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-262ae670dc82ab51b693146e7a0ab438c675ffd4db68837c6a217c85850ebdba | Build a function \( f : \mathbb{Q} \to \mathbb{R}_+ \) such that every point is a local, strict minimum point and \( f \) is unbounded on \( I \cap \mathbb{Q} \), where \( I \) is a non-degenerate interval. | [
"\\( f(x) = g(q) \\) for \\( x = \\frac{p}{q} \\) with \\( (p, q) = 1 \\) and \\( p \\geq 0 \\), and \\( f(x) = 2 \\) for \\( x \\) irrational, where \\( g(x) = \\arctan(x) + 1 \\)."
] | [
"Build a function \\( f : \\mathbb{Q} \\to \\mathbb{R}_+ \\) such that every point is a local, strict minimum point and \\( f \\) is unbounded on \\( I \\cap \\mathbb{Q} \\), where \\( I \\) is a non-degenerate interval."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9911303d4f4ee56f4ff632e1dae69108dbac72e2e754d4d60df075cc7a270c12 | Build a function \( f : \mathbb{R} \to \mathbb{R}_+ \) such that all \( x \in \mathbb{Q} \) are local, strict minimum points. | [
"\\( f(x) = \\begin{cases} \n\\mathcal{C} - \\frac{1}{q} & \\text{if } x = \\frac{p}{q}, \\text{ where } p, q \\in \\mathbb{Z}, (p,q) = 1, q > 0, \\\\\n\\mathcal{C} & \\text{if } x \\text{ is irrational.}\n\\end{cases} \\)"
] | [
"Build a function \\( f : \\mathbb{R} \\to \\mathbb{R}_+ \\) such that all \\( x \\in \\mathbb{Q} \\) are local, strict minimum points."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-8c20a03b179c3ff6f1eba909ea9e08ac65089ab10e923eb8bbd0ba79d86835eb | Build a function \( f: \mathbb{R}^+ \rightarrow \mathbb{R}^+ \) such that \[ 2f(x^2) \geq xf(x) + x \] for every \( x \in \mathbb{R}^+ \). | [
"\\( f(x) = x + 1 \\)"
] | [
"Build a function \\( f: \\mathbb{R}^+ \\rightarrow \\mathbb{R}^+ \\) such that\n\\[ 2f(x^2) \\geq xf(x) + x \\]\nfor every \\( x \\in \\mathbb{R}^+ \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 40,
"genselect": 6,
"tir": 12
} | 58 | 0 | 0 |
omr-ws-00538b3073516ad5daf747fb3242a90e6ec1f9bdbb02a24aabba90a9deb80979 | Build a function \( f: \mathbb{Z}^+ \to \mathbb{Z}^+ \) such that: i) \( f(1) = 1 \) ii) \( \forall x \in \mathbb{Z}^+: f(f(x)) = f(x) \) iii) \( \forall p \in \mathbb{P}, x \in \mathbb{Z}^+, 2 \mid v_p(x) : f(px) = p^2 f(x) \) | [
"\\( f(x) = \\prod_{i=1}^k p_i^{2 \\lfloor \\frac{e_i}{2} \\rfloor} \\)"
] | [
"Build a function \\( f: \\mathbb{Z}^+ \\to \\mathbb{Z}^+ \\) such that:\ni) \\( f(1) = 1 \\)\nii) \\( \\forall x \\in \\mathbb{Z}^+: f(f(x)) = f(x) \\)\niii) \\( \\forall p \\in \\mathbb{P}, x \\in \\mathbb{Z}^+, 2 \\mid v_p(x) : f(px) = p^2 f(x) \\)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 25
} | 45 | 0 | 0 |
omr-ws-dcdcd77a6e6c59ae884e85349564f0d0d3732faf7b6d6962df8ccc797d042400 | Build a parallelepipedical container with a volume of $V = 20 \text{ m}^3$. The cost of the material for the lateral faces is $1 \text{ dollar/m}^2$, for the bottom face it is $2 \text{ dollars/m}^2$, and for the upper face it is $3 \text{ dollars/m}^2$. What should be the dimensions of the container to minimize the to... | [
"\\(x = 2\\), \\(y = 2\\), \\(z = 5\\)"
] | [
"Build a parallelepipedical container with a volume of $V = 20 \\text{ m}^3$. The cost of the material for the lateral faces is $1 \\text{ dollar/m}^2$, for the bottom face it is $2 \\text{ dollars/m}^2$, and for the upper face it is $3 \\text{ dollars/m}^2$. What should be the dimensions of the container to minimi... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 27
} | 37 | 0 | 0 |
omr-ws-928c7b7f10c262df074fdda76ec0cfae9192e9411bff344c240abe16f9e5a5a3 | Build the addition and multiplication operation tables for $\mathbb{Z}_5$. | [
"\\text{Addition and multiplication tables for } \\mathbb{Z}_5 \\text{ as shown above.}"
] | [
"Build the addition and multiplication operation tables for $\\mathbb{Z}_5$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-3ab6537d9d173943cdc270970a4b1532fe39e38135d3498311e13cf148e9b823 | Build the addition and multiplication operation tables for $\mathbb{Z}_6$. | [
"\\text{Addition and multiplication tables for } \\mathbb{Z}_6 \\text{ as shown above.}"
] | [
"Build the addition and multiplication operation tables for $\\mathbb{Z}_6$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-317fc528bb2c60c9a05d5ede48a0f49dad272a6c6b46011cc557d0fd0ab1fefd | Build the lowest sigma-algebra generated by the set \(\{A_1, A_2, \ldots, A_N\}\). | [
"\\sigma(\\{A_1, A_2, \\ldots, A_N\\}) = \\left\\{ \\bigcup_{J \\subseteq \\{1,2,\\ldots,N\\}} \\left( \\bigcap_{j \\in J} A_j \\cap \\bigcap_{j \\notin J} A_j^c \\right) \\mid \\text{non-empty intersections only} \\right\\}"
] | [
"Build the lowest sigma-algebra generated by the set \\(\\{A_1, A_2, \\ldots, A_N\\}\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-be33e7e27b95075e30a605f42eb52ae0e720b3c099613efb5e0ed257ef649467 | Build the set of points $(x, y)$ on the coordinate plane that satisfies the equality: \[ \sqrt{1-x^2}+\sqrt{1-y^2}=2-x^2-y^2. \] | [
"\\{(0,0), (0,1), (0,-1), (1,0), (-1,0), (1,1), (1,-1), (-1,1), (-1,-1)\\}"
] | [
"Build the set of points $(x, y)$ on the coordinate plane that satisfies the equality:\n\\[ \\sqrt{1-x^2}+\\sqrt{1-y^2}=2-x^2-y^2. \\]"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 40,
"tir": 24
} | 64 | 0 | 0 |
omr-ws-16978fccc3ac9c7e989d010cee7eae42e436ebff42a498c0fa00ba1d1e110bb5 | Bulbs produced by a factory $F_i$ have lifetimes (in months) distributed as $exp(\frac{1}{3^i})$ for $i=1,2,3$. A firm randomly procures 40% of its required bulbs from $F_1$, 30% from $F_2$, and 30% from $F_3$. A randomly selected bulb from the firm is found to be working after 27 months. What is the probability that i... | [
"\\(\\frac{0.3 e^{-1}}{0.4 e^{-9} + 0.3 e^{-3} + 0.3 e^{-1}}\\)"
] | [
"Bulbs produced by a factory $F_i$ have lifetimes (in months) distributed as $exp(\\frac{1}{3^i})$ for $i=1,2,3$. A firm randomly procures 40% of its required bulbs from $F_1$, 30% from $F_2$, and 30% from $F_3$. A randomly selected bulb from the firm is found to be working after 27 months. What is the probability ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 1,
"tir": 2
} | 13 | 0 | 0 |
omr-ws-552dc60457e8b02b1fa9cdd7f0f717bc1ea1e7cc4b7ebf4419391cb123200488 | Bumps and Grimes play a game with a standard six-sided die. The first player places the die on the table, and the score is given by its top face. Players alternately roll the die through a quarter turn to bring a new, adjacent face to the top, adding the value of the top face to the total. The first to make the total e... | [
"Bumps"
] | [
"Bumps and Grimes play a game with a standard six-sided die. The first player places the die on the table, and the score is given by its top face. Players alternately roll the die through a quarter turn to bring a new, adjacent face to the top, adding the value of the top face to the total. The first to make the to... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-f61dfdfdad42aa617648d72edc6572865dfff4be8ade6e664db983bd0f1b3d45 | Burrito Bear has a white unit square. She inscribes a circle inside the square and paints it black. She then inscribes a square inside the black circle and paints it white. She repeats this process indefinitely. The total black area can be expressed as $\frac{a\pi+b}{c}$. Find $a+b+c$. | [
"1"
] | [
"Burrito Bear has a white unit square. She inscribes a circle inside the square and paints it black. She then inscribes a square inside the black circle and paints it white. She repeats this process indefinitely. The total black area can be expressed as $\\frac{a\\pi+b}{c}$. Find $a+b+c$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-725c363e50d095d9a7043394fc77a397d12f3b3c5f87c952efd2a731a7d5a6fa | Bus $A$ comes every 20 minutes and bus $B$ comes every 25 minutes. If you arrive at the bus stop at a random time and can take either bus to get home, what is the probability that your wait will be less than 5 minutes? | [
"\\(\\frac{2}{5}\\)"
] | [
"Bus $A$ comes every 20 minutes and bus $B$ comes every 25 minutes. If you arrive at the bus stop at a random time and can take either bus to get home, what is the probability that your wait will be less than 5 minutes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-0b3c3a0cf8d66f9307382af2d918846b7a559a748b0d0d96d288d3a2dbdde9af | Buses from Dallas to Houston leave every hour on the hour, and buses from Houston to Dallas leave every hour on the half hour. The trip between the two cities takes 5 hours. How many Dallas-bound buses does a Houston-bound bus pass on the highway (not in the station)? | [
"10"
] | [
"Buses from Dallas to Houston leave every hour on the hour, and buses from Houston to Dallas leave every hour on the half hour. The trip between the two cities takes 5 hours. How many Dallas-bound buses does a Houston-bound bus pass on the highway (not in the station)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-152af6e882a2bee5d9c831bf66278cd1a1b1a2d42c6834e1c602e47352977959 | Buses leave the airport every 3 minutes to drive to the city centre. A car leaves the airport at the same time as one bus and drives to the city centre by the same route. It takes each bus 60 minutes and the car 35 minutes to drive from the airport to the city centre. How many buses does the car pass on its way to the ... | [
"8"
] | [
"Buses leave the airport every 3 minutes to drive to the city centre. A car leaves the airport at the same time as one bus and drives to the city centre by the same route. It takes each bus 60 minutes and the car 35 minutes to drive from the airport to the city centre. How many buses does the car pass on its way to... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-3fc9bf3917143cd5efbdbabdd8881111df88faf73e9f5ffad003fc01693fb8aa | Business sales are at \$5,000,000 in March. Sales are expected to increase by 10 percent in April, 15 percent in May, and 30 percent in June. What would be the expected total earnings in June? | [
"8,222,500"
] | [
"Business sales are at \\$5,000,000 in March. Sales are expected to increase by 10 percent in April, 15 percent in May, and 30 percent in June. What would be the expected total earnings in June?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 53
} | 85 | 0 | 0 |
omr-ws-2b0340795c72dc9ff69322aeb5fa1425948eecbbaa9254f693ccde1007468666 | Butch and Sundance need to get out of Dodge by alternating walking and riding their only horse, Sparky. Butch walks at 4 miles per hour, Sundance walks at 2.5 miles per hour, and Sparky rides at 6 miles per hour. They meet at a milepost $n$ miles from Dodge after traveling for $t$ minutes. Find $n + t$ given that the f... | [
"279."
] | [
"Butch and Sundance need to get out of Dodge by alternating walking and riding their only horse, Sparky. Butch walks at 4 miles per hour, Sundance walks at 2.5 miles per hour, and Sparky rides at 6 miles per hour. They meet at a milepost $n$ miles from Dodge after traveling for $t$ minutes. Find $n + t$ given that ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 8,
"tir": 1
} | 46 | 0 | 0 |
omr-ws-9872bdb284fd1269c2a329e904b55400b53c916df8fecb3b188ec48f3aa9c0ad | Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins by walking while Sundance rides. When Sundance reaches the first of the hitching posts that are conveniently located at one-mile intervals along their rout... | [
"279"
] | [
"Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins by walking while Sundance rides. When Sundance reaches the first of the hitching posts that are conveniently located at one-mile intervals along their... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-43bd039ec673e7c6f62919708ae20d60e19d217b40a1a1e7ed9c8f08097ce1a5 | Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins by walking while Sundance rides. When Sundance reaches the first of the hitching posts that are conveniently located at one-mile intervals along their rout... | [
"96"
] | [
"Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins by walking while Sundance rides. When Sundance reaches the first of the hitching posts that are conveniently located at one-mile intervals along their... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-c94730e7a656b482ee2507b8939266569181ddaae3e59c400d32d890dfe984f4 | Butch and Sundance need to travel as quickly as possible by alternating walking and riding their only horse, Sparky. Butch walks at 4 miles per hour, Sundance walks at 2.5 miles per hour, and Sparky rides at 6 miles per hour. They use hitching posts located at one-mile intervals. The first time Butch and Sundance meet ... | [
"279"
] | [
"Butch and Sundance need to travel as quickly as possible by alternating walking and riding their only horse, Sparky. Butch walks at 4 miles per hour, Sundance walks at 2.5 miles per hour, and Sparky rides at 6 miles per hour. They use hitching posts located at one-mile intervals. The first time Butch and Sundance ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 8,
"tir": 2
} | 35 | 0 | 0 |
omr-ws-49a1c7ab94f491ad1cfeaa121b72ac4c2d75cfaef7b79769d553f050e0636a94 | Butch and Sundance need to travel as quickly as possible by alternating walking and riding their only horse, Sparky. They use hitching posts located at one-mile intervals. Butch begins walking as Sundance rides. When Sundance reaches the first hitching post, he ties Sparky and starts walking. When Butch reaches Sparky,... | [
"\\dfrac{(b + 1)(c + 1) - (a + 1)^2}{c - b}"
] | [
"Butch and Sundance need to travel as quickly as possible by alternating walking and riding their only horse, Sparky. They use hitching posts located at one-mile intervals. Butch begins walking as Sundance rides. When Sundance reaches the first hitching post, he ties Sparky and starts walking. When Butch reaches Sp... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-b0a0b43b6096421fb0d93cf484572bdec5878674e39db55d9b306c5ac34ac03b | Button B now adds n to the four-digit display, while button A remains the same. For how many positive integers n ≤ 20 (including 11) can every possible four-digit output be reached? | [
"8"
] | [
"Button B now adds n to the four-digit display, while button A remains the same. For how many positive integers n ≤ 20 (including 11) can every possible four-digit output be reached?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-ae59baf1310a30e2002fa1a6ea4db1808bce69c50dbd4a5fe3a89d2f1abc22b9 | By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called nice if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers? | [
"182"
] | [
"By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called nice if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9a2df4d24be7ea4a73532f2d1f3d797a6a001f9584aa123642c6825d4c4900e6 | By a signal of the trainer two ponies started simultaneously and ran uniformly along the outer circumference of the circus ring in opposite directions. The first pony ran faster than the second and by the moment of their meeting it covered 5 m more than the second. Continuing the run, the first pony ran to the trainer ... | [
"11 m approx"
] | [
"By a signal of the trainer two ponies started simultaneously and ran uniformly along the outer circumference of the circus ring in opposite directions. The first pony ran faster than the second and by the moment of their meeting it covered 5 m more than the second. Continuing the run, the first pony ran to the tra... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9dac12f2712600a7e309608d9dc9b63c1fa68d8f951900740aed966f840c0c5b | By an integer triangle we shall simply mean a triangle the lengths of whose sides are expressed by whole numbers. Find all integer triangles each of which has a perimeter equal to its area. Problem 171 belongs to an important division of the theory of integral solutions of equations, and we do not go into detail here b... | [
"Denote $p - a= x$, $p- b = y$, $p- c = z$ where $a$, $b$ and $c$ are the sides of the triangle and $p$ is half the perimeter of the triangle: $p = (a+ b + c) /2$. Then the problem reduces to the determination of the integral solutions of the equation $xyz=4(x+y+z)$ or of the equation $x= (4y+4z)/(yz-4)$. The condi... | [
"By an integer triangle we shall simply mean a triangle the lengths of whose sides are expressed by whole numbers. Find all integer triangles each of which has a perimeter equal to its area. Problem 171 belongs to an important division of the theory of integral solutions of equations, and we do not go into detail h... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9667bae2587580c85e793188a72b16fdeff943ddde2734e027364a8d0315bf5a | By approximately what percentage does the volume of a sphere increase if the radius increases from 10 to 11 units? | [
"30%"
] | [
"By approximately what percentage does the volume of a sphere increase if the radius increases from 10 to 11 units?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 22
} | 44 | 0 | 0 |
omr-ws-8ee61776f1845602539a9a8c5f25ee8ec829c226f3691dda84e2a040a33e49df | By approximating using straight lines, find the area between the curve \( y = a^{\sin(\pi e^x)} \) and the x-axis between the \( k \)th and \( (k+1) \)th maxima. | [
"\\(\\frac{a^{2}+1}{2a}\\cdot \\ln \\frac{4k+5}{4k+1}\\)"
] | [
"By approximating using straight lines, find the area between the curve \\( y = a^{\\sin(\\pi e^x)} \\) and the x-axis between the \\( k \\)th and \\( (k+1) \\)th maxima."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-a72c2ad02a558ea53eaa36910828f907711efc010109b53e26e485b87808a4d6 | By choosing a suitable value of $n$, deduce that $2.5 < e < 2.99$. | [
"\\( 2.5 < e < 2.99 \\)"
] | [
"By choosing a suitable value of $n$, deduce that $2.5 < e < 2.99$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 4,
"tir": 3
} | 22 | 0 | 0 |
omr-ws-adbf35196959ace43fa6d550640023c62e68956304f8587b39edd290ff5e9815 | By choosing from the 10 women at the party, how many ways are there to assign the 5 men a dancing partner each such that exactly two men are dancing with their wives? | [
"2270"
] | [
"By choosing from the 10 women at the party, how many ways are there to assign the 5 men a dancing partner each such that exactly two men are dancing with their wives?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-9fa5927f631a91a9815b3a19e4de09fcd19de43c363aba106f908592f9041a8e | By choosing the function \( R(x) \) to be of the form \( a + bx + cx^2 \), find \(\int \frac{5x^2 - 4x - 3}{(1 + 2x + 3x^2)^2} \, dx\). | [
"\\(\\frac{a + (-3+2a)x + (3a-2)x^2}{1+2x+3x^2} + C\\)"
] | [
"By choosing the function \\( R(x) \\) to be of the form \\( a + bx + cx^2 \\), find \\(\\int \\frac{5x^2 - 4x - 3}{(1 + 2x + 3x^2)^2} \\, dx\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-c5f4b51a8ac8776a4687f26536d5b0279d69463ef34dc1ecd6d1d96dbb9ec4e2 | By combining the three basic colors red, yellow, and blue, other colors can be formed. Suppose there are $5$ cans of red paint, $5$ cans of yellow, and $5$ cans of blue. Budi can choose any can to mix colors, and all paint in a can must be used all. How many color choices are there? | [
"215"
] | [
"By combining the three basic colors red, yellow, and blue, other colors can be formed. Suppose there are $5$ cans of red paint, $5$ cans of yellow, and $5$ cans of blue. Budi can choose any can to mix colors, and all paint in a can must be used all. How many color choices are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-a5743dccc9c4a6fe11019a673446d890e11e6adec16850041c64c039f1b2d3a8 | By comparing with minimal further calculation, find the inverse of the matrix \(\begin{pmatrix} 3 & 1 & 2 \\ 6 & 1 & 3 \\ -3 & 2 & 2 \end{pmatrix}\). | [
"\\(\\begin{pmatrix} \\frac{4}{3} & -\\frac{2}{3} & -\\frac{1}{3} \\\\ 7 & -4 & -1 \\\\ -5 & 3 & 1 \\end{pmatrix}\\)"
] | [
"By comparing with minimal further calculation, find the inverse of the matrix \\(\\begin{pmatrix} 3 & 1 & 2 \\\\ 6 & 1 & 3 \\\\ -3 & 2 & 2 \\end{pmatrix}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 25
} | 39 | 0 | 0 |
omr-ws-b6f9b2a68c5738370cb27cd7e7b8b650d9a33605f21471e4943d8251dfdd67e4 | By composing a symmetry of axis $r$ with a right angle rotation around a point $P$ that does not belong to the line, another movement $M$ results. Is $M$ an axis symmetry? Is there any line invariant under $M$? | [
"\\text{Yes}"
] | [
"By composing a symmetry of axis $r$ with a right angle rotation around a point $P$ that does not belong to the line, another movement $M$ results. Is $M$ an axis symmetry? Is there any line invariant under $M$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-427f4eaa2584808b72203b544ba54f1c5a9115599987aad89dd3a1ed851555b0 | By considering another set of \( n \) rectangles of equal width, find a similar expression for an upper bound, \( U \), for the area given by \( \int_{3}^{6} \ln (\ln(x)) \, dx \). | [
"U = \\frac{3}{n} \\sum_{i=1}^{n} \\ln\\left(\\ln\\left(3 + \\frac{3i}{n}\\right)\\right)"
] | [
"By considering another set of \\( n \\) rectangles of equal width, find a similar expression for an upper bound, \\( U \\), for the area given by \\( \\int_{3}^{6} \\ln (\\ln(x)) \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0e51ed740e5c34d12e4aed9c1bdefe09374de943c6f63e77dddd1865e038f36f | By considering the function \( f(z) = \frac{e^{az}}{z} \) and the contour \( C: z = e^{i\theta}, \ (-\pi \le \theta \le \pi) \), calculate the integral \( \int_{0}^\pi e^{a\cos\theta} \cos(a\sin\theta) \ d\theta \). | [
"\\(\\pi\\)"
] | [
"By considering the function \\( f(z) = \\frac{e^{az}}{z} \\) and the contour \\( C: z = e^{i\\theta}, \\ (-\\pi \\le \\theta \\le \\pi) \\), calculate the integral \\( \\int_{0}^\\pi e^{a\\cos\\theta} \\cos(a\\sin\\theta) \\ d\\theta \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-6f0aaa768743759f085374e7f7f44c9eab6c94fc6d78cf613a455eb2c729643b | By considering the graphs of \( y = x^3 \) and \( y = s - rx \), where \( r \) and \( s \) are positive, and by differentiating \( y = x^3 + rx - s \) and using the factor theorem, find the conditions for which the equation \( x^3 + rx - s = 0 \) has only one real root. | [
"\\( r > 0 \\) and \\( s > 0 \\)"
] | [
"By considering the graphs of \\( y = x^3 \\) and \\( y = s - rx \\), where \\( r \\) and \\( s \\) are positive, and by differentiating \\( y = x^3 + rx - s \\) and using the factor theorem, find the conditions for which the equation \\( x^3 + rx - s = 0 \\) has only one real root."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 26
} | 44 | 0 | 0 |
omr-ws-694adc19149a72c0273388d7aa185d8540f1d37528c118943d5b7c2fc3133fb2 | By considering the imaginary part of the equation \( z^7 = 1 \), find all the roots of the equation \( t^6 - 21t^4 + 35t^2 - 7 = 0 \). | [
"\\( t = -\\tan\\frac{k\\pi}{7} \\), where \\( k \\in \\{1,2,3,4,5,6\\} \\)"
] | [
"By considering the imaginary part of the equation \\( z^7 = 1 \\), find all the roots of the equation \\( t^6 - 21t^4 + 35t^2 - 7 = 0 \\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-bbae4a83203068bd93e8213bb2cb2920261762f8329ceea3220b5481cc0f93bf | By considering the three cases \( 4p^3 + 27q^2 < 0 \), \( 4p^3 + 27q^2 = 0 \), and \( 4p^3 + 27q^2 > 0 \), where \( p < 0 \) and \( q < 0 \), what can be deduced about the number and nature of the roots of the equation \( x^3 + px + q = 0 \)? | [
"- For \\( 4p^3 + 27q^2 < 0 \\): Three distinct real roots.\n- For \\( 4p^3 + 27q^2 = 0 \\): One real root with multiplicity 2 and one distinct real root.\n- For \\( 4p^3 + 27q^2 > 0 \\): One real root and two complex conjugate roots."
] | [
"By considering the three cases \\( 4p^3 + 27q^2 < 0 \\), \\( 4p^3 + 27q^2 = 0 \\), and \\( 4p^3 + 27q^2 > 0 \\), where \\( p < 0 \\) and \\( q < 0 \\), what can be deduced about the number and nature of the roots of the equation \\( x^3 + px + q = 0 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 26
} | 44 | 0 | 0 |
omr-ws-8243ce4e4ed102deef302d574ed9f916e7d9f37feeec977aefcaee6a584c7cee | By dividing 100,000 by an integer with 3 different digits, Mathias obtains a quotient and an integer residue. The quotient is written with the same digits as the divisor, but in reverse order. What is the divisor? | [
"320"
] | [
"By dividing 100,000 by an integer with 3 different digits, Mathias obtains a quotient and an integer residue. The quotient is written with the same digits as the divisor, but in reverse order. What is the divisor?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-55f8daab1e2af19599ea2f31135a950ebe4eab539825189310ad3d4a91a3a732 | By dividing 100000 by an integer with 3 different digits, Mathias obtains a quotient and an integer residue. The quotient is written with the same digits as the divisor, but in reverse order. What is the divisor? | [
"175"
] | [
"By dividing 100000 by an integer with 3 different digits, Mathias obtains a quotient and an integer residue. The quotient is written with the same digits as the divisor, but in reverse order. What is the divisor?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 6,
"tir": 32
} | 59 | 0 | 0 |
omr-ws-504351e2d65088bf0975c3476a6cc64240d3abd4edaf7aed86840ec482a45ed7 | By dividing the integer $m$ by the integer $n$, the quotient is $22$ and the remainder is $5$. When the remainder $5$ is divided by $n$, the new quotient is $0.4$ and the new remainder is $0.2$. Find the values of $m$ and $n$. | [
"\\( m = 489 \\), \\( n = 22 \\)"
] | [
"By dividing the integer $m$ by the integer $n$, the quotient is $22$ and the remainder is $5$. When the remainder $5$ is divided by $n$, the new quotient is $0.4$ and the new remainder is $0.2$. Find the values of $m$ and $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b676ba03a5ddc90ffbb5ca6ea4ab0d35d08138a1bdda5d0f7151d926c0fcbe1 | By evaluating the expression for expected profit for different values of \( s \), determine how many papers the newspaper seller should stock. | [
"26"
] | [
"By evaluating the expression for expected profit for different values of \\( s \\), determine how many papers the newspaper seller should stock."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-551951645ad7f564bf8c626f170b371df77d0c932ea820eb1d510219ca12d9c7 | By expanding $(1-4x^{2})^{\frac{-1}{2}}$ and integrating term by term, find the series expansion for $\sin^{-1}2x$ when $\mid x \mid < \frac{1}{2}$ as far as the term in $x^{7}$. | [
"\\(2x+\\frac{4x^3}{3}+\\frac{12x^5}{5}+\\frac{40x^7}{7}\\)"
] | [
"By expanding $(1-4x^{2})^{\\frac{-1}{2}}$ and integrating term by term, find the series expansion for $\\sin^{-1}2x$ when $\\mid x \\mid < \\frac{1}{2}$ as far as the term in $x^{7}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-f5a11609839455d693f5f9be9fe1ff8aef062831084e1818e1c037ef9cc728a4 | By expanding (x − 1.95)², find a number k such that P = −20(x − 1.95)² + k. | [
"64.8"
] | [
"By expanding (x − 1.95)², find a number k such that P = −20(x − 1.95)² + k."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-666aa80b5465f1e6fb62698f588ef11ac0b6687b96ae550058be331e6c43cde2 | By expanding \( f_1(z) \) as a Laurent series valid for \( |z| > 2n \pi \), and \( f_2(z) \) as a Taylor series valid for \( |z| < 2(n + 1)\pi \), find the coefficients of \( z^j \) for \( -1 \le j \le 1 \) in the Laurent series for \( f(z) \) valid for \( 2n\pi < |z| < 2(n+1) \pi \). | [
"0"
] | [
"By expanding \\( f_1(z) \\) as a Laurent series valid for \\( |z| > 2n \\pi \\), and \\( f_2(z) \\) as a Taylor series valid for \\( |z| < 2(n + 1)\\pi \\), find the coefficients of \\( z^j \\) for \\( -1 \\le j \\le 1 \\) in the Laurent series for \\( f(z) \\) valid for \\( 2n\\pi < |z| < 2(n+1) \\pi \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-d7ab803705fdd338a75e5fa53d5d0e77ffb614409dd69cca440b4fe5b6d7cd6c | By folding a rectangular sheet of paper with length \( x \) and width \( y \) in half along one of its diagonals, the "visible" area (which becomes a pentagon) is reduced by 30%. What is the ratio \( \frac{x}{y} \)? | [
"\\( \\sqrt{5} \\)"
] | [
"By folding a rectangular sheet of paper with length \\( x \\) and width \\( y \\) in half along one of its diagonals, the \"visible\" area (which becomes a pentagon) is reduced by 30%. What is the ratio \\( \\frac{x}{y} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 7
} | 21 | 0 | 0 |
omr-ws-7166a33da7497abb82910904eb5e37af806368a08b76f1e475a6503dd402b2a8 | By fusing together two ingots of pig iron of equal mass and different chromium contents, a new alloy was obtained containing 12 kg of chromium. If the mass of the first ingot had been doubled, the alloy would have contained 16 kg of chromium. The chromium content of the second ingot exceeded that of the first by 5\%. F... | [
"5\\% and 10\\%"
] | [
"By fusing together two ingots of pig iron of equal mass and different chromium contents, a new alloy was obtained containing 12 kg of chromium. If the mass of the first ingot had been doubled, the alloy would have contained 16 kg of chromium. The chromium content of the second ingot exceeded that of the first by 5... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5cf115d4e57af38d710afaddb5df67b03dc6140ecbc0c17e757815d977401abb | By hitting a target, Juanita can score 16, 17, 23, 26, 29, or 40 points with each arrow she shoots. What is the fewest number of arrows she can shoot to score exactly 100 points? | [
"4"
] | [
"By hitting a target, Juanita can score 16, 17, 23, 26, 29, or 40 points with each arrow she shoots. What is the fewest number of arrows she can shoot to score exactly 100 points?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-530d12fcecb3c36e09275969da4693684fe3cc6cb70ed2f2ce8b7d8302573d4f | By how many different paths can a rumor travel through a group of 10 people in $n$ calls, where each person can pass the rumor to anyone except the individual who just called? | [
"$10\\cdot9\\cdot8^{n-1}$"
] | [
"By how many different paths can a rumor travel through a group of 10 people in $n$ calls, where each person can pass the rumor to anyone except the individual who just called?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 8,
"tir": 5
} | 18 | 0 | 0 |
omr-ws-f25e9e2f06e1df5c45393fe891a0ecf33149bc981ebbc0699a40d0b69f1cba80 | By how many different paths can a rumor travel through a group of 10 people in \( n \) calls, given that a person can pass the rumor on to anyone except the individual who just called? | [
"10 \\times 9 \\times 8^{n-1}"
] | [
"By how many different paths can a rumor travel through a group of 10 people in \\( n \\) calls, given that a person can pass the rumor on to anyone except the individual who just called?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-3ba639aa059b4ad7ff15f61f621bd50660e9b81e28deed2cf4ec9a99eb775c24 | By how many different paths can a rumor travel through a group of 10 people in three calls, given that a person can pass the rumor on to anyone except the individual who just called? | [
"5760"
] | [
"By how many different paths can a rumor travel through a group of 10 people in three calls, given that a person can pass the rumor on to anyone except the individual who just called?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-fb8f1e560e1c4bc8aa86486f323c7c781615a20d73e807d64ca327b787e448aa | By how many maximum number of partitions can the plane be divided by zigzags, where a zigzag is defined as the line on the plane formed by two parallel rays and the segment connecting the beginning of these rays? | [
"2n^2 - n + 1"
] | [
"By how many maximum number of partitions can the plane be divided by zigzags, where a zigzag is defined as the line on the plane formed by two parallel rays and the segment connecting the beginning of these rays?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-e52bc5ed186427a16dcdccf6ff24058028398f03ac55437ba8d2c40bbacfb92e | By how much does Elka's electric bill go up if her usage increases from 350 kwh to 450 kwh, given the following pricing structure: 11 cents per kwh for the first 320 kwh, 10 cents per kwh for the next 100 kwh, and 17 cents per kwh for anything over that? | [
"12.10"
] | [
"By how much does Elka's electric bill go up if her usage increases from 350 kwh to 450 kwh, given the following pricing structure: 11 cents per kwh for the first 320 kwh, 10 cents per kwh for the next 100 kwh, and 17 cents per kwh for anything over that?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-78ed93ec9238b437bc623b0ae921286569a6ae3de20c6160114731a809d2413f | By how much does \( r - s \) exceed \( s - r \)? | [
"-2"
] | [
"By how much does \\( r - s \\) exceed \\( s - r \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-43e7ebddfbb7d6eb41538ea7a1bcf1fea38b6667189ba8037048241eed9ca6b8 | By how much must the parabola \( y = x^2 - 3x + 2 \) be translated along the x-axis and along the y-axis so that it becomes the parabola \( y = x^2 + x + 1 \)? | [
"\\(-2\\) units along the x-axis and \\(1\\) unit along the y-axis."
] | [
"By how much must the parabola \\( y = x^2 - 3x + 2 \\) be translated along the x-axis and along the y-axis so that it becomes the parabola \\( y = x^2 + x + 1 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-67b694cacccc3cd20dba919b84e6cd262cfb6b92ba4eab4b94ecaef5f602f412 | By inserting parentheses in the expression \(1:2:3:4:5:6:7:8\), how many different numerical values can be obtained? | [
"56"
] | [
"By inserting parentheses in the expression \\(1:2:3:4:5:6:7:8\\), how many different numerical values can be obtained?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2b6db8bdc0d123e8d0504090dc2b749f70a498be8b23918f606ceb374d8e484e | By inserting parentheses in the expression \(1:2:3:4:5:6:7:8\), what are the maximum and minimum numerical values that can be obtained? | [
"Minimum = \\((8!)^{-1}\\), Maximum = \\(\\frac{8!}{4}\\)"
] | [
"By inserting parentheses in the expression \\(1:2:3:4:5:6:7:8\\), what are the maximum and minimum numerical values that can be obtained?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 20
} | 20 | 0 | 0 |
omr-ws-1c2b1228b12480e1a3d77e43bfc6ba9e2c9bc0fe54c98ab98d7a0b4f501af5d0 | By inserting parentheses, how many different values can be obtained for the expression \(2 \times 3 + 4 \times 5\)? | [
"4"
] | [
"By inserting parentheses, how many different values can be obtained for the expression \\(2 \\times 3 + 4 \\times 5\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c054c914dd1b4698e7308be1744e063f195888ddd33aaab2ffa849a94eab57fc | By inserting parentheses, how many different values can be obtained from the expression \(2 \times 3 + 4 \times 5\)? | [
"4"
] | [
"By inserting parentheses, how many different values can be obtained from the expression \\(2 \\times 3 + 4 \\times 5\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff3910781e7b9b8534e34044531464b657c52ea1648a0e78f7e651030aa762b7 | By making a straight cut through one vertex of an isosceles triangle, one can dissect the triangle into two smaller isosceles triangles. Find the angle sizes of the original triangle. There is more than one possibility. How can you be sure that you have found them all? | [
"\\(72^\\circ, 72^\\circ, 36^\\circ\\) or \\(90^\\circ, 45^\\circ, 45^\\circ\\)"
] | [
"By making a straight cut through one vertex of an isosceles triangle, one can dissect the triangle into two smaller isosceles triangles. Find the angle sizes of the original triangle. There is more than one possibility. How can you be sure that you have found them all?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-0c1523b19aae95adeec30dafa304990aeb705c335dba1067eea829fa73d2262e | By partial fractions, \[\frac{1}{x(x + 1)(x + 2)(x + 3)(x + 4)} = \frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x + 2} + \frac{D}{x + 3} + \frac{E}{x + 4}\]for some constants $A,$ $B,$ $C,$ $D,$ and $E.$ Find $A + B + C + D + E.$ | [
"0"
] | [
"By partial fractions,\n\\[\\frac{1}{x(x + 1)(x + 2)(x + 3)(x + 4)} = \\frac{A}{x} + \\frac{B}{x + 1} + \\frac{C}{x + 2} + \\frac{D}{x + 3} + \\frac{E}{x + 4}\\]for some constants $A,$ $B,$ $C,$ $D,$ and $E.$ Find $A + B + C + D + E.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-193a4fe8103a124b414fc4924fd10c66c2d30776ab339ec4e6be3b4f62cdaa71 | By placing a 2 at both ends of a number, its value increases by 2137. What is the number? | [
"15"
] | [
"By placing a 2 at both ends of a number, its value increases by 2137. What is the number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-21079155eb73f1c3635c53751bd523b385ab29eac899ea39e036cdff2284223b | By placing a 2 at both ends of a number, its value is increased by 2317. What is the sum of the digits of the original number? | [
"8"
] | [
"By placing a 2 at both ends of a number, its value is increased by 2317. What is the sum of the digits of the original number?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-35899ba8cc6219c316d67aacfd14e82d6baaf48943b8f31507c299d56649f653 | By placing either a $+$ or $-$ sign between each pair of consecutive numbers $0$, $1$, $2$, $3$, $\dots$, $30$, $31$ in the expression $0 \ast 1 \ast 2 \ast 3 \ast \cdots \ast 30 \ast 31$, how many different positive integers can be obtained? | [
"248"
] | [
"By placing either a $+$ or $-$ sign between each pair of consecutive numbers $0$, $1$, $2$, $3$, $\\dots$, $30$, $31$ in the expression $0 \\ast 1 \\ast 2 \\ast 3 \\ast \\cdots \\ast 30 \\ast 31$, how many different positive integers can be obtained?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d4a4e67bae78db2bf13738f9224561693b17eba3e8f004d95830832789a2a539 | By purchasing rupees 50 shares for rupees 80 each, a man gets a 4% profit on his investment. What is his dividend if he buys 60 shares? | [
"} 192 \\text{ rupees} \\]"
] | [
"By purchasing rupees 50 shares for rupees 80 each, a man gets a 4% profit on his investment. What is his dividend if he buys 60 shares?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-90ac885ddedc918d40768006d143c5f2710acb2d5cfb54cb6e4b36eb2892d023 | By selling shares at rupees 92, where each share has a face value of rupees 100 and a dividend rate of 5/2%, and then investing the proceeds in 5% shares with a face value of rupees 100 selling at rupees 115, a person's income increased by rupees 90 a year. Determine: i) The number of shares sold. ii) The number of sha... | [
"i) 60\nii) 48\niii) 150"
] | [
"By selling shares at rupees 92, where each share has a face value of rupees 100 and a dividend rate of 5/2%, and then investing the proceeds in 5% shares with a face value of rupees 100 selling at rupees 115, a person's income increased by rupees 90 a year. Determine:\ni) The number of shares sold.\nii) The number... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-1bc87c52894308ccc72184e04d606619067dd4a118b69c2f7ccc1a07fd0caa85 | By setting $f(x) = p \sin \frac{1}{2} x + q \cos \frac{1}{2} x + r$, where $p, q,$ and $r$ are suitably chosen, obtain another inequality for $\pi$. | [
"\\(\\pi < \\frac{22}{7}\\)"
] | [
"By setting $f(x) = p \\sin \\frac{1}{2} x + q \\cos \\frac{1}{2} x + r$, where $p, q,$ and $r$ are suitably chosen, obtain another inequality for $\\pi$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-62fd5d1c74bdd19f12a5b9d71bfdcf4962ce35dd3c7697bc29e0d3c438b47782 | By setting \( k=1 \) in the result from Problem 1, find the solution of the differential equation \( 2xy \frac{dy}{dx} = y^2 + x^2 + 1 \) for which \( y=2 \) when \( x=1 \). Describe geometrically this solution. | [
"\\( y(x) = \\sqrt{x^2 + 4x - 1} \\)"
] | [
"By setting \\( k=1 \\) in the result from Problem 1, find the solution of the differential equation \\( 2xy \\frac{dy}{dx} = y^2 + x^2 + 1 \\) for which \\( y=2 \\) when \\( x=1 \\). Describe geometrically this solution."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 4,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-509e1c3cb22782c4f6c6d6555d7eaf2bf6349047a54be34e19a5170d375cdf03 | By setting \( x + \sqrt{x^2 - 1} = t \), find the indefinite integral \( \int \sqrt{x^2 - 1} \, dx \). | [
"\\( \\frac{(x + \\sqrt{x^2 - 1})^2}{8} - \\frac{\\log(|x + \\sqrt{x^2 - 1}|)}{2} - \\frac{1}{2(x + \\sqrt{x^2 - 1})^2} + C \\)"
] | [
"By setting \\( x + \\sqrt{x^2 - 1} = t \\), find the indefinite integral \\( \\int \\sqrt{x^2 - 1} \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"genselect"
] | {
"genselect": 3
} | 3 | 0 | 0 |
omr-ws-79ee6c46c1f53332418aeb523021e5029c6f6a72652394a142640ed656d86bf4 | By setting one variable constant, find a plane that intersects the graph of $z=4x^2-y^2+1$ in a pair of intersecting lines. | [
"z=1"
] | [
"By setting one variable constant, find a plane that intersects the graph of $z=4x^2-y^2+1$ in a pair of intersecting lines."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7ec1ea5b058e4632f590a226d38a7063a4c4b2795277e00aeab7a93ffbc7807b | By starting with a million and alternatively dividing by 2 and multiplying by 5, Anisha created a sequence of integers that starts 1000000, 500000, 2500000, 1250000, and so on. What is the last integer in her sequence? Express your answer in the form $a^b$, where $a$ and $b$ are positive integers and $a$ is as small as... | [
"5^{12}"
] | [
"By starting with a million and alternatively dividing by 2 and multiplying by 5, Anisha created a sequence of integers that starts 1000000, 500000, 2500000, 1250000, and so on. What is the last integer in her sequence? Express your answer in the form $a^b$, where $a$ and $b$ are positive integers and $a$ is as sma... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8f955c80d5bcf51c30cfa87ba290c939517d6963aa043589976debfed700b440 | By switching any amount of plus signs to minus signs in the expression $1+2+3+\dots+49+50$, how many possible values can be made? | [
"638"
] | [
"By switching any amount of plus signs to minus signs in the expression $1+2+3+\\dots+49+50$, how many possible values can be made?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c664c43e84b3802c2dd11dfbf2c2487a71ec74747396048622eea623477922a9 | By the end of the night, there are 20 people left, 8 of which are Venusian, 10 of which are Martian, and 2 of which are attendance takers. At least 2 Venusians, \( C \) Martians, and 1 attendance taker leaves. If \( \frac{m}{n} \) is the likelihood that there are exactly 7 people left at the party, what is \( m + n \)? | [
"9697"
] | [
"By the end of the night, there are 20 people left, 8 of which are Venusian, 10 of which are Martian, and 2 of which are attendance takers. At least 2 Venusians, \\( C \\) Martians, and 1 attendance taker leaves. If \\( \\frac{m}{n} \\) is the likelihood that there are exactly 7 people left at the party, what is \\... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-bf47cba41a105cddc3bb34bfbc81221d67bbe04753f6739cc9e60aaf9747fae0 | By using digits from $1$ to $9$, how many [i]heavy[/i] $5$-digit numbers are there? A number is considered [i]heavy[/i] if the higher digit value appears strictly more frequently than the lower digit value. (Note: A number like $333$ is not considered [i]heavy[/i] as it has only one unique digit value.) | [
"549"
] | [
"By using digits from $1$ to $9$, how many [i]heavy[/i] $5$-digit numbers are there? A number is considered [i]heavy[/i] if the higher digit value appears strictly more frequently than the lower digit value. (Note: A number like $333$ is not considered [i]heavy[/i] as it has only one unique digit value.)"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-83cf893e28b40f8ff7a4fe93e3fea363f4dd2e16bd6fd7449b6b932604767c98 | By weight, raspberries are 85% water, raspberry jam is 30% water, and sugar contains no water. I make raspberry jam by mixing equal weights of raspberries and sugar and then boiling them to evaporate off some of the water. What weight of jam can I make with 2.8 kg of raspberries? | [
"4.6 kg"
] | [
"By weight, raspberries are 85% water, raspberry jam is 30% water, and sugar contains no water. I make raspberry jam by mixing equal weights of raspberries and sugar and then boiling them to evaporate off some of the water. What weight of jam can I make with 2.8 kg of raspberries?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-faedc31aebed7d08bfbc845655332f3900b63e4076fedcd3dd07a61c055a2f11 | By what common fraction does $0.\overline{81}$ exceed $0.81$? | [
"\\frac{9}{1100}"
] | [
"By what common fraction does $0.\\overline{81}$ exceed $0.81$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fd567f39c8cd4f7ea9b717ce6d795ae3fc206e13c3baba3a22fbde420ed867b7 | By what factor does the intensity increase when the car moves from 86 feet away to 43 feet away? By what factor does the intensity increase when the car moves from 1043 feet away to 1000 feet away? | [
"2, 2"
] | [
"By what factor does the intensity increase when the car moves from 86 feet away to 43 feet away? By what factor does the intensity increase when the car moves from 1043 feet away to 1000 feet away?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eb56fba8cc6ed4cee488a5c8573775a68189a817285396461ccbade9b695a259 | By what factor should the number 999999999 be multiplied in order to obtain a number consisting only of ones? | [
"Since the given number $A$ is equal to $10^9-1$, for any number $$X=\\overline{x_1x_2\\ldots x_k},$$ we have $AX=\\overline{x_1x_2\\ldots x_k000000000}-\\overline{x_1x_2\\ldots x_k}=M-N$. On writing the numbers $M$ and $N$ in a column consider the process of subtraction of $N$ from~$M$"
] | [
"By what factor should the number 999999999 be multiplied in order to obtain a number consisting only of ones?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-69496c0d855a5757df002391e15b65244928edbb12e16cd4fc288fa74249eef9 | By what percentage will a bacteria colony have grown in 8 hours if it has a growth rate of 15% per hour? | [
"205.9%"
] | [
"By what percentage will a bacteria colony have grown in 8 hours if it has a growth rate of 15% per hour?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-1692343c3c5260452ecff0efcc9ae5ba17df227b51263a81a6ad9595146d8cdb | By writing the Maclaurin series for the functions $\exp \left( \frac{x}{a} \right)$ and $\sqrt{\frac{a+x}{a-x}}$ to a proper degree, find the largest $p \in \mathbb{N}$ such that: \[ \sqrt{\frac{a+x}{a-x}} = \exp \left( \frac{x}{a} \right) + O(x^p) \] where $O$ is the big $O$ notation. | [
"3"
] | [
"By writing the Maclaurin series for the functions $\\exp \\left( \\frac{x}{a} \\right)$ and $\\sqrt{\\frac{a+x}{a-x}}$ to a proper degree, find the largest $p \\in \\mathbb{N}$ such that:\n\\[\n\\sqrt{\\frac{a+x}{a-x}} = \\exp \\left( \\frac{x}{a} \\right) + O(x^p)\n\\]\nwhere $O$ is the big $O$ notation."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2dddb6cce0de48c7989233a701305b77db76b26da4718bf0d6e3ae5a32506827 | Byan Rai has 1 red cup, 4 blue cups, 1 orange cup, 2 yellow cups, 3 green cups, 3 purple cups, and 8 black cups in a box. Every second, Byan will pull out a random cup from the box, and all other cups of the same color will disappear. What is the expected number of seconds it will take for Byan to pick a blue cup? | [
"5.5"
] | [
"Byan Rai has 1 red cup, 4 blue cups, 1 orange cup, 2 yellow cups, 3 green cups, 3 purple cups, and 8 black cups in a box. Every second, Byan will pull out a random cup from the box, and all other cups of the same color will disappear. What is the expected number of seconds it will take for Byan to pick a blue cup?... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d2163144c41c819dfc8259db1ab570af993b4e5006244f8fadbdad5bebc6298b | Byan Rai is standing at the origin of a 2D plane. In each second, he jumps: - one unit up with probability $\frac{6}{11}$, - three units down with probability $\frac{2}{11}$, - four units right with probability $\frac{3}{22}$, - four units left with probability $\frac{3}{22}$. After 2024 seconds, Byan ends up at $(x, y... | [
"45889"
] | [
"Byan Rai is standing at the origin of a 2D plane. In each second, he jumps:\n- one unit up with probability $\\frac{6}{11}$,\n- three units down with probability $\\frac{2}{11}$,\n- four units right with probability $\\frac{3}{22}$,\n- four units left with probability $\\frac{3}{22}$.\n\nAfter 2024 seconds, Byan e... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-35924ab8f67480ca1a3abcfbcc184b81dcf28e501a4990441cd5dad623c8bcc7 | Byan has $999, 998, \dots, 2, 1$ balls in $999$ bins from left to right, respectively. In one move, he selects two adjacent bins where the left bin has an even number of balls and the right bin has an odd number of balls and moves one ball from the left bin to the right bin. Byan keeps making moves until he is unable t... | [
"499"
] | [
"Byan has $999, 998, \\dots, 2, 1$ balls in $999$ bins from left to right, respectively. In one move, he selects two adjacent bins where the left bin has an even number of balls and the right bin has an odd number of balls and moves one ball from the left bin to the right bin. Byan keeps making moves until he is un... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-0d11544c6392584bab614cbbbdc0fff0f821c1f4b19f84670570b2016f7eb2a9 | Byan is playing a game with his three friends, walking clockwise around them and tapping each friend on the head, saying 'raven' or 'falcon' with equal probability. The game ends after Byan has said 'falcon' twice. The probability that one of his friends will be called a falcon twice can be expressed as $\frac{m}{n}$, ... | [
"107"
] | [
"Byan is playing a game with his three friends, walking clockwise around them and tapping each friend on the head, saying 'raven' or 'falcon' with equal probability. The game ends after Byan has said 'falcon' twice. The probability that one of his friends will be called a falcon twice can be expressed as $\\frac{m}... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 10
} | 32 | 0 | 0 |
omr-ws-a124f2d7b91849010c3d8dedcdc57ce0b2c9e791743196d337b8a9e805cf2a72 | Byan rolls a $12$-sided die, a $14$-sided die, a $20$-sided die, and a $24$-sided die. The probability that the sum of the numbers the dice landed on is divisible by $7$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"8"
] | [
"Byan rolls a $12$-sided die, a $14$-sided die, a $20$-sided die, and a $24$-sided die. The probability that the sum of the numbers the dice landed on is divisible by $7$ can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-103cef065aba71d9c8fe44f0ef670405c2307fa8ca9cb79661eb676e697e52ba | Bárbara and Catarina bought a square frame measuring $1$ m in length and placed a watercolor in its center, also square. Knowing that the area of each green rectangular triangle is $1$ dm$^2$, calculate the area of the watercolor. | [
"0.32 m^2"
] | [
"Bárbara and Catarina bought a square frame measuring $1$ m in length and placed a watercolor in its center, also square. Knowing that the area of each green rectangular triangle is $1$ dm$^2$, calculate the area of the watercolor."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-600214033ac05e9c8a7a3fcfc753d85fd273a11c4fca0037b7b6b8742b3007a3 | C and D are points on a semicircle. The tangent at C meets the extended diameter of the semicircle at B, and the tangent at D meets it at A, so that A and B are on opposite sides of the center. The lines AC and BD meet at E. F is the foot of the perpendicular from E to AB. If the radius of the semicircle is 10, calcula... | [
"5"
] | [
"C and D are points on a semicircle. The tangent at C meets the extended diameter of the semicircle at B, and the tangent at D meets it at A, so that A and B are on opposite sides of the center. The lines AC and BD meet at E. F is the foot of the perpendicular from E to AB. If the radius of the semicircle is 10, ca... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c63c8fe4f99b7cb50bb224fbeda6616e8616eca4dc2bbca73ac0c074d9b1cd81 | C and D are two events such that \( P(D|C) = \frac{1}{5} \) and \( P(D|D) = \frac{1}{4} \). Given that \( P(C \cap D) = p \), find (i) \( P(C) \) and (ii) \( P(D) \). | [
"\\( P(C) = 5p \\), \\( P(D) = 4p \\)"
] | [
"C and D are two events such that \\( P(D|C) = \\frac{1}{5} \\) and \\( P(D|D) = \\frac{1}{4} \\). Given that \\( P(C \\cap D) = p \\), find (i) \\( P(C) \\) and (ii) \\( P(D) \\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-150e8e070c72750078e599a729c0822f5c09039406d52df27828f08b75b76032 | C.A can catch 4 bass in 15 minutes and J.O can catch 7 bass in an hour. If they both go fishing for 2 hours, including a half-hour lunch break, how many fish can they catch in total? Assume the odds of catching fish remain constant. | [
"34"
] | [
"C.A can catch 4 bass in 15 minutes and J.O can catch 7 bass in an hour. If they both go fishing for 2 hours, including a half-hour lunch break, how many fish can they catch in total? Assume the odds of catching fish remain constant."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e5787e5d8da03249353a1c906e5582d61e1ee44720fe83327a52433c7a98bd5 | CD is the diameter of a semicircle with center O. If AB = OD, find the measure of angle BAO. | [
"15 degrees"
] | [
"CD is the diameter of a semicircle with center O. If AB = OD, find the measure of angle BAO."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e3730ccf3f238ee1d557970f49e6f901921713d2c89567e6d175b9ce5dd4f966 | CK's car runs out of gas as it is going up a hill. The car rolls to a stop then starts rolling backwards. The displacement \( d(t) \) in meters from the bottom of the hill at \( t \) seconds since CK's car ran out of gas is given by \( d(t) = 99 + 30t - t^2 \). At what time did CK's car stop rolling up and start rollin... | [
"\\( t = 15 \\) seconds, \\( d = 324 \\) meters"
] | [
"CK's car runs out of gas as it is going up a hill. The car rolls to a stop then starts rolling backwards. The displacement \\( d(t) \\) in meters from the bottom of the hill at \\( t \\) seconds since CK's car ran out of gas is given by \\( d(t) = 99 + 30t - t^2 \\). At what time did CK's car stop rolling up and s... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 21
} | 24 | 0 | 0 |
omr-ws-b99aedc82bd36605d969cb3fe5f64cdaf81eabbc59b51748cfb1393c07ab19d6 | Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes? | [
"25"
] | [
"Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9f3c9653fe827e3133d388648bf99fee2541dd7d11d966cc448bcd2c6a340d8b | Cai is racing a train traveling east at 40 mph. He can run at 8 mph for 2 hours before resting for 15 minutes at a time, while the train stops for 20 minutes between stations that are 100 miles apart. If Cai departs at 7:00 AM going eastward, and the train departs at 10:00 AM from a station, at what time will the train... | [
"10:44"
] | [
"Cai is racing a train traveling east at 40 mph. He can run at 8 mph for 2 hours before resting for 15 minutes at a time, while the train stops for 20 minutes between stations that are 100 miles apart. If Cai departs at 7:00 AM going eastward, and the train departs at 10:00 AM from a station, at what time will the ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d1ce917c74e77538fcb8edb7991ccbb5baa396133f9716a925341d18d246a276 | Caitlin draws 6 line segments on a piece of paper without lifting her pencil off the paper and without retracing her path, such that each segment crosses exactly one other segment (not counting intersections at vertices) and she ends up back where she started. If the segments form a closed loop and each intersection is... | [
"3"
] | [
"Caitlin draws 6 line segments on a piece of paper without lifting her pencil off the paper and without retracing her path, such that each segment crosses exactly one other segment (not counting intersections at vertices) and she ends up back where she started. If the segments form a closed loop and each intersecti... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e4761f00b03c33352100d6b6874483db94bf4882d5eb50572ad7501d79c4f9fb | Caitlin draws \( n \) line segments on a piece of paper without lifting her pencil or retracing her path, such that each segment crosses exactly one other segment (not counting intersections at vertices), and she ends up back where she started. Find the sum of all possible values of \( n \) for which this is possible. | [
"10"
] | [
"Caitlin draws \\( n \\) line segments on a piece of paper without lifting her pencil or retracing her path, such that each segment crosses exactly one other segment (not counting intersections at vertices), and she ends up back where she started. Find the sum of all possible values of \\( n \\) for which this is p... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-30df4d55407b8610b6efef76bc410abb5442774ffbc115d0e92edec26722d821 | Cal Q. Later was given a pair of two-digit integers to multiply. The four digits which formed these two integers were all distinct. Cal mistakenly reversed the digits of both numbers but still obtained the correct product. How many pairs of integers could have been given? | [
"14"
] | [
"Cal Q. Later was given a pair of two-digit integers to multiply. The four digits which formed these two integers were all distinct. Cal mistakenly reversed the digits of both numbers but still obtained the correct product. How many pairs of integers could have been given?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7c633ade50860485c7d808a989863ffdde75c1afa839247e62e84cf8c30383fa | Calculate $ \arcsin ( \sin 2019).$ | [
"\\( 643\\pi - 2019 \\)"
] | [
"Calculate $ \\arcsin ( \\sin 2019).$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 8
} | 26 | 0 | 0 |
omr-ws-6b49f310641e1e6b31e2f9113bb3f13d16ee4d8a1ce4347f74344a75fe1c6eb7 | Calculate $ \lim_{n\to\infty }\lim_{t\to\infty }\int_{1}^t\frac{\sin x}{x^n} . $ | [
"0"
] | [
"Calculate $ \\lim_{n\\to\\infty }\\lim_{t\\to\\infty }\\int_{1}^t\\frac{\\sin x}{x^n} . $"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e4d56e7418fed27b1b5852caffa281c7b7e119f305f9a818deed8d5d38e2dc17 | Calculate $ \sin{\frac {3\pi}{7}}\sin{\frac {6\pi}{7}}\sin{\frac {9\pi}{7}}$. | [
"\\(-\\frac{\\sqrt{7}}{8}\\)"
] | [
"Calculate $ \\sin{\\frac {3\\pi}{7}}\\sin{\\frac {6\\pi}{7}}\\sin{\\frac {9\\pi}{7}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-a38b1cece699cc8c2da359c6bd87277cab3e05a381669da1586519a7aef1430e | Calculate $ {\sec }^{4}\frac{\pi }{9}+{\sec }^{4}\frac{3\pi }{9}+{\sec }^{4}\frac{5\pi }{9}+{\sec }^{4}\frac{7\pi }{9} $. | [
"1120"
] | [
"Calculate $ {\\sec }^{4}\\frac{\\pi }{9}+{\\sec }^{4}\\frac{3\\pi }{9}+{\\sec }^{4}\\frac{5\\pi }{9}+{\\sec }^{4}\\frac{7\\pi }{9} $."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
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