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-d5a862700bf13dbc11226a128571f49446023eedf08376d25428665cf7a07274 | An interval of length 1 is broken at a point uniformly distributed over $(0,1)$. Let $x$ be a point in $(0,1)$. Find the expected length of the subinterval that contains the point $x$. Then, calculate the value of $x$ that maximizes this expected length. Express your answer as a common fraction. | [
"\\dfrac{1}{2}"
] | [
"An interval of length 1 is broken at a point uniformly distributed over $(0,1)$. Let $x$ be a point in $(0,1)$. Find the expected length of the subinterval that contains the point $x$. Then, calculate the value of $x$ that maximizes this expected length. Express your answer as a common fraction."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
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"additional_problems": 1
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omr-ws-eca8c5256d46e4b4eb490e4c81e29a4ea4d4b5b1d8ee1d22bfd28dd933dee8eb | An intramural quarterback throws a football with an initial speed of 9.7 m/s at an upward angle of 38 degrees above a horizontal playing field. The ball leaves his hand at a point 1.5 m above the ground. How long will the ball remain in the air before falling to the ground? | [
"1.4"
] | [
"An intramural quarterback throws a football with an initial speed of 9.7 m/s at an upward angle of 38 degrees above a horizontal playing field. The ball leaves his hand at a point 1.5 m above the ground. How long will the ball remain in the air before falling to the ground?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
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"cot": 16
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omr-ws-b06dcb3e68e51c3eb03018a6caf41f6fdd6474f177143f032affd6e4915d327e | An intramural quarterback throws a football with an initial speed of 9.7 m/s at an upward angle of 38 degrees above a horizontal playing field. The ball leaves his hand at a point 1.5 m above the ground. What horizontal distance will the ball travel before striking the ground? | [
"11"
] | [
"An intramural quarterback throws a football with an initial speed of 9.7 m/s at an upward angle of 38 degrees above a horizontal playing field. The ball leaves his hand at a point 1.5 m above the ground. What horizontal distance will the ball travel before striking the ground?"
] | [
"aops_c4_high_school_math"
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"cot"
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"cot": 16
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omr-ws-b616a1344bc7ca04164fc882c5c8c8e1b3b3a12df2fe4a89ae7472461de6f3bb | An inverted cone with base radius $12 \text{ cm}$ and height $18 \text{ cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has radius of $24 \text{ cm}$. What is the height in centimeters of the water in the cylinder? | [
"\\dfrac{3}{2}"
] | [
"An inverted cone with base radius $12 \\text{ cm}$ and height $18 \\text{ cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has radius of $24 \\text{ cm}$. What is the height in centimeters of the water in the cylinder?"
] | [
"aops_c5_contests_amp_programs"
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"cot"
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omr-ws-b22af4263ceca03c7e6443c420389da66afbe06e6282e96051fe4d9cd0c2cedf | An inverted cone with base radius $12 \text{cm}$ and height $18 \text{cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of $24\text{cm}$. What is the height in centimeters of the water in the cylinder? | [
"\\dfrac{3}{2}"
] | [
"An inverted cone with base radius $12 \\text{cm}$ and height $18 \\text{cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of $24\\text{cm}$. What is the height in centimeters of the water in the cylinder?"
] | [
"aops_c5_contests_amp_programs"
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"cot"
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"cot": 15
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omr-ws-6e72fab6444a18b6973499e6885bb9b46d757b34903cf18ad2a222a9658749e0 | An inverted cone with base radius $12 \text{cm}$ and height $18\text{cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of $24\text{cm}$. What is the height in centimeters of the water in the cylinder? | [
"\\dfrac{3}{2}"
] | [
"An inverted cone with base radius $12 \\text{cm}$ and height $18\\text{cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of $24\\text{cm}$. What is the height in centimeters of the water in the cylinder?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
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"cot": 15
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omr-ws-1ed91f213c8fafb4daf3adeb083dde78eff96b81c84015ab6be38a221807282b | An inverted cone with base radius $12~{\rm cm}$ and height $18~{\rm cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of $24~{\rm cm}.$ What is the height in centimeters of the water in the cylinder? | [
"\\dfrac{3}{2}"
] | [
"An inverted cone with base radius $12~{\\rm cm}$ and height $18~{\\rm cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of $24~{\\rm cm}.$ What is the height in centimeters of the water in the cylinder?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
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"cot": 15
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omr-ws-fa0b03a5e65eca543fe026a549669fd287beabdbb0d671d0a3449b0bb98f83e8 | An invertible number is one which can be read upside down on a standard calculator. For example, 6592 can be read upside down as 2659 on a calculator. The first invertible number is $1$, the 10th invertible number is $15$, and the 15th invertible number is $21$. What is the 2007th invertible number? | [
"9686"
] | [
"An invertible number is one which can be read upside down on a standard calculator. For example, 6592 can be read upside down as 2659 on a calculator. The first invertible number is $1$, the 10th invertible number is $15$, and the 15th invertible number is $21$. What is the 2007th invertible number?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9728ae1571d30cb01aff1831fa51f75573d0c26550cdf4ac5eb8bd4683b85ec7 | An investigator interviewed 100 students to determine their preferences for three drinks: milk (M), coffee (C), and tea (T). The reported data are as follows: 10 students had all three drinks, 20 had M and C, 30 had C and T, 25 had M and T, 12 had M only, 5 had C only, and 8 had T only. How many students did not have a... | [
"20"
] | [
"An investigator interviewed 100 students to determine their preferences for three drinks: milk (M), coffee (C), and tea (T). The reported data are as follows: 10 students had all three drinks, 20 had M and C, 30 had C and T, 25 had M and T, 12 had M only, 5 had C only, and 8 had T only. How many students did not h... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-696c8b7d5ac7877e840c7e9bcfa7d90e81733ddbc111c3c45c093d2565a66b8b | An investment firm has $100,000 to invest in two stocks, A and B. The expected annual rate of return for stock A is 15%, but the client does not wish to invest more than $50,000 in this stock. The annual rate of return for stock B is 10%. Determine how much should be invested in each stock so that the annual interest f... | [
"\\text{Invest \\$40,000 in stock A and \\$60,000 in stock B}"
] | [
"An investment firm has $100,000 to invest in two stocks, A and B. The expected annual rate of return for stock A is 15%, but the client does not wish to invest more than $50,000 in this stock. The annual rate of return for stock B is 10%. Determine how much should be invested in each stock so that the annual inter... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-84d9b4fc4ef987e8705272202f96c127c6076faca8c18ff6b3f1468b68130b41 | An investment in a lease offers returns of $3713 per month due at the beginning of each month for 4.5 years. If the returns are deferred for two years and the interest required is 6.12% compounded monthly, what investment is justified? | [
"$155,565.967$"
] | [
"An investment in a lease offers returns of $3713 per month due at the beginning of each month for 4.5 years. If the returns are deferred for two years and the interest required is 6.12% compounded monthly, what investment is justified?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
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"tir": 19
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omr-ws-3216a23b65e05e98e3645daa78076dd5d7b20cb007576c5586667d89c5412183 | An investment of $5000, earning interest compounded semi-annually, grew to $9000 in 15 years. What was the nominal annual interest rate? | [
"\\( r \\approx 3.957\\% \\)"
] | [
"An investment of $5000, earning interest compounded semi-annually, grew to $9000 in 15 years. What was the nominal annual interest rate?"
] | [
"aops_c4_high_school_math"
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"cot",
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omr-ws-35bc6b471b8103cef969f5e1355d4384c455cd5986b9d39617ac30532a18fc9e | An investment of $600 will increase to $2400 in 20 years. Find the sum of the present values of 3 payments of $8000 each at the end of 30, 40, and 50 years. (Do not calculate the interest rate directly.) | [
"1750"
] | [
"An investment of $600 will increase to $2400 in 20 years. Find the sum of the present values of 3 payments of $8000 each at the end of 30, 40, and 50 years. (Do not calculate the interest rate directly.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-14244e0f4c4a896323b7f40d557e4b7df35684a8140312108dcded1809f28d10 | An investment of $\$$10,000 is made in a government bond that will pay 6$\%$ interest compounded annually. At the end of five years, what is the total number of dollars in this investment? Express your answer to the nearest whole number. | [
"13382"
] | [
"An investment of $\\$$10,000 is made in a government bond that will pay 6$\\%$ interest compounded annually. At the end of five years, what is the total number of dollars in this investment? Express your answer to the nearest whole number."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-892ed680440898be685b662ec696e0fc0cf4fd818a66b949e73cd3cbe1cb40a8 | An investment of 85,000,000 loses value each year according to the formula $10\% + 2.5\%(n-1)$, where $n$ is the year of decrement. In which year does the investment lose all of its value? | [
"7"
] | [
"An investment of 85,000,000 loses value each year according to the formula $10\\% + 2.5\\%(n-1)$, where $n$ is the year of decrement. In which year does the investment lose all of its value?"
] | [
"aops_c4_high_school_math"
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"cot",
"genselect"
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omr-ws-e997ebf5f7f42c68fbcddcef15517118f0710acdac133d3463459465a5573485 | An investor bought a number of shares of stock for $1,600. After the price dropped by $10 per share, the investor sold all but 4 of her shares for $1,120. How many shares did she originally buy? | [
"32"
] | [
"An investor bought a number of shares of stock for $1,600. After the price dropped by $10 per share, the investor sold all but 4 of her shares for $1,120. How many shares did she originally buy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
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omr-ws-16e8af6971ba236a90a463c7df1d4ce8e2d663ddb5f0d9b8b52ac9599f5c4110 | An investor has a total of $25,100 deposited in three different accounts, which earn annual interest rates of 8%, 6%, and 4%. The amount deposited in the 8% account is twice the amount in the 6% account. If the three accounts earn a total annual interest of $1534, how much money is deposited in each account? | [
"$5300, $10600, $9200"
] | [
"An investor has a total of $25,100 deposited in three different accounts, which earn annual interest rates of 8%, 6%, and 4%. The amount deposited in the 8% account is twice the amount in the 6% account. If the three accounts earn a total annual interest of $1534, how much money is deposited in each account?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 26
} | 26 | 0 | 0 |
omr-ws-cde1fd95795c6f432e02c68d58d66cf4f45cf3a4bf928125ddd2580891b595c0 | An investor has shares in Delta Airlines, Hilton Hotels, and McDonald's. Two days ago, the shares collectively lost $350, and yesterday they gained $600. The changes in share prices were as follows: - Two days ago: Delta lost $1 per share, Hilton lost $1.5 per share, and McDonald's gained $0.5 per share. - Yesterday: D... | [
"100 shares of Hilton and 300 shares of Delta."
] | [
"An investor has shares in Delta Airlines, Hilton Hotels, and McDonald's. Two days ago, the shares collectively lost $350, and yesterday they gained $600. The changes in share prices were as follows:\n- Two days ago: Delta lost $1 per share, Hilton lost $1.5 per share, and McDonald's gained $0.5 per share.\n- Yeste... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fdaf1bc09a26cf264b59db197d962b6d244aa09adca11cf744b8d3de438a88c2 | An investor has shares in Delta Airlines, Hilton Hotels, and McDonald's. Two days ago, the shares collectively lost $350, and yesterday they gained $600. The changes in share prices were as follows: two days ago, Delta lost $1 per share, Hilton lost $1.5 per share, and McDonald's gained $0.5 per share. Yesterday, Delta... | [
"\\text{No}"
] | [
"An investor has shares in Delta Airlines, Hilton Hotels, and McDonald's. Two days ago, the shares collectively lost $350, and yesterday they gained $600. The changes in share prices were as follows: two days ago, Delta lost $1 per share, Hilton lost $1.5 per share, and McDonald's gained $0.5 per share. Yesterday, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-67663bbe0ef04651d2827136130e8db6adcb1634c5edbe4d21cc400e19a79ade | An invisible tank is on a $100 \times 100$ table. A cannon can fire at any $k$ cells of the board. After each shot, the tank moves to one of the adjacent cells (by side). Determine the smallest value of $k$ such that the cannon can definitely hit the tank after some time. | [
"50"
] | [
"An invisible tank is on a $100 \\times 100$ table. A cannon can fire at any $k$ cells of the board. After each shot, the tank moves to one of the adjacent cells (by side). Determine the smallest value of $k$ such that the cannon can definitely hit the tank after some time."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-84b8ed40c4055f665f69c9c56ffc676218d33833db2169a4941a39be2b06aaa1 | An irregular hexagon has sides in arithmetic progression. Its circumradius is twice the smallest side. Find the perimeter of such a hexagon. | [
"12a"
] | [
"An irregular hexagon has sides in arithmetic progression. Its circumradius is twice the smallest side. Find the perimeter of such a hexagon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-93f574e1237f7f94195da191a46f68d985e601500fd7e5ec053b1721be57024f | An island has 10 cities, where some pairs of cities are connected by roads. A tour route is a loop that starts from a city, passes exactly eight out of the other nine cities exactly once each, and returns to the starting city. For each city, there exists a tour route that does not pass through that city. Find the minim... | [
"15"
] | [
"An island has 10 cities, where some pairs of cities are connected by roads. A tour route is a loop that starts from a city, passes exactly eight out of the other nine cities exactly once each, and returns to the starting city. For each city, there exists a tour route that does not pass through that city. Find the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-26750c1d8293ddd72458c903e65493cb5a5f6803b4470d23d7a7825b2b08dd3c | An island is a disc of radius $100$ km. Helicopters can fly at $300$ km/h. Determine the minimum number of helicopters needed to ensure that every point on the island can be reached within $10$ minutes. Assume that takeoff and landing times are negligible. | [
"4"
] | [
"An island is a disc of radius $100$ km. Helicopters can fly at $300$ km/h. Determine the minimum number of helicopters needed to ensure that every point on the island can be reached within $10$ minutes. Assume that takeoff and landing times are negligible."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ecb15f94535daeb0a6c23f3528fe805c69fa7906788cdb8d5d88258faae047d2 | An island is inhabited by five men and a pet monkey. One afternoon, the men gathered a large pile of coconuts, which they proposed to divide equally among themselves the next morning. During the night, each of the five men, one at a time, took a turn dividing the pile into five equal parts and found one coconut left ov... | [
"15621"
] | [
"An island is inhabited by five men and a pet monkey. One afternoon, the men gathered a large pile of coconuts, which they proposed to divide equally among themselves the next morning. During the night, each of the five men, one at a time, took a turn dividing the pile into five equal parts and found one coconut le... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7580491a611134e744db5a2608bf58eafb9e70056c84fa5c89874e494f5aafe2 | An island is located 4 miles from a coastal village and a small forest is located 11 miles up the coast from the village. If it takes only 3/4 as much energy for the birds to fly over land as it does to fly over water, how far from the forest should the birds make landfall in order to preserve as much energy as possibl... | [
"\\( 11 - \\frac{12\\sqrt{7}}{7} \\)"
] | [
"An island is located 4 miles from a coastal village and a small forest is located 11 miles up the coast from the village. If it takes only 3/4 as much energy for the birds to fly over land as it does to fly over water, how far from the forest should the birds make landfall in order to preserve as much energy as po... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-19bc3bb4299a81e11b4ac266ae9a484b50ce1bcbe838b65db452e78f737edaaf | An island nation has two banana companies. Sally's banana company produces $b_s$ crates of bananas at a cost of $4b_s^2 + 10$. Nick's banana company produces $b_n$ crates of bananas at a cost of $b_n^2 + 30b_n$. If there are $b$ (total) crates of bananas being produced on the island, then they can be sold for $120 - b$... | [
"\\( (b_s, b_n) = (10, 20) \\)"
] | [
"An island nation has two banana companies. Sally's banana company produces $b_s$ crates of bananas at a cost of $4b_s^2 + 10$. Nick's banana company produces $b_n$ crates of bananas at a cost of $b_n^2 + 30b_n$. If there are $b$ (total) crates of bananas being produced on the island, then they can be sold for $120... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 27
} | 33 | 0 | 0 |
omr-ws-f158b8c261750d2a5be8adf34464927bb2d6c1b95e4f16d921f3014f594fe645 | An isosceles right triangle ABC is given with the right angle at B. Point D is on side BC. The angle bisector AD has length 4. Determine the area of triangle ABC. | [
"4 + 2\\sqrt{2}"
] | [
"An isosceles right triangle ABC is given with the right angle at B. Point D is on side BC. The angle bisector AD has length 4. Determine the area of triangle ABC."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0f91e66d58097ec108c590f399d5b72a25798277b2af5adf2e6f9f2a26397543 | An isosceles right triangle has a leg of length $1$. It is allowed to replace one vertex of the triangle with another vertex symmetric with respect to it. After several such operations, a triangle with sides $a$, $b$, and $c$ (where $c$ is the longest side) is obtained. Calculate the maximum possible value of $a + b - ... | [
"2 - \\sqrt{2}"
] | [
"An isosceles right triangle has a leg of length $1$. It is allowed to replace one vertex of the triangle with another vertex symmetric with respect to it. After several such operations, a triangle with sides $a$, $b$, and $c$ (where $c$ is the longest side) is obtained. Calculate the maximum possible value of $a +... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-bbd5c402481e026289c89395ed4329f2f0521276e6678c341ec6fc0d9dbd9cd1 | An isosceles right-angled triangle \(ABC\) is given with \(AB = AC\). On the extensions of sides \(AB\) and \(AC\) behind vertices \(B\) and \(C\), equal segments \(BK\) and \(CL\) are laid. \(E\) and \(F\) are the points of intersection of the segment \(KL\) and the lines perpendicular to \(KC\), passing through the p... | [
"1"
] | [
"An isosceles right-angled triangle \\(ABC\\) is given with \\(AB = AC\\). On the extensions of sides \\(AB\\) and \\(AC\\) behind vertices \\(B\\) and \\(C\\), equal segments \\(BK\\) and \\(CL\\) are laid. \\(E\\) and \\(F\\) are the points of intersection of the segment \\(KL\\) and the lines perpendicular to \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2b23f4bf46958c5b2c070c9c3d39d1ae45d7f3f58439c994bc6a016daf62df1e | An isosceles trapezium ABCD is given with AB parallel to CD. The side AB is a diameter of a circle which touches side CD and passes through the midpoint of AD. Find the value of angle ABC. | [
"75^\\circ"
] | [
"An isosceles trapezium ABCD is given with AB parallel to CD. The side AB is a diameter of a circle which touches side CD and passes through the midpoint of AD. Find the value of angle ABC."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-feba7cb17e1e6314131a390bbc0a2126399992d3f612f22fd532c8d2c47fe64b | An isosceles trapezium ABCD is given, with AB || CD. The side AB is a diameter of a circle which touches side CD and passes through the midpoint of AD. Find the value of angle ABC. | [
"15°"
] | [
"An isosceles trapezium ABCD is given, with AB || CD. The side AB is a diameter of a circle which touches side CD and passes through the midpoint of AD. Find the value of angle ABC."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
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"cot": 1,
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omr-ws-c5e714ae11e2a05fed0a0ffa63502f8361242da318b4e9ddef80b8fe8f135419 | An isosceles trapezoid $ABCD$ is inscribed in a circle $\omega$ with $AB=CD$, $AD<BC$, and $AD<CD$. A circle centered at $D$ and passing through $A$ intersects $BD$, $CD$, and $\omega$ at points $E$, $F$, and $P$ (where $P \neq A$), respectively. Let $AP \cap EF = Q$, and let $\omega$ intersect $CQ$ and the circumcircl... | [
"1"
] | [
"An isosceles trapezoid $ABCD$ is inscribed in a circle $\\omega$ with $AB=CD$, $AD<BC$, and $AD<CD$. A circle centered at $D$ and passing through $A$ intersects $BD$, $CD$, and $\\omega$ at points $E$, $F$, and $P$ (where $P \\neq A$), respectively. Let $AP \\cap EF = Q$, and let $\\omega$ intersect $CQ$ and the c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-abb7a69411b64bd1b23771c30bbf9b01eec6badff3b812550deaf653e876be95 | An isosceles trapezoid $ABCD$ is inscribed in a circle with $AC$ perpendicular to $BD$. There exists a point $O$ in the triangle such that $OA + OB + OC + OD = 12$. What is the maximum area of $ABCD$? | [
"18"
] | [
"An isosceles trapezoid $ABCD$ is inscribed in a circle with $AC$ perpendicular to $BD$. There exists a point $O$ in the triangle such that $OA + OB + OC + OD = 12$. What is the maximum area of $ABCD$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
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omr-ws-b6fc53fa8b122a5e3efd87834f9806d05097a9d20be56705678b41f48cb16b04 | An isosceles trapezoid $ABCD$ is placed inside a semicircle such that they share the same base $AD$, where $AD=4$ (and $AD$ is the diameter of the semicircle). The lengths $AB=DC=1$ are chords. Find the length of $BC$. | [
"\\(\\frac{7}{2}\\)"
] | [
"An isosceles trapezoid $ABCD$ is placed inside a semicircle such that they share the same base $AD$, where $AD=4$ (and $AD$ is the diameter of the semicircle). The lengths $AB=DC=1$ are chords. Find the length of $BC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-5e27501818af9afca277d1caf7ccc01f49d8b6d0a041e8e166b40ead2ac4d5a6 | An isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ is given. A circle $\omega$ passes through $B$ and $C$ and intersects $AB$ and diagonal $BD$ at points $X$ and $Y$, respectively. The tangent to $\omega$ at $C$ intersects $AD$ at $Z$. If $AB = 10$, $BC = 6$, $AD = 18$, and $BD = 15$, calculate the length of segmen... | [
"12"
] | [
"An isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ is given. A circle $\\omega$ passes through $B$ and $C$ and intersects $AB$ and diagonal $BD$ at points $X$ and $Y$, respectively. The tangent to $\\omega$ at $C$ intersects $AD$ at $Z$. If $AB = 10$, $BC = 6$, $AD = 18$, and $BD = 15$, calculate the length of... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-4522c7ec18f7087ecfb727a75094b269cfc63a0eb2bc896a20f875bbe3c51c9f | An isosceles trapezoid $ABCD$, with parallel sides $AB$ and $CD$, is such that diagonal $BD = 100$ m and $\angle BDC = 30^\circ$. Let $S$ be the area of the trapezoid in m$^2$. Determine $S\sqrt{3}$. | [
"7500"
] | [
"An isosceles trapezoid $ABCD$, with parallel sides $AB$ and $CD$, is such that diagonal $BD = 100$ m and $\\angle BDC = 30^\\circ$. Let $S$ be the area of the trapezoid in m$^2$. Determine $S\\sqrt{3}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-131cdd653b9927a327f334c69c77c1a824153701f795389c4238b2e07a2757b3 | An isosceles trapezoid \(ABCD\) is circumscribed about a circle \(O\). The longer base \(AB = 16\), and \(\angle ABC = \arcsin(0.8)\). Find the area of the trapezoid. | [
"80"
] | [
"An isosceles trapezoid \\(ABCD\\) is circumscribed about a circle \\(O\\). The longer base \\(AB = 16\\), and \\(\\angle ABC = \\arcsin(0.8)\\). Find the area of the trapezoid."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-3d79b41a963401e3ecc74698b5a488182f1c593ed1811ca5992f49d2167e31ef | An isosceles trapezoid \(PQRS\) has the properties \(\dfrac{PQ}{PS} = 2\) and \(\dfrac{PQ}{RS} = 4\). The trapezoid is rotated \(45^\circ\) about its circumcenter to form the image \(P'Q'R'S'\). What is the ratio of the area of the intersection \(PQRS \cap P'Q'R'S'\) to the area of \(PQRS\)? | [
"\\dfrac{1}{2}"
] | [
"An isosceles trapezoid \\(PQRS\\) has the properties \\(\\dfrac{PQ}{PS} = 2\\) and \\(\\dfrac{PQ}{RS} = 4\\). The trapezoid is rotated \\(45^\\circ\\) about its circumcenter to form the image \\(P'Q'R'S'\\). What is the ratio of the area of the intersection \\(PQRS \\cap P'Q'R'S'\\) to the area of \\(PQRS\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-593e092ed0f5fa9cf99d85bf9e277be0b37519bbb42c40397e2f9a4eebdbc31c | An isosceles trapezoid has a larger base of 10 cm, sides of 6 cm, and diagonals of 8 cm. Draw a line parallel to the bases that divides the trapezoid into two regions of equal area. Find the distance from this line to the lower base of the trapezoid in centimeters. (Note: The expected answer is \(\frac{12\sqrt{2}}{5}\)... | [
"\\dfrac{12\\sqrt{2}}{5}"
] | [
"An isosceles trapezoid has a larger base of 10 cm, sides of 6 cm, and diagonals of 8 cm. Draw a line parallel to the bases that divides the trapezoid into two regions of equal area. Find the distance from this line to the lower base of the trapezoid in centimeters. \n\n(Note: The expected answer is \\(\\frac{12\\s... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-2c752f07d1359e352ebd22691c2b23316e21879a8a6c4aef663ed0bc3e5728a8 | An isosceles trapezoid has legs of length 30 cm each, two diagonals of length 40 cm each and the longer base is 50 cm. What is the trapezoid's area in sq cm? | [
"768"
] | [
"An isosceles trapezoid has legs of length 30 cm each, two diagonals of length 40 cm each and the longer base is 50 cm. What is the trapezoid's area in sq cm?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8aabd12c59d1922316f2e63726e7963761007cecb7c6a808384ec3a5bae62d59 | An isosceles trapezoid has parallel sides of length 15 and 60. A circle is inscribed in the trapezoid, touching all four sides. What is the radius of the circle? | [
"15"
] | [
"An isosceles trapezoid has parallel sides of length 15 and 60. A circle is inscribed in the trapezoid, touching all four sides. What is the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d1982fd3efe483c4a508861deab0e6c0ac454cd097e719ab1888047e00cf819d | An isosceles trapezoid has parallel sides of length 16 and 28 and non-parallel sides of length $T$. Compute the area of the trapezoid. | [
"176"
] | [
"An isosceles trapezoid has parallel sides of length 16 and 28 and non-parallel sides of length $T$. Compute the area of the trapezoid."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-844967f9c152d958ac69ef72d4ab4c58933b3b94ec76370108a3acd9a05c014a | An isosceles trapezoid has side lengths of 5, 5, 5, and 11. Marty draws in the two diagonals of the trapezoid and receives four small triangles. What is the area of the largest of these four triangles? | [
"$\\frac{121}{8}$"
] | [
"An isosceles trapezoid has side lengths of 5, 5, 5, and 11. Marty draws in the two diagonals of the trapezoid and receives four small triangles. What is the area of the largest of these four triangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-7171ed6f3bfd04d5c7989de9faac98ca3ccecff8d6f5e4b6a714a0a1cf4bd9bb | An isosceles trapezoid has side lengths of 5, 5, 5, and 8. What is the sum of the lengths of its diagonals? | [
"\\(2 \\sqrt{65}\\)"
] | [
"An isosceles trapezoid has side lengths of 5, 5, 5, and 8. What is the sum of the lengths of its diagonals?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 7,
"tir": 63
} | 101 | 0 | 0 |
omr-ws-3939ca4ca2187199b4b2d5dc306f8c66b4c50626eab878fa9756fdceef2cead1 | An isosceles trapezoid is circumscribed about a circle of radius $2$ cm and the area of the trapezoid is $20$ $cm^2$. The equal sides of the trapezoid have length $\dots$ | [
"5"
] | [
"An isosceles trapezoid is circumscribed about a circle of radius $2$ cm and the area of the trapezoid is $20$ $cm^2$. The equal sides of the trapezoid have length $\\dots$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-1c651ceed5d676e8937475fd4c45937e9fc3b8feb50dd7c642c232ac16c964d1 | An isosceles trapezoid is circumscribed about a circle with a radius of $2cm$ and has an area of $20cm^2$. What is the length of the equal sides of the trapezoid? | [
"5"
] | [
"An isosceles trapezoid is circumscribed about a circle with a radius of $2cm$ and has an area of $20cm^2$. What is the length of the equal sides of the trapezoid?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-abc0b46397b967af9bfbf62944548b0756dfc9f48467b60968eb57f5caa5ff3f | An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is 16, and one of the base angles is \(\arcsin(0.8)\). Find the area of the trapezoid. | [
"80"
] | [
"An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is 16, and one of the base angles is \\(\\arcsin(0.8)\\). Find the area of the trapezoid."
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 32
} | 79 | 0 | 0 |
omr-ws-7cdc55c207347e1253c8a47e26a8bc9cf175c08c0abb1d99527dedd7eef34e8b | An isosceles trapezoid is divided by each diagonal into two isosceles triangles. Determine the angles of the trapezoid. | [
"108^\\circ"
] | [
"An isosceles trapezoid is divided by each diagonal into two isosceles triangles. Determine the angles of the trapezoid."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-b2d6d202657b060da32ffe306be7a598b9207d524f9b2f7b26a6c3248da9a81b | An isosceles trapezoid is divided by its diagonal into two isosceles triangles. Find the angles of the trapezoid. | [
"\\(72^{\\circ}\\) and \\(108^{\\circ}\\)"
] | [
"An isosceles trapezoid is divided by its diagonal into two isosceles triangles. Find the angles of the trapezoid."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-0f62638e208a2d671b3caec34b9f1ef2e1c9bd2e0c60b013d8c6003397b056b7 | An isosceles trapezoid is inscribed in a circle so that a diameter of the circle serves as a base of the trapezoid. Find the ratio of the areas of the circle and the trapezoid if the obtuse angle of the trapezoid is equal to $\alpha $. | [
"$\\frac{S_{tr}}{S_{cir}}=-\\frac{4R^2\\sin ^3 \\alpha \\cos \\alpha }{\\pi }$"
] | [
"An isosceles trapezoid is inscribed in a circle so that a diameter of the circle serves as a base of the trapezoid. Find the ratio of the areas of the circle and the trapezoid if the obtuse angle of the trapezoid is equal to $\\alpha $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b5907f562ab1d8da198e1e79bcfce11b2369db963936d54994308fb5e83cef2 | An isosceles trapezoid is inscribed in a circle with bases of lengths 10 and 18, and a height of 4. The center of the circle lies outside the trapezoid. Find the radius of the circle. | [
"\\(\\sqrt{106}\\)"
] | [
"An isosceles trapezoid is inscribed in a circle with bases of lengths 10 and 18, and a height of 4. The center of the circle lies outside the trapezoid. Find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-058cb6be418f3477918184c067225c1b896f3a88712ca064a56a40442d9898f4 | An isosceles trapezoid of area \( A \) is circumscribed about a circle with radius \( r \). Find the lengths of the parallel sides in terms of \( A \) and \( r \). | [
"\\( b_1 = \\dfrac{A}{2r} - \\dfrac{\\sqrt{A^2 - 16r^4}}{2r} \\) and \\( b_2 = \\dfrac{A}{2r} + \\dfrac{\\sqrt{A^2 - 16r^4}}{2r} \\)"
] | [
"An isosceles trapezoid of area \\( A \\) is circumscribed about a circle with radius \\( r \\). Find the lengths of the parallel sides in terms of \\( A \\) and \\( r \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 18
} | 22 | 0 | 0 |
omr-ws-9eb2f9a7a3f09db6069777634d9c2eeb696a03c77db7878d69820ecfc57a3b8d | An isosceles trapezoid which is circumscribed about a circle has two adjacent angles of $120$ degrees and two of $60$ degrees. What is the ratio of its smaller base to its greater base? | [
"\\(\\frac{1}{3}\\)"
] | [
"An isosceles trapezoid which is circumscribed about a circle has two adjacent angles of $120$ degrees and two of $60$ degrees. What is the ratio of its smaller base to its greater base?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 13
} | 34 | 0 | 0 |
omr-ws-7edab7ebd8e5bbf9fe384a4fba9e1c9435014e6c7770745d1311c07764187c6d | An isosceles trapezoid with a perimeter of $64$ cm is both inscribable and circumscribable. If the radius of the circumscribed circle is $16$ cm, find the lengths of the sides of the trapezoid. | [
"\\( \\frac{192 + 16\\sqrt{134}}{25} \\), \\( \\frac{208 - 16\\sqrt{134}}{25} \\), \\( 8 \\), \\( 8 \\)"
] | [
"An isosceles trapezoid with a perimeter of $64$ cm is both inscribable and circumscribable. If the radius of the circumscribed circle is $16$ cm, find the lengths of the sides of the trapezoid."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-45b21d513755ed8adf9877a7ae5f37557b691e6c46bd9b59556990c4e6e87814 | An isosceles trapezoid with an area of 20 cm$^2$ is circumscribed about a circle of a radius of 2 cm. Find the sides of the trapezoid. | [
"$AD=8$ cm, $BC=2$ cm, $AB=CD=5$ cm"
] | [
"An isosceles trapezoid with an area of 20 cm$^2$ is circumscribed about a circle of a radius of 2 cm. Find the sides of the trapezoid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5979dd5ded8315c182a6e9cdc3d50b7634dea7408e75f7653142c7085a31721a | An isosceles trapezoid with base angle $60^\circ$ is such that it is possible to inscribe in it two circles tangent to each other, each one of which is tangent to the bases of the trapezoid and to one of the nonparallel sides. Each nonparallel side of the trapezoid is equal to 2. Find the area of the trapezoid. | [
"$3+2\\sqrt 3$"
] | [
"An isosceles trapezoid with base angle $60^\\circ$ is such that it is possible to inscribe in it two circles tangent to each other, each one of which is tangent to the bases of the trapezoid and to one of the nonparallel sides. Each nonparallel side of the trapezoid is equal to 2. Find the area of the trapezoid."
... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4da10b175a67f2e37b7eac7d0e90be7706d165af4bee363f709f3601ce301b89 | An isosceles trapezoid with base lengths 16 and 36 is circumscribed about a circle. Find the circumference of the circle. | [
"\\( 24\\pi \\)"
] | [
"An isosceles trapezoid with base lengths 16 and 36 is circumscribed about a circle. Find the circumference of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-fcaf17afd66337f4a9da5434c09695e237efac3f19a7b684553fc2753a5c6c32 | An isosceles trapezoid with bases of lengths 10 and 4 and legs of length 5 rotates around the longer base. Find the surface area and the volume of the obtained body. | [
"\\[\n\\text{Surface Area} = 72\\pi\n\\]\n\\[\n\\text{Volume} = \\frac{320\\pi}{3}\n\\]"
] | [
"An isosceles trapezoid with bases of lengths 10 and 4 and legs of length 5 rotates around the longer base. Find the surface area and the volume of the obtained body."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ed0f6ccac50693dc42bd00092d1534fa7bb64d3cffd38f1bb9f381abea37a642 | An isosceles triangle $ABC$ has a perimeter of $90$ cm. Find the side lengths that will maximize the area of the triangle. | [
"30 cm"
] | [
"An isosceles triangle $ABC$ has a perimeter of $90$ cm. Find the side lengths that will maximize the area of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-0416bf558a426ed47ab2aacd9b0dac07d668275dbea0360e83cb6687d6bc1d19 | An isosceles triangle $ABC$ has angle $BAC = 135^\circ$ and $AB = 2.$ What is its area? | [
"\\(\\sqrt{2}\\)"
] | [
"An isosceles triangle $ABC$ has angle $BAC = 135^\\circ$ and $AB = 2.$ What is its area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-2ec78233666af1cfcfa8279853c447acd5353f8104306ce41e323514e747a8e7 | An isosceles triangle $ABC$ is given with the angle at the vertex $\angle B=20^\circ$. On its extension of the base $AC$ beyond the point $C$, a point $D$ is chosen such that $CB = CD$, and on side $BC$ a point $E$ is chosen such that $\angle CDE = 10^\circ$. Find the angle $\angle CAE$. | [
"30^\\circ"
] | [
"An isosceles triangle $ABC$ is given with the angle at the vertex $\\angle B=20^\\circ$. On its extension of the base $AC$ beyond the point $C$, a point $D$ is chosen such that $CB = CD$, and on side $BC$ a point $E$ is chosen such that $\\angle CDE = 10^\\circ$. Find the angle $\\angle CAE$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-ff771adf5103b6c82cc151dff77d39ab34e3061fac18f6201930326a209b272c | An isosceles triangle $T$ has the property that a line through one of its vertices splits it into two smaller isosceles triangles $R$ and $S$, neither of which are similar to $T$. Find all possible values of the vertex (apex) angle of $T$. | [
"108^\\circ"
] | [
"An isosceles triangle $T$ has the property that a line through one of its vertices splits it into two smaller isosceles triangles $R$ and $S$, neither of which are similar to $T$. Find all possible values of the vertex (apex) angle of $T$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-332cf862f5c6a6828e61655b2ccabab8879ad0b41c20fa637d1f93839a7bd748 | An isosceles triangle $\triangle ABC$ with $\overline{AB} = \overline{AC}$ is inscribed in a circle. The angle bisector of point $B$ intersects $\overline{AC}$ at point $D$ and the circle at point $E$. Given that $\overline{AE} = 6$, $\overline{ID} = 4$, and $\frac{AB}{BC} = \frac{3}{4}$, find the length of side $\over... | [
"321"
] | [
"An isosceles triangle $\\triangle ABC$ with $\\overline{AB} = \\overline{AC}$ is inscribed in a circle. The angle bisector of point $B$ intersects $\\overline{AC}$ at point $D$ and the circle at point $E$. Given that $\\overline{AE} = 6$, $\\overline{ID} = 4$, and $\\frac{AB}{BC} = \\frac{3}{4}$, find the length o... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8b15553f4f1dee7f155cae25124b19578eee9b191785bb80b14dbaa374370716 | An isosceles triangle \( ABC \) is right-angled at \( B \) with \( AB = 3 \). A circle of unit radius is drawn with its center at any of the vertices of this triangle. Find the maximum value of the area of the part of the triangle that is not shared by the circle. | [
"\\(\\frac{36 - \\pi}{8}\\)"
] | [
"An isosceles triangle \\( ABC \\) is right-angled at \\( B \\) with \\( AB = 3 \\). A circle of unit radius is drawn with its center at any of the vertices of this triangle. Find the maximum value of the area of the part of the triangle that is not shared by the circle."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 8,
"tir": 18
} | 64 | 0 | 0 |
omr-ws-d05caff5000849d99ce4e62a22c1ad7ece0828e02f1a965dd209894ffe35711b | An isosceles triangle \(ABC\) (\(AB = AC\)) is inscribed in a circle \(\omega\). Let \(D\) be an arbitrary point on segment \(BC\) (\(BD \neq DC\)). \(AD\) intersects \(\omega\) again at \(E\). Point \(F\) is selected on \(\omega\) such that \(\angle DFE = 90^\circ\). \(FE\) intersects \(AB\) and \(AC\) at \(X\) and \(... | [
"\\tan 20^\\circ"
] | [
"An isosceles triangle \\(ABC\\) (\\(AB = AC\\)) is inscribed in a circle \\(\\omega\\). Let \\(D\\) be an arbitrary point on segment \\(BC\\) (\\(BD \\neq DC\\)). \\(AD\\) intersects \\(\\omega\\) again at \\(E\\). Point \\(F\\) is selected on \\(\\omega\\) such that \\(\\angle DFE = 90^\\circ\\). \\(FE\\) interse... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-86d0f1c0d0c21cc786befbc56e0cde79784df5bd8fee3d4a1f51a724affa1dd3 | An isosceles triangle \(ABC\) (\(AB = AC\)) is inscribed in a circle. The tangents to the circle at points \(A\) and \(C\) intersect each other at point \(D\). If \(\angle DBC = 30^\circ\), find the measure of \(\angle BAC\). | [
"60^\\circ"
] | [
"An isosceles triangle \\(ABC\\) (\\(AB = AC\\)) is inscribed in a circle. The tangents to the circle at points \\(A\\) and \\(C\\) intersect each other at point \\(D\\). If \\(\\angle DBC = 30^\\circ\\), find the measure of \\(\\angle BAC\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-52f4eb2d4e42ea43048f981ee9c20b71fcf0376e41ca122050c9ef83b1c8e09e | An isosceles triangle \(ABC\) (\(AB = BC\)) is given. Circles \(\omega_1\) and \(\omega_2\) with centers \(O_1\) and \(O_2\) lie in the angle \(ABC\) and touch the sides \(AB\) and \(CB\) at \(A\) and \(C\) respectively, and touch each other externally at point \(X\). The side \(AC\) meets the circles again at points \... | [
"6"
] | [
"An isosceles triangle \\(ABC\\) (\\(AB = BC\\)) is given. Circles \\(\\omega_1\\) and \\(\\omega_2\\) with centers \\(O_1\\) and \\(O_2\\) lie in the angle \\(ABC\\) and touch the sides \\(AB\\) and \\(CB\\) at \\(A\\) and \\(C\\) respectively, and touch each other externally at point \\(X\\). The side \\(AC\\) me... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-9d31e2dd0f756792646e49c23c62de7145fc18e178270a9772cdbaa99380e34d | An isosceles triangle \(ABC\) (\(AC = BC\)) is inscribed in a circle \(k\). Point \(M\) is on side \(BC\), and point \(N\) is on the ray \(AM\) such that \(AN = AC\). The circumcircle of \(\triangle MCN\) intersects \(k\) at \(C\) and \(P\), where \(P\) is on the arc \(BC\) not containing \(A\). The lines \(AB\) and \(... | [
"1"
] | [
"An isosceles triangle \\(ABC\\) (\\(AC = BC\\)) is inscribed in a circle \\(k\\). Point \\(M\\) is on side \\(BC\\), and point \\(N\\) is on the ray \\(AM\\) such that \\(AN = AC\\). The circumcircle of \\(\\triangle MCN\\) intersects \\(k\\) at \\(C\\) and \\(P\\), where \\(P\\) is on the arc \\(BC\\) not contain... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-15117277d05295ce69a32a35b1817a14a1d973f422ea8e21801a4bed49d71f0c | An isosceles triangle \(ABC\) has a vertex angle \(x\) and its sides are \(\sin x\), \(\sqrt{\sin x}\), and \(\sqrt{\sin x}\). What is its area? | [
"\\(\\frac{8}{25}\\)"
] | [
"An isosceles triangle \\(ABC\\) has a vertex angle \\(x\\) and its sides are \\(\\sin x\\), \\(\\sqrt{\\sin x}\\), and \\(\\sqrt{\\sin x}\\). What is its area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 24
} | 35 | 0 | 0 |
omr-ws-ff3906ba6906ffbeb574161574ab8d0f4f86adc6cd5e929bb255dacbbf4eba75 | An isosceles triangle \(ABC\) is inscribed in a circle with \(\angle ACB = 90^\circ\). A chord \(EF\) of the circle is such that neither \(E\) nor \(F\) coincide with \(C\). Lines \(CE\) and \(CF\) meet \(AB\) at \(D\) and \(G\) respectively. If \(|CE| = 5\), \(|EF| = 7\), and \(|CG| = 3\), calculate the length of \(|D... | [
"4"
] | [
"An isosceles triangle \\(ABC\\) is inscribed in a circle with \\(\\angle ACB = 90^\\circ\\). A chord \\(EF\\) of the circle is such that neither \\(E\\) nor \\(F\\) coincide with \\(C\\). Lines \\(CE\\) and \\(CF\\) meet \\(AB\\) at \\(D\\) and \\(G\\) respectively. If \\(|CE| = 5\\), \\(|EF| = 7\\), and \\(|CG| =... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-956369baae5d7d38d60ce826fe54784aa7f645ec40234e6bbb5fa660785172fb | An isosceles triangle \(ABC\) is inscribed in a circle with radius \(r_1\) such that side \(AB\) is a diameter of the circle. If a circle inscribed within triangle \(ABC\) has radius \(r_2\), what is \(\frac{r_2}{r_1}\)? | [
"\\(\\sqrt{2} - 1\\)"
] | [
"An isosceles triangle \\(ABC\\) is inscribed in a circle with radius \\(r_1\\) such that side \\(AB\\) is a diameter of the circle. If a circle inscribed within triangle \\(ABC\\) has radius \\(r_2\\), what is \\(\\frac{r_2}{r_1}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-40590ee67bf6c66746c3018165586c27225347ad1f6d8e9d5a3d7dc547652baf | An isosceles triangle \(ABC\) with \(AB = AC = 6\) and \(BC = 4\) is inscribed in a circle. Line \(MN\) is tangent to the circle at point \(C\), and chord \(BD\) is parallel to the tangent \(MN\). Lines \(AC\) and \(BD\) intersect at point \(E\). Find the length of segment \(AE\). | [
"4"
] | [
"An isosceles triangle \\(ABC\\) with \\(AB = AC = 6\\) and \\(BC = 4\\) is inscribed in a circle. Line \\(MN\\) is tangent to the circle at point \\(C\\), and chord \\(BD\\) is parallel to the tangent \\(MN\\). Lines \\(AC\\) and \\(BD\\) intersect at point \\(E\\). Find the length of segment \\(AE\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19fc4970ad0c271caa72cb0e29cf2fc1d3666963f8628a25e6322e84eaa36dff | An isosceles triangle \(ABC\) with \(AB = AC\) has an incircle of radius \(r\). The centroid \(M\) of the triangle lies on the incircle. Find the distance from the vertex \(A\) to the point of intersection of the angle bisector of \(\angle BAC\). | [
"\\(3r\\)"
] | [
"An isosceles triangle \\(ABC\\) with \\(AB = AC\\) has an incircle of radius \\(r\\). The centroid \\(M\\) of the triangle lies on the incircle. Find the distance from the vertex \\(A\\) to the point of intersection of the angle bisector of \\(\\angle BAC\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1a9e4452478a661ccf8d1eed47a69db559288874e460f17a0c355afe30e4880 | An isosceles triangle \(ABC\) with base \(AC\) is inscribed in a circle \(\omega\). The radius of the inscribed circle of triangle \(ABC\) is equal to the radius of the circle tangent to the smaller arc \(BC\) of the circle \(\omega\) and the side \(BC\) at its midpoint. Find the ratio of the sides of the triangle \(AB... | [
"\\(\\frac{1}{2}\\)"
] | [
"An isosceles triangle \\(ABC\\) with base \\(AC\\) is inscribed in a circle \\(\\omega\\). The radius of the inscribed circle of triangle \\(ABC\\) is equal to the radius of the circle tangent to the smaller arc \\(BC\\) of the circle \\(\\omega\\) and the side \\(BC\\) at its midpoint. Find the ratio of the sides... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4fbda316b6be1ff49575e4b43127b8c282281c1daa1ab6edf50d94967544c914 | An isosceles triangle has AB=AC=6. If D is a point on AB such that ADC is a right angle, and DC=4, find the area of the triangle. | [
"12"
] | [
"An isosceles triangle has AB=AC=6. If D is a point on AB such that ADC is a right angle, and DC=4, find the area of the triangle."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-133785586d76035ef842b06d7fe30ab101828285a83b7258d6d292cc1fa8b4de | An isosceles triangle has a base length $a$ and two equal sides of length $b$. It satisfies the equation $a^3 + \sqrt{2}b^3 = 3ab^2$. Find the measure of the vertex angle of the triangle. | [
"\\( 90^\\circ \\) or \\( 30^\\circ \\)"
] | [
"An isosceles triangle has a base length $a$ and two equal sides of length $b$. It satisfies the equation $a^3 + \\sqrt{2}b^3 = 3ab^2$. Find the measure of the vertex angle of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-07d24ce1468c43a0c3c8532782ee15e743e44b6953227d968145fd213489be9e | An isosceles triangle has a base of 12 and legs of length 10. If point P is on the angle bisector of one of the base angles and the distance from P to the base is 2, then what is the sum of the squares of the distances of P to each of the vertices? | [
"160"
] | [
"An isosceles triangle has a base of 12 and legs of length 10. If point P is on the angle bisector of one of the base angles and the distance from P to the base is 2, then what is the sum of the squares of the distances of P to each of the vertices?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-192a402a33356f2fc6cbb5e4bd254f564b96eeb8afc3a48f15703df629074319 | An isosceles triangle has a base of length 4 and sides of length 3. A line is drawn through the base and one side (but not through any vertex) such that it divides both the perimeter and the area in half. Find the lengths of the segments of the base defined by this line. | [
"2 and 2"
] | [
"An isosceles triangle has a base of length 4 and sides of length 3. A line is drawn through the base and one side (but not through any vertex) such that it divides both the perimeter and the area in half. Find the lengths of the segments of the base defined by this line."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-942564d40ba73caa3afdc23cf2aad5dedf85f04a0c4688d50b060673d6802ba2 | An isosceles triangle has a perimeter of 18 and integer side lengths. What is the smallest possible angle measure at one of its corners? | [
"\\(36.87^\\circ\\)"
] | [
"An isosceles triangle has a perimeter of 18 and integer side lengths. What is the smallest possible angle measure at one of its corners?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3da2ed11d32a19fd4523503753bb421d8564e156f07059499b69dc388cbf28bd | An isosceles triangle has a perimeter of 18 and integer side lengths. What is the smallest possible area of the triangle? | [
"\\(3\\sqrt{7}\\)"
] | [
"An isosceles triangle has a perimeter of 18 and integer side lengths. What is the smallest possible area of the triangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-403d22eb9e3d806091f11e81da6d1ad6777578f1e267c415fc620387f25b6f92 | An isosceles triangle has integer side lengths. A cevian separates the base into two segments of lengths 41 and 49. Find the smallest possible integer length of the cevian. | [
"140"
] | [
"An isosceles triangle has integer side lengths. A cevian separates the base into two segments of lengths 41 and 49. Find the smallest possible integer length of the cevian."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-ece8ac1e35ddbfcd166912ce73e0807af94753d6f9a27b4d4f43608f900094a8 | An isosceles triangle has its orthocenter lying on its incircle. Find the angles of the triangle. | [
"\\(\\cos^{-1} \\left(\\frac{2}{3}\\right), \\cos^{-1} \\left(\\frac{2}{3}\\right), 180^\\circ - 2 \\cos^{-1} \\left(\\frac{2}{3}\\right)\\)"
] | [
"An isosceles triangle has its orthocenter lying on its incircle. Find the angles of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-81da79d82d0bb1557bae35ee51206ab093b17dba11bb26e434a6ee5538a04c94 | An isosceles triangle has lateral sides of length $2$ and the angle between them is $120^\circ$. This triangle can be folded to form a regular tetrahedron with all edges equal to $1$. Calculate the volume of this regular tetrahedron. | [
"\\dfrac{\\sqrt{2}}{12}"
] | [
"An isosceles triangle has lateral sides of length $2$ and the angle between them is $120^\\circ$. This triangle can be folded to form a regular tetrahedron with all edges equal to $1$. Calculate the volume of this regular tetrahedron."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ee365a084c56ca143d443e5eb4c1ec13f5fb7071f6a669babd69fc6ce47f9c5a | An isosceles triangle has one angle equal to \( n \) where \( n < 90 \). Find all possible values of the other three angles in terms of \( n \). | [
"\\( n \\) and \\( 180 - 2n \\), or \\( \\frac{180 - n}{2} \\) and \\( \\frac{180 - n}{2} \\)"
] | [
"An isosceles triangle has one angle equal to \\( n \\) where \\( n < 90 \\). Find all possible values of the other three angles in terms of \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 20
} | 37 | 0 | 0 |
omr-ws-b52ab44c6831a8c33d86e01f7b2b6613c8008b7aa3c247e912c39be4d212ddc0 | An isosceles triangle has side lengths 8 cm, 8 cm and 10 cm. The longest side of a similar triangle is 25 cm. What is the perimeter of the larger triangle, in centimeters? | [
"65"
] | [
"An isosceles triangle has side lengths 8 cm, 8 cm and 10 cm. The longest side of a similar triangle is 25 cm. What is the perimeter of the larger triangle, in centimeters?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b11a3f00ae286933fed5fc4143792f32d3a13b37e5a489fb673624150fae753 | An isosceles triangle is inscribed into a circle of radius $R$. For what magnitude of the vertex angle $\alpha $ of the triangle is the height $H$ drawn to the lateral side, of the greatest length? Find that length. | [
"$\\alpha =2\\arcsin \\frac{\\sqrt 3}{3},\\ H=\\frac{8R}{3\\sqrt 3}$"
] | [
"An isosceles triangle is inscribed into a circle of radius $R$. For what magnitude of the vertex angle $\\alpha $ of the triangle is the height $H$ drawn to the lateral side, of the greatest length? Find that length."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-52cddb148552384199d7442b2e1b85ab6875dbae4cfe07d1e83ef47729bc825a | An isosceles triangle with a vertex angle of $120^\circ$ is circumscribed about a circle of radius $R$. Find its sides. | [
"$AB=BC=\\frac{2R\\left(\\sqrt 3+2\\right)}{\\sqrt 3},\\ AC=2R\\left(\\sqrt 3+2\\right)$"
] | [
"An isosceles triangle with a vertex angle of $120^\\circ$ is circumscribed about a circle of radius $R$. Find its sides."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b248d1710fa735e05c3d7586846c1e715b4b8cf8b50d4106bc493611f8572f6 | An isosceles triangle with base \( B \) and height \( H \) has an infinite sequence of squares inscribed within it. Each square has its top two vertices on the sides of the triangle, with the base of the bottom square on the base of the triangle and the bases of the remaining squares on the top sides of the previous sq... | [
"\\(\\frac{H^2B}{2H+B}\\)"
] | [
"An isosceles triangle with base \\( B \\) and height \\( H \\) has an infinite sequence of squares inscribed within it. Each square has its top two vertices on the sides of the triangle, with the base of the bottom square on the base of the triangle and the bases of the remaining squares on the top sides of the pr... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-2cb77efd7bf003d2818543e295fc2f2e95ccb386a82935c4a6467740d4879c6d | An isosceles triangle with some 30-degree angles has its base angles quintisected (divided into 5 parts). One of the quintisection intersections forms a 90-degree angle. Perpendiculars are drawn from the points where each quintisection hits a leg of the triangle, forming an inner rectangle. If the area of this rectangl... | [
"2"
] | [
"An isosceles triangle with some 30-degree angles has its base angles quintisected (divided into 5 parts). One of the quintisection intersections forms a 90-degree angle. Perpendiculars are drawn from the points where each quintisection hits a leg of the triangle, forming an inner rectangle. If the area of this rec... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-17211fc31e01283b534cc551592c5dfb0cd8392aa0bf55a7f36086bf213022ea | An isosceles, obtuse triangle has one angle with a degree measure that is 50$\%$ larger than the measure of a right angle. What is the measure, in degrees, of one of the two smallest angles in the triangle? Express your answer as a decimal to the nearest tenth. | [
"22.5^{\\circ}"
] | [
"An isosceles, obtuse triangle has one angle with a degree measure that is 50$\\%$ larger than the measure of a right angle. What is the measure, in degrees, of one of the two smallest angles in the triangle? Express your answer as a decimal to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d0e7d00d421249fa0be06031a342bb39e6280188c5f82b6a0aeafdd1965c665b | An item at a department store is priced at $189.99 and can be bought by making 20 payments of $10.50. Find the interest rate, assuming that interest is compounded monthly. | [
"11.68%"
] | [
"An item at a department store is priced at $189.99 and can be bought by making 20 payments of $10.50. Find the interest rate, assuming that interest is compounded monthly."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 21
} | 23 | 0 | 0 |
omr-ws-4e6211375291567a2ab48ad12921efb2ad10fc792361dbae1861549ff9bf3d53 | An iterative average of the numbers $1$, $2$, $3$, $4$, and $5$ is computed by arranging the five numbers in some order, finding the mean of the first two numbers, then the mean of that with the third number, and so on until the fifth number. What is the difference between the largest and smallest possible values that ... | [
"\\(\\frac{17}{8}\\)"
] | [
"An iterative average of the numbers $1$, $2$, $3$, $4$, and $5$ is computed by arranging the five numbers in some order, finding the mean of the first two numbers, then the mean of that with the third number, and so on until the fifth number. What is the difference between the largest and smallest possible values ... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d2a18d128410337c480c0122df889695d63d4d3ee060ace9c78d6fc3f4ea7536 | An ivy plant grows to a height of twelve feet after one year, eighteen feet after two years, and twenty-seven feet after three years. How high was it after four years? | [
"40.5"
] | [
"An ivy plant grows to a height of twelve feet after one year, eighteen feet after two years, and twenty-seven feet after three years. How high was it after four years?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-66eeb8730540ae8fe1b7e0e7b121a9cacfb3c46871c35d28639d0fdbfcbd67de | An n-tuple of real numbers \(a_1, a_2, \dots, a_n\) is called "good" if \(a_i \geq \frac{a_{i-1} + a_{i+1}}{2}\) for all \(2 \leq i \leq n-1\). Find the maximum value \(c > 0\) (in terms of \(n\)) such that for all good n-tuples of non-negative real numbers, the inequality \(\sum_{i=1}^{n} i a_i^2 \geq c \sum_{i=1}^{n}... | [
"\\dfrac{n(n+1)}{2(2n-1)}"
] | [
"An n-tuple of real numbers \\(a_1, a_2, \\dots, a_n\\) is called \"good\" if \\(a_i \\geq \\frac{a_{i-1} + a_{i+1}}{2}\\) for all \\(2 \\leq i \\leq n-1\\). Find the maximum value \\(c > 0\\) (in terms of \\(n\\)) such that for all good n-tuples of non-negative real numbers, the inequality \\(\\sum_{i=1}^{n} i a_i... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-d5cfd0935789b8f7bbf437ae80e1e11c3923d08390c6178bb0bd88c1e2c0876b | An o-Pod MP3 player stores and plays entire songs. Celeste has 10 songs stored on her o-Pod. The time length of each song is different. When the songs are ordered by length, the shortest song is only 30 seconds long and each subsequent song is 30 seconds longer than the previous song. Her favorite song is 3 minutes, 30... | [
"\\dfrac{79}{90}"
] | [
"An o-Pod MP3 player stores and plays entire songs. Celeste has 10 songs stored on her o-Pod. The time length of each song is different. When the songs are ordered by length, the shortest song is only 30 seconds long and each subsequent song is 30 seconds longer than the previous song. Her favorite song is 3 minute... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2e9e53499c4ce5cf6bcb48f5f22c4adb5f65661f1c6a14cf0c8c032640f808a2 | An object begins to move from a point 1 cm from the origin with a velocity given by \( \sqrt{t} - \frac{1}{\sqrt{t}} \). Find the distance traveled after 4 seconds. | [
"3"
] | [
"An object begins to move from a point 1 cm from the origin with a velocity given by \\( \\sqrt{t} - \\frac{1}{\\sqrt{t}} \\). Find the distance traveled after 4 seconds."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-138a40d653a7e3b3b60471d87eece8d366fbce82f767d1b9f87ba30371f37d46 | An object has a chance \( p \) of moving 1 step left and a chance \( (1-p) \) of moving 1 step right. What is the expected number of steps the object must take to move to a point \( n \) steps to the right of its starting point if the object cannot move left of its starting point? | [
"\\(\\frac{n}{1 - p}\\)"
] | [
"An object has a chance \\( p \\) of moving 1 step left and a chance \\( (1-p) \\) of moving 1 step right. What is the expected number of steps the object must take to move to a point \\( n \\) steps to the right of its starting point if the object cannot move left of its starting point?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-549f0012bd702673921c3f0e53e230cf5cdf8c0e060ebcd76d3e7024eaa7276f | An object has a chance \( p \) of moving 1 step left and a chance \( (1-p) \) of moving 1 step right. What is the expected number of steps the object must take to move to a point \( n \) steps to the right of its starting point if the object's movement is unrestricted? | [
"\\(\\frac{n}{1-2p}\\)"
] | [
"An object has a chance \\( p \\) of moving 1 step left and a chance \\( (1-p) \\) of moving 1 step right. What is the expected number of steps the object must take to move to a point \\( n \\) steps to the right of its starting point if the object's movement is unrestricted?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-169bad8ae2ae59a928eb8002000e87e8b64e01d311108e23b3f78fe473cd1e55 | An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? | [
"121"
] | [
"An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 5
} | 53 | 0 | 0 |
omr-ws-63a82018cc0704ae832bd6dc2d339eddba1d18a7d9ef1599d985fac4373b5186 | An object is at \((x(t), y(t))\) at time \(t \geq 0\) with \(x'(t) = 12t - 3t^2\) and \(y'(t) = \ln(1 + (t-4)^4)\). If \((x(0), y(0)) = (-13, 5)\) and the object has an \(x\)-coordinate of 3 when \(t = 2\), find its \(y\)-coordinate at \(t = 2\). | [
"13.671"
] | [
"An object is at \\((x(t), y(t))\\) at time \\(t \\geq 0\\) with \\(x'(t) = 12t - 3t^2\\) and \\(y'(t) = \\ln(1 + (t-4)^4)\\). If \\((x(0), y(0)) = (-13, 5)\\) and the object has an \\(x\\)-coordinate of 3 when \\(t = 2\\), find its \\(y\\)-coordinate at \\(t = 2\\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 23
} | 33 | 0 | 0 |
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