frac{5}{\\sqrt{29}}x + \\frac{2}{\\sqrt{29}}z\\right)^2\\)"
] | [
"Transform the cone given by the equation \\(x^2 + y^2 = 3z^2\\) such that its axis of symmetry is parallel to the vector \\(\\langle -5, 0, 2 \\rangle\\). Identify the correct linear transformation and resulting equation of the cone."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 2
} | 9 | 0 | 0 |
omr-ws-15a4842e6552f616f4031a30da8e793e5ddfbd946d03858d3fe64cc43ed7a41f | Transform the cubic equation \(x^3 + px^2 + qx + r = 0\) with roots \(a, b, c\) into a cubic equation whose roots are \(\left( \frac{1}{a^2} + \frac{1}{b^2} \right)\), \(\left( \frac{1}{a^2} + \frac{1}{c^2} \right)\), and \(\left( \frac{1}{b^2} + \frac{1}{c^2} \right)\). | [
"\\( x^{3} + 2 \\cdot \\frac{2pr - q^{2}}{r^{2}} x^{2} + \\left[ \\left( \\frac{q^{2} - 2pr}{r^{2}} \\right)^{2} + \\frac{p^{2} - 2q}{r^{2}} \\right] x - \\left[ \\frac{q^{2} - 2pr}{r^{2}} \\cdot \\frac{p^{2} - 2q}{r^{2}} - \\frac{1}{r^{2}} \\right] = 0 \\)"
] | [
"Transform the cubic equation \\(x^3 + px^2 + qx + r = 0\\) with roots \\(a, b, c\\) into a cubic equation whose roots are \\(\\left( \\frac{1}{a^2} + \\frac{1}{b^2} \\right)\\), \\(\\left( \\frac{1}{a^2} + \\frac{1}{c^2} \\right)\\), and \\(\\left( \\frac{1}{b^2} + \\frac{1}{c^2} \\right)\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-ef1eb8228974a47cb694600227fcb8676c9183e921b1f77886cd0e3080b7d437 | Transform the cubic equation \(x^3 - px^2 + qx - r = 0\), whose roots are \(a, b, c\), into a cubic equation whose roots are \(\frac{1}{a+b}\), \(\frac{1}{a+c}\), and \(\frac{1}{b+c}\). | [
"\\((pq-r)x^3-(p^2+q)x^2+2px-1=0\\)"
] | [
"Transform the cubic equation \\(x^3 - px^2 + qx - r = 0\\), whose roots are \\(a, b, c\\), into a cubic equation whose roots are \\(\\frac{1}{a+b}\\), \\(\\frac{1}{a+c}\\), and \\(\\frac{1}{b+c}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-840c69a5b99b38c46f385c5263e05fa777d3ae55844ea2e971f97ea23376dccc | Transform the determinant \(\begin{vmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 1 & 2 \\ 2 & 1 & 0 & 1 \\ 3 & 2 & 1 & 0 \end{vmatrix}\) into another determinant with zeroes everywhere below the main diagonal, and find its value. | [
"-12"
] | [
"Transform the determinant \\(\\begin{vmatrix} 0 & 1 & 2 & 3 \\\\ 1 & 0 & 1 & 2 \\\\ 2 & 1 & 0 & 1 \\\\ 3 & 2 & 1 & 0 \\end{vmatrix}\\) into another determinant with zeroes everywhere below the main diagonal, and find its value."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb93aa86d638cda16d0be016580dccbe62b3f572dd135ae4de2d568f7ea372d0 | Transform the determinant \(\begin{vmatrix} 1 & 3 & -1 \\ 4 & 7 & 2 \\ 3 & -6 & 5 \end{vmatrix}\) into another determinant with zeroes everywhere below the main diagonal, and find its value. | [
"50"
] | [
"Transform the determinant \\(\\begin{vmatrix} 1 & 3 & -1 \\\\ 4 & 7 & 2 \\\\ 3 & -6 & 5 \\end{vmatrix}\\) into another determinant with zeroes everywhere below the main diagonal, and find its value."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-00b11cb8bef6662d46ab52781bf53baeb4bddeefce0708192b5a004208d24749 | Transform the differential equation $\frac{d^2y}{dx^2} - y = x^2$ with boundary conditions $y(0) = 0$ and $y'(1) = 0$ into an integral equation. | [
"y(x) = \\int_{0}^{1} G(x, \\xi) \\xi^2 \\, d\\xi"
] | [
"Transform the differential equation $\\frac{d^2y}{dx^2} - y = x^2$ with boundary conditions $y(0) = 0$ and $y'(1) = 0$ into an integral equation."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-5a81ffc3ccfd3a532d078b11308f1dc681996a9b9568a7a74e7c492f78ade73b | Transform the double integral \(\int_0^5 \int_0^{\sqrt{25 - y^2}} xe^y \, dx \, dy\) to polar coordinates and determine the new integration limits. | [
"\\( 0 \\leq \\theta \\leq \\frac{\\pi}{2} \\) and \\( 0 \\leq r \\leq 5 \\)"
] | [
"Transform the double integral \\(\\int_0^5 \\int_0^{\\sqrt{25 - y^2}} xe^y \\, dx \\, dy\\) to polar coordinates and determine the new integration limits."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 18
} | 32 | 0 | 0 |
omr-ws-720b5238c63f2eb85a10507545fa7cfc3e12a253a8b0377a090249bfc4c69558 | Transform the equation $5x^2 + 4xy + 2y^2 - 2x + 4y + 4 = 0$ into one without $xy$, $x$, and $y$ terms. | [
"\\(6X^2 + Y^2 - 1 = 0\\)"
] | [
"Transform the equation $5x^2 + 4xy + 2y^2 - 2x + 4y + 4 = 0$ into one without $xy$, $x$, and $y$ terms."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-f596c764532f9392b9e69c8f49da7bd5bc4a7775fcac1ab96a12ea711a1f536c | Transform the equation $AX + BY = C$ (where $A$, $B$, and $C$ are constants) into a polar equation in the form $r = f(\theta)$. | [
"\\(r = \\frac{C}{A \\cos \\theta + B \\sin \\theta}\\)"
] | [
"Transform the equation $AX + BY = C$ (where $A$, $B$, and $C$ are constants) into a polar equation in the form $r = f(\\theta)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-a2b08edf239985cd8089551be685fc7ff144e0b8005e398f8cc1828ce78acaba | Transform the equation $\dfrac{d^2 z}{dx^2}+2\dfrac{d^2 z}{dxdy}+\dfrac{d^2 z}{dy^2}=0$ using the substitutions $u=\dfrac{1}{2}(x+y)$ and $v=\dfrac{1}{2}(x-y)$. | [
"\\dfrac{\\partial^2 z}{\\partial u^2} = 0"
] | [
"Transform the equation $\\dfrac{d^2 z}{dx^2}+2\\dfrac{d^2 z}{dxdy}+\\dfrac{d^2 z}{dy^2}=0$ using the substitutions $u=\\dfrac{1}{2}(x+y)$ and $v=\\dfrac{1}{2}(x-y)$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c53521e048045b6bc161047a7fffe78f94c21351ad34c30e8dc5c5f8e5ba6d08 | Transform the equation $\sqrt{2x+9}-\sqrt{x+1}=\sqrt{x+4}$ into a quadratic equation. | [
"\\(x^2 + 5x = 0\\)"
] | [
"Transform the equation $\\sqrt{2x+9}-\\sqrt{x+1}=\\sqrt{x+4}$ into a quadratic equation."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-4d5608397fda9628713b09f6e92d72b5d7cb52138027b601c7708a958d3cae46 | Transform the equation $x^6 + x^5 + x^4 + x^3 + x^2 + 1 = 0$ into a problem involving roots of unity. | [
"The roots of \\(x^6 + x^5 + x^4 + x^3 + x^2 + 1 = 0\\) are the 7th roots of unity, excluding 1."
] | [
"Transform the equation $x^6 + x^5 + x^4 + x^3 + x^2 + 1 = 0$ into a problem involving roots of unity."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 2,
"tir": 2
} | 6 | 0 | 0 |
omr-ws-5bec85c8d8a4ca74e24e3e131284f94b677c4c9d156a0427b183ae9b0d7d1e46 | Transform the equation \( x^5 + 25x^4 + 250x^3 + 1250x^2 + 3125x + 3125 = 0 \) so that its roots are increased by \( 5 \frac{1}{2} \). Find the sum of the numerical coefficients of the third degree term and the first degree term if the transformed equation has integral coefficients. | [
"90"
] | [
"Transform the equation \\( x^5 + 25x^4 + 250x^3 + 1250x^2 + 3125x + 3125 = 0 \\) so that its roots are increased by \\( 5 \\frac{1}{2} \\). Find the sum of the numerical coefficients of the third degree term and the first degree term if the transformed equation has integral coefficients."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 25
} | 48 | 0 | 0 |
omr-ws-e0898518ef6c98522f1cfc76d41dc64448637602d2bac51c0be525fbef423849 | Transform the expression $\cos \alpha \cos \beta \cos \gamma $ into a sum of sines provided that $\alpha +\beta +\gamma=\pi /2$. | [
"$(\\sin 2\\alpha +\\sin 2\\beta +\\sin 2\\gamma )/4$"
] | [
"Transform the expression $\\cos \\alpha \\cos \\beta \\cos \\gamma $ into a sum of sines provided that $\\alpha +\\beta +\\gamma=\\pi /2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dd40fd6777d1deaf290aec5f3aed1f3d3e468ab0e8ca1af2ce6f7e841768ff81 | Transform the following equation using the variables \((u,v,w(u,v)) = (yz - x, xz - y, xy - z)\): \[ (xy + z)\frac{\partial z}{\partial x} + (1 - y^2)\frac{\partial z}{\partial y} = x + yz. \] | [
"\\(\\frac{\\partial w}{\\partial v} = 0\\)"
] | [
"Transform the following equation using the variables \\((u,v,w(u,v)) = (yz - x, xz - y, xy - z)\\):\n\\[ (xy + z)\\frac{\\partial z}{\\partial x} + (1 - y^2)\\frac{\\partial z}{\\partial y} = x + yz. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 2
} | 22 | 0 | 0 |
omr-ws-d23ba6e63cff767a386f69735bfb9454053f5a538a557f84f2fb84459f96ebdf | Transform the following system of differential equations to polar coordinates: \[ \begin{cases} \frac{dx}{dt} = y + kx(x^2 + y^2) \\ \frac{dy}{dt} = -x + ky(x^2 + y^2). \end{cases} \] | [
"\\(\\left\\{ \\begin{array} {c} \\frac{dr}{dt} = kr^3 \\\\ \\frac{d\\theta}{dt} = -1 \\end{array} \\right.\\)"
] | [
"Transform the following system of differential equations to polar coordinates:\n\\[\n\\begin{cases}\n\\frac{dx}{dt} = y + kx(x^2 + y^2) \\\\\n\\frac{dy}{dt} = -x + ky(x^2 + y^2).\n\\end{cases}\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-2eb6f09ba0d58f5e5c85a82bc74ee35578f313e2b81bd653d29d9d4620c11ee3 | Transform the following system of equations into a diagonally dominant form so that it can be solved using the Gauss-Zeidel method: \[ \begin{cases} 2.7x_1 + 3.3x_2 + 1.3x_3 = 2.1 \\ 3.5x_1 - 1.7x_2 + 2.8x_3 = 1.7 \\ 4.1x_1 + 5.8x_2 - 1.7x_3 = 0.8 \end{cases} \] | [
"\n\\begin{cases}\n4.1x_1 + 5.8x_2 - 1.7x_3 = 0.8 \\\\\n3.5x_1 - 1.7x_2 + 2.8x_3 = 1.7 \\\\\n2.7x_1 + 3.3x_2 + 1.3x_3 = 2.1\n\\end{cases}\n"
] | [
"Transform the following system of equations into a diagonally dominant form so that it can be solved using the Gauss-Zeidel method:\n\n\\[\n\\begin{cases}\n2.7x_1 + 3.3x_2 + 1.3x_3 = 2.1 \\\\\n3.5x_1 - 1.7x_2 + 2.8x_3 = 1.7 \\\\\n4.1x_1 + 5.8x_2 - 1.7x_3 = 0.8\n\\end{cases}\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 1
} | 13 | 0 | 0 |
omr-ws-15a1b856177cab8b2567fd69ef97864470b69c3e39debaa9674280a4a9e1bd86 | Transform the function \( f(k) = k^9 + k^8 - 7k^7 - 5k^6 + 21k^5 + 11k^4 - 27k^3 - 5k^2 + 20k + 4 \) into a form where it is immediately clear that there are no positive solutions. | [
"f(k) = k(k^8 - 7k^6 + 21k^4 - 27k^2 + 20) + (k^8 - 5k^6 + 11k^4 - 5k^2 + 4)"
] | [
"Transform the function \\( f(k) = k^9 + k^8 - 7k^7 - 5k^6 + 21k^5 + 11k^4 - 27k^3 - 5k^2 + 20k + 4 \\) into a form where it is immediately clear that there are no positive solutions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-f31cb68b934c48a5e4ad3675a76e220ecaf0b9e3e39ff4b3319f3673d56d5699 | Transform the function \( f(x) = 5 \cdot |x-3| + 2 \cdot |x-5| + 3 \cdot |x-9| \) into a linear programming form for minimizing \( f(x) \). | [
"\n\\begin{aligned}\n\\text{minimize} \\quad & 5t_1 + 2t_2 + 3t_3 \\\\\n\\text{subject to} \\quad & t_1 \\geq x - 3, \\\\\n& t_1 \\geq 3 - x, \\\\\n& t_2 \\geq x - 5, \\\\\n& t_2 \\geq 5 - x, \\\\\n& t_3 \\geq x - 9, \\\\\n& t_3 \\geq 9 - x, \\\\\n& t_1, t_2, t_3 \\geq 0.\n\\end{aligned}\n"
] | [
"Transform the function \\( f(x) = 5 \\cdot |x-3| + 2 \\cdot |x-5| + 3 \\cdot |x-9| \\) into a linear programming form for minimizing \\( f(x) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-cc9a1430cbb47f1faf9ec635d56463c5746d0bc6ee54d20a2cbd1fcefc225ed4 | Transform the function \( f(x) = \sin x + 1 \) into a product form. | [
"2 \\sin\\left(\\frac{x}{2} + \\frac{\\pi}{4}\\right) \\cos\\left(\\frac{x}{2} - \\frac{\\pi}{4}\\right)"
] | [
"Transform the function \\( f(x) = \\sin x + 1 \\) into a product form."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-66b1f6c7156e83562039e32216fb2d82cdcc1a505b570fd81b2c4dca430c2e50 | Transform the given ellipse equation \(3x^2 + 2xy + 5y^2 = 1\) into the standard form of an ellipse. | [
"\\((x')^2(4-\\sqrt{2}) + (y')^2(4+\\sqrt{2}) = 1\\)"
] | [
"Transform the given ellipse equation \\(3x^2 + 2xy + 5y^2 = 1\\) into the standard form of an ellipse."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 11
} | 21 | 0 | 0 |
omr-ws-a9852697e2cf5c2007d0a6354df1442eb8fc5c840fb34e6c626c1b1d359e2b6b | Transform the given ellipse equation \(\Omega: \left(\frac{x - \frac{2y}{3}}{2} - 1\right)^2 + \left(\frac{y}{3} - 1\right)^2 = 1\) into the standard form of an ellipse. | [
"\\left(\\frac{x - \\frac{2y}{3} - 2}{2}\\right)^2 + \\left(\\frac{y - 3}{3}\\right)^2 = 1"
] | [
"Transform the given ellipse equation \\(\\Omega: \\left(\\frac{x - \\frac{2y}{3}}{2} - 1\\right)^2 + \\left(\\frac{y}{3} - 1\\right)^2 = 1\\) into the standard form of an ellipse."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-05ce9130c60eeae655c7d40875634b53670c9f78bbf0f9e5c3ec5544e91dad82 | Transform the inequality $\sum_{cyc}a^5c^4 + \sum_{cyc}a^5b^3c + 9a^3b^3c^3 \geq 5\sum_{cyc}a^4b^2c^3$ into the form $\sum_{cyc}(b-c)^2S_{a} \geq 0$ using the SOS (Sum of Squares) technique. | [
"\\sum_{cyc} (b - c)^2 \\left( a^3b^2 + a^3c^2 + a^2b^3 + a^2c^3 + ab^3c + abc^3 \\right) \\geq 0"
] | [
"Transform the inequality $\\sum_{cyc}a^5c^4 + \\sum_{cyc}a^5b^3c + 9a^3b^3c^3 \\geq 5\\sum_{cyc}a^4b^2c^3$ into the form $\\sum_{cyc}(b-c)^2S_{a} \\geq 0$ using the SOS (Sum of Squares) technique."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-9718b290fa4d884ab89516c6dce915383a6ec859866b4f01335c8deb8afdf277 | Transform the inequality \[ \frac{a+b}{a+b+2}+\frac{b+c}{b+c+2}+\frac{c+a}{c+a+2} \ge \frac{3}{2} \] into the form \[ \sum \frac{a+b}{a+b+2} \ge \frac{\left[ \sum (a+b)\right]^2}{\sum (a+b)(a+b+2)}. \] | [
"\\sum \\frac{a+b}{a+b+2} \\ge \\frac{\\left[ \\sum (a+b)\\right]^2}{\\sum (a+b)(a+b+2)}"
] | [
"Transform the inequality \n\n\\[ \\frac{a+b}{a+b+2}+\\frac{b+c}{b+c+2}+\\frac{c+a}{c+a+2} \\ge \\frac{3}{2} \\]\n\ninto the form\n\n\\[ \\sum \\frac{a+b}{a+b+2} \\ge \\frac{\\left[ \\sum (a+b)\\right]^2}{\\sum (a+b)(a+b+2)}. \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-213dee0aa602e2bb9b7b97276250def9fc559c5891ebde00a3ce176520361326 | Transform the integral \( T = \sqrt{\frac{8l}{g}} \int_0^{\theta_0} \frac{d\theta}{\sqrt{\cos \theta - \cos \theta_0}} \) using the substitution \( \cos \theta = 1 - 2 \sin^2(\theta/2) \) and \( \sin x = \frac{\sin(\theta/2)}{\sin(\theta_0/2)} \) to expand it into a Taylor series. | [
"T = 2\\pi \\sqrt{\\frac{l}{g}} \\left( 1 + \\frac{1}{4} \\sin^2\\left(\\frac{\\theta_0}{2}\\right) + \\frac{9}{64} \\sin^4\\left(\\frac{\\theta_0}{2}\\right) + \\cdots \\right)"
] | [
"Transform the integral \\( T = \\sqrt{\\frac{8l}{g}} \\int_0^{\\theta_0} \\frac{d\\theta}{\\sqrt{\\cos \\theta - \\cos \\theta_0}} \\) using the substitution \\( \\cos \\theta = 1 - 2 \\sin^2(\\theta/2) \\) and \\( \\sin x = \\frac{\\sin(\\theta/2)}{\\sin(\\theta_0/2)} \\) to expand it into a Taylor series."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-f3290d54d571e002e2515a3edeaf17c25ba8a2b30cecf39923aec7c0741eb7a3 | Transform the number \( 1989^{1989} \) by taking the sum of its digits four times. What is the result? | [
"9"
] | [
"Transform the number \\( 1989^{1989} \\) by taking the sum of its digits four times. What is the result?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0291d97621174595b90b7bbb23b2cc03c4e84e64ab0438cc510469bd95a40094 | Transform the partial differential equation and its initial and boundary conditions to the interval $[0,6]$ using the change of variables $w = x - 1$. Given: \[ u_t - k u_{xx} = x + 2t, \quad 1 < x < 7, \quad t > 0 \] Boundary Conditions (BC): \[ u(1,t) = u(7,t) = 0 \] Initial Condition (IV): \[ u(x,0) = x + 5 \] Trans... | [
"\\( u_t - k u_{ww} = w + 1 + 2t, \\quad 0 < w < 6, \\quad t > 0 \\), BC: \\( u(0,t) = u(6,t) = 0 \\), IV: \\( u(w,0) = w + 6 \\)"
] | [
"Transform the partial differential equation and its initial and boundary conditions to the interval $[0,6]$ using the change of variables $w = x - 1$.\n\nGiven:\n\\[ u_t - k u_{xx} = x + 2t, \\quad 1 < x < 7, \\quad t > 0 \\]\nBoundary Conditions (BC):\n\\[ u(1,t) = u(7,t) = 0 \\]\nInitial Condition (IV):\n\\[ u(x... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 17
} | 31 | 0 | 0 |
omr-ws-b6433edcf3f583f2ae55d26e40c8dabb850d175c5dc1b77253c62066d83ae020 | Transform the polar coordinate equation \( r = \frac{L}{1 - e \cos(a)} \) into a Cartesian coordinate equation, where \( e \) and \( L \) are constants (the eccentricity and semilatus rectum, respectively). | [
"\\((1 - e^2)x^2 - 2eLx + y^2 = L^2\\)"
] | [
"Transform the polar coordinate equation \\( r = \\frac{L}{1 - e \\cos(a)} \\) into a Cartesian coordinate equation, where \\( e \\) and \\( L \\) are constants (the eccentricity and semilatus rectum, respectively)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-7925ae9799a7bd6f64be5c55ad4a29cb2703f154fc5d6a73b4115cc89d5e9623 | Transform the quadratic congruence \(a x^2 \equiv b \pmod{c}\) (where \(c\) is an odd prime) into the form \(x^2 \equiv d \pmod{c}\). | [
"x^2 \\equiv b \\cdot a^{-1} \\pmod{c}"
] | [
"Transform the quadratic congruence \\(a x^2 \\equiv b \\pmod{c}\\) (where \\(c\\) is an odd prime) into the form \\(x^2 \\equiv d \\pmod{c}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-107a223a54cb00218f6597ae3520b35e4af3e886d2da43c72bb9c641c2760b0b | Transform the quadratic equation \( y = 3x^2 + 24x + 43 \) to vertex form. | [
"\\( y = 3(x + 4)^2 - 5 \\)"
] | [
"Transform the quadratic equation \\( y = 3x^2 + 24x + 43 \\) to vertex form."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-d67762655da485c1b6fd3705f085e4236a967333d1b8c28321feb62acc9e73d2 | Trapezoid $ABCD$ ($AB \parallel CD$) is inscribed in a circle $\omega$. On the ray $DC$ beyond the point $C$, point $E$ is marked such that $BC = BE$. Line $BE$ intersects circle $\omega$ again at point $F$ outside the segment $BE$. Let $O$ be the center of the circumcircle of triangle $CEF$. If the radius of $\omega$ ... | [
"10"
] | [
"Trapezoid $ABCD$ ($AB \\parallel CD$) is inscribed in a circle $\\omega$. On the ray $DC$ beyond the point $C$, point $E$ is marked such that $BC = BE$. Line $BE$ intersects circle $\\omega$ again at point $F$ outside the segment $BE$. Let $O$ be the center of the circumcircle of triangle $CEF$. If the radius of $... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-bfde60c85631cbf42cbceac2202788050dc1d84a79538021ace605748cbe3d4b | Trapezoid $ABCD$ ($AD \parallel BC$) is inscribed in the circle $\omega$. The incircles of triangles $ABC$ and $ABD$ touch $BC$ and $AD$ at points $P$ and $Q$, respectively. Points $X$ and $Y$ are the midpoints of arcs $BC$ and $AD$ of $\omega$, not containing points $A$ and $B$, respectively. If $PX$ and $YQ$ intersec... | [
"180^\\circ"
] | [
"Trapezoid $ABCD$ ($AD \\parallel BC$) is inscribed in the circle $\\omega$. The incircles of triangles $ABC$ and $ABD$ touch $BC$ and $AD$ at points $P$ and $Q$, respectively. Points $X$ and $Y$ are the midpoints of arcs $BC$ and $AD$ of $\\omega$, not containing points $A$ and $B$, respectively. If $PX$ and $YQ$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-72b9c3a84d876c81669138cf83592aa26e2ecc9a8a08a6179e4f5d93bfbbf928 | Trapezoid $ABCD$ has $AB = BC = 15$, $CD = 40$, and $AD = 20$. Point $E$ is the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. What is the ratio of the area of $\vartriangle ADE$ to the area of $\vartriangle BCE$? | [
"1"
] | [
"Trapezoid $ABCD$ has $AB = BC = 15$, $CD = 40$, and $AD = 20$. Point $E$ is the intersection of diagonals $\\overline{AC}$ and $\\overline{BD}$. What is the ratio of the area of $\\vartriangle ADE$ to the area of $\\vartriangle BCE$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5005dd45813a1ebed8249c22745049e6f03200263f4538cb2fe96f393dd25f66 | Trapezoid $ABCD$ has $AB \parallel CD$ and $BC=CD=AD$. If $\angle ACB=123^{\circ}$, what is the measure of $\angle B$? | [
"38"
] | [
"Trapezoid $ABCD$ has $AB \\parallel CD$ and $BC=CD=AD$. If $\\angle ACB=123^{\\circ}$, what is the measure of $\\angle B$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-002520aca3c958208e6e329ee1347b3fa75a381c015fefa999b52e959d8f1992 | Trapezoid $ABCD$ has $AB$ parallel to $CD$ and $AD$ perpendicular to $AB$. If $AB = 8$, $AD = 4$, and $CD = BC$, find the area of the trapezoid. | [
"26"
] | [
"Trapezoid $ABCD$ has $AB$ parallel to $CD$ and $AD$ perpendicular to $AB$. If $AB = 8$, $AD = 4$, and $CD = BC$, find the area of the trapezoid."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5576d92e44f5a475cdf8e8537d75a57c88d668bc594fbe5fb347c64421dbc136 | Trapezoid $ABCD$ has $AB$ parallel to $CD$. Let $M$ be the midpoint of diagonal $AC$, $N$ the midpoint of diagonal $BD$, and $P$ the midpoint of side $AB$. If $AB = 15$, $CD = 24$, and the height of the trapezoid is $h = 14$, calculate the area of the triangle $MNP$. | [
"$\\frac{63}{4}$"
] | [
"Trapezoid $ABCD$ has $AB$ parallel to $CD$. Let $M$ be the midpoint of diagonal $AC$, $N$ the midpoint of diagonal $BD$, and $P$ the midpoint of side $AB$. If $AB = 15$, $CD = 24$, and the height of the trapezoid is $h = 14$, calculate the area of the triangle $MNP$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-463e11e43e2673e891f1bb11dd48245e7672206d576ac85a75a4797ec2185408 | Trapezoid $ABCD$ has $AD \parallel BC$, $BD=1$, $\angle DBA=23$ degrees, $\angle BDA=46$ degrees, and $\frac{BC}{AD}=\frac{9}{5}$. Find the length of $CD$. | [
"\\dfrac{9}{5}"
] | [
"Trapezoid $ABCD$ has $AD \\parallel BC$, $BD=1$, $\\angle DBA=23$ degrees, $\\angle BDA=46$ degrees, and $\\frac{BC}{AD}=\\frac{9}{5}$. Find the length of $CD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-f216f57ad5ff5ffd7e42b8040b34b477073a77631e81d6ce44fe7f285e971599 | Trapezoid $ABCD$ has $AD \parallel BC$, $BD=1$, $\angle DBA=23^\circ$, and $\angle BDC=46^\circ$. The ratio $BC:AD$ is $9:5$. What is the length of $CD$? | [
"\\(\\frac{4}{5}\\)"
] | [
"Trapezoid $ABCD$ has $AD \\parallel BC$, $BD=1$, $\\angle DBA=23^\\circ$, and $\\angle BDC=46^\\circ$. The ratio $BC:AD$ is $9:5$. What is the length of $CD$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 8,
"tir": 12
} | 26 | 0 | 0 |
omr-ws-17d1699c93d36407115582e0d8c01675027207fdff533376e01764aa4e7c657d | Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}$ such that its diagonals intersect at point $E$. The bisector of $\angle AEB$ intersects sides $\overline{AB}$ and $\overline{CD}$ at points $I_1$ and $I_3$ respectively. The bisector of $\angle AED$ intersects sides $\overline{BC}$ and $\overline{AD}$ at poin... | [
"656"
] | [
"Trapezoid $ABCD$ has $\\overline{AB} \\parallel \\overline{CD}$ such that its diagonals intersect at point $E$. The bisector of $\\angle AEB$ intersects sides $\\overline{AB}$ and $\\overline{CD}$ at points $I_1$ and $I_3$ respectively. The bisector of $\\angle AED$ intersects sides $\\overline{BC}$ and $\\overlin... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-ec06ad3069b9d6dcb3a73150c9b8b92743fde645739521bd7f5ed4594d307e92 | Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}$, $BC = CD = 43$, and $\overline{AB} \perp \overline{CD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$ and let $P$ be the midpoint of $\overline{BD}$. Given that $OP = 11$, the length of $AD$ can be written in the form $m\s... | [
"45"
] | [
"Trapezoid $ABCD$ has $\\overline{AB} \\parallel \\overline{CD}$, $BC = CD = 43$, and $\\overline{AB} \\perp \\overline{CD}$. Let $O$ be the intersection of the diagonals $\\overline{AC}$ and $\\overline{BD}$ and let $P$ be the midpoint of $\\overline{BD}$. Given that $OP = 11$, the length of $AD$ can be written in... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca3765106bd5f7abc9d30d06f2f54bec307fc13da3d90889fe37cfede74659b9 | Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}$, $BC = CD = 43$, and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP = 11$, the length $AD$ can be written in the form $m\sqr... | [
"194"
] | [
"Trapezoid $ABCD$ has $\\overline{AB} \\parallel \\overline{CD}$, $BC = CD = 43$, and $\\overline{AD} \\perp \\overline{BD}$. Let $O$ be the intersection of the diagonals $\\overline{AC}$ and $\\overline{BD}$, and let $P$ be the midpoint of $\\overline{BD}$. Given that $OP = 11$, the length $AD$ can be written in t... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 108,
"tir": 40
} | 148 | 0 | 0 |
omr-ws-b0609f4a109448b5156ec2e252848165b53057752facf5173b355e3fb115427e | Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}$, $BC = CD = 43$, and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP = 11$, the length of $AD$ can be written in the form $m\... | [
"194"
] | [
"Trapezoid $ABCD$ has $\\overline{AB} \\parallel \\overline{CD}$, $BC = CD = 43$, and $\\overline{AD} \\perp \\overline{BD}$. Let $O$ be the intersection of the diagonals $\\overline{AC}$ and $\\overline{BD}$, and let $P$ be the midpoint of $\\overline{BD}$. Given that $OP = 11$, the length of $AD$ can be written i... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 8,
"tir": 16
} | 71 | 0 | 0 |
omr-ws-e2f5064948bdcb2303c88b0aaaa2e17ccb0dd8a8b79df51276f92e90064a080d | Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}$, $BC=CD=43,$ and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length $AD$ can be written in the form $m\sqrt{n}$,... | [
"194"
] | [
"Trapezoid $ABCD$ has $\\overline{AB} \\parallel \\overline{CD}$, $BC=CD=43,$ and $\\overline{AD} \\perp \\overline{BD}$. Let $O$ be the intersection of the diagonals $\\overline{AC}$ and $\\overline{BD}$, and let $P$ be the midpoint of $\\overline{BD}$. Given that $OP=11$, the length $AD$ can be written in the for... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ceae1a8b376aa861d55b190ca41a2c5a72c023d116b421f1a0c7546414acfb9 | Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD},$ $BC=CD=43,$ and $\overline{AD} \perp \overline{BD}.$ Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD},$ and let $P$ be the midpoint of $\overline{BD}.$ Given that $OP=11,$ the length $AD$ can be written in the form $m\sqrt{n},$... | [
"194"
] | [
"Trapezoid $ABCD$ has $\\overline{AB} \\parallel \\overline{CD},$ $BC=CD=43,$ and $\\overline{AD} \\perp \\overline{BD}.$ Let $O$ be the intersection of the diagonals $\\overline{AC}$ and $\\overline{BD},$ and let $P$ be the midpoint of $\\overline{BD}.$ Given that $OP=11,$ the length $AD$ can be written in the for... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 21
} | 41 | 0 | 0 |
omr-ws-26aba576b3b502c4d975955e584b6ac946c65c719540246290a5a85a74d3fc32 | Trapezoid $ABCD$ has base $AB = 20$ units and base $CD = 30$ units. Diagonals $AC$ and $BD$ intersect at $X$. If the area of trapezoid $ABCD$ is $300$ square units, what is the area of triangle $BXC$? | [
"72"
] | [
"Trapezoid $ABCD$ has base $AB = 20$ units and base $CD = 30$ units. Diagonals $AC$ and $BD$ intersect at $X$. If the area of trapezoid $ABCD$ is $300$ square units, what is the area of triangle $BXC$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-431480b64e67c602feba5db9fdaa8e7c7104d874032b76bd79923945f5da4a4a | Trapezoid $ABCD$ has bases $AB=4$ and $CD=29$, and sides $BC=15$ and $DA=20$. Two circles of equal radius, $\omega_1$ and $\omega_2$, are inscribed in the trapezoid such that $\omega_1$ is tangent to $\overline{AB}$, $\overline{BC}$, and $\omega_2$, while $\omega_2$ is tangent to $\overline{CD}$, $\overline{DA}$, and $... | [
"968"
] | [
"Trapezoid $ABCD$ has bases $AB=4$ and $CD=29$, and sides $BC=15$ and $DA=20$. Two circles of equal radius, $\\omega_1$ and $\\omega_2$, are inscribed in the trapezoid such that $\\omega_1$ is tangent to $\\overline{AB}$, $\\overline{BC}$, and $\\omega_2$, while $\\omega_2$ is tangent to $\\overline{CD}$, $\\overli... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 3
} | 5 | 0 | 0 |
omr-ws-700e56ad81d816de3713a67d49d1da264381d9156f57025bc1553f4828e16246 | Trapezoid $ABCD$ has bases $\overline{AB}$ and $\overline{CD}$ and diagonals intersecting at $K$. Suppose that $AB = 9$, $DC = 12$, and the area of $\triangle AKD$ is $24$. What is the area of trapezoid $ABCD$? | [
"98"
] | [
"Trapezoid $ABCD$ has bases $\\overline{AB}$ and $\\overline{CD}$ and diagonals intersecting at $K$. Suppose that $AB = 9$, $DC = 12$, and the area of $\\triangle AKD$ is $24$. What is the area of trapezoid $ABCD$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 2
} | 50 | 0 | 0 |
omr-ws-5682adcf352ee00bf03f9ce300ba8dbd5f45fbfc7294510fd38d2c310ef3e8b4 | Trapezoid $ABCD$ has legs (non-parallel sides) $BC$ and $DA$ of length 5 and 6 respectively, and there exists a point $X$ on $CD$ such that $\angle XBC = \angle XAD = \angle AXB = 90^\circ$. Find the area of trapezoid $ABCD$. | [
"45"
] | [
"Trapezoid $ABCD$ has legs (non-parallel sides) $BC$ and $DA$ of length 5 and 6 respectively, and there exists a point $X$ on $CD$ such that $\\angle XBC = \\angle XAD = \\angle AXB = 90^\\circ$. Find the area of trapezoid $ABCD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-632e41afe1d6b3d3cdbab5332bc0315641ff50796c4f129cccaf463b08f4e988 | Trapezoid $ABCD$ has parallel sides $AB$ and $CD$, with $BC$ perpendicular to them. Given $AB=13$, $BC=16$, and $DC=11$. Let $E$ be the midpoint of $AD$ and $F$ the point on $BC$ so that $EF$ is perpendicular to $AD$. Find the area of quadrilateral $AEFB$. | [
"91"
] | [
"Trapezoid $ABCD$ has parallel sides $AB$ and $CD$, with $BC$ perpendicular to them. Given $AB=13$, $BC=16$, and $DC=11$. Let $E$ be the midpoint of $AD$ and $F$ the point on $BC$ so that $EF$ is perpendicular to $AD$. Find the area of quadrilateral $AEFB$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-d0fc8652a08810549054813b83c2f41def6db7620fca17bd6b16fe926f45bfac | Trapezoid $ABCD$ has parallel sides $\overline{AB}$ of length $33$ and $\overline{CD}$ of length $21$. The other two sides are of lengths $10$ and $14$. The angles $A$ and $B$ are acute. What is the length of the shorter diagonal of $ABCD$? | [
"25"
] | [
"Trapezoid $ABCD$ has parallel sides $\\overline{AB}$ of length $33$ and $\\overline{CD}$ of length $21$. The other two sides are of lengths $10$ and $14$. The angles $A$ and $B$ are acute. What is the length of the shorter diagonal of $ABCD$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-b0d253dacd43260079cb24f13d452d64b14b1e4512550d4cf685757ac8074340 | Trapezoid $ABCD$ has parallel sides $\overline{AB}$ of length $33$ and $\overline{CD}$ of length $21$. The other two sides are of lengths $10$ and $14$. The angles at $A$ and $B$ are acute. What is the length of the shorter diagonal of $ABCD$? | [
"25"
] | [
"Trapezoid $ABCD$ has parallel sides $\\overline{AB}$ of length $33$ and $\\overline{CD}$ of length $21$. The other two sides are of lengths $10$ and $14$. The angles at $A$ and $B$ are acute. What is the length of the shorter diagonal of $ABCD$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-c093511cae9d89f5e654694f8a7c67474f59a90348125cb66638516cc599fe91 | Trapezoid $ABCD$ has sides $AB = 92$, $BC = 50$, $CD = 19$, $AD = 70$. The side $AB$ is parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP = \frac{m}{n}$ where $m$ and $n$ are relatively prime integers, what is $m+n$? | [
"167"
] | [
"Trapezoid $ABCD$ has sides $AB = 92$, $BC = 50$, $CD = 19$, $AD = 70$. The side $AB$ is parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP = \\frac{m}{n}$ where $m$ and $n$ are relatively prime integers, what is $m+n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c68aa8022bbd416d1ab4094d12b9c4fca4a6324d158980e711ed1b1c66577585 | Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP = \frac{m}{n}$ where $m$ and $n$ are relatively prime, what is $m+n$? | [
"164"
] | [
"Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP = \\frac{m}{n}$ where $m$ and $n$ are relatively prime, what is $m+n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f65759080073765ab06f63599163da80122415026cd2bb5d80c34f71766c520a | Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. | [
"164"
] | [
"Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP = \\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8ef452c7e2fa6040e7d372eb85f3b3538157ffddb696c60a4e1a26cf6eb022d0 | Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=\frac mn$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. | [
"164"
] | [
"Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=\\frac mn$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9b9d927b2b3b4d43aa5e2432175ea1e2dc2d01c9570e3db5a20b2ce0001ece6e | Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. | [
"164"
] | [
"Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-c877962d8ce6775dfb8d788afd2e97d9225b85439778fe28234884a12dfad246 | Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$. The side $AB$ is parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Find the value of $AP$. | [
"\\(\\frac{161}{3}\\)"
] | [
"Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$. The side $AB$ is parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Find the value of $AP$."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-3a442906079ae886fb4099ccb36f2cb03e1a3a5ca3c5b8186ae1af3b52abf550 | Trapezoid $ABCD$ is circumscribed around a circle with radius $r$ and inscribed in a circle with radius $R$, with bases $AB$ and $CD$ such that $AB > CD$, and $BC = AD = R$. If $k = \frac{R}{AB}$ can be expressed as $a - \sqrt{b}$, find $a + b$. | [
"9"
] | [
"Trapezoid $ABCD$ is circumscribed around a circle with radius $r$ and inscribed in a circle with radius $R$, with bases $AB$ and $CD$ such that $AB > CD$, and $BC = AD = R$. If $k = \\frac{R}{AB}$ can be expressed as $a - \\sqrt{b}$, find $a + b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-673ba02713457a9cf7245fe70620a2870da4b39e9599efbe6a1521431208b2f8 | Trapezoid $ABCD$ is inscribed in a circle of radius $100$. If $AB = 200$ and $BC = 50$, what is the perimeter of $ABCD$? | [
"475"
] | [
"Trapezoid $ABCD$ is inscribed in a circle of radius $100$. If $AB = 200$ and $BC = 50$, what is the perimeter of $ABCD$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-465af3344e00f36d2f54b1dc6ca49b1793e813ec834d2cef6fd51ade24ed2553 | Trapezoid $ABCD$ is inscribed in a semicircle with $AB$ as the base and diameter of the semicircle. Given $AB = 4$, $AD = BC = 1$, find the length of $DC$ expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime, and determine the value of $a+b$. | [
"9"
] | [
"Trapezoid $ABCD$ is inscribed in a semicircle with $AB$ as the base and diameter of the semicircle. Given $AB = 4$, $AD = BC = 1$, find the length of $DC$ expressed as $\\frac{a}{b}$ where $a$ and $b$ are relatively prime, and determine the value of $a+b$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a8e07a6d7bceb4380d63904d022ba12ae75ef3f98587bbc845c8ea4fae70488b | Trapezoid $ABCD$ is isosceles, with $AB = 15$, $BC=AD=10$, and $CD=27$. Points $X_1$ and $X_2$ are on the median of $ABCD$ such that $\triangle{AX_1D}$ and $\triangle{BX_2C}$ are both right triangles. Let the foot of the altitude from $X_1$ to $\overline{AD}$ be $H_1$, and the foot of the altitude from $X_2$ to $\overl... | [
"92"
] | [
"Trapezoid $ABCD$ is isosceles, with $AB = 15$, $BC=AD=10$, and $CD=27$. Points $X_1$ and $X_2$ are on the median of $ABCD$ such that $\\triangle{AX_1D}$ and $\\triangle{BX_2C}$ are both right triangles. Let the foot of the altitude from $X_1$ to $\\overline{AD}$ be $H_1$, and the foot of the altitude from $X_2$ to... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-fbb3f51770b85718f16b70790586fcb038b320f428e924e01174f93f9033aefd | Trapezoid $ABCD$ with $AB \parallel CD$ has diagonals that intersect at right angles. Additionally, $AD$ and $BC$ intersect at a $45^{\circ}$ angle. Given that $AB=4$ and $CD=10$, find the area of trapezoid $ABCD$. | [
"\\(\\frac{140}{3}\\)"
] | [
"Trapezoid $ABCD$ with $AB \\parallel CD$ has diagonals that intersect at right angles. Additionally, $AD$ and $BC$ intersect at a $45^{\\circ}$ angle. Given that $AB=4$ and $CD=10$, find the area of trapezoid $ABCD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-388be98d241a78c974504fac17ec312686cace8be71bf8c371c08884b479eee2 | Trapezoid $ABCD$ with $\overline{AB} \parallel \overline{CD}$ is inscribed in circle $\omega$, and point $G$ lies inside triangle $BCD$. Rays $AG$ and $BG$ meet $\omega$ again at points $P$ and $Q$, respectively. The line through $G$ parallel to $\overline{AB}$ intersects $\overline{BD}$ and $\overline{BC}$ at points $... | [
"2"
] | [
"Trapezoid $ABCD$ with $\\overline{AB} \\parallel \\overline{CD}$ is inscribed in circle $\\omega$, and point $G$ lies inside triangle $BCD$. Rays $AG$ and $BG$ meet $\\omega$ again at points $P$ and $Q$, respectively. The line through $G$ parallel to $\\overline{AB}$ intersects $\\overline{BD}$ and $\\overline{BC}... | [
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-77ab33222ecc739038d05e6d5480723986b0d6a0c4878a23538ff074da465f28 | Trapezoid $ABCD$ with bases $AD$ and $BC$ is inscribed in a circle. Let $M$ be the intersection of its diagonals. A line through $M$ perpendicular to the bases intersects $BC$ at point $K$ and the circle at point $L$, where $M$ lies on the segment $KL$. Given that $MK = a$ and $LM = b$, find the radius of the circle ta... | [
"\\sqrt{ab}"
] | [
"Trapezoid $ABCD$ with bases $AD$ and $BC$ is inscribed in a circle. Let $M$ be the intersection of its diagonals. A line through $M$ perpendicular to the bases intersects $BC$ at point $K$ and the circle at point $L$, where $M$ lies on the segment $KL$. Given that $MK = a$ and $LM = b$, find the radius of the circ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-f136ede5a72256a2642075f51b503d397195dd5f9e4ed0b7eb831568816b2213 | Trapezoid $ABCD^{}_{}$ has sides $AB=92^{}_{}$, $BC=50^{}_{}$, $CD=19^{}_{}$, and $AD=70^{}_{}$, with $AB^{}_{}$ parallel to $CD^{}_{}$. A circle with center $P^{}_{}$ on $AB^{}_{}$ is drawn tangent to $BC^{}_{}$ and $AD^{}_{}$. Given that $AP^{}_{}=\frac mn$, where $m^{}_{}$ and $n^{}_{}$ are relatively prime positive... | [
"164"
] | [
"Trapezoid $ABCD^{}_{}$ has sides $AB=92^{}_{}$, $BC=50^{}_{}$, $CD=19^{}_{}$, and $AD=70^{}_{}$, with $AB^{}_{}$ parallel to $CD^{}_{}$. A circle with center $P^{}_{}$ on $AB^{}_{}$ is drawn tangent to $BC^{}_{}$ and $AD^{}_{}$. Given that $AP^{}_{}=\\frac mn$, where $m^{}_{}$ and $n^{}_{}$ are relatively prime po... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-f1b69635a00f2aa3766a003d32e676a41aafbf69813addd9989fffe5d7bf8752 | Trapezoid $KLMN$ has sides $KL=80$ units, $LM=60$, $MN=22$, and $KN=65$, with $KL$ parallel to $MN$. A semicircle with center $A$ on $KL$ is drawn tangent to both sides $KN$ and $ML$. What is the length of segment $KA$? | [
"\\(\\frac{208}{5}\\)"
] | [
"Trapezoid $KLMN$ has sides $KL=80$ units, $LM=60$, $MN=22$, and $KN=65$, with $KL$ parallel to $MN$. A semicircle with center $A$ on $KL$ is drawn tangent to both sides $KN$ and $ML$. What is the length of segment $KA$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
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