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-cb60486b1baf81cfbd1854152ddf81f546937039f631e08aa121dfd65dee54c6 | Write \( \left(\frac{1 + i \sqrt{3}}{1 - i \sqrt{3}}\right)^{20} \) in algebraic form. | [
"\\( -\\frac{1}{2} - \\frac{\\sqrt{3}}{2}i \\)"
] | [
"Write \\( \\left(\\frac{1 + i \\sqrt{3}}{1 - i \\sqrt{3}}\\right)^{20} \\) in algebraic form."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-bce94f18029f7a5328ca30d2dd1321a2ad31e0673bf76eae1dd4ffeefb34875a | Write \( \left| \sum_{k=0}^{N} C_{k} + \sum_{k=0}^{N} X_{k} \right|^{2} \) as a sum of terms where each term depends only on \( C_{k} \) or \( X_{k} \). Here, \( C_{k} \) and \( X_{k} \) are complex variables. The solution can include inequalities. The current form of the solution is: \[ \left| \sum_{k=0}^{N} C_{k} \ri... | [
"\\(\\left| \\sum_{k=0}^{N} C_{k} \\right|^{2} + \\left| \\sum_{k=0}^{N} X_{k} \\right|^{2} + 2 \\left| \\sum_{k=0}^{N} C_{k} \\right| \\left| \\sum_{k=0}^{N} X_{k} \\right|\\)"
] | [
"Write \\( \\left| \\sum_{k=0}^{N} C_{k} + \\sum_{k=0}^{N} X_{k} \\right|^{2} \\) as a sum of terms where each term depends only on \\( C_{k} \\) or \\( X_{k} \\). Here, \\( C_{k} \\) and \\( X_{k} \\) are complex variables. The solution can include inequalities.\n\nThe current form of the solution is:\n\\[ \\left|... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-6806f9bb1172f86ea984a4b0315287823b8209eea2326a3d47325c321cf95500 | Write \( \sin^4 x \) in terms of \( \cos 2x \) and \( \cos 4x \). | [
"\\( \\frac{3}{8} - \\frac{1}{2}\\cos 2x + \\frac{1}{8}\\cos 4x \\)"
] | [
"Write \\( \\sin^4 x \\) in terms of \\( \\cos 2x \\) and \\( \\cos 4x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-ece0f40a6949ec67d20119cf9a08cd8a232d5d70d0a0f6e6c55dd39bbc83e030 | Write \( \sqrt{10001} \) to the sixth decimal place, rounding down. | [
"100.004999"
] | [
"Write \\( \\sqrt{10001} \\) to the sixth decimal place, rounding down."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94f200f137804e6cc6a512a36951693c97a69c6c55e83f99203a68bda8d98720 | Write \( e^x \) as a sum of an even function \( h(x) \) and an odd function \( g(x) \). Sketch the graphs of \( g(x) \) and \( h(x) \). | [
"\\( g(x) = \\frac{e^x + e^{-x}}{2} \\), \\( h(x) = \\frac{e^x - e^{-x}}{2} \\)"
] | [
"Write \\( e^x \\) as a sum of an even function \\( h(x) \\) and an odd function \\( g(x) \\). Sketch the graphs of \\( g(x) \\) and \\( h(x) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-e3bd449f69c1bc58677197549316be377b5498995f64b0e012fcc9f791cf2315 | Write \( k \) real numbers (\( k \geq 5 \)) on a line such that (a) the sum of any three consecutive numbers is positive, and (b) the sum of any five consecutive numbers is negative. Determine the highest value of \( k \) for which this is possible. | [
"6"
] | [
"Write \\( k \\) real numbers (\\( k \\geq 5 \\)) on a line such that (a) the sum of any three consecutive numbers is positive, and (b) the sum of any five consecutive numbers is negative. Determine the highest value of \\( k \\) for which this is possible."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-29f65fbd764c2427320616f63a54a0e59f50960204aa3fc012488c094f85a380 | Write \( x^4 + 6x^3 + 9x^2 - 2x - 7 \) as a product of two quadratic factors. | [
"\\(\\left[x^{2}+(3+\\sqrt{2})x+1+2\\sqrt{2}\\right]\\left[x^{2}+(3-\\sqrt{2})x+1-2\\sqrt{2}\\right]\\)"
] | [
"Write \\( x^4 + 6x^3 + 9x^2 - 2x - 7 \\) as a product of two quadratic factors."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 22,
"tir": 21
} | 43 | 0 | 0 |
omr-ws-8b7e78c3dd62333be5828c9af79a064387f45e032cf4b3debdbd712e2178668e | Write \( x^5 + x^4 + x^2 + 2x \) in \( \mathbb{Z}_3 \) as a product of irreducible polynomials. | [
"\\( x \\cdot (x^2 + 1) \\cdot (x^2 + x + 2) \\)"
] | [
"Write \\( x^5 + x^4 + x^2 + 2x \\) in \\( \\mathbb{Z}_3 \\) as a product of irreducible polynomials."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-b3949b3428e2bb75fef4f7ed0277945f7df41062fd23eb27b12574d2153ef4d0 | Write \( |-5x^2y^3| \) in two different ways. | [
"\\( 5x^2|y^3| \\) and \\( |5x^2y^3| \\)"
] | [
"Write \\( |-5x^2y^3| \\) in two different ways."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-e79121ebbe6e1aef58c65eb4dce646d26a2504d9b76b893377f053fed3af5ccb | Write \((1+\frac{1}{3})(1+\frac{1}{3^2})(1+\frac{1}{3^{2^2}})\cdots(1+\frac{1}{3^{2^{100}}})\) in the form \(a(1-b^{c})\), where \(a\), \(b\), and \(c\) are constants. | [
"\\(\\frac{3}{2}(1-\\frac{1}{3^{2^{101}}})\\)"
] | [
"Write \\((1+\\frac{1}{3})(1+\\frac{1}{3^2})(1+\\frac{1}{3^{2^2}})\\cdots(1+\\frac{1}{3^{2^{100}}})\\) in the form \\(a(1-b^{c})\\), where \\(a\\), \\(b\\), and \\(c\\) are constants."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-959b7f9153ea9dab2ac9ece37aca85edca6aa3c048a64b40701f829505862675 | Write \((2+\sqrt{3})^{2n-1} = a_n + b_n \sqrt{3}\) for integers \(a_n\) and \(b_n\) where \(n > 0\). Find the value of \(a_5 - 1\). | [
"70225"
] | [
"Write \\((2+\\sqrt{3})^{2n-1} = a_n + b_n \\sqrt{3}\\) for integers \\(a_n\\) and \\(b_n\\) where \\(n > 0\\). Find the value of \\(a_5 - 1\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d479d3cc5de18e8dd41a61a5eda562eb0dd825e3f20d9047002abd7e830c2885 | Write \((a^2 + a + 1)^2\) as the sum of three squares. | [
"\\((a(a+1))^2 + (a+1)^2 + a^2\\)"
] | [
"Write \\((a^2 + a + 1)^2\\) as the sum of three squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 19
} | 66 | 0 | 0 |
omr-ws-61a6d496ecb1c30be472fd898a4901a1feb0178b0cede46447aacdb73c4c5b7b | Write \((a^2 + b^2 + c^2)(x^2 + y^2 + z^2)\) as a sum of 3 squares. | [
"(ax + by + cz)^2 + (bz - cy)^2 + (cx - az)^2 + (ay - bx)^2"
] | [
"Write \\((a^2 + b^2 + c^2)(x^2 + y^2 + z^2)\\) as a sum of 3 squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-c43c11479e096730c101e84bf7ac9df9972950c4cf6a4e2027e1ca73ef12394a | Write \(1,000,000\) as the sum of a prime number and a perfect square. | [
"\\(1999\\) and \\(999^2\\)"
] | [
"Write \\(1,000,000\\) as the sum of a prime number and a perfect square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-979a06fee2cfa8f0b952dd318ea9278a977b2bd057a58ea01378e961c10f79f6 | Write \(2012^{2013}\) as a sum of 2013 positive integers \(a_1, a_2, \ldots, a_{2013}\). Let \(T = a_1^{13} + a_2^{13} + \cdots + a_{2013}^{13}\). Calculate the remainder when \(T + 2012^{2013}\) is divided by 4. | [
"no_valid_answer_found"
] | [
"Write \\(2012^{2013}\\) as a sum of 2013 positive integers \\(a_1, a_2, \\ldots, a_{2013}\\). Let \\(T = a_1^{13} + a_2^{13} + \\cdots + a_{2013}^{13}\\). Calculate the remainder when \\(T + 2012^{2013}\\) is divided by 4."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c930933a492a1ac4b772c9ed7ba69942eda19941b9f9eac00eabb43f7e48289c | Write \(2018^{2019} + 2018\) as the sum of two perfect squares. | [
"(13 \\times 2018^{1009} - 43)^2 + (43 \\times 2018^{1009} + 13)^2"
] | [
"Write \\(2018^{2019} + 2018\\) as the sum of two perfect squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-647bcc12cc760dd1e02eed18653d88c2b71e6e86012063ca536aa8c290eed166 | Write \(2a(b^3) + (a^2)(c^2) + (b^2)(c^2) - (a^2)(b^2) - 2ab(c^2) - (b^4)\) as a product of linear factors. | [
"\\((a-b)^2(c-b)(c+b)\\)"
] | [
"Write \\(2a(b^3) + (a^2)(c^2) + (b^2)(c^2) - (a^2)(b^2) - 2ab(c^2) - (b^4)\\) as a product of linear factors."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-41d1e8740b24f2647c3f20eb16a3cf5088891e8978c6ea921983f592684e0c7e | Write \(\frac{1}{x} + \frac{1}{y}\) as a single fraction. | [
"\\dfrac{x + y}{xy}"
] | [
"Write \\(\\frac{1}{x} + \\frac{1}{y}\\) as a single fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7dc8e7f250c03e3722931e9ecd58706a6acef3086c88f30dec42d1e87dc39523 | Write a Riemann sum and then a definite integral representing the area of the region using the strip shown. The region is bounded by the curves \( y = \sqrt{x} \) and \( y = x \) from \( x = 0 \) to \( x = 1 \). | [
"\\int_{0}^{1} \\left( \\sqrt{x} - x \\right) \\, dx"
] | [
"Write a Riemann sum and then a definite integral representing the area of the region using the strip shown. The region is bounded by the curves \\( y = \\sqrt{x} \\) and \\( y = x \\) from \\( x = 0 \\) to \\( x = 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-11300b10fe8b4f3b7dc07ef23f6076f7466f62248fa339b9378604e8c097851a | Write a construction for the problem assuming the relationship is linear. Given a fixed cost of $500 and the cost to produce 5 items is $3210, determine the linear relationship between the number of items produced and the total cost. | [
"C = 542x + 500"
] | [
"Write a construction for the problem assuming the relationship is linear. Given a fixed cost of $500 and the cost to produce 5 items is $3210, determine the linear relationship between the number of items produced and the total cost."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b8f0376dee369067f95a92122683c30e5ac31522ce6ecef2c969ae04f25245c6 | Write a construction for the problem. Assume that the relationship is linear. Given a fixed cost of \$500 and that producing 5 items costs \$3210, determine the linear equation that represents the total cost as a function of the number of items produced. | [
"\\( C(n) = 542n + 500 \\)"
] | [
"Write a construction for the problem. Assume that the relationship is linear. Given a fixed cost of \\$500 and that producing 5 items costs \\$3210, determine the linear equation that represents the total cost as a function of the number of items produced."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-4ba4e172db5e03ac8b1f152d887c92b2159538cf1d322ea3ede9cecb43fd0ffe | Write a cost equation where shirts cost 10 dollars each and dresses cost 20 dollars each. | [
"\\( 10x + 20y \\)"
] | [
"Write a cost equation where shirts cost 10 dollars each and dresses cost 20 dollars each."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-63b29c6e34385f9c2014b6f3dc02f5f7e22ce15eb341dc9213491f592553100e | Write a cubic polynomial equation with integer coefficients that has the roots $ \cot^2 \frac{\pi}{7} $, $ \cot^2 \frac{2\pi}{7} $, and $ \cot^2 \frac{3\pi}{7} $. | [
"\\( 7x^3 - 35x^2 + 21x - 1 = 0 \\)"
] | [
"Write a cubic polynomial equation with integer coefficients that has the roots $ \\cot^2 \\frac{\\pi}{7} $, $ \\cot^2 \\frac{2\\pi}{7} $, and $ \\cot^2 \\frac{3\\pi}{7} $."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-523e901292e4c6b40d7b88cdc21951f38fddc8f541b43da8abd042c09e3c3d85 | Write a cyclic block matrix \( L \) as \( L = AB \) where \( A \) and \( B \) are matrices such that \( \det(AB - BA) \neq 0 \). (This result is true at least for \( 2 \times 2 \) matrices.) | [
"L = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}"
] | [
"Write a cyclic block matrix \\( L \\) as \\( L = AB \\) where \\( A \\) and \\( B \\) are matrices such that \\( \\det(AB - BA) \\neq 0 \\). (This result is true at least for \\( 2 \\times 2 \\) matrices.)"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-6a87c2f8e44423019c1254f583c9dc164259efc8fcbdef3f82fb4d5ccb713a59 | Write a degree three equation \(x^3 + bx^2 + cx + d = 0\) with roots \(x_1\), \(x_2\), \(x_3 \not\in \mathbb{Q}\) such that \(|\arcsin x_1| + |\arcsin x_2| + |\arcsin x_3| = \pi\). The coefficients should be rational. | [
"x^3 - \\frac{3}{4}x + \\frac{1}{4} = 0"
] | [
"Write a degree three equation \\(x^3 + bx^2 + cx + d = 0\\) with roots \\(x_1\\), \\(x_2\\), \\(x_3 \\not\\in \\mathbb{Q}\\) such that \\(|\\arcsin x_1| + |\\arcsin x_2| + |\\arcsin x_3| = \\pi\\). The coefficients should be rational."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-93e08ab8ccc638009853198324fa1592d34454b46ff207ec467c34c522c6eb39 | Write a divisibility rule for 144. | [
"To check if a number is divisible by 144, the number must satisfy both:\n1. The sum of its digits must be divisible by 9.\n2. The last four digits must satisfy the equation \\(4a_2 - 6a_1 + a_0 \\mod 16\\) if the thousands digit is even, or \\(8 + 4a_2 - 6a_1 + a_0 \\mod 16\\) if the thousands digit is odd."
] | [
"Write a divisibility rule for 144."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-4bf0b99d4d8f345154a77015aab677f6eccab099d2027fe390291998084702b2 | Write a formula for a function whose plot in the coordinate system forms a regular hexagon centered at the origin (0,0) with a side length of $a$. | [
"\\( |y| + \\frac{\\sqrt{3}}{2} |x + a| + \\frac{\\sqrt{3}}{2} |x - a| = 2\\sqrt{3}a \\)"
] | [
"Write a formula for a function whose plot in the coordinate system forms a regular hexagon centered at the origin (0,0) with a side length of $a$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-1455f515aaf18ffb8a3a514967da14b0b09adde4b6737068215c10a8482bd360 | Write a function for the area of the rectangular region in terms of \( x \), given that the total fencing material is 900 meters and the garage side does not require fencing. | [
"\\( x(900 - 3x) \\)"
] | [
"Write a function for the area of the rectangular region in terms of \\( x \\), given that the total fencing material is 900 meters and the garage side does not require fencing."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d112614b73592fd69a036122e8dfe4dcde93d431d43b982121482a18e0aa62d5 | Write a function for the sinusoid with a maximum at \( A \left( \frac{2\pi}{3}, 9 \right) \) and a minimum at \( B (2\pi, 5) \). Determine the period of the sinusoid. | [
"\\(\\frac{8\\pi}{3}\\)"
] | [
"Write a function for the sinusoid with a maximum at \\( A \\left( \\frac{2\\pi}{3}, 9 \\right) \\) and a minimum at \\( B (2\\pi, 5) \\). Determine the period of the sinusoid."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 22
} | 36 | 0 | 0 |
omr-ws-2a21c31af294c48e9c6ae9026171607199ee716d55793950dabe303630849062 | Write a function relating the camera purchase price after a \$50 coupon, \( y \), to the retail price, \( x \). Give the domain of this function. | [
"y = x - 50 \\text{ with domain } x \\geq 50"
] | [
"Write a function relating the camera purchase price after a \\$50 coupon, \\( y \\), to the retail price, \\( x \\). Give the domain of this function."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e8ebe18776b03adda1c8a14085b82c782259929286782915dcaf4ac893efbff | Write a function relating the camera purchase price, \( y \), to the retail price, \( x \), given a 40% discount. Give the domain of this function. | [
"y = 0.6x \\text{ with domain } x \\geq 0"
] | [
"Write a function relating the camera purchase price, \\( y \\), to the retail price, \\( x \\), given a 40% discount. Give the domain of this function."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-21b964200537a1e42c048e859264efe18aefb4191a5078b9bc29d8295de19338 | Write a function to represent the advertising costs given the following data: - 10 words cost $7.50 - 15 words cost $8.75 - 20 words cost $10.00 - 25 words cost $11.25 | [
"\\( C(w) = 0.25w + 5.00 \\)"
] | [
"Write a function to represent the advertising costs given the following data:\n- 10 words cost $7.50\n- 15 words cost $8.75\n- 20 words cost $10.00\n- 25 words cost $11.25"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-b721904fdb2be96fd50e562f36610616acfa19ba934aaa563c2aa46327f4bccb | Write a general term to show how to multiply a number by 2 in base 2. | [
"b \\times 10_2"
] | [
"Write a general term to show how to multiply a number by 2 in base 2."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bdcb26a774ddf0ff9ef6b522cb2aa431b8d52514a175e8add55ae4dc51a0bd1f | Write a generating function using power series for the problem of distributing seven stuffed animals among four children. | [
"\\((1 + x + x^2 + \\cdots + x^7)^4\\)"
] | [
"Write a generating function using power series for the problem of distributing seven stuffed animals among four children."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-aa4592b675a3a90e0b7690f69b6d69f5219d514805cbda44c0eb28269faf5e5f | Write a limit using summations that would equal $\int ^4_2 \sqrt{x} \, dx$. | [
"$$\\lim_{n\\to \\infty} \\frac{1}{n}\\sum_{r=2n}^{4n}\\sqrt{\\frac{r}{n}}$$"
] | [
"Write a limit using summations that would equal $\\int ^4_2 \\sqrt{x} \\, dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 1,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-e3558fe928c18c9737b34d0d508ec488dd5b06fc5dfa79060a8944249ce69aa1 | Write a linear equation that gives the height \( y \) of the first bullet in terms of its distance \( x \) from the tower, given that the plane began firing at an altitude of 1379 ft when 336 ft east of the tower, and the first bullet hit the ground 7231 ft east of the tower. | [
"y = −1/5x + 1446.2"
] | [
"Write a linear equation that gives the height \\( y \\) of the first bullet in terms of its distance \\( x \\) from the tower, given that the plane began firing at an altitude of 1379 ft when 336 ft east of the tower, and the first bullet hit the ground 7231 ft east of the tower."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e1df36aadbfdc78a7f734d21886498897eb5a32ac3bf89b2d0e122534918a5a6 | Write a list of all 4-digit numbers that do not use the digit 7. For each number, multiply its digits. Then, add all the multiplication results together. What is the final sum? | [
"2085136"
] | [
"Write a list of all 4-digit numbers that do not use the digit 7. For each number, multiply its digits. Then, add all the multiplication results together. What is the final sum?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6e5ae6115a5e8767207d7a1c59aa4e1c5e7af91e8e3b3fbe41b56f86a227fc77 | Write a mathematical name given to the curve described by the equation \( y = \frac{e^{ax} + e^{-ax}}{2a} \), which is known as an inverted "hanging chain" and was studied by mathematicians including the Bernoulli brothers, Christian Huygens, and Gottfried Leibniz. | [
"Catenary"
] | [
"Write a mathematical name given to the curve described by the equation \\( y = \\frac{e^{ax} + e^{-ax}}{2a} \\), which is known as an inverted \"hanging chain\" and was studied by mathematicians including the Bernoulli brothers, Christian Huygens, and Gottfried Leibniz."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-9deee155357f6aff7d82150bde400d8670ebea727a60dad247922b53ab50a9ab | Write a number from 1 to 12 on each face of a dodecahedron. A face is called a 'good face' if the sum of the numbers on the 5 adjacent faces is divisible by 5. Given that the face with 1 and the face with 5 are opposite each other, and the face with 5 is adjacent to the face with 6, how many 'good faces' can there be? | [
"6"
] | [
"Write a number from 1 to 12 on each face of a dodecahedron. A face is called a 'good face' if the sum of the numbers on the 5 adjacent faces is divisible by 5. Given that the face with 1 and the face with 5 are opposite each other, and the face with 5 is adjacent to the face with 6, how many 'good faces' can there... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-09763355d117e5f007e2d32a6d11231679f6ff51da89dd1c26a177a4c8192130 | Write a number in each circle such that the sum of the numbers lined up on each of the nine straight lines joining 3 or 4 circles is always the same. Use all integers between 0 and 14, except three. The numbers written in circles a, b, and c must be in increasing order. | [
"8"
] | [
"Write a number in each circle such that the sum of the numbers lined up on each of the nine straight lines joining 3 or 4 circles is always the same. Use all integers between 0 and 14, except three. The numbers written in circles a, b, and c must be in increasing order."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-96a9c2b66df39705a5416d95128fbb5e1bd0ac54f354a9259d229b4b287a9432 | Write a number in each square of an $n \times n$ table such that every number written is divisible by the number written below or on the left of it. The number in the top right corner square of the table is 4. How many ways are there to fill in the table? For example, when $n=3$: \[ \begin{array}{|c|c|c|} \hline 1 & 2 ... | [
"2^{n-1}"
] | [
"Write a number in each square of an $n \\times n$ table such that every number written is divisible by the number written below or on the left of it. The number in the top right corner square of the table is 4. How many ways are there to fill in the table?\n\nFor example, when $n=3$:\n\\[\n\\begin{array}{|c|c|c|}\... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-095f92615a5511e090394404e89b12e7b8de337fd8d258ff65eda169400811f2 | Write a partial sum for the power series which represents the function $ln(1+6x)$ consisting of the first 5 nonzero terms. Also, indicate the radius of convergence. | [
"\\( 6x - \\frac{(6x)^2}{2} + \\frac{(6x)^3}{3} - \\frac{(6x)^4}{4} + \\frac{(6x)^5}{5} \\), Radius of Convergence: \\( \\frac{1}{6} \\)"
] | [
"Write a partial sum for the power series which represents the function $ln(1+6x)$ consisting of the first 5 nonzero terms. Also, indicate the radius of convergence."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 3,
"tir": 25
} | 29 | 0 | 0 |
omr-ws-f979b292af02d018e1e5d0a1f72b9973e2ae66cdcc5659c99a16fe89c67e248c | Write a polynomial equation with integer coefficients such that the real roots are all the values of $c$ for which the integral $\int_1^\infty \! \frac{1}{c^x - c^{-x}} \, dx$ converges to $1$. | [
"c^3 - c^2 - c - 1 = 0"
] | [
"Write a polynomial equation with integer coefficients such that the real roots are all the values of $c$ for which the integral $\\int_1^\\infty \\! \\frac{1}{c^x - c^{-x}} \\, dx$ converges to $1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-5ba3451d0a60896dcb90e64c22ef653ede563d44f8d1a6ddf07d1c2637075e8d | Write a positive integer on each of the 16 small triangles in such a way that every number having at least two neighbors is equal to the difference of two of its neighbors. Two triangles are said to be neighbors if they have a common side. | [
"1"
] | [
"Write a positive integer on each of the 16 small triangles in such a way that every number having at least two neighbors is equal to the difference of two of its neighbors. Two triangles are said to be neighbors if they have a common side."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ccc0c001eaef2cc8f4d914f3bf77632355166df51331a924f3dbc3781e18a344 | Write a possible formula for the desert temperature $A$, which oscillates daily between $30$ F at 5 am and $70$ F at 5 pm, in terms of $t$, measured in hours from 5 am. | [
"\\( A(t) = -20\\cos\\left(\\frac{\\pi t}{12}\\right) + 50 \\)"
] | [
"Write a possible formula for the desert temperature $A$, which oscillates daily between $30$ F at 5 am and $70$ F at 5 pm, in terms of $t$, measured in hours from 5 am."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-e1002bd04d549a0a4c38525ffc254e77dc3d019fb7dc4db48c849bad7693d25f | Write a power function whose graph passes through the points (1,3) and (6,10). | [
"y = 3x^{\\log_6\\left(\\frac{10}{3}\\right)}"
] | [
"Write a power function whose graph passes through the points (1,3) and (6,10)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8016c1a9f8f77303e434252be073e5d308d90f14571a7549fa86a882b698ba38 | Write a profit equation where the shirts are sold for 70 dollars each and the dresses are sold for 50 dollars each. (Note: Profit = Sale price - Cost.) | [
"\\(60x + 30y\\)"
] | [
"Write a profit equation where the shirts are sold for 70 dollars each and the dresses are sold for 50 dollars each. (Note: Profit = Sale price - Cost.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-353b9e4b70fe5b52124fa22d707c8efc5e64b08d3eb02c8da1fdbca7b37b352a | Write a program to implement the generator \( x_{i+1} = (25173 \cdot x_i + 13849) \mod 65536 \) with initial value \( x_0 = 10000 \) and obtain the first 10 values generated. | [
"25928"
] | [
"Write a program to implement the generator \\( x_{i+1} = (25173 \\cdot x_i + 13849) \\mod 65536 \\) with initial value \\( x_0 = 10000 \\) and obtain the first 10 values generated."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7cd48f2ae94d818e52455c6ff63350f8b83b58260f29694fcd2ffd0b12717052 | Write a quadratic equation that has two roots $2016\alpha + \beta$ and $2016\beta + \alpha$, knowing that $\alpha$ and $\beta$ are the roots of the equation $x^2 - 5x + 3 = 0$. | [
"\\( x^2 - 10085x + 12231075 = 0 \\)"
] | [
"Write a quadratic equation that has two roots $2016\\alpha + \\beta$ and $2016\\beta + \\alpha$, knowing that $\\alpha$ and $\\beta$ are the roots of the equation $x^2 - 5x + 3 = 0$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-fc2c457829863c53e34fbd682f1c423bf17943bdf36be4e70c6ab9b12bd97144 | Write a quadratic equation, whose roots are $\frac{a}{\sqrt a\pm \sqrt {a-b}}$. | [
"$bx^2-2a\\sqrt {ax}+a^2=0$"
] | [
"Write a quadratic equation, whose roots are $\\frac{a}{\\sqrt a\\pm \\sqrt {a-b}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-282a6cbeafb0b080cfa94c187c1f63be5f06968f36749160e52b07e95593670f | Write a quadratic polynomial having as its two roots the numbers obtained by increasing each root of \(x^2 - 2x - 5 = 0\) by the reciprocal of the other. | [
"\\(5x^2 - 8x - 16\\)"
] | [
"Write a quadratic polynomial having as its two roots the numbers obtained by increasing each root of \\(x^2 - 2x - 5 = 0\\) by the reciprocal of the other."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-3059d12ac4d5510119a7177d28b6960ac8ebc7f717c72e42583738802ff07e43 | Write a rational function that satisfies the following conditions: - A vertical asymptote at \( x = 2 \) - A slant asymptote at \( y = x + 1 \) - A zero of the function at \( x = -2 \) | [
"\\( \\frac{x^2 - x - 6}{x-2} \\)"
] | [
"Write a rational function that satisfies the following conditions:\n- A vertical asymptote at \\( x = 2 \\)\n- A slant asymptote at \\( y = x + 1 \\)\n- A zero of the function at \\( x = -2 \\)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-53b8ccc84969c4d72e5c80645ea59ed623b93ab7afc5d8d87ca82c515c742c99 | Write a recursive definition for the sequence: 1, 3, 6, 10. | [
"\\( a_1 = 1, a_{n} = a_{n-1} + n, n \\ge 2 \\)"
] | [
"Write a recursive definition for the sequence: 1, 3, 6, 10."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-5b4916e0b13847454dbac6b2cc6650cd7ec7c62a75dc7676478642782645c5bb | Write a recursive equation describing Larry's balance just after he makes each payment, given that he owes $5000 initially, is charged 1.75% interest on the remaining balance every month, and pays $200 on the first day of every month. | [
"\\( a_n = 1.0175(a_{n-1} - 200) \\)"
] | [
"Write a recursive equation describing Larry's balance just after he makes each payment, given that he owes $5000 initially, is charged 1.75% interest on the remaining balance every month, and pays $200 on the first day of every month."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-39490c885b266edd2373b1e3e3f0d8d6fe09b0b8e4600771ba4d27e80c55cab3 | Write a recursive formula for the following scenario: On October 1, 2002, Sal invested $24,000 in a bank account earning 3.4 percent interest annually, compounded monthly. A month later, he withdrew $100 and continued to withdraw $100 on the first of every month thereafter. | [
"A_0 = 24,000; \\ A_n = A_{n-1} \\left(1 + \\frac{0.034}{12}\\right) - 100 \\text{ for } n \\geq 1"
] | [
"Write a recursive formula for the following scenario: On October 1, 2002, Sal invested $24,000 in a bank account earning 3.4 percent interest annually, compounded monthly. A month later, he withdrew $100 and continued to withdraw $100 on the first of every month thereafter."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c6d1aebd207cae8c65ad8bb588367e8ab23e1308e6a52cb1a11bb58a0c9d6e50 | Write a recursive formula for the number of ancestors Egor has if we go back \( n \) generations. | [
"\\( a_{1} = 2 \\); \\( a_{n} = 2 a_{n-1} \\)"
] | [
"Write a recursive formula for the number of ancestors Egor has if we go back \\( n \\) generations."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 22
} | 36 | 0 | 0 |
omr-ws-d628010706590aa62bad9ee15c06d99c6835325f2303a29bc920b6b27411c49e | Write a recursive formula for the sequence 3, 3(sqrt{3}), 9, 9(sqrt{3}), ... | [
"\\( a_1 = 3 \\) and \\( a_n = \\sqrt{3} \\cdot a_{n-1} \\)"
] | [
"Write a recursive formula for the sequence 3, 3(sqrt{3}), 9, 9(sqrt{3}), ..."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-a30f5c184cdd4fbfabdc4000bce923840d50826ba98fd4aa0deed8e7d927c9f1 | Write a recursive formula for the sequence: $3$, $12$, $24$, $33$, $66$, $75$, $\dots$ | [
"\\( a_{n+1} = \\begin{cases} \na_n + 9 & \\text{if } n \\text{ is odd} \\\\\n2a_n & \\text{if } n \\text{ is even}\n\\end{cases} \\)"
] | [
"Write a recursive formula for the sequence: $3$, $12$, $24$, $33$, $66$, $75$, $\\dots$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 14
} | 26 | 0 | 0 |
omr-ws-305a9f75577de8dc41116641512a9d50ec25ca741a9519f1c165a1c5712e601c | Write a rule giving your salary \( a(n) \) for the \( n \)-th year of employment, given that the initial salary is $45,000 and it increases by 3.5% each year. | [
"\\( S = 45000(1.035)^{(n - 1)} \\)"
] | [
"Write a rule giving your salary \\( a(n) \\) for the \\( n \\)-th year of employment, given that the initial salary is $45,000 and it increases by 3.5% each year."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-ccb003ccf8a80c5560703f3b69a688068513db893c87bf693c029be568b590e8 | Write a sine function that relates the depth of the water in the harbour, \( y \), to the time, \( t \) in hours, after low tide, given that the average depth at high tide is 22m and at low tide is 10m, with the tides completing one cycle every 12 hours. | [
"\\( y = 16 + 6\\sin\\left(\\frac{\\pi}{6}t + \\frac{3\\pi}{2}\\right) \\)"
] | [
"Write a sine function that relates the depth of the water in the harbour, \\( y \\), to the time, \\( t \\) in hours, after low tide, given that the average depth at high tide is 22m and at low tide is 10m, with the tides completing one cycle every 12 hours."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-d382ce2d36519a268fe3d7ed111d7ddb3137552a00b0e8a07bc4026b4b2a40d9 | Write a summation with lower bound $k$ and upper bound $n$ such that when $k$ is inserted into the function, it yields $n$, when $k+1$ is inserted, it yields $n-1$, and so on, until when $n$ is inserted, it yields $k$. | [
"\\(\\sum_{i=k}^n f(n+k-i)\\)"
] | [
"Write a summation with lower bound $k$ and upper bound $n$ such that when $k$ is inserted into the function, it yields $n$, when $k+1$ is inserted, it yields $n-1$, and so on, until when $n$ is inserted, it yields $k$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 5
} | 23 | 0 | 0 |
omr-ws-ce0e80f5fc3281c594626aad792a578963944c26e4274b044272fddd22a61a59 | Write a system of inequalities representing the range of heights and ages for qualifying astronauts in NASA's Mercury program. The initial qualifications specified males between 25 and 40 years of age, who were at least 5 foot 4 inches tall, but no taller than 5 foot 11 inches. Let \( h \) represent the height in inche... | [
"\\( 25 \\leq a \\leq 40 \\) and \\( 64 \\leq h \\leq 71 \\)"
] | [
"Write a system of inequalities representing the range of heights and ages for qualifying astronauts in NASA's Mercury program. The initial qualifications specified males between 25 and 40 years of age, who were at least 5 foot 4 inches tall, but no taller than 5 foot 11 inches. Let \\( h \\) represent the height i... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-ee5988327d99a1f10772de3ec1bb2eaffe5d6fb207b0533544f9a81d96f30b00 | Write a system of three equations showing the interaction among three species $X$, $Y$, and $Z$ such that: (a) $X$ and $Y$ compete with each other for a limited amount of food, (b) $Z$ feeds on both $X$ and $Y$. Given the equation for the evolution of $Z$ as: \[ Z' = -Z (a - b (X + Y)) \] Determine the equations for th... | [
"\\[X' = (a - bX - cY - dZ)X\\]\n\\[Y' = (e - fY - gX - hZ)Y\\]\n\\[Z' = (i(X + Y) - j)Z\\]"
] | [
"Write a system of three equations showing the interaction among three species $X$, $Y$, and $Z$ such that:\n(a) $X$ and $Y$ compete with each other for a limited amount of food,\n(b) $Z$ feeds on both $X$ and $Y$.\n\nGiven the equation for the evolution of $Z$ as:\n\\[ Z' = -Z (a - b (X + Y)) \\]\n\nDetermine the ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3498a3ec64da44bb84280ec617f0efc63418e1ca2d8728ce2206c308fd4cbd32 | Write a third degree polynomial $p$ with complex coefficients such that $p(-i)=-1-i$, $p(i)=-1+i$, $p(2i)=2-4i$, and $p(1+i)=-2+2i$. | [
"z^3 - z^2 + 2z - 2"
] | [
"Write a third degree polynomial $p$ with complex coefficients such that $p(-i)=-1-i$, $p(i)=-1+i$, $p(2i)=2-4i$, and $p(1+i)=-2+2i$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-60dee0fe959b0393e79b6154bbc71285f51e93fa1e9f6e209c8d2a8eebdd5641 | Write a triple integral, including limits of integration, that gives the volume under the sphere \( x^2 + y^2 + z^2 = 9 \) and above the region between \( y = x \) and \( y = 2x - 2 \) in the \( xy \)-plane in the first quadrant. | [
"\\int_{1}^{2} \\int_{2x-2}^{x} \\int_{0}^{\\sqrt{9 - x^2 - y^2}} \\, dz \\, dy \\, dx"
] | [
"Write a triple integral, including limits of integration, that gives the volume under the sphere \\( x^2 + y^2 + z^2 = 9 \\) and above the region between \\( y = x \\) and \\( y = 2x - 2 \\) in the \\( xy \\)-plane in the first quadrant."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-640ba0fb767d2dbad063bb95f3c48ed621b7e2ae763e0275e3c9cc08f9421fcd | Write all Pythagorean triples that contain the number 37. | [
"(12, 35, 37), (37, 684, 685)"
] | [
"Write all Pythagorean triples that contain the number 37."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-11b5e054c622af6bef089d53bcf663540134faaef89eda1ec5c06bbf8399ebd5 | Write all numbers from 1 to 20 using only the digit 1 and mathematical operations. | [
"The expressions for numbers from 1 to 20 using only the digit 1 and mathematical operations are provided above."
] | [
"Write all numbers from 1 to 20 using only the digit 1 and mathematical operations."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 13
} | 23 | 0 | 0 |
omr-ws-807534eb6949f069de34ec8a193f9a35d4e25f6f535ca503c809dca48bc56676 | Write all pairs of three-digit positive integers whose product is equal to 222222. | [
"(481, 462), (462, 481), (407, 546), (546, 407), (962, 231), (231, 962), (814, 273), (273, 814), (518, 429), (429, 518), (777, 286), (286, 777)"
] | [
"Write all pairs of three-digit positive integers whose product is equal to 222222."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 27
} | 29 | 0 | 0 |
omr-ws-f2336a1d890c12f086045a2a5a706b2a92486a979dc7dbae2849514fa1cd8a9c | Write an algebraic equation to represent the English sentence. (Let \( x \) represent the unknown number.) Then solve the equation. The quotient of a number and seven is negative ten. | [
"\\(\\frac{x}{7} = -10\\) and \\( x = -70 \\)"
] | [
"Write an algebraic equation to represent the English sentence. (Let \\( x \\) represent the unknown number.) Then solve the equation.\nThe quotient of a number and seven is negative ten."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-36d964d3756af71f2053dafb8619a748c1f6ebff1fdcc3d74c977659a97e898d | Write an algebraic expression for the interest Mitch earned at 7.5% in one year. | [
"\\(0.075n\\)"
] | [
"Write an algebraic expression for the interest Mitch earned at 7.5% in one year."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 5
} | 27 | 0 | 0 |
omr-ws-f8968b271a001ca96180f7e999f862fd4a2f2b3df456461bcab14e52305518e3 | Write an algebraic expression that represents the final number used to calculate a birthdate given the following steps: - Let \( A \) be the day of the month you were born. - Let \( B \) be the number formed by the month you were born. - Let \( C \) be the number formed by the last two digits of the year of your birth.... | [
"\\( 10000a + 100b + c \\)"
] | [
"Write an algebraic expression that represents the final number used to calculate a birthdate given the following steps:\n- Let \\( A \\) be the day of the month you were born.\n- Let \\( B \\) be the number formed by the month you were born.\n- Let \\( C \\) be the number formed by the last two digits of the year ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-217ca6264d64597d162d54a6d176e0789ecc95ff90104f216784358aa317b25b | Write an analogue of the division algorithm for the Gaussian integers \( \mathbb{Z}[i] \). | [
"\\text{For any } \\alpha, \\beta \\in \\mathbb{Z}[i] \\text{ with } \\beta \\neq 0, \\text{ there exist } \\gamma, \\rho \\in \\mathbb{Z}[i] \\text{ such that } \\alpha = \\beta \\gamma + \\rho \\text{ and } N(\\rho) < N(\\beta)."
] | [
"Write an analogue of the division algorithm for the Gaussian integers \\( \\mathbb{Z}[i] \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-08d065f40b0149099cb024ffe4148284a23e2c1060beb7051ba3b8eca1726c98 | Write an arithmetic progression whose first term is 1, the sum of the first five terms being equal to $\frac{1}{4}$ of that of the next five terms. | [
"$+1;-2;-5;-8;\\ldots $"
] | [
"Write an arithmetic progression whose first term is 1, the sum of the first five terms being equal to $\\frac{1}{4}$ of that of the next five terms."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a146b8ceb5c2e4e0a4e814f58d76c1fab4415958d129ba0d33ade4308a654133 | Write an equation connecting the points where the slope of the function \( f(x) = \tan(2x) - \cot(x) \) is \(\infty\). | [
"\\cos(2x) \\sin(x) = 0"
] | [
"Write an equation connecting the points where the slope of the function \\( f(x) = \\tan(2x) - \\cot(x) \\) is \\(\\infty\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-0013f10de97b03cae4ba066c1906f6049aeed234f98f53ba2b5cd3b4e813c568 | Write an equation describing the temperature of an object placed into a $160^{\circ}$ oven, given that the rate of change of temperature is proportional to the difference between the environment temperature and the object's temperature with a proportionality constant $k = 2$. | [
"\\( T(t) = 160 - 135e^{-2t} \\)"
] | [
"Write an equation describing the temperature of an object placed into a $160^{\\circ}$ oven, given that the rate of change of temperature is proportional to the difference between the environment temperature and the object's temperature with a proportionality constant $k = 2$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13b72017ce161dcbd8f3e1d0b80cafb00237efe6140de8eae7a4346d858a5f70 | Write an equation for an ellipse where the major axis lies on the line \( y = x \) and the minor axis lies on the line \( y = -x \). | [
"\\(\\frac{(x+y)^2}{a^2} + \\frac{(x-y)^2}{b^2} = 2\\)"
] | [
"Write an equation for an ellipse where the major axis lies on the line \\( y = x \\) and the minor axis lies on the line \\( y = -x \\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4728cd173c49a8a316ea66de7d5c674614c84264ba80318770a9b7a907f49719 | Write an equation for the line tangent to the curve \(-x^2 + 3xy + y^3 = 3\) at the point \((2, 1)\). | [
"y = \\frac{1}{9}x + \\frac{7}{9}"
] | [
"Write an equation for the line tangent to the curve \\(-x^2 + 3xy + y^3 = 3\\) at the point \\((2, 1)\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-aa94a0a22162cef19f6910c60c420590c3475409eba1b124cc2632fc50fba36d | Write an equation for the locus of points that include (2,0) and are equidistant from the line \( y = 3x - 1 \). | [
"3x - y - 6 = 0"
] | [
"Write an equation for the locus of points that include (2,0) and are equidistant from the line \\( y = 3x - 1 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-0eb677ad0600de998e8db151deb3635ce853e757d93cb7baa641dab91052ac4a | Write an equation for the set of points that are twice as far from the point $(2,1)$ as from the line $x=8$. | [
"\\((y-1)^2 = 3(x-10)^2 - 48\\)"
] | [
"Write an equation for the set of points that are twice as far from the point $(2,1)$ as from the line $x=8$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0ed4eca0dae25aa4d806f84231486f188f9b7ccdeddd751d4472e60d2cfac933 | Write an equation for the tangent line to the elliptic curve \( y^2 = x^3 + ax + b \) at the point \( A(m, n) \). | [
"\\( y = \\frac{3m^2 + a}{2n}(x - m) + n \\)"
] | [
"Write an equation for the tangent line to the elliptic curve \\( y^2 = x^3 + ax + b \\) at the point \\( A(m, n) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-36d1f4ba8e7de61ea1296d8a57a2b6e5ea4ecb5a47bbccefc5f78590f366daa1 | Write an equation for the total cost of the meals including a 25% tip, given that the total amount of money available is $20 and both meals cost the same amount. | [
"\\( 1.25c = 20 \\)"
] | [
"Write an equation for the total cost of the meals including a 25% tip, given that the total amount of money available is $20 and both meals cost the same amount."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 3
} | 20 | 0 | 0 |
omr-ws-f6dfc03a5c46ac35295ed1aeb369051f005cc1aaa363e8edef83eac84029d7a5 | Write an equation in standard form for the line that contains the point (2, -3) and has a slope of 5. | [
"\\( 5x - y = 13 \\)"
] | [
"Write an equation in standard form for the line that contains the point (2, -3) and has a slope of 5."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fd6a9b5e0589995c97441d69ce65e9ab65e657f5b70b41c5f326b99d5b15fb8f | Write an equation involving an integral expression whose solution would give the number of days that sand must be pumped to restore the original distance between the road and the edge of the water, given that the distance between the road and the edge of the water is growing at a rate of $g(p)$ meters per day, where $p... | [
"\\( \\int_{0}^{p} g(p) \\, dp = D \\)"
] | [
"Write an equation involving an integral expression whose solution would give the number of days that sand must be pumped to restore the original distance between the road and the edge of the water, given that the distance between the road and the edge of the water is growing at a rate of $g(p)$ meters per day, whe... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 5,
"tir": 4
} | 13 | 0 | 0 |
omr-ws-96ced5ba6a1c7547552bfc63d0b4bfde62c026e920374b3ad47cc26ede9727f8 | Write an equation of the tangent line to the curve \( y = \frac{6}{1 - x^2} \) at the point \( P(2, -2) \). Express the answer in the form \( ax + by = c \). | [
"\\( 8x - 3y = 22 \\)"
] | [
"Write an equation of the tangent line to the curve \\( y = \\frac{6}{1 - x^2} \\) at the point \\( P(2, -2) \\). Express the answer in the form \\( ax + by = c \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-8d025c5e9a678a4a6a2094793e9d01507df6133a9487d2efabdde44217410b0e | Write an equation showing the dependence of the number of shirts sold (n) on the price charged (x). What does this equation tell you in general? Graph the relationship. | [
"\\( n = -10x + 150 \\)"
] | [
"Write an equation showing the dependence of the number of shirts sold (n) on the price charged (x). What does this equation tell you in general? Graph the relationship."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b426e784a44f1767946cca7a19dc3a7ae61a9267ee6486769d86090fdaa06767 | Write an equation that can be solved to find the actual value of \(k\) such that the point \((2.1, k)\) is on the curve \(-x^2 + 3xy + y^3 = 3\). | [
"\\(-(2.1)^2 + 3(2.1)k + k^3 = 3\\)"
] | [
"Write an equation that can be solved to find the actual value of \\(k\\) such that the point \\((2.1, k)\\) is on the curve \\(-x^2 + 3xy + y^3 = 3\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-e1408978a3469c1c3d7d41836a8c619bad1628cc0b8bebd58be4b9d4854da508 | Write an equation that models the sound of a guitar string plucked at time \( t = 0 \), which fades to inaudibility 12 seconds later. The initial volume is represented by an amplitude of 52, and the threshold of inaudibility is represented by an amplitude of 1. The note is an A, which vibrates 110 times per second. | [
"\\( V(t) = 52^{1 - \\frac{t}{12}} \\cos(220\\pi t) \\)"
] | [
"Write an equation that models the sound of a guitar string plucked at time \\( t = 0 \\), which fades to inaudibility 12 seconds later. The initial volume is represented by an amplitude of 52, and the threshold of inaudibility is represented by an amplitude of 1. The note is an A, which vibrates 110 times per seco... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 17
} | 39 | 0 | 0 |
omr-ws-db0bac383b0c1a956118f5709eade0b2c719807e42a837daa16cb5b420a930d7 | Write an equation that relates the number of loops made by the gears to the number of teeth before it comes back to its starting point, using \( R \) (number of teeth on the fixed spirograph) and \( Q \) (number of teeth on the rotating spirograph). | [
"L = \\dfrac{R - Q}{\\gcd(R, Q)}",
"\\dfrac{R - Q}{\\gcd(R, Q)}"
] | [
"Write an equation that relates the number of loops made by the gears to the number of teeth before it comes back to its starting point, using \\( R \\) (number of teeth on the fixed spirograph) and \\( Q \\) (number of teeth on the rotating spirograph)."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 27,
"genselect": 1
} | 28 | 0 | 0 |
omr-ws-a5698769f5bc20b2e26f0d7eed5f895eb15105a79422c9c586f37713668f8da6 | Write an equation that shows how many times the rotating wheel rotates around its axis before it comes back to its starting point, using \( R \) (number of teeth on the fixed spirograph) and \( Q \) (number of teeth on the rotating spirograph). | [
"\\dfrac{R - Q}{\\gcd(R, Q)}"
] | [
"Write an equation that shows how many times the rotating wheel rotates around its axis before it comes back to its starting point, using \\( R \\) (number of teeth on the fixed spirograph) and \\( Q \\) (number of teeth on the rotating spirograph)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-27dec86754d88ee43326b65c916f367a664d85aae916df5678cd2d282fc26322 | Write an equation that shows how many times the rotating wheel will revolve around the fixed wheel before it comes back to its starting point, using \( R \) (number of teeth on the fixed spirograph) and \( Q \) (number of teeth on the rotating spirograph). | [
"\\dfrac{Q}{\\gcd(R, Q)}"
] | [
"Write an equation that shows how many times the rotating wheel will revolve around the fixed wheel before it comes back to its starting point, using \\( R \\) (number of teeth on the fixed spirograph) and \\( Q \\) (number of teeth on the rotating spirograph)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2556ea985390bd926d789fb1cbcf5f7b16b57e5f1056faacf385b8d56efcf4c7 | Write an equation that uses pi four times and equals any integer number \( n \) (positive or negative). You can use common mathematical operations: sin, cos, tan, ln, log, square root, ^, (, ), !, %. | [
"\\(\\frac{\\pi}{\\pi} + \\pi - \\pi = 1\\) for \\( n = 1 \\)"
] | [
"Write an equation that uses pi four times and equals any integer number \\( n \\) (positive or negative). You can use common mathematical operations: sin, cos, tan, ln, log, square root, ^, (, ), !, %."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-173f22c65acf9136fad8d033369ccad7338a09cc82ceee9fe3de0b1ab0108219 | Write an equation to show a rider's position at a given time \( t \), to the left or right of an imaginary vertical line drawn down the center of a Ferris Wheel with a radius of 60 feet and a center 75 feet off the ground, which takes 120 seconds to make one full rotation, starting at the 3:00 position and rotating cou... | [
"\\( x = 60 \\cos \\left(\\frac{\\pi t}{60}\\right) \\)"
] | [
"Write an equation to show a rider's position at a given time \\( t \\), to the left or right of an imaginary vertical line drawn down the center of a Ferris Wheel with a radius of 60 feet and a center 75 feet off the ground, which takes 120 seconds to make one full rotation, starting at the 3:00 position and rotat... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-e1c5b4b7c08e1ded05f35cfafb5a62b6f77769939b8fd153dbb94ff8df69c0f8 | Write an equation using four pi's ($\pi$), the number $n$ (positive or negative), and the following mathematical operations: sin, cos, tan, ln, log, square root, ^, (, ), !, %, to find any integer number $n$. | [
"\\( n + \\sqrt{\\log\\frac{\\sin\\left(\\pi\\left(\\tan(\\sin\\ln\\cos\\tan\\pi)\\%\\right)!\\right)}{\\cos\\tan\\pi^\\pi\\%}} \\)"
] | [
"Write an equation using four pi's ($\\pi$), the number $n$ (positive or negative), and the following mathematical operations: sin, cos, tan, ln, log, square root, ^, (, ), !, %, to find any integer number $n$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3252079d00e9e3d46d74d3a030f5e307e9a465a63a0f325a7ab509f8b7ad9d5a | Write an explicit formula for the number of ancestors Egor has if we go back \( n \) generations. | [
"\\( 2^n \\)"
] | [
"Write an explicit formula for the number of ancestors Egor has if we go back \\( n \\) generations."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-3994254380b561f60857a6b4d4b8241cdc4aecf38937104111cc2aeef7995121 | Write an exponential function that passes through the points (2,1) and (4,8). Identify the horizontal asymptote. | [
"y = 0"
] | [
"Write an exponential function that passes through the points (2,1) and (4,8). Identify the horizontal asymptote."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b649c50ad7723b9da65730ea6ff735e38b0690a55476a0cfac697e9320a56061 | Write an expression that could be used to find the surface area of the curve \(y = \frac{1}{x}\) when it is revolved around the x-axis on the interval \([1, 3]\). | [
"\\( 2\\pi \\int_{1}^{3} \\frac{\\sqrt{x^4 + 1}}{x^3} \\, dx \\)"
] | [
"Write an expression that could be used to find the surface area of the curve \\(y = \\frac{1}{x}\\) when it is revolved around the x-axis on the interval \\([1, 3]\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-7ecda4d5acd9a739d42dbc9e2baf40df4d2c5ca040de2df2243572e6d21ba842 | Write an inequality representing the sponsor's requirement that at most 300 dollars can be spent on the project. Then find the minimum cost. | [
"\\( 10x + 20y \\leq 300 \\)"
] | [
"Write an inequality representing the sponsor's requirement that at most 300 dollars can be spent on the project. Then find the minimum cost."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8d6c77c6546aabcb87c4b08a0e1a0fc6a4600bb23290921e0caca02233ec705b | Write an inequality whose solution does not exist. | [
"y > 1 and y < 1"
] | [
"Write an inequality whose solution does not exist."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-0f357f9354b55b7ee67e6ab421e34d56f4eed951f2fe5c98535d2aef4c753942 | Write an inequality whose solution lies on exactly one point. | [
"\\( y \\geq x^2 \\) and \\( y \\leq 0 \\)"
] | [
"Write an inequality whose solution lies on exactly one point."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c31d0ffd7aefd8a68077e46ac224b9ce67382429fae892bd1e48d765ff85a815 | Write an integer between $1$ and $9$ in each square of a $5 \times 7$ grid such that any two squares that share a vertex have different numbers. Find the maximum possible value of the sum of the $35$ numbers written. | [
"272"
] | [
"Write an integer between $1$ and $9$ in each square of a $5 \\times 7$ grid such that any two squares that share a vertex have different numbers. Find the maximum possible value of the sum of the $35$ numbers written."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 7,
"tir": 8
} | 60 | 0 | 0 |
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