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-4d8b5701ab12e34c7b59c5c13488c3de7b3b3c3233ebd115f521ca88b4c0cea6 | A farmer has a flock of \( n \) sheep, where \( 2000 \leq n \leq 2100 \). The farmer puts some number of the sheep into one barn and the rest into a second barn. The probability that two randomly selected sheep are in different barns is exactly \( \frac{1}{2} \). Determine the value of \( n \). | [
"2025"
] | [
"A farmer has a flock of \\( n \\) sheep, where \\( 2000 \\leq n \\leq 2100 \\). The farmer puts some number of the sheep into one barn and the rest into a second barn. The probability that two randomly selected sheep are in different barns is exactly \\( \\frac{1}{2} \\). Determine the value of \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-e7e46d1321d54f4842d9eaa4e9b5c7c13ac77f2468e2c61e55aa60eb32688b85 | A farmer has a machine that tests whether an animal is a cow or a sheep. If a cow walks through the machine, it correctly identifies the cow 90% of the time and incorrectly identifies it as a sheep 10% of the time. If a sheep walks through the machine, it correctly identifies the sheep 95% of the time and incorrectly i... | [
"12"
] | [
"A farmer has a machine that tests whether an animal is a cow or a sheep. If a cow walks through the machine, it correctly identifies the cow 90% of the time and incorrectly identifies it as a sheep 10% of the time. If a sheep walks through the machine, it correctly identifies the sheep 95% of the time and incorrec... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-9becca90bccdbda6dd7fe7b7195c4bcfd5c3bcc5ada3ce503728b09dc9187f5d | A farmer has a rectangular field with dimensions $2m+7$ and $m-2$. If the field has an area of 51 square units, what is the value of $m$? | [
"5"
] | [
"A farmer has a rectangular field with dimensions $2m+7$ and $m-2$. If the field has an area of 51 square units, what is the value of $m$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0b6ad48c6b21bab1e71674df7b59429ced5956aad764e8c7698211c82714c6de | A farmer has an orchard that can support 144 trees. He sells as many apples as he does peaches and pears combined, and he sells twice as many peaches as pears. If all the trees bear about the same amount of fruit, how many of each kind of tree should he plant? | [
"72 apples, 48 peaches, 24 pears"
] | [
"A farmer has an orchard that can support 144 trees. He sells as many apples as he does peaches and pears combined, and he sells twice as many peaches as pears. If all the trees bear about the same amount of fruit, how many of each kind of tree should he plant?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-752c1df734e07c430a19ccc4390b357f74489a26bfbb70675f9f206badef7682 | A farmer has determined that a crop of strawberries yields a yearly profit of $4.50 per square yard. If strawberries are planted on a triangular piece of land whose sides are 500 ft, 700 ft, and 550 ft, how much profit, to the nearest dollar, would the farmer expect to make from this piece of land during the next harve... | [
"Profit = 81,192 dollars, Smallest angle = 38 degrees"
] | [
"A farmer has determined that a crop of strawberries yields a yearly profit of $4.50 per square yard. If strawberries are planted on a triangular piece of land whose sides are 500 ft, 700 ft, and 550 ft, how much profit, to the nearest dollar, would the farmer expect to make from this piece of land during the next ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04aa194cdd9be128a1c4abbf8e10bba6bd27733db9c4430961a7ed43c52c57ff | A farmer has four fences with lengths 4m, 3m, 2m, and 1m. What is the maximum area that can be enclosed by forming a closed shape (such as a trapezoid) using all four fences? | [
"\\(\\sqrt{24}\\)"
] | [
"A farmer has four fences with lengths 4m, 3m, 2m, and 1m. What is the maximum area that can be enclosed by forming a closed shape (such as a trapezoid) using all four fences?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-3862812b602ab6fea33c03cf316442ac1ebac3dd630172b2ce1f714d9fc98063 | A farmer has sheep that multiply by \( \frac{103}{101} \) each day. His utility is given by \( ab \), where \( a \) is the money made by selling the sheep and \( b \) is the number of days left until he dies (day 365). On which day should he sell all his sheep to maximize his utility? | [
"314th day"
] | [
"A farmer has sheep that multiply by \\( \\frac{103}{101} \\) each day. His utility is given by \\( ab \\), where \\( a \\) is the money made by selling the sheep and \\( b \\) is the number of days left until he dies (day 365). On which day should he sell all his sheep to maximize his utility?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-60a542814d124d48cf27d2834bf2c26bd3ef28282bd0bbab386ab2af7787b5da | A farmer intends to plant crops A and B on a 100-acre field. The seed cost for crop A is $6 per acre, and labor and equipment cost $20 per acre. For crop B, the seed cost is $9 per acre, and labor and equipment cost $15 per acre. The farmer cannot spend more than $810 on seed and $1800 on labor and equipment. The incom... | [
"45 acres of crop A and 60 acres of crop B to maximize total income of $17,250."
] | [
"A farmer intends to plant crops A and B on a 100-acre field. The seed cost for crop A is $6 per acre, and labor and equipment cost $20 per acre. For crop B, the seed cost is $9 per acre, and labor and equipment cost $15 per acre. The farmer cannot spend more than $810 on seed and $1800 on labor and equipment. The ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-aacd26afbe8d3203647ee2e1edbff176759a227152b605c66c023baa90b1c9ad | A farmer needs to provide each animal with 15 grams of protein and 7.5 grams of carbohydrates using two food mixes. Mix A contains 12% protein and 9% carbohydrates, while Mix B contains 15% protein and 5% carbohydrates. How many grams of each mix should be used to provide the correct nutrients for each animal? | [
"\\( x = 50 \\) grams of Mix A, \\( y = 60 \\) grams of Mix B."
] | [
"A farmer needs to provide each animal with 15 grams of protein and 7.5 grams of carbohydrates using two food mixes. Mix A contains 12% protein and 9% carbohydrates, while Mix B contains 15% protein and 5% carbohydrates. How many grams of each mix should be used to provide the correct nutrients for each animal?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 31
} | 40 | 0 | 0 |
omr-ws-72b4fb9d479beb18cb4be59bbb67b20c02a522c919d2cf1dfeef6570cf557de2 | A farmer noticed that the number of calves born in the last year is equal to the sum of the calves born in the two preceding years. Additionally, the number of pigs born in the last year is one more than the sum of the pigs born in the two preceding years. Will there be a year in which at least twice as many pigs as ca... | [
"Yes"
] | [
"A farmer noticed that the number of calves born in the last year is equal to the sum of the calves born in the two preceding years. Additionally, the number of pigs born in the last year is one more than the sum of the pigs born in the two preceding years. Will there be a year in which at least twice as many pigs ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-726e6067f33ea8cdf04c0312fdd1a23c5dc8071ee9fcc40d015a1f713a977aea | A farmer notices that the number of calves born each year is equal to the sum of the number of calves born in the two preceding years. The number of lambs born each year is one more than the sum of the number of lambs born in the two preceding years. Will the farmer eventually have at least twice as many lambs as calve... | [
"No"
] | [
"A farmer notices that the number of calves born each year is equal to the sum of the number of calves born in the two preceding years. The number of lambs born each year is one more than the sum of the number of lambs born in the two preceding years. Will the farmer eventually have at least twice as many lambs as ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-43e1105dbea8d74e931db4305dec1ca2cf4541b332d1df26b836ec2ef5ddd6be | A farmer owns 50 papaya trees, each producing 600 papayas per year. For each additional tree planted, the output of each tree decreases by 5 papayas. How many additional trees should be planted to maximize the total production of papayas? | [
"35"
] | [
"A farmer owns 50 papaya trees, each producing 600 papayas per year. For each additional tree planted, the output of each tree decreases by 5 papayas. How many additional trees should be planted to maximize the total production of papayas?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 6
} | 20 | 0 | 0 |
omr-ws-e4fc61c9a9a538417ec03cae24886ad8239390448595d2ed2ba6422f217e32c1 | A farmer owns a 200-acre farm and can plant any combination of two crops, 1 and 2. Crop 1 requires 1 man-day of labor and USD10 of capital per acre planted, while crop 2 requires 4 man-days of labor and USD20 capital per acre. The farmer has USD2200 of capital and 320 man-days of labor available for the year. Crop 1 pr... | [
"80"
] | [
"A farmer owns a 200-acre farm and can plant any combination of two crops, 1 and 2. Crop 1 requires 1 man-day of labor and USD10 of capital per acre planted, while crop 2 requires 4 man-days of labor and USD20 capital per acre. The farmer has USD2200 of capital and 320 man-days of labor available for the year. Crop... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-1d78b10be67f358e9731cb869d019fc14626408c1a82717113940cbee38277d5 | A farmer picks 2017 apples with an average weight of 100 grams. The average weight of all the apples heavier than 100 grams is 122 grams, while the average weight of all the apples lighter than 100 grams is 77 grams. At least how many apples weighing exactly 100 grams did the farmer pick? | [
"37"
] | [
"A farmer picks 2017 apples with an average weight of 100 grams. The average weight of all the apples heavier than 100 grams is 122 grams, while the average weight of all the apples lighter than 100 grams is 77 grams. At least how many apples weighing exactly 100 grams did the farmer pick?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-5a1678ad3945c9d4ac4926b9c144c5651b0839693c8fc185f812d84e45d4ac50 | A farmer plans to use 21 meters of fencing to enclose a rectangular pen with an area of 55 square meters. Only three sides of the pen need fencing because part of the existing wall will form the fourth side. Find the dimensions of the pen. | [
"\\( x = 11 \\) meters, \\( y = 5 \\) meters"
] | [
"A farmer plans to use 21 meters of fencing to enclose a rectangular pen with an area of 55 square meters. Only three sides of the pen need fencing because part of the existing wall will form the fourth side. Find the dimensions of the pen."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 6,
"tir": 11
} | 20 | 0 | 0 |
omr-ws-243846c897e711700f83062339339427d5975453e42870661f9fec22afbdebfd | A farmer purchased 100 head of livestock for a total cost of $4000. The prices were $120 for calves, $50 for lambs, and $25 for piglets. How many of each animal did the farmer buy, assuming that he must buy at least one of each? | [
"(5, 41, 54), (10, 22, 68), (15, 3, 82)"
] | [
"A farmer purchased 100 head of livestock for a total cost of $4000. The prices were $120 for calves, $50 for lambs, and $25 for piglets. How many of each animal did the farmer buy, assuming that he must buy at least one of each?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-337efd78207fe111938c3b9cb7ae8bfe55afc7e1fac9393f2160c3f708ae09c1 | A farmer sells bean seeds in packets of 100 and refunds the price if fewer than 75 seeds germinate. On average, 80% of the seeds germinate, and it costs the farmer Rs. 2 per packet. How should the farmer set the price of the packet to ensure an average profit of 25%? | [
"2.8"
] | [
"A farmer sells bean seeds in packets of 100 and refunds the price if fewer than 75 seeds germinate. On average, 80% of the seeds germinate, and it costs the farmer Rs. 2 per packet. How should the farmer set the price of the packet to ensure an average profit of 25%?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-67c6671a054db4dbb2375b2ddb3491b47097941e9aa2f161eb9c387a6ed47b5b | A farmer spent 3 equal sums of money on buying hens, pigeons, and parrots. Each hen cost $1 more than a pigeon and $2 more than a parrot. Altogether, he bought 47 birds, and the number of pigeons exceeded that of the hens by as many parrots as he could have bought for $9. Find the number of birds of each kind and the t... | [
"Number of parrots = 20, Number of hens = 12, Number of pigeons = 15, Total amount of money spent = 180."
] | [
"A farmer spent 3 equal sums of money on buying hens, pigeons, and parrots. Each hen cost $1 more than a pigeon and $2 more than a parrot. Altogether, he bought 47 birds, and the number of pigeons exceeded that of the hens by as many parrots as he could have bought for $9. Find the number of birds of each kind and ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-c0b6076fcd9e2fbbbd9e771f9fbfa00779af7de7793bcfe7aeb16315ed955a5f | A farmer starts with 2 rabbits. After 1 month, each pair of rabbits can produce 4 baby rabbits, and they do so every month. After 2 months, the baby rabbits become mature and can also produce 4 baby rabbits each month. How many rabbits will the farmer have after 1 year? | [
"2730"
] | [
"A farmer starts with 2 rabbits. After 1 month, each pair of rabbits can produce 4 baby rabbits, and they do so every month. After 2 months, the baby rabbits become mature and can also produce 4 baby rabbits each month. How many rabbits will the farmer have after 1 year?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2839335489eefb141c56f61a5e107bfc7c1317a7605d9dd921493f99916de259 | A farmer wants to build a rectangular paddock with the largest possible area. a) Let the width of the paddock be \( x \) meters. What is the length of the paddock in terms of \( x \)? b) Show that the area of the paddock, \( A \) square meters, is given by \( A = 100x - x^2 \). d) Find the maximum area of the paddock a... | [
"2500 square meters, dimensions 50 meters by 50 meters."
] | [
"A farmer wants to build a rectangular paddock with the largest possible area.\na) Let the width of the paddock be \\( x \\) meters. What is the length of the paddock in terms of \\( x \\)?\nb) Show that the area of the paddock, \\( A \\) square meters, is given by \\( A = 100x - x^2 \\).\nd) Find the maximum area ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 26
} | 26 | 0 | 0 |
omr-ws-83b971c337d04018b44a572d9820c4fbec55653764dfe0c859d8c036a18f04b8 | A farmer wants to fence an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can this be done to minimize the cost of the fence? | [
"\\( x = 1000 \\) feet, \\( y = 1500 \\) feet"
] | [
"A farmer wants to fence an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can this be done to minimize the cost of the fence?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 10
} | 27 | 0 | 0 |
omr-ws-362f6a25e92c61dc87cd6170b49ab01272e5d8ba7a6d6be89d4ec92bbb397688 | A farmer wants to fence in a pigpen along a stone wall which will serve as part of a boundary. The rest of the pen is circular with a radius of 2 yards and with the center of the circle 1 yard away from the wall and within the pen. Find the amount of fence required for the circular part of the boundary. | [
"\\(\\frac{8\\pi}{3}\\)"
] | [
"A farmer wants to fence in a pigpen along a stone wall which will serve as part of a boundary. The rest of the pen is circular with a radius of 2 yards and with the center of the circle 1 yard away from the wall and within the pen. Find the amount of fence required for the circular part of the boundary."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-38d5831d5961491f22c11dd9b3204813ec76a75f98d45696b7ad5b6e0d2efebe | A farmer wants to fence in a rectangular garden plot on a roadside. The fencing on the side toward the road costs \$5/ft, while that on the other 3 sides costs \$3/ft. If the area of the plot is to be 1200 sq. ft., how much of each type of fencing should the farmer use to minimize the cost of materials? | [
"\\( w = 30 \\) ft, \\( l = 40 \\) ft"
] | [
"A farmer wants to fence in a rectangular garden plot on a roadside. The fencing on the side toward the road costs \\$5/ft, while that on the other 3 sides costs \\$3/ft. If the area of the plot is to be 1200 sq. ft., how much of each type of fencing should the farmer use to minimize the cost of materials?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 10
} | 13 | 0 | 0 |
omr-ws-bf905c56604351f4355946d754d3992a3e81dd1d2495c2c5b3126aab8abd0bd8 | A farmer wants to transport 2500 apples to a market 800 miles away using a truck that can carry a maximum of 1000 apples. For each mile traveled, one apple falls out of the truck. The farmer can make as many trips as needed and can store apples anywhere along the route. What is the maximum number of apples the farmer c... | [
"866"
] | [
"A farmer wants to transport 2500 apples to a market 800 miles away using a truck that can carry a maximum of 1000 apples. For each mile traveled, one apple falls out of the truck. The farmer can make as many trips as needed and can store apples anywhere along the route. What is the maximum number of apples the far... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-a23f83daf4814f598e08ab3c961da6cdc500c99b070dff93221f9738c239fc4c | A farmer wishes to build a rectangular paddock using 200 meters of fencing, he wishes to make the paddock as large as possible (that is, have the largest possible area). a) Let the width of his paddock be \( x \) meters. What is the length of the paddock in terms of \( x \)? b) Show that the area of the paddock \( A \)... | [
"Maximum area = 2500 m², dimensions = 50m x 50m; Smallest area when \\( 20 \\leq x \\leq 70 \\) = 1600 m², dimensions = 20m x 80m."
] | [
"A farmer wishes to build a rectangular paddock using 200 meters of fencing, he wishes to make the paddock as large as possible (that is, have the largest possible area).\na) Let the width of his paddock be \\( x \\) meters. What is the length of the paddock in terms of \\( x \\)?\nb) Show that the area of the padd... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-69a642ad343f8b5dc0cad22f37c4844556ce9b994358807631ed2959599af3e8 | A farmer wishes to build a rectangular sheep pen using one side of an existing 20m x 20m shed as the width of the pen and a 160m fence for the other three sides. Find the dimensions and the area of the sheep pen. | [
"1400"
] | [
"A farmer wishes to build a rectangular sheep pen using one side of an existing 20m x 20m shed as the width of the pen and a 160m fence for the other three sides. Find the dimensions and the area of the sheep pen."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-6bd3f0b4d0efa06536ef09bef1c261d7208f2fb112c3219b5fbd799259a8d90a | A farmer with 2000 meters of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed? | [
"500000"
] | [
"A farmer with 2000 meters of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d17d5f76eaeba7f1ba635bddf4f5b350d2dd01c9d1281b9a8f72ed5c3391c01e | A farmer's house is in the shape of a convex pentagon with perimeter \( P \) and area \( A \). The yard around the house includes all points that are at a distance at most 20 meters from the house. Find the area of the farmer's lot (yard plus house). | [
"A + 20P + 400\\pi"
] | [
"A farmer's house is in the shape of a convex pentagon with perimeter \\( P \\) and area \\( A \\). The yard around the house includes all points that are at a distance at most 20 meters from the house. Find the area of the farmer's lot (yard plus house)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-30bc0001a51a8932b2c218d319ce5a04fc6c59c78888bf2f499ed6ab1a4a28dd | A farmer's market sells baskets of oranges and baskets of apples for fixed whole number prices. Abigail paid 23 dollars for five baskets of fruit. Benedict paid 24 dollars for four baskets of fruit. If Charles paid 20 dollars, how many baskets of fruit did he buy? | [
"4"
] | [
"A farmer's market sells baskets of oranges and baskets of apples for fixed whole number prices. Abigail paid 23 dollars for five baskets of fruit. Benedict paid 24 dollars for four baskets of fruit. If Charles paid 20 dollars, how many baskets of fruit did he buy?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e403881255db0d817c5929447a921e60002b11985f6d52649b866e270a66491c | A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections. The farmer will plant one crop in each section: wheat, corn, soybeans, or potatoes. The farmer does not want to grow corn and wheat in two sections that share a border and does not want to grow soybeans and potatoes in two s... | [
"84"
] | [
"A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections. The farmer will plant one crop in each section: wheat, corn, soybeans, or potatoes. The farmer does not want to grow corn and wheat in two sections that share a border and does not want to grow soybeans and potatoes in ... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 23,
"tir": 23
} | 46 | 0 | 0 |
omr-ws-a5f9c297d7c5f6b22b7b877aa13e62643f6b5121fd4d2a7a6c0d73909bd8f2bc | A farmer's rectangular field is partitioned into a 2 by 2 grid of 4 rectangular sections. The farmer will plant one of four crops (corn, wheat, soybeans, or potatoes) in each section. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans a... | [
"84"
] | [
"A farmer's rectangular field is partitioned into a 2 by 2 grid of 4 rectangular sections. The farmer will plant one of four crops (corn, wheat, soybeans, or potatoes) in each section. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybe... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 24,
"tir": 24
} | 48 | 0 | 0 |
omr-ws-91e51ddc2e5d773e8d4056d8f540541b383b3e87a57a5cae9e0c6fc1d32e0007 | A fast train covers the distance between two towns 4 hours faster than a goods train and 1 hour faster than a passenger train. It is known that the speed of the goods train is $\frac{5}{8}$ of the speed of the passenger train and 50 km/h less than the speed of the fast train. Find the speeds of the goods and fast train... | [
"50 km/h and 100 km/h"
] | [
"A fast train covers the distance between two towns 4 hours faster than a goods train and 1 hour faster than a passenger train. It is known that the speed of the goods train is $\\frac{5}{8}$ of the speed of the passenger train and 50 km/h less than the speed of the fast train. Find the speeds of the goods and fast... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d6a9232374ebf06381788ee7e63e2abe3da8bcc1d37571763a60296f23be05d | A fat muffin likes to eat jelly beans. He has an infinite amount of them. Every day he rolls 5 pairs of normal 6-sided dice. Each jelly bean weighs 5 pounds, and the muffin never loses weight. He starts weighing 5 pounds. After 10 days, what is the expected value of his weight in pounds? | [
"5 pounds"
] | [
"A fat muffin likes to eat jelly beans. He has an infinite amount of them. Every day he rolls 5 pairs of normal 6-sided dice. Each jelly bean weighs 5 pounds, and the muffin never loses weight. He starts weighing 5 pounds. After 10 days, what is the expected value of his weight in pounds?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-90074d977c7ee9a15a684207a8ac920a93da8cc1de9297f9e4ec64399f785690 | A father distributes 83 diamonds to his 5 sons according to the following rules: - No diamond is to be cut. - No two sons are to receive the same number of diamonds. - None of the differences between the numbers of diamonds received by any two sons is to be the same. - Any 3 sons receive more than half of the total dia... | [
"11"
] | [
"A father distributes 83 diamonds to his 5 sons according to the following rules:\n- No diamond is to be cut.\n- No two sons are to receive the same number of diamonds.\n- None of the differences between the numbers of diamonds received by any two sons is to be the same.\n- Any 3 sons receive more than half of the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-815541c0ac4cebb8c31ba1d52af83e29d3ade0e1c467c2bf8f37b8dbd27fa831 | A father divides 20 candies among his three children, Tom, Marry, and Max, such that none of them receives all 20 candies. Each minute, each child gives 1 candy to each of the other two children. After some time, the number of candies each child has is \(a\), \(b\), and \(c\) respectively. Find the possible values of \... | [
"(a, b, c) \\text{ where } a + b + c = 20 \\text{ and } 0 \\leq a, b, c < 20"
] | [
"A father divides 20 candies among his three children, Tom, Marry, and Max, such that none of them receives all 20 candies. Each minute, each child gives 1 candy to each of the other two children. After some time, the number of candies each child has is \\(a\\), \\(b\\), and \\(c\\) respectively. Find the possible ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-f5471fbc7b7a70ebb281b3581f63f4717dbf39350c9bbf27db1cefa98735ab0d | A father has three daughters whose ages are integers and the product of their ages is 36. The sum of their ages does not provide enough information to determine their ages uniquely. However, knowing that the oldest daughter is named Annie (implying there is a unique oldest daughter) allows the determination of their ag... | [
"2, 2, 9"
] | [
"A father has three daughters whose ages are integers and the product of their ages is 36. The sum of their ages does not provide enough information to determine their ages uniquely. However, knowing that the oldest daughter is named Annie (implying there is a unique oldest daughter) allows the determination of the... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-0a246b883c0396d67f5e827bf67cf450268f5f7e9633e964bf269e2e1c800d44 | A father in his will left all his money to his children in the following manner: - 1000 dollars to the first born and \(\frac{1}{10}\) of what then remains, - 2000 dollars to the second born and \(\frac{1}{10}\) of what then remains, - 3000 dollars to the third born and \(\frac{1}{10}\) of what then remains, and so on.... | [
"9"
] | [
"A father in his will left all his money to his children in the following manner:\n- 1000 dollars to the first born and \\(\\frac{1}{10}\\) of what then remains,\n- 2000 dollars to the second born and \\(\\frac{1}{10}\\) of what then remains,\n- 3000 dollars to the third born and \\(\\frac{1}{10}\\) of what then re... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-624fa036cd214a2d6eda18c289b2034cc5696a27e8ec236ff3c907066ec27ef5 | A father is making financial plans to send his small son to college. Each January 1 he will deposit a sum of money into an account which pays 10% interest compounded annually each December 31. The amount deposited will be increased 10% each year. If \( P \) is the amount deposited the first year then: (a) Give the valu... | [
"9100"
] | [
"A father is making financial plans to send his small son to college. Each January 1 he will deposit a sum of money into an account which pays 10% interest compounded annually each December 31. The amount deposited will be increased 10% each year. If \\( P \\) is the amount deposited the first year then:\n(a) Give ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f9a100b3b4a0304b9667638b3e22582dc8c5ab249bb84528fb87ba194bd8919 | A father left his money to his children in the following manner: 1000$ to the first born and 1/10 of what remains, 2000$ to the second born and 1/10 of what remains, 3000$ to the third born and 1/10 of what remains, and so on. Each child received the same amount. How many children were there? | [
"9"
] | [
"A father left his money to his children in the following manner: 1000$ to the first born and 1/10 of what remains, 2000$ to the second born and 1/10 of what remains, 3000$ to the third born and 1/10 of what remains, and so on. Each child received the same amount. How many children were there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb4c899d987ce9ae963bfc376dafc92025b734821303c13a7925c2be45184964 | A faucet fills a tank such that each new hour it fills with twice the volume of the previous hour, and it takes 10 hours to fill the tank completely. If two such faucets are turned on simultaneously in an empty tank, in how many hours will the tank be filled? | [
"9"
] | [
"A faucet fills a tank such that each new hour it fills with twice the volume of the previous hour, and it takes 10 hours to fill the tank completely. If two such faucets are turned on simultaneously in an empty tank, in how many hours will the tank be filled?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-a7bbc5433fafbc24186b8eca7dc006c42e320bfed6f389f024bfb8eaf5cdb34b | A faulty calculator displayed \(\Diamond 38 \Diamond 1625\) as an output of a calculation. We know that two of the digits of this number are missing and these are replaced with the symbol \(\Diamond\). Furthermore, we know that 9 and 11 divide the computed output. What are the missing digits and the complete output of ... | [
"73841625"
] | [
"A faulty calculator displayed \\(\\Diamond 38 \\Diamond 1625\\) as an output of a calculation. We know that two of the digits of this number are missing and these are replaced with the symbol \\(\\Diamond\\). Furthermore, we know that 9 and 11 divide the computed output. What are the missing digits and the complet... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-322d055bf85b69929b7dfd87652a67270fbd21349af4ed94f035bfc82a2c6509 | A faulty car odometer proceeds from 3 to 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled? No other problems identified. | [
"1462"
] | [
"A faulty car odometer proceeds from 3 to 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?\n\nNo other problems identified."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-d5200ea0026b0011b0dd39dada5f8121bd34610d735e2c3408d9aceea6260dd7 | A faulty car odometer proceeds from digit $3$ to digit $5$, always skipping the digit $4$, regardless of position. If the odometer now reads $002005$, how many miles has the car actually traveled? | [
"1462"
] | [
"A faulty car odometer proceeds from digit $3$ to digit $5$, always skipping the digit $4$, regardless of position. If the odometer now reads $002005$, how many miles has the car actually traveled?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-e70fa597838b9b176746ce05ebd473cb166b2a74b84dcf63344dd9b125912f5b | A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. For example, after traveling one mile the odometer changed from 000039 to 000050. If the odometer now reads 002005, how many miles has the car actually traveled? | [
"1462"
] | [
"A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. For example, after traveling one mile the odometer changed from 000039 to 000050. If the odometer now reads 002005, how many miles has the car actually traveled?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-917bde92d579acb431892827eb73e08e4e6298049f783940f1b8043e9c0ead76 | A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled? | [
"1462"
] | [
"A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 4
} | 35 | 0 | 0 |
omr-ws-1a1b71773141806bae440e4d1817337cbd5e940f0e7ed7c782ca16a9272e1d29 | A faulty car odometer skips the digit 4 and proceeds from digit 3 to digit 5. If the odometer now reads 002005, how many miles has the car actually traveled? | [
"1462"
] | [
"A faulty car odometer skips the digit 4 and proceeds from digit 3 to digit 5. If the odometer now reads 002005, how many miles has the car actually traveled?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-620c2c5cebf1773b6696f6192771e52150aeee5175eacb95a8406f17e9cd40f1 | A fence 8 feet tall runs parallel to a tall building at a distance of 4 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? | [
"\\( 4\\left(1 + 2^{2/3}\\right)^{3/2} \\)"
] | [
"A fence 8 feet tall runs parallel to a tall building at a distance of 4 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 19
} | 36 | 0 | 0 |
omr-ws-833ab19017de1c8f5f4fd7a696eef32a290ba8d598a330444b17147d085581ea | A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? (Give your answer correct to two decimal places.) | [
"16.65"
] | [
"A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? (Give your answer correct to two decimal places.)"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b6f97a95889853ac64f3d21d700032d7f6fdd01f0efc3abf180910c4ecfd6869 | A fence is to be built to enclose a rectangular area. The fence along three sides costs $5 per foot, and the fourth side costs $15 per foot. If $3,000 is available for the fencing, find the dimensions of the rectangle that will enclose the most area. | [
"\\( x = 75 \\) feet, \\( y = 150 \\) feet"
] | [
"A fence is to be built to enclose a rectangular area. The fence along three sides costs $5 per foot, and the fourth side costs $15 per foot. If $3,000 is available for the fencing, find the dimensions of the rectangle that will enclose the most area."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 31
} | 44 | 0 | 0 |
omr-ws-30f83e425fc7cdcdb4779291517b2895f27d5a01d0abf52312c1d2de27dd2c7d | A fence is to be built to enclose a rectangular area. The fence along three sides costs $5 per foot, and the fourth side costs $15 per foot. If the area is 5000 square feet, find the dimensions of the rectangle that will allow the most economical fence to be built. | [
"\\( x = 50 \\) feet, \\( y = 100 \\) feet"
] | [
"A fence is to be built to enclose a rectangular area. The fence along three sides costs $5 per foot, and the fourth side costs $15 per foot. If the area is 5000 square feet, find the dimensions of the rectangle that will allow the most economical fence to be built."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 29
} | 39 | 0 | 0 |
omr-ws-9ce8a2f12a70e6cf56b47d104bc303bbcfde708ac39832a3d0bfa15550a427e5 | A fence is to be built to enclose a rectangular area. The fence along three sides costs 5 dollars per foot, and the fourth side costs 15 dollars per foot. If 3,000 dollars is available for the fencing, find the dimensions of the rectangle that will enclose the most area. | [
"\\( x = 75 \\) feet, \\( y = 150 \\) feet"
] | [
"A fence is to be built to enclose a rectangular area. The fence along three sides costs 5 dollars per foot, and the fourth side costs 15 dollars per foot. If 3,000 dollars is available for the fencing, find the dimensions of the rectangle that will enclose the most area."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 24
} | 34 | 0 | 0 |
omr-ws-bae72f94162457600ab0f3bbc12350b60a27293509b3a6b8ea778498cb12b9c3 | A fence is to be built to enclose a rectangular area. The fence along three sides costs 5 dollars per foot, and the fourth side costs 15 dollars per foot. If the area is 5000 square feet, find the dimensions of the rectangle that will allow the most economical fence to be built. | [
"\\( x = 50, y = 100 \\)"
] | [
"A fence is to be built to enclose a rectangular area. The fence along three sides costs 5 dollars per foot, and the fourth side costs 15 dollars per foot. If the area is 5000 square feet, find the dimensions of the rectangle that will allow the most economical fence to be built."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 25
} | 36 | 0 | 0 |
omr-ws-1ab37e91db16aee4872cf246d2a619555582ac27c552de619bda677ef9d61090 | A fenced, rectangular field measures $24$ meters by $52$ meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the fie... | [
"702"
] | [
"A fenced, rectangular field measures $24$ meters by $52$ meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of th... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 15,
"tir": 14
} | 74 | 0 | 0 |
omr-ws-43c6605092ab17ac71a154dbfa5940963bb8e5df9f219fdb859d4be1a2a77e76 | A ferris wheel with a diameter of 100 feet takes five minutes to make one complete revolution. How many times would a person reach the maximum height in one hour? | [
"12"
] | [
"A ferris wheel with a diameter of 100 feet takes five minutes to make one complete revolution. How many times would a person reach the maximum height in one hour?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-371d47dd88c83653faccc49c923e4d0b4710286d8f99bf36904860eba7e50063 | A ferris wheel with a diameter of 100 feet takes five minutes to make one complete revolution. If the platform to enter the ferris wheel is 5 feet off the ground, find the maximum and minimum distance from the ground that a passenger could reach while riding the ferris wheel. | [
"Maximum distance = 105 feet, Minimum distance = 5 feet"
] | [
"A ferris wheel with a diameter of 100 feet takes five minutes to make one complete revolution. If the platform to enter the ferris wheel is 5 feet off the ground, find the maximum and minimum distance from the ground that a passenger could reach while riding the ferris wheel."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-c22b6f8a76ccf1058c8d4c571241d5a22d196a26253513042cdbbec9b187de81 | A ferris wheel with a radius of 32 meters makes 2 revolutions every minute. If the ferris wheel rotates for 3 minutes, what is the distance the rider travels? | [
"\\(384\\pi\\)"
] | [
"A ferris wheel with a radius of 32 meters makes 2 revolutions every minute. If the ferris wheel rotates for 3 minutes, what is the distance the rider travels?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-c35b2ede9448a0324ae07572afa998bd210bd2daaa68ad7d52f26b9921a4f6f2 | A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. On the 10 AM trip, there were 100 tourists, and on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day? | [
"585"
] | [
"A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. On the 10 AM trip, there were 100 tourists, and on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-299b7ec306bccc0f20f6b4fa5aac93ab36a8cbc51953f116fa52f921b0bfc1d7 | A ferry boat travels at 10 mph and has to travel repeatedly across a river, arriving at a port 4 miles downstream from the starting port. If this trip takes 26 minutes, how many minutes will be saved by moving one of the ports directly opposite the river from the other? | [
"16"
] | [
"A ferry boat travels at 10 mph and has to travel repeatedly across a river, arriving at a port 4 miles downstream from the starting port. If this trip takes 26 minutes, how many minutes will be saved by moving one of the ports directly opposite the river from the other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 9
} | 22 | 0 | 0 |
omr-ws-b349c5c574c1921cd0413e786c774eefb51efc19fa863900ca01416836071ff7 | A fertilizer makes plants grow such that on the first day, the plant's height increases by one-half, on the second day by one-third of its height from the previous day, on the third day by one-fourth of its height from the previous day, and so on. How many days does it take for the plant to grow to 100 times its origin... | [
"198"
] | [
"A fertilizer makes plants grow such that on the first day, the plant's height increases by one-half, on the second day by one-third of its height from the previous day, on the third day by one-fourth of its height from the previous day, and so on. How many days does it take for the plant to grow to 100 times its o... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-59b67bbf9e585bc3d9150d93c29d79a962a7a52616b681d98ba3152280d75837 | A few (at least 5) integers are placed on a circle such that each integer is divisible by the sum of its two neighbors. If the sum of all the numbers is positive, what is the minimal possible value of this sum? | [
"2"
] | [
"A few (at least 5) integers are placed on a circle such that each integer is divisible by the sum of its two neighbors. If the sum of all the numbers is positive, what is the minimal possible value of this sum?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 23,
"genselect": 8
} | 31 | 0 | 0 |
omr-ws-8977c6d1954008eb713cd1d15c3334c0c825dfad1264e6bbff8bc73c87b2b8fd | A fidget spinner is modeled with a series of arcs on circles of radii $1$. What is the area swept out by the fidget spinner as it is turned $60^\circ$? | [
"\\dfrac{\\pi}{2}"
] | [
"A fidget spinner is modeled with a series of arcs on circles of radii $1$. What is the area swept out by the fidget spinner as it is turned $60^\\circ$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-f042735668d74ce3b4933afcb90e2fcc6a3e536eed4fd5397e96a5fa5355433a | A field has a shape of a 41x41 checkboard. A tank is concealed in one of the cells. By one shot, a fighter airplane can fire one of the cells. If a shot hits the tank, the tank moves to a neighboring cell (cells are neighbors if they share a side). Otherwise, the tank stays in its cell. The pilot has no information abo... | [
"2521"
] | [
"A field has a shape of a 41x41 checkboard. A tank is concealed in one of the cells. By one shot, a fighter airplane can fire one of the cells. If a shot hits the tank, the tank moves to a neighboring cell (cells are neighbors if they share a side). Otherwise, the tank stays in its cell. The pilot has no informatio... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-4e8012c8c15bfb97e092fe3ea41b807762dd1799ec2cac7bd893aac082b42ed1 | A field is made of $2017 \times 2017$ unit squares. Luffy has $k$ gold detectors, which he places on some of the unit squares. Sanji then chooses a $1500 \times 1500$ area and buries a gold coin on each unit square in this area. A gold detector beeps if and only if there is a gold coin underneath it. Determine the mini... | [
"1034"
] | [
"A field is made of $2017 \\times 2017$ unit squares. Luffy has $k$ gold detectors, which he places on some of the unit squares. Sanji then chooses a $1500 \\times 1500$ area and buries a gold coin on each unit square in this area. A gold detector beeps if and only if there is a gold coin underneath it. Determine t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-9da4c579fc0402fe3e30a095fd7035bba479f5785ed83657205c985144099ca3 | A field trip group has $3$ adults, $3$ boys, and $4$ girls. How many (possibly empty) subsets of the $10$ people can pose for a photo if at least one adult is left out to take the photo, one adult can't be alone with one child, and each child refuses to pose with only children of the opposite gender? | [
"448"
] | [
"A field trip group has $3$ adults, $3$ boys, and $4$ girls. How many (possibly empty) subsets of the $10$ people can pose for a photo if at least one adult is left out to take the photo, one adult can't be alone with one child, and each child refuses to pose with only children of the opposite gender?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-ac393e6d8e730a5c084bedaaba0f2027968af1cc931ad18a5c42e0e2c457e054 | A fifth-degree polynomial has roots that form an arithmetic progression. Given the polynomial \( x^5 - 5x^4 - 35x^3 + \ldots \), find the roots. | [
"\\( 1 - 3\\sqrt{2}, 1 - \\frac{3}{\\sqrt{2}}, 1, 1 + \\frac{3}{\\sqrt{2}}, 1 + 3\\sqrt{2} \\)"
] | [
"A fifth-degree polynomial has roots that form an arithmetic progression. Given the polynomial \\( x^5 - 5x^4 - 35x^3 + \\ldots \\), find the roots."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e4a9326bdb8009771890ea2dffc01441a65f3e58b78514bbe866f5b9e187f9b9 | A figure consisting of five equal-sized squares is placed in a rectangle of size $7 \times 8$ units. Find the side length of the squares. | [
"\\(\\sqrt{5}\\)"
] | [
"A figure consisting of five equal-sized squares is placed in a rectangle of size $7 \\times 8$ units. Find the side length of the squares."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5fdbb17223561068fa406cf5bb32e1182241e5f282a2fa57e07b1014ece3b2c6 | A figure consists of two circles, each of which passes through the center of the other. The smallest region formed has an area of one square unit. Find the radii of the circles. | [
"\\sqrt{\\dfrac{6}{4\\pi - 3\\sqrt{3}}}"
] | [
"A figure consists of two circles, each of which passes through the center of the other. The smallest region formed has an area of one square unit. Find the radii of the circles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-689091a1b50588ddc5d961f6e1ac320ced0ab2095954f590b4a9fe4417c03b86 | A figure is made out of unit squares joined along their sides. It has the property that if the squares of an \( m \times n \) rectangle are filled with real numbers with a positive sum, then the figure can be placed over the rectangle (possibly after being rotated, but with each square of the figure coinciding with a s... | [
"12"
] | [
"A figure is made out of unit squares joined along their sides. It has the property that if the squares of an \\( m \\times n \\) rectangle are filled with real numbers with a positive sum, then the figure can be placed over the rectangle (possibly after being rotated, but with each square of the figure coinciding ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-982eaa6d2243bf2ad829fa6288de223ed43188c651adec423cf52549057f1146 | A figure is made up of small squares enclosed in a rectangle. The horizontal side of the rectangle measures 73 units and the vertical side measures 94 units. Calculate the side length of each small square. | [
"13"
] | [
"A figure is made up of small squares enclosed in a rectangle. The horizontal side of the rectangle measures 73 units and the vertical side measures 94 units. Calculate the side length of each small square."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-32241e6130c4ea3e8256fee165b22894ac5b16de9bbf330c04f44d2cbe2eb0dd | A figure of area $1$ is cut out from a sheet of paper and divided into $10$ parts, each of which is colored in one of $10$ colors. The figure is then turned over and divided into $10$ parts (not necessarily the same way). Color these parts in the $10$ colors such that the total area of the portions of the figure both o... | [
"1"
] | [
"A figure of area $1$ is cut out from a sheet of paper and divided into $10$ parts, each of which is colored in one of $10$ colors. The figure is then turned over and divided into $10$ parts (not necessarily the same way). Color these parts in the $10$ colors such that the total area of the portions of the figure b... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e8b0f6c5e49700282ee99e6115581f9bd06af1ddfe00d8ddc248c15c364feb1 | A figure with area $1$ is divided into $10$ parts and each part is colored with one of $10$ different colors. The figure is then turned over and divided into $10$ parts again (in a different way). Let $S$ be the maximum possible sum of the areas of all parts of the figure colored with the same color on both sides. Find... | [
"\\dfrac{1}{10}"
] | [
"A figure with area $1$ is divided into $10$ parts and each part is colored with one of $10$ different colors. The figure is then turned over and divided into $10$ parts again (in a different way). Let $S$ be the maximum possible sum of the areas of all parts of the figure colored with the same color on both sides.... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-53a2d12100ab4f4f6c82e4f18a3bcd7f734057b0a05712dc1ef4ebd4459dbbb7 | A financial advisor suggests to a client that she invest part of her portfolio in municipal bonds earning 6.5% annual simple interest, and the remainder in corporate bonds earning 8.5% interest. How much should she be putting in the corporate bonds if she has $2,190 in interest each year? | [
"12000"
] | [
"A financial advisor suggests to a client that she invest part of her portfolio in municipal bonds earning 6.5% annual simple interest, and the remainder in corporate bonds earning 8.5% interest. How much should she be putting in the corporate bonds if she has $2,190 in interest each year?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-89aaa40f02e94cb05cdc182bc5c929ef49b2effe58c3d18bd54c1780ef0dac8d | A financial correspondent writes about insurance, pensions, or investments each day according to the following rules: - She never writes about insurance on two consecutive days; after writing about insurance, she writes about pensions the next day with probability $\frac{3}{5}$. - After writing about pensions, she writ... | [
"\\(\\frac{1}{3}\\)"
] | [
"A financial correspondent writes about insurance, pensions, or investments each day according to the following rules:\n- She never writes about insurance on two consecutive days; after writing about insurance, she writes about pensions the next day with probability $\\frac{3}{5}$.\n- After writing about pensions, ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-31ad255ce0d4742af159025739e7c5041f912573450659d23307f56fef4af8ca | A financial market experiences growth (G) 20% of the time, stagnation (S) 70% of the time, and recession (R) 10% of the time. A consulting firm forecasts the market with the following accuracy: it correctly forecasts growth 80% of the time, stagnation 80% of the time, and recession 80% of the time. If the firm forecast... | [
"a) 0.667, b) 0.292, c) 0.042"
] | [
"A financial market experiences growth (G) 20% of the time, stagnation (S) 70% of the time, and recession (R) 10% of the time. A consulting firm forecasts the market with the following accuracy: it correctly forecasts growth 80% of the time, stagnation 80% of the time, and recession 80% of the time. If the firm for... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-f67108c6eb8cf99adb3610f8bdf6153bf90650f6e35ce523e1d359318f68267b | A finite amount of cards is placed in two towers, with more cards in the left tower than in the right. Each card has one or more distinct names written on it, and different cards can share some names. For each name, define an action as moving every card that has that name written on it to the opposite tower. After perf... | [
"3"
] | [
"A finite amount of cards is placed in two towers, with more cards in the left tower than in the right. Each card has one or more distinct names written on it, and different cards can share some names. For each name, define an action as moving every card that has that name written on it to the opposite tower. After... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-1e49bb3659381558459ec040f83099388e077421b85a7816744f53f487fbbdf8 | A finite collection of digits $0$ and $1$ is written around a circle. An arc of length $L \ge 0$ consists of $L$ consecutive digits around the circle. For each arc $w$, let $Z(w)$ and $N(w)$ denote the number of $0$'s and $1$'s in $w$, respectively. Assume that $|Z(w) - Z(w')| \le 1$ for any two arcs $w, w'$ of the sam... | [
"25"
] | [
"A finite collection of digits $0$ and $1$ is written around a circle. An arc of length $L \\ge 0$ consists of $L$ consecutive digits around the circle. For each arc $w$, let $Z(w)$ and $N(w)$ denote the number of $0$'s and $1$'s in $w$, respectively. Assume that $|Z(w) - Z(w')| \\le 1$ for any two arcs $w, w'$ of ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-97ee7c70a0c328d79d060d5bd203d775b6a3034a1ee05c35efcbee3bb80d347f | A finite collection of squares has a total area of $4$. Determine the maximum possible side length $s$ of a square that can always be completely covered by these squares, regardless of their individual sizes and how they are arranged. | [
"2"
] | [
"A finite collection of squares has a total area of $4$. Determine the maximum possible side length $s$ of a square that can always be completely covered by these squares, regardless of their individual sizes and how they are arranged."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-30fffb20eb3d0c089ac49de5e0f73ad6827c2d4ae8f670f8b0d1f344270f1429 | A finite collection of squares has a total area of 4. Determine the maximum side length of a square that can always be completely covered by these squares, regardless of their individual sizes. | [
"\\sqrt{2}"
] | [
"A finite collection of squares has a total area of 4. Determine the maximum side length of a square that can always be completely covered by these squares, regardless of their individual sizes."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-6626860118d3cf94b9046c7ebd79bd4f139ce27a3672ebb6b165c21c434dcc33 | A finite family of disjoint rectangles with a total area of 4 is given in the coordinate plane such that the projection of their union onto the horizontal axis is an interval. Determine the minimum possible area of the largest triangle that can be formed by any three points in the union of these rectangles. | [
"2"
] | [
"A finite family of disjoint rectangles with a total area of 4 is given in the coordinate plane such that the projection of their union onto the horizontal axis is an interval. Determine the minimum possible area of the largest triangle that can be formed by any three points in the union of these rectangles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e17630f2291817d0a3ed4011ffa1146394a7ec0fe1c22e4893ed9047118330f6 | A finite family of red and blue lines in the plane is given, with no two lines being parallel. Assume that for every intersection point of two lines of the same color, a line of the other color passes through the intersection point. Let \( P \) be the point where all lines intersect. If the coordinates of \( P \) are \... | [
"0"
] | [
"A finite family of red and blue lines in the plane is given, with no two lines being parallel. Assume that for every intersection point of two lines of the same color, a line of the other color passes through the intersection point. Let \\( P \\) be the point where all lines intersect. If the coordinates of \\( P ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-57c3ba033ecb4f7c4ae9c9c612c015a0039fc18db39ac0905287f55e6bc07561 | A finite graph \( G(V,E) \) on \( n \) points is drawn in the plane. For an edge \( e \) of the graph, let \( \chi(e) \) denote the number of edges that cross over edge \( e \). Let \( S \) be the sum of \( \frac{1}{\chi(e) + 1} \) over all edges \( e \in E \). If \( n = 10 \), find the maximum possible value of \( S \... | [
"24"
] | [
"A finite graph \\( G(V,E) \\) on \\( n \\) points is drawn in the plane. For an edge \\( e \\) of the graph, let \\( \\chi(e) \\) denote the number of edges that cross over edge \\( e \\). Let \\( S \\) be the sum of \\( \\frac{1}{\\chi(e) + 1} \\) over all edges \\( e \\in E \\). If \\( n = 10 \\), find the maxim... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-a14960b725af41309c2cfc524ac73d46c7ac75a880c1b3d8075c64e3fb137ed3 | A finite lattice \( L \) has \( n \) join-irreducible elements. What is the maximum number \( f(n) \) of meet-irreducible elements that \( L \) can have? | [
"2^n - 1"
] | [
"A finite lattice \\( L \\) has \\( n \\) join-irreducible elements. What is the maximum number \\( f(n) \\) of meet-irreducible elements that \\( L \\) can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-87494040645666534744c814fc78ce82078fa360b5ee9b296b421bf2663efb30 | A finite number of coins are placed on an infinite row of squares. A sequence of moves is performed as follows: at each stage, a square containing more than one coin is chosen. Two coins are taken from this square; one is placed on the square immediately to the left, and the other is placed on the square immediately to... | [
"55"
] | [
"A finite number of coins are placed on an infinite row of squares. A sequence of moves is performed as follows: at each stage, a square containing more than one coin is chosen. Two coins are taken from this square; one is placed on the square immediately to the left, and the other is placed on the square immediate... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-23669bcada26e51cf53c47bef67e6cd6580c62ec75a91f92fc4e5cc7ab0466d3 | A finite number of points are plotted such that the distances between any two points are distinct. Each point is connected to the point closest to it. Find the maximum number of segments that can start from a single point. | [
"5"
] | [
"A finite number of points are plotted such that the distances between any two points are distinct. Each point is connected to the point closest to it. Find the maximum number of segments that can start from a single point."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-67b2809b81f6e4f0abdb7c814f29978224a90429551eae620434de7345711abd | A finite number of points in the plane are chosen, and each is colored either red or blue. Suppose that for any line $\ell$ in the plane, the difference between the number of red points on $\ell$ and the number of blue points on $\ell$ is at most $1$. Let $N$ be the total number of points. Find the maximum possible val... | [
"4"
] | [
"A finite number of points in the plane are chosen, and each is colored either red or blue. Suppose that for any line $\\ell$ in the plane, the difference between the number of red points on $\\ell$ and the number of blue points on $\\ell$ is at most $1$. Let $N$ be the total number of points. Find the maximum poss... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-6d9915eff4922adde51b4a765d50c69234c89575010777fd78dd5d1dde8c51b2 | A finite number of points is marked on the plane such that no three points are collinear. A circle is circumscribed around each triangle formed by these points. If there are exactly 10 marked points, what is the maximum number of these circumscribed circle centers that can also be among the marked points? | [
"no_valid_answer_found"
] | [
"A finite number of points is marked on the plane such that no three points are collinear. A circle is circumscribed around each triangle formed by these points. If there are exactly 10 marked points, what is the maximum number of these circumscribed circle centers that can also be among the marked points?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c673d36488f53e73f813d25c36de486752d90226f53af397e86e274b3d397a31 | A finite number of real numbers are written along a circle. If some four successive numbers \(a, b, c, d\) satisfy \((a-d)(b-c) > 0\), we can exchange \(b\) and \(c\). Let \(S\) be the sum of all such \((a-d)(b-c)\) before any exchanges. After performing a sequence of exchanges, let \(T\) be the sum of all such \((a-d)... | [
"0"
] | [
"A finite number of real numbers are written along a circle. If some four successive numbers \\(a, b, c, d\\) satisfy \\((a-d)(b-c) > 0\\), we can exchange \\(b\\) and \\(c\\). Let \\(S\\) be the sum of all such \\((a-d)(b-c)\\) before any exchanges. After performing a sequence of exchanges, let \\(T\\) be the sum ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-de212756a7078c13232485521d41d36bec2e7eb87d9fd54a9edf64aee68e96d5 | A finite number of rectangular tablecloths are placed on a square desk such that they do not protrude from the desk. Each side of the tablecloth is parallel to the side of the desk. Let $A$ be the area of the overlapping parts of all tablecloths, and $B$ be the area of the overlapping parts of any two tablecloths. If $... | [
"1"
] | [
"A finite number of rectangular tablecloths are placed on a square desk such that they do not protrude from the desk. Each side of the tablecloth is parallel to the side of the desk. Let $A$ be the area of the overlapping parts of all tablecloths, and $B$ be the area of the overlapping parts of any two tablecloths.... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-da4b5d397fdd73602e7abb1ff56ce0b4dd7d45c2bb0d15cb06629ecb918d0d6c | A finite number of red and blue lines are drawn on the plane such that no two lines are parallel to each other. Additionally, through any point where two lines of the same color meet, there also passes a line of the other color. If there are 10 red lines and 15 blue lines, find the number of distinct points where at le... | [
"150"
] | [
"A finite number of red and blue lines are drawn on the plane such that no two lines are parallel to each other. Additionally, through any point where two lines of the same color meet, there also passes a line of the other color. If there are 10 red lines and 15 blue lines, find the number of distinct points where ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-09b699c29ec80149ab3ce77b213ddc6e473081ed89dc2916723202a0e583c8c7 | A finite sequence \(a_0, a_1, \ldots, a_n\) of integers is called a curious sequence if for every \(k = 0, 1, 2, \ldots, n\), the number of times \(k\) appears in the sequence is \(a_k\). Given a curious sequence \(a_0, a_1, \ldots, a_{2011}\), what is the value of the sum \(\sum_{k=0}^{2011} a_k\)? | [
"2012"
] | [
"A finite sequence \\(a_0, a_1, \\ldots, a_n\\) of integers is called a curious sequence if for every \\(k = 0, 1, 2, \\ldots, n\\), the number of times \\(k\\) appears in the sequence is \\(a_k\\). Given a curious sequence \\(a_0, a_1, \\ldots, a_{2011}\\), what is the value of the sum \\(\\sum_{k=0}^{2011} a_k\\)... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c83dc60a4fb8b01e2c20f1b71d7fe5e3da98716afbd4cdf67c646b3949526d79 | A finite sequence of integers \(a_1, a_2, \dots, a_n\) is called regular if there exists a real number \(x\) such that \(\left\lfloor kx \right\rfloor = a_k\) for \(1 \le k \le n\). Given a regular sequence \(a_1, a_2, \dots, a_n\), a term \(a_k\) is called forced if the sequence \(a_1, a_2, \dots, a_{k-1}, b\) is regu... | [
"985"
] | [
"A finite sequence of integers \\(a_1, a_2, \\dots, a_n\\) is called regular if there exists a real number \\(x\\) such that \\(\\left\\lfloor kx \\right\\rfloor = a_k\\) for \\(1 \\le k \\le n\\). Given a regular sequence \\(a_1, a_2, \\dots, a_n\\), a term \\(a_k\\) is called forced if the sequence \\(a_1, a_2, \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-15e503cd954c273b3901abf0d382f46d9659c700fd438b6f39db90e1331d4e50 | A finite sequence of natural numbers \(a_1, a_2, \dots, a_n\) is given. A sub-sequence \(a_{k+1}, a_{k+2}, \dots, a_l\) is called a repetition if there exists a natural number \(p \leq \frac{l-k}{2}\) such that \(a_i = a_{i+p}\) for \(k+1 \leq i \leq l-p\), but \(a_i \neq a_{i+p}\) for \(i = k\) (if \(k > 0\)) and \(i ... | [
"50"
] | [
"A finite sequence of natural numbers \\(a_1, a_2, \\dots, a_n\\) is given. A sub-sequence \\(a_{k+1}, a_{k+2}, \\dots, a_l\\) is called a repetition if there exists a natural number \\(p \\leq \\frac{l-k}{2}\\) such that \\(a_i = a_{i+p}\\) for \\(k+1 \\leq i \\leq l-p\\), but \\(a_i \\neq a_{i+p}\\) for \\(i = k\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-ed8c10bfc5fe555321ebda0003c5ce4fef6a1fe5cc5e1a123bc68513530e1b0c | A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might be... | [
"37"
] | [
"A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence mig... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7429e008ff60b9a8ed9641c9e5b01160635a658dea0e3b2df31bb114054c6308 | A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. Let $S$ be the sum of all the terms i... | [
"37"
] | [
"A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. Let $S$ be the sum of all the te... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-93c827c2d4711ba4bb4b9e3ff0f58f917df351562c14bd9f77e694625194acbc | A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. Let \( S \) be the sum of all the ter... | [
"37"
] | [
"A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. Let \\( S \\) be the sum of all ... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 51,
"tir": 1
} | 52 | 0 | 0 |
omr-ws-e2f35b1f7909b07902a65f01ea77df6ffc5f0d334b222ef6fc324f3bb2cde578 | A finite set $M$ of circles of the same radius is given on the plane. It is known that for any three circles from $M$, there is a line that intersects them all. If the radii of each circle are doubled, find the minimum number of circles in $M$ such that there is guaranteed to be a line that intersects all circles of se... | [
"5"
] | [
"A finite set $M$ of circles of the same radius is given on the plane. It is known that for any three circles from $M$, there is a line that intersects them all. If the radii of each circle are doubled, find the minimum number of circles in $M$ such that there is guaranteed to be a line that intersects all circles ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-ddd17238689891c6bf4722e61cf15f43cffb4c8396b02ad13a98848f7000f317 | A finite set $S$ of points in the plane is called tri-separable if for every subset $A \subseteq S$ of the points in the given set, we can find a triangle $T$ such that (i) every point of $A$ is inside $T$, and (ii) every point of $S$ that is not in $A$ is outside $T$. What is the smallest positive integer $n$ such tha... | [
"8"
] | [
"A finite set $S$ of points in the plane is called tri-separable if for every subset $A \\subseteq S$ of the points in the given set, we can find a triangle $T$ such that (i) every point of $A$ is inside $T$, and (ii) every point of $S$ that is not in $A$ is outside $T$. What is the smallest positive integer $n$ su... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-adb209f090cff285e6ec1fb24d6a7d5b5e23c89a67ba26e154e25e697a4c8221 | A finite set $\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\mathcal{S} \cup \{1\}$ is $13$ less than the mean of $\mathcal{S}$, and the mean of $\mathcal{S} \cup \{2001\}$ is $27$ more than the mean of $\mathcal{S}$. Find the mean of $\mathcal{S}$. | [
"651"
] | [
"A finite set $\\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\\mathcal{S} \\cup \\{1\\}$ is $13$ less than the mean of $\\mathcal{S}$, and the mean of $\\mathcal{S} \\cup \\{2001\\}$ is $27$ more than the mean of $\\mathcal{S}$. Find the mean of $\\mathcal{S}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-edd0642a2418d1d2bb46ddd37bc1801b2aadd755cd1768f2d95d61574503944f | A finite set $\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\mathcal{S}\cup\{1\}$ is $13$ less than the mean of $\mathcal{S}$, and the mean of $\mathcal{S}\cup\{2001\}$ is $27$ more than the mean of $\mathcal{S}$. Find the mean of $\mathcal{S}$. | [
"651"
] | [
"A finite set $\\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\\mathcal{S}\\cup\\{1\\}$ is $13$ less than the mean of $\\mathcal{S}$, and the mean of $\\mathcal{S}\\cup\\{2001\\}$ is $27$ more than the mean of $\\mathcal{S}$. Find the mean of $\\mathcal{S}$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-d9899e575b06aa1dad338df1b85cbfe972d4f67b570c8f1ca7a759f55ad98eaa | A finite set $\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\mathcal{S}\cup\{1\}$ is 13 less than the mean of $\mathcal{S}$, and the mean of $\mathcal{S}\cup\{2001\}$ is 27 more than the mean of $\mathcal{S}.$ Find the mean of $\mathcal{S}.$ | [
"651"
] | [
"A finite set $\\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\\mathcal{S}\\cup\\{1\\}$ is 13 less than the mean of $\\mathcal{S}$, and the mean of $\\mathcal{S}\\cup\\{2001\\}$ is 27 more than the mean of $\\mathcal{S}.$ Find the mean of $\\mathcal{S}.$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
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