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-9f91befd3ed9a4a5821d8d0b6a7bcc2ab0cbf9e5b411dbbcc467c55f08300737 | Write an 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, 8, 5, 3, 18, 13, 5, 8, 11, 49, 31, 18, 13, 6, 19, 30"
] | [
"Write an 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"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-76ad39e80bc840d7fb68f48bbec7d1096e16ce58c8c492f38b2f8e72281fc27f | Write an integral expression for the average temperature \( T(x) \) of the wire and estimate the average temperature using the trapezoidal sum with 4 subintervals indicated by the data points: \[ \begin{array}{c|c} x (\text{cm}) & t(x) (\text{Celsius}) \\ \hline 0 & 100 \\ 1 & 93 \\ 5 & 70 \\ 6 & 62 \\ 8 & 55 \\ \end{a... | [
"75.6875"
] | [
"Write an integral expression for the average temperature \\( T(x) \\) of the wire and estimate the average temperature using the trapezoidal sum with 4 subintervals indicated by the data points:\n\\[\n\\begin{array}{c|c}\nx (\\text{cm}) & t(x) (\\text{Celsius}) \\\\\n\\hline\n0 & 100 \\\\\n1 & 93 \\\\\n5 & 70 \\\\... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-79204742215f8e8083716f71a2d92a1cf3410b86e124b98b42a8f23ba3ae5bb9 | Write and simplify an expression for the area of a triangle with a base of length \(2x^2y\) and a height of \(x^3 - xy - 2\). Then, find the area of the triangle when \(x = 2\) and \(y = 1\). | [
"\\( x^5y - x^3y^2 - 2x^2y \\) for the expression, and 16 for the area when \\( x = 2 \\) and \\( y = 1 \\)."
] | [
"Write and simplify an expression for the area of a triangle with a base of length \\(2x^2y\\) and a height of \\(x^3 - xy - 2\\). Then, find the area of the triangle when \\(x = 2\\) and \\(y = 1\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-80884960fa774bc8d6492718e6b055dcfb423b45cb838cc33b9d9cd55dfdb3f9 | Write down 20 positive integers ≤ 14 in the first row and 14 positive integers ≤ 20 in the second row. Calculate the minimum possible value of the largest sum that can be formed by taking some numbers from the first row and some numbers from the second row, such that no two sums are equal. | [
"210"
] | [
"Write down 20 positive integers ≤ 14 in the first row and 14 positive integers ≤ 20 in the second row. Calculate the minimum possible value of the largest sum that can be formed by taking some numbers from the first row and some numbers from the second row, such that no two sums are equal."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-2257f687d9f8c3504fd6f2c6436551e79f3499fb05411a82abac14fe6978fa1a | Write down 2006 integers from 1 to 2006 at random in a circle. We call each set of three adjacent numbers a string. If there are 500 strings with all even integers and 450 strings with exactly two even integers, find the number of strings that have exactly two odd integers. | [
"1056"
] | [
"Write down 2006 integers from 1 to 2006 at random in a circle. We call each set of three adjacent numbers a string. If there are 500 strings with all even integers and 450 strings with exactly two even integers, find the number of strings that have exactly two odd integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cc1172708dd1fec16798623d61c9f56752c3c2086f9356e939f48ff8bda91640 | Write down a $2 \times 2$ matrix which has a saddle point, and indicate which entry is the saddle point. | [
"\\(\\begin{pmatrix} 2 & 1 \\\\ 3 & 4 \\end{pmatrix}\\) with the saddle point at \\(3\\)."
] | [
"Write down a $2 \\times 2$ matrix which has a saddle point, and indicate which entry is the saddle point."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-81834204d2cc38c8a1e01860d0e0a8a55037ed766e89b4ac3a8bfcffbcd6acbe | Write down a $2 \times 2$ matrix which has no saddle point. | [
"\\(\\begin{pmatrix} 2 & 2 \\\\ 2 & 2 \\end{pmatrix}\\)"
] | [
"Write down a $2 \\times 2$ matrix which has no saddle point."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c626d5d3b33f8639b7621c46210927b9f6e296a4db938c070a0c21e40346030b | Write down a composition series for: (a) $_RR$, where $R=T_2(\mathbb{R})$, the ring of all $2\times 2$ upper triangular matrices with real entries. (b) $M=\mathbb{C}^2$ as an $R$-module, where $R=\left\{\left(\begin{array}{cc}a&b\\b&a\end{array}\right):a,b\in\mathbb{C}\right\}$. (c) $M_2(\mathbb{C})$ as an $M_2(\mathbb... | [
"M_2(\\mathbb{C}) \\supset M_2(\\mathbb{R}) \\supset 0"
] | [
"Write down a composition series for:\n(a) $_RR$, where $R=T_2(\\mathbb{R})$, the ring of all $2\\times 2$ upper triangular matrices with real entries.\n(b) $M=\\mathbb{C}^2$ as an $R$-module, where $R=\\left\\{\\left(\\begin{array}{cc}a&b\\\\b&a\\end{array}\\right):a,b\\in\\mathbb{C}\\right\\}$.\n(c) $M_2(\\mathbb... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-b219001c976d547ed0282241c4f82641299b31f0bd2f8940d033e47b3a16432d | Write down a definite integral that will give the value of the area under the curve \( y = x^2 \cos\left(\frac{1}{5}x\right) \) between \( x = \frac{1}{2}\pi \) and \( x = \frac{5}{2}\pi \). (The expression \( x^2 \cos\left(\frac{1}{5}x\right) \) takes no negative values for \( \frac{1}{2}\pi \leq x \leq \frac{5}{2}\pi... | [
"\\(\\int_{\\frac{1}{2}\\pi}^{\\frac{5}{2}\\pi} x^2 \\cos\\left(\\frac{1}{5}x\\right) \\, dx\\)"
] | [
"Write down a definite integral that will give the value of the area under the curve \\( y = x^2 \\cos\\left(\\frac{1}{5}x\\right) \\) between \\( x = \\frac{1}{2}\\pi \\) and \\( x = \\frac{5}{2}\\pi \\). (The expression \\( x^2 \\cos\\left(\\frac{1}{5}x\\right) \\) takes no negative values for \\( \\frac{1}{2}\\p... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 26
} | 58 | 0 | 0 |
omr-ws-7069f232c1ae82a1280a4d2a7e61839eb275fe22ea94cfd0b5ce3b17c47dc601 | Write down a double iterated integral that represents the volume of the solid under the surface \( \sqrt{x^3 + y^3} - z + 5 = 0 \) and above the triangle with vertices (0, 0), (2, 0), and (0, 3). | [
"\\(\\int_{0}^{2}\\int_{0}^{-\\frac{3x}{2}+3} \\left( \\sqrt{x^3 + y^3} + 5 \\right) \\, dy \\, dx\\)"
] | [
"Write down a double iterated integral that represents the volume of the solid under the surface \\( \\sqrt{x^3 + y^3} - z + 5 = 0 \\) and above the triangle with vertices (0, 0), (2, 0), and (0, 3)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-1ec2e5edaa8151140829bc48cd6a8f326a426116b7367c000788ef879a803395 | Write down a fifth degree polynomial \( Q(x) \) such that \( Q(1) = Q(-1) \) and \( Q(2) = Q(-2) \) but \( Q(3) \neq Q(-3) \). | [
"x^5 - 5x^3 + 4x"
] | [
"Write down a fifth degree polynomial \\( Q(x) \\) such that \\( Q(1) = Q(-1) \\) and \\( Q(2) = Q(-2) \\) but \\( Q(3) \\neq Q(-3) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-8e7d116292f9d962a42dd757822f24fe17399931db1f39b5d294a01997014d81 | Write down a formula for the total profit £P Marcus makes, in terms of the price per cake £x and the number of cakes sold n. | [
"\\( P = n(x - 0.15) \\)"
] | [
"Write down a formula for the total profit £P Marcus makes, in terms of the price per cake £x and the number of cakes sold n."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18ee942933ebeb44f6722ea7b644ce86fd534e0a9933c7383f5c88ff2dc05069 | Write down a fourth degree polynomial \( P(x) \) such that \( P(1) = P(-1) \) but \( P(2) \neq P(-2) \). | [
"\\( x^4 - 2x^3 - x^2 + 2x \\)"
] | [
"Write down a fourth degree polynomial \\( P(x) \\) such that \\( P(1) = P(-1) \\) but \\( P(2) \\neq P(-2) \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-97e72a6e9f23bcbc41dfacccf544855c8192d0f6cc10f6893271dc5fd3a0344b | Write down a polynomial equation of degree 5 which is satisfied by \( w \), where \( w = \cos \frac{2}{5}\pi + i \sin \frac{2}{5}\pi \). | [
"\\( z^5 - 1 = 0 \\)"
] | [
"Write down a polynomial equation of degree 5 which is satisfied by \\( w \\), where \\( w = \\cos \\frac{2}{5}\\pi + i \\sin \\frac{2}{5}\\pi \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-7316100d2dbc0dda826d116efebd81996337f828368170adf1f6b626b7b56953 | Write down a solution of the equation \(x^2 - 2y^2 = 1\) for which \(x\) and \(y\) are non-negative integers. | [
"\\( (x, y) = (3, 2) \\)"
] | [
"Write down a solution of the equation \\(x^2 - 2y^2 = 1\\) for which \\(x\\) and \\(y\\) are non-negative integers."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-389749a41ad310a1e1d15810210c5514376eac9f3b917926fb4ceaa19191999c | Write down all finite abelian groups of order $7^6$ that are not isomorphic to each other. | [
"$\\mathbb{Z}_7^6, \\mathbb{Z}_7^4 \\times \\mathbb{Z}_{7^2}, \\mathbb{Z}_7^3 \\times \\mathbb{Z}_{7^3}, \\mathbb{Z}_7^2 \\times \\mathbb{Z}_{7^2}^2, \\mathbb{Z}_7^2 \\times \\mathbb{Z}_{7^4}, \\mathbb{Z}_7 \\times \\mathbb{Z}_{7^2} \\times \\mathbb{Z}_{7^3}, \\mathbb{Z}_7 \\times \\mathbb{Z}_{7^5}, \\mathbb{Z}_{7^... | [
"Write down all finite abelian groups of order $7^6$ that are not isomorphic to each other."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 5,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-4e80e180ad21f26c4541ad1b8393ee1d58525b9a14c42344ae0d531b68d6eadf | Write down all the positive integers from 1 to $10^{10000}$. Start by adding the first 3 numbers in the sequence. Cross out the sum in the sequence and write it in a list. Then cross out each of the terms you included in the sum but don't write it in the list. Repeat this process with the next 3 numbers in the sequence... | [
"Yes"
] | [
"Write down all the positive integers from 1 to $10^{10000}$. Start by adding the first 3 numbers in the sequence. Cross out the sum in the sequence and write it in a list. Then cross out each of the terms you included in the sum but don't write it in the list. Repeat this process with the next 3 numbers in the seq... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-46cecfe223f2c3b8c623cb4ccd537f3a0847fff34d71c10725802b86e3094a32 | Write down an arrangement for a chess tournament with 8 players (Ann, Bob, Con, Dot, Eve, Fay, Guy, and Hal) such that there are 24 games, no pair plays together more than once, and no group of five people in which each one plays against all of the other four. | [
"Ann-Bob, Ann-Con, Ann-Dot, Ann-Eve, Ann-Fay, Ann-Guy, Ann-Hal, Bob-Con, Bob-Dot, Bob-Eve, Bob-Fay, Bob-Guy, Bob-Hal, Con-Dot, Con-Eve, Con-Fay, Con-Guy, Con-Hal, Dot-Eve, Dot-Fay, Dot-Guy, Dot-Hal, Eve-Fay, Eve-Guy, Eve-Hal, Fay-Guy, Fay-Hal, Guy-Hal (excluding Ann-Bob, Con-Dot, Eve-Fay, Guy-Hal)"
] | [
"Write down an arrangement for a chess tournament with 8 players (Ann, Bob, Con, Dot, Eve, Fay, Guy, and Hal) such that there are 24 games, no pair plays together more than once, and no group of five people in which each one plays against all of the other four."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 21
} | 41 | 0 | 0 |
omr-ws-0e62676c4a532c12bd36bde6aaacd7cbadb8c479132ec021b7ce2fa3903e7a04 | Write down an explicit isomorphism from $ l^2(\mathbb{Z})$ to $ l^2$. | [
"(x_k)_{k \\in \\mathbb{Z}} \\mapsto \\left( x_0, x_1, x_{-1}, x_2, x_{-2}, x_3, x_{-3}, \\ldots \\right)"
] | [
"Write down an explicit isomorphism from $ l^2(\\mathbb{Z})$ to $ l^2$."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-df26dcd5f47bab651e838b0bb0079ce15f02d5261477cccaa7a846b8fa25ed86 | Write down an integer 2-rational point and obtain a non-integer 2-rational point on the circle \(x^2 + y^2 = 3\). | [
"Integer 2-rational point: \\((1, \\sqrt{2})\\), Non-integer 2-rational point: \\(\\left(\\frac{2+5\\sqrt{2}}{6}, \\frac{2-5\\sqrt{2}}{6}\\right)\\)"
] | [
"Write down an integer 2-rational point and obtain a non-integer 2-rational point on the circle \\(x^2 + y^2 = 3\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f3697cac952174612673cc60c7b2af27c989b61094f11eab740311b59633f485 | Write down an integer rational point and a non-integer rational point on the circle \(x^2 + y^2 = 1\). | [
"Integer rational point: \\((1, 0)\\), Non-integer rational point: \\(\\left(\\frac{3}{5}, \\frac{4}{5}\\right)\\)"
] | [
"Write down an integer rational point and a non-integer rational point on the circle \\(x^2 + y^2 = 1\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-eca75004ee4ec6b3d7854a949583bf866e8bbd6b5c5760ad7717cc2c48528efc | Write down and calculate an iterated double integral in rectangular coordinates that represents the volume of the solid bounded below by the xy-plane and bounded above by the sphere \( z = 9 - x^2 - y^2 \). | [
"\\(\\frac{81\\pi}{2}\\)"
] | [
"Write down and calculate an iterated double integral in rectangular coordinates that represents the volume of the solid bounded below by the xy-plane and bounded above by the sphere \\( z = 9 - x^2 - y^2 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-b9af365bf76c8ef9aa7c44a9407f8556c84418f5826c35fd14c584b912d70958 | Write down explicitly the expression for the n-th derivative of the function $f(x) = x^2e^{3x}$. | [
"\\( 3^n e^{3x} \\left( x^2 + \\frac{2nx}{3} + \\frac{n^2 - n}{9} \\right) \\)"
] | [
"Write down explicitly the expression for the n-th derivative of the function $f(x) = x^2e^{3x}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-de7cd144287b58fad8d66a485762bfd0ba7eb7bb4ada6d2e8eb611492bfc54e5 | Write down explicitly the first five equations for the sum of the cubes of the first \( n \) positive integers: \[ 1^3 = \left( \frac{1(1+1)}{2} \right)^2, \] \[ 1^3 + 2^3 = \left( \frac{2(2+1)}{2} \right)^2, \] \[ 1^3 + 2^3 + 3^3 = \left( \frac{3(3+1)}{2} \right)^2, \] \[ 1^3 + 2^3 + 3^3 + 4^3 = \left( \frac{4(4+1)}{2... | [
"\\[ 1^3 = 1, \\]\n\\[ 1^3 + 2^3 = 9, \\]\n\\[ 1^3 + 2^3 + 3^3 = 36, \\]\n\\[ 1^3 + 2^3 + 3^3 + 4^3 = 100, \\]\n\\[ 1^3 + 2^3 + 3^3 + 4^3 + 5^3 = 225. \\]"
] | [
"Write down explicitly the first five equations for the sum of the cubes of the first \\( n \\) positive integers:\n\\[ 1^3 = \\left( \\frac{1(1+1)}{2} \\right)^2, \\]\n\\[ 1^3 + 2^3 = \\left( \\frac{2(2+1)}{2} \\right)^2, \\]\n\\[ 1^3 + 2^3 + 3^3 = \\left( \\frac{3(3+1)}{2} \\right)^2, \\]\n\\[ 1^3 + 2^3 + 3^3 + 4... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-e883109abafa54ffd0474da0bbfc23e082e524aee6d2095a4ddedddf6c910055 | Write down expressions for \( T_{2m} - T_{2m-1} \) and \( T_{2m} - T_{2m-2} \). | [
"4m - 1"
] | [
"Write down expressions for \\( T_{2m} - T_{2m-1} \\) and \\( T_{2m} - T_{2m-2} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4275287be28421ba6498e3c0c4252e472c0d1b360104fa0c8c350bac365d260e | Write down one of the figures by selecting among the numbers $1, 2, 3, 4, 5$ at random. Doing this procedure repeatedly, let $P_n\ (n \geq 5)$ be the probability that all five kinds of figures are presented for the first time in the $n$th round. Find the expected value $E_n$ of $n$. | [
"11.4165"
] | [
"Write down one of the figures by selecting among the numbers $1, 2, 3, 4, 5$ at random. Doing this procedure repeatedly, let $P_n\\ (n \\geq 5)$ be the probability that all five kinds of figures are presented for the first time in the $n$th round. Find the expected value $E_n$ of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 31
} | 78 | 0 | 0 |
omr-ws-e433ba66d4855bae9b5d2f80242f13431912cea4ee975b6a239b364c3f5a17a8 | Write down six consecutive natural numbers such that each of them is a multiple of a perfect square (other than 1). Please list all those perfect squares and show your calculations. | [
"4, 9, 25, 49, 4, 121 or 121, 4, 49, 25, 9, 4"
] | [
"Write down six consecutive natural numbers such that each of them is a multiple of a perfect square (other than 1). Please list all those perfect squares and show your calculations."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-69622ef21cbf83d34440049b392ed782fe30e657bc60eb9088c000a26f9a0fac | Write down six different ideals of the ring $$R=\left\{\left(\begin{array}{cc}a&b\\0&a\end{array}\right)\mid a,b\in\mathbb{Z}/4\mathbb{Z}\right\}.$$ | [
"\\[\n\\left\\{\\left(\\begin{array}{cc}0&I_1\\\\0&0\\end{array}\\right)\\right\\}, \\left\\{\\left(\\begin{array}{cc}0&I_2\\\\0&0\\end{array}\\right)\\right\\}, \\left\\{\\left(\\begin{array}{cc}I_1&I_1\\\\0&I_1\\end{array}\\right)\\right\\}, \\left\\{\\left(\\begin{array}{cc}I_2&I_2\\\\0&I_2\\end{array}\\right)\\... | [
"Write down six different ideals of the ring\n$$R=\\left\\{\\left(\\begin{array}{cc}a&b\\\\0&a\\end{array}\\right)\\mid a,b\\in\\mathbb{Z}/4\\mathbb{Z}\\right\\}.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 7,
"tir": 8
} | 19 | 0 | 0 |
omr-ws-52bad003499b5dba27e5092c9d3361c9344d4d0473f4171aa4afaa3c1e6b3fa8 | Write down the 5 nice representations of 10, where a nice representation of a positive integer \( n \) is defined as a sum of powers of 2 with at most two equal powers in the sum. | [
"\\( 2^0 + 2^0 + 2^3, 2^2 + 2^1 + 2^1 + 2^0 + 2^0, 2^2 + 2^2 + 2^1, 2^1 + 2^3, 2^0 + 2^0 + 2^2 + 2^2 \\)"
] | [
"Write down the 5 nice representations of 10, where a nice representation of a positive integer \\( n \\) is defined as a sum of powers of 2 with at most two equal powers in the sum."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 20
} | 20 | 0 | 0 |
omr-ws-533e0d07d2c6f1a417f7a9ef667bfc9fd6e604d537d45a901d2dc0b87818d833 | Write down the Cartesian equation of the face ABC of a tetrahedron with vertices at A(0, 0, 1), B(3, 0, 2), and C(2, 3, 1) in standard form. | [
"-3x + 2y + 9z = 9"
] | [
"Write down the Cartesian equation of the face ABC of a tetrahedron with vertices at A(0, 0, 1), B(3, 0, 2), and C(2, 3, 1) in standard form."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7ebdedb635352360f0c92d9eca568f55d87f26a2477e42d6e723eaa8e8157501 | Write down the Cartesian equation of the line CD in standard form for a tetrahedron with vertices at C(2, 3, 1) and D(1, 1, 2). | [
"\\dfrac{x - 2}{-1} = \\dfrac{y - 3}{-2} = \\dfrac{z - 1}{1}"
] | [
"Write down the Cartesian equation of the line CD in standard form for a tetrahedron with vertices at C(2, 3, 1) and D(1, 1, 2)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-682466b61c70c4fe443774755ba5a2078fcabb9eb035e9b9519a235ed248192c | Write down the Cauchy-Schwarz inequality for \( n \) elements. | [
"\\(\\left(\\sum_{k=1}^n a_k^2\\right )\\left(\\sum_{k=1}^n b_k^2\\right )\\geq \\left(\\sum_{k=1}^n a_k b_k\\right )^2\\)"
] | [
"Write down the Cauchy-Schwarz inequality for \\( n \\) elements."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 11
} | 25 | 0 | 0 |
omr-ws-bd84c060f75afbe90d8ccebd4cb39936f37d32347d55760983a31a1ea25a6cd1 | Write down the Green's function for the domain \(\{(x,t): a \le t < x \le b\}\) given the boundary value problem \((py')' + Vy = r\) with boundary conditions \(y(a) = 0\) and \(ky(b) + my'(b) = 0\). The Green's function \(G(x,t)\) on the domain \(\{(x,t): a \le x < t \le b\}\) is given by \(G(x,t) = -\frac{u(x)v(t)}{p(... | [
"\\( G(x,t) = \\frac{v(x)u(t)}{p(x)W(u,v)(t)} \\)"
] | [
"Write down the Green's function for the domain \\(\\{(x,t): a \\le t < x \\le b\\}\\) given the boundary value problem \\((py')' + Vy = r\\) with boundary conditions \\(y(a) = 0\\) and \\(ky(b) + my'(b) = 0\\). The Green's function \\(G(x,t)\\) on the domain \\(\\{(x,t): a \\le x < t \\le b\\}\\) is given by \\(G(... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-18aa9cbe15ecb1b58a24512fbf566177caa73ed49fa66664c0783f00818a8c70 | Write down the Green's function in terms of \( u \) and \( v \) for the Sturm-Liouville operator \[ \frac{d}{dx}\left[ p(x) \frac{dy}{dx} \right] + V(x) \] where \( u \) and \( v \) are the two solutions of the differential equation \[ \frac{d}{dx}\left[ p(x) \frac{dy}{dx} \right] + V(x)y = 0. \] | [
"G(x, \\xi) = \\frac{u(x_<) v(x_>)}{p(\\xi) [u(\\xi) v'(\\xi) - u'(\\xi) v(\\xi)]}"
] | [
"Write down the Green's function in terms of \\( u \\) and \\( v \\) for the Sturm-Liouville operator\n\n\\[\n\\frac{d}{dx}\\left[ p(x) \\frac{dy}{dx} \\right] + V(x)\n\\]\n\nwhere \\( u \\) and \\( v \\) are the two solutions of the differential equation\n\n\\[\n\\frac{d}{dx}\\left[ p(x) \\frac{dy}{dx} \\right] + ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87784c61ecf3d21d136a520753cd32534c3289092c6be5cc7a866749c9210e60 | Write down the Taylor series at \( x = 0 \) for \( \sqrt{1-x^{2}} \). | [
"\\( 1 - \\frac{x^2}{2} - \\frac{x^4}{8} + \\mathcal{O}(x^5) \\)"
] | [
"Write down the Taylor series at \\( x = 0 \\) for \\( \\sqrt{1-x^{2}} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 27
} | 34 | 0 | 0 |
omr-ws-28da58f31e4a194ed62d3d150d3015b007a6055ebd70bae70fcb5fc6f2d317a4 | Write down the Taylor series at \( x = 0 \) for \( x^{2}\cos(\sqrt{x}) \). | [
"\\( x^2 - \\frac{x^3}{2} + \\frac{x^4}{24} + \\mathcal{O}(x^5) \\)"
] | [
"Write down the Taylor series at \\( x = 0 \\) for \\( x^{2}\\cos(\\sqrt{x}) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 20
} | 37 | 0 | 0 |
omr-ws-b27c907a26389afbd1cad90045a79d0eaefe4fbd92e1c0f8e715bb554125e384 | Write down the Taylor series at \( x = 0 \) for \( x^{2}e^{x} \). | [
"\\(\\sum_{n=0}^{\\infty} \\frac{x^{n+2}}{n!}\\)"
] | [
"Write down the Taylor series at \\( x = 0 \\) for \\( x^{2}e^{x} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-343b10ce851f9b2794610bc1185019bc40344f3fbc4d106842e572d87bd400e0 | Write down the Taylor series expansion about the origin, up to the term involving \( x^7 \), for the function \( f(x) = \frac{1}{2} \left[ x\sqrt{1-x^2} + \sin^{-1}x \right] \). | [
"\\( x - \\frac{1}{6}x^3 - \\frac{1}{40}x^5 - \\frac{1}{112}x^7 \\)"
] | [
"Write down the Taylor series expansion about the origin, up to the term involving \\( x^7 \\), for the function \\( f(x) = \\frac{1}{2} \\left[ x\\sqrt{1-x^2} + \\sin^{-1}x \\right] \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-d26fa2a74429d024f27a2846cf124d464e018054c0396ddf0cf13b4572396ac3 | Write down the automorphism of order $2$ of a cyclic group of order $p$, where $p > 2$ is a prime. | [
"\\( \\phi(x) = -x \\)"
] | [
"Write down the automorphism of order $2$ of a cyclic group of order $p$, where $p > 2$ is a prime."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 26
} | 40 | 0 | 0 |
omr-ws-ff0b52c0051d2b04ef77c5e02970eefab8cb2058645ede2fb12e18178fd70b2c | Write down the automorphism of order 2 of a cyclic group of order \( p \). | [
"\\( f(x) = -x \\)"
] | [
"Write down the automorphism of order 2 of a cyclic group of order \\( p \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-fa833f8e4deae1248a1f8ee9877e91ba5500b7c0ef0674c047dec5121502637c | Write down the binomial expansion of $(1+\frac{k}{100})^{0.5}$ in ascending powers of $k$, up to and including the $k^3$ term. | [
"\\(1 + \\frac{k}{200} - \\frac{k^2}{80000} + \\frac{k^3}{16000000}\\)"
] | [
"Write down the binomial expansion of $(1+\\frac{k}{100})^{0.5}$ in ascending powers of $k$, up to and including the $k^3$ term."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-3c1efe6ccc1641d244a54aed1c2e2e4349ead13a1d989273c17c4cf8ab4f4908 | Write down the coefficient of $x^n$ in the binomial expansion for $(1-x)^{-N}$, where $N$ is a positive integer, and write down the expansion using the $\Sigma$ summation notation. | [
"\\(\\binom{N+n-1}{n}\\)"
] | [
"Write down the coefficient of $x^n$ in the binomial expansion for $(1-x)^{-N}$, where $N$ is a positive integer, and write down the expansion using the $\\Sigma$ summation notation."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 15
} | 23 | 0 | 0 |
omr-ws-2e75e8273d28d7ef219034d581a5d22c0a7468a5e8fd3d9418337fabf0aeedc9 | Write down the coordinates of the minimum point on the curve with equation \( y = 13 \cosh x - 5 \sinh x \). | [
"\\( \\left( \\ln \\left( \\frac{3}{2} \\right), 12 \\right) \\)"
] | [
"Write down the coordinates of the minimum point on the curve with equation \\( y = 13 \\cosh x - 5 \\sinh x \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5fdf09f5792b4ba3adbe79cd10030c8cba884db2e6d72422682ba5174bc3455b | Write down the different ways of getting \( x^7 \) from the infinite polynomial \( Q(x) = (1+x)(1+x^2)(1+x^3)(1+x^4)\cdots \). | [
"5"
] | [
"Write down the different ways of getting \\( x^7 \\) from the infinite polynomial \\( Q(x) = (1+x)(1+x^2)(1+x^3)(1+x^4)\\cdots \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-9eadfdd819e28eb416146d4a4d85339fdc68bf1d9af53d489e2c42dcfe1bdba0 | Write down the differential equation governing the motion of a body falling under gravity, where the resisting force of air is proportional to the square of the velocity of the body. The distance travelled by the body in time \( t \) is denoted by \( y(t) \). | [
"\\( y'' = g - \\frac{\\beta}{m} y'^2 \\)"
] | [
"Write down the differential equation governing the motion of a body falling under gravity, where the resisting force of air is proportional to the square of the velocity of the body. The distance travelled by the body in time \\( t \\) is denoted by \\( y(t) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 7
} | 21 | 0 | 0 |
omr-ws-6660ab6066d066fd72d540147ff8cf6f13b8536f77bc2d2ada98eea2c7d11afd | Write down the differential equation satisfied by \( y_{1}(x) \) and verify that \( y_{1}(x) = x \). | [
"\\((1-x^{2})(\\frac{dy}{dx})^{2} + y^{2} = 1\\)"
] | [
"Write down the differential equation satisfied by \\( y_{1}(x) \\) and verify that \\( y_{1}(x) = x \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-afff301f32e99283a8cd5476596452d1dcdb9cffbe96815a40d981db712489a6 | Write down the differential equation satisfied by \( y_{2}(x) \) and verify that \( y_{2}(x) = 2x^{2} - 1 \). | [
"\\((1-x^{2})(\\frac{dy}{dx})^{2} + 4y^{2} = 4\\)"
] | [
"Write down the differential equation satisfied by \\( y_{2}(x) \\) and verify that \\( y_{2}(x) = 2x^{2} - 1 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b9108932ca3ba544a73dead4143ff9fb5b3e73073a053ba8424a5af2bb11ec3b | Write down the elements of a subgroup of order 2 in the group $M$ defined by the matrices $A, B, C, D, E, F$ where $\omega$ and $\omega^2$ are the complex roots of unity. | [
"\\(\\{A, D\\}\\)"
] | [
"Write down the elements of a subgroup of order 2 in the group $M$ defined by the matrices $A, B, C, D, E, F$ where $\\omega$ and $\\omega^2$ are the complex roots of unity."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 8
} | 25 | 0 | 0 |
omr-ws-1fd0d6e6a478ef2d9d9513b6c8bd34908d350f6a47f20d97f708c53b9adda49e | Write down the equation of the chord of contact of the parabola \(x^2 = 4ay\) from the point \((x_0, y_0)\). | [
"\\(xx_0 = 2a(y + y_0)\\)"
] | [
"Write down the equation of the chord of contact of the parabola \\(x^2 = 4ay\\) from the point \\((x_0, y_0)\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-4cd6cab7cadf2372fefabcbd74233690d4f9d9f90d77299718b8127aad0d56cd | Write down the exact values of \( \cos \left( \frac{2}{5} \pi \right) \) and \( \cos \left( \frac{4}{5} \pi \right) \). | [
"\\(\\cos \\left( \\frac{2}{5} \\pi \\right) = -\\frac{1}{4}(1 - \\sqrt{5})\\) and \\(\\cos \\left( \\frac{4}{5} \\pi \\right) = -\\frac{1}{4}(1 + \\sqrt{5})\\)"
] | [
"Write down the exact values of \\( \\cos \\left( \\frac{2}{5} \\pi \\right) \\) and \\( \\cos \\left( \\frac{4}{5} \\pi \\right) \\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 26
} | 26 | 0 | 0 |
omr-ws-87efd0e998fae3f755f0248dfb3697a0f244f826e46654e112ed3d673e61c348 | Write down the five roots of the equation $z^{5} - 1 = 0$. | [
"\\( 1, \\cos \\frac{2\\pi}{5} + i \\cdot \\sin \\frac{2\\pi}{5}, \\cos \\frac{4\\pi}{5} + i \\cdot \\sin \\frac{4\\pi}{5}, \\cos \\frac{6\\pi}{5} + i \\cdot \\sin \\frac{6\\pi}{5}, \\cos \\frac{8\\pi}{5} + i \\cdot \\sin \\frac{8\\pi}{5} \\)"
] | [
"Write down the five roots of the equation $z^{5} - 1 = 0$."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 29
} | 36 | 0 | 0 |
omr-ws-d0719bf4e24a7345625d0fd021b70563bfdc63b2b36b2aaaac2890c3d4ec39c2 | Write down the general term in the expansion in powers of $x$ of $(1-x^{6})^{-2}$. | [
"\\( (n+1)x^{6n} \\)"
] | [
"Write down the general term in the expansion in powers of $x$ of $(1-x^{6})^{-2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-dff5ef00b8268c19514fdd5c19c7d36794eb6b2f5770f93954796ebc71856195 | Write down the matrix that represents the linear transformation \( L: \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \( L(i) = 3j \) and \( L(j) = -6i + 5j \). | [
"\\(\\begin{pmatrix} 0 & -6 \\\\ 3 & 5 \\end{pmatrix}\\)"
] | [
"Write down the matrix that represents the linear transformation \\( L: \\mathbb{R}^2 \\to \\mathbb{R}^2 \\) defined by \\( L(i) = 3j \\) and \\( L(j) = -6i + 5j \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-a28614dfdd762341ee69af2a07b0e51e05b141736d92deef467b2c961591bc59 | Write down the matrix that represents the linear transformation \( R : \mathbb{R}^5 \to \mathbb{R}^4 \) defined by \( R(e_1) = 2e_3 \), \( R(e_2) = 3e_4 \), \( R(e_3) = 4e_1 \), \( R(e_4) = 5e_2 \), and \( R(e_5) = 6e_3 \). | [
"\\(\\begin{pmatrix} 0 & 0 & 4 & 0 & 0 \\\\ 0 & 0 & 0 & 5 & 0 \\\\ 2 & 0 & 0 & 0 & 6 \\\\ 0 & 3 & 0 & 0 & 0 \\end{pmatrix}\\)"
] | [
"Write down the matrix that represents the linear transformation \\( R : \\mathbb{R}^5 \\to \\mathbb{R}^4 \\) defined by \\( R(e_1) = 2e_3 \\), \\( R(e_2) = 3e_4 \\), \\( R(e_3) = 4e_1 \\), \\( R(e_4) = 5e_2 \\), and \\( R(e_5) = 6e_3 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-c9747a2e8979371b8ff744163a3365ef644c029bc7df48aea03582d7bf9faa85 | Write down the negation, the converse, and the contrapositive of the implication "If $a=0$ and $b=0$, then $ab=0$." Which of these statements is equivalent to the given implication? | [
"\\text{The contrapositive}"
] | [
"Write down the negation, the converse, and the contrapositive of the implication \"If $a=0$ and $b=0$, then $ab=0$.\" Which of these statements is equivalent to the given implication?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6b86bc84b7212257497490f2c48d3e18c6fc9fa499a908e0eb0e94a60a0ed5f7 | Write down the parametric equations of the locus of the point P on a unit circle centered at C, which moves without slipping on the positive x-axis with C moving in the upper half-plane. The point P coincides with the origin at the initial position of the circle, and the parameter θ is the angle through which the radiu... | [
"\\( x = \\theta - \\sin \\theta \\), \\( y = 1 - \\cos \\theta \\)"
] | [
"Write down the parametric equations of the locus of the point P on a unit circle centered at C, which moves without slipping on the positive x-axis with C moving in the upper half-plane. The point P coincides with the origin at the initial position of the circle, and the parameter θ is the angle through which the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 13
} | 17 | 0 | 0 |
omr-ws-51b98bec3821bb6411ff07649153ea82e866fa8b6e79cf55a4ba95efa65281f0 | Write down the set of all natural numbers \( 1 \le x \le 200 \) such that: \[ \begin{cases} x \equiv 1 \pmod{2} \\ x \equiv 2 \pmod{3} \\ x \equiv 3 \pmod{4} \\ x \equiv 4 \pmod{5} \\ x \equiv 5 \pmod{6} \\ x \equiv 6 \pmod{7} \\ x \equiv 7 \pmod{8} \\ x \equiv 8 \pmod{9} \end{cases} \] | [
"\\emptyset"
] | [
"Write down the set of all natural numbers \\( 1 \\le x \\le 200 \\) such that:\n\\[ \\begin{cases} \nx \\equiv 1 \\pmod{2} \\\\ \nx \\equiv 2 \\pmod{3} \\\\ \nx \\equiv 3 \\pmod{4} \\\\ \nx \\equiv 4 \\pmod{5} \\\\ \nx \\equiv 5 \\pmod{6} \\\\ \nx \\equiv 6 \\pmod{7} \\\\ \nx \\equiv 7 \\pmod{8} \\\\ \nx \\equiv 8... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-034bd962b65431f658453adcbe76d641c3fa504273fcc11e0fc100fe518e117d | Write down the subgroups of \( \text{Gal}(E/\mathbb{Q}) \) and the corresponding extensions of \( \mathbb{Q} \), and indicate which ones are Galois. | [
"E"
] | [
"Write down the subgroups of \\( \\text{Gal}(E/\\mathbb{Q}) \\) and the corresponding extensions of \\( \\mathbb{Q} \\), and indicate which ones are Galois."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-f1107d3bf224a02c936200ce70bc46a21f89bf42ccb987dbab9a8dcec67f7574 | Write down the two primitive fourth roots of unity. | [
"\\( i \\) and \\( -i \\)"
] | [
"Write down the two primitive fourth roots of unity."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 26
} | 26 | 0 | 0 |
omr-ws-71811d2cfbf25bca03e0f8badac7bb6e6247a72118adaf4857697dc17e891fb9 | Write down the vector equation of the line through $A$ perpendicular to $BC$, in terms of $u, a$ and a parameter $\lambda$. | [
"\\(\\vec{r} = \\vec{a} + \\lambda \\vec{u}\\)"
] | [
"Write down the vector equation of the line through $A$ perpendicular to $BC$, in terms of $u, a$ and a parameter $\\lambda$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-8593a0a7a8666a38f912d60ee57b276eb527b8d71c5f4233830b1c3734e21165 | Write each system of differential equations in matrix form \( x' = Ax + b \). State whether the given system is autonomous or nonautonomous, and also whether it is homogeneous or inhomogeneous. (a) \( x' = y \); \( y' = x + 4 \) (b) \( x' = x + 2y + \sin t \); \( y' = x + y - \cos t \) (c) \( x' = 2tx + y \); \( y' = 3... | [
"\\begin{pmatrix} x' \\\\ y' \\end{pmatrix} = \\begin{pmatrix} 3 & -4 \\\\ 1 & 3 \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\end{pmatrix}"
] | [
"Write each system of differential equations in matrix form \\( x' = Ax + b \\). State whether the given system is autonomous or nonautonomous, and also whether it is homogeneous or inhomogeneous.\n(a) \\( x' = y \\); \\( y' = x + 4 \\)\n(b) \\( x' = x + 2y + \\sin t \\); \\( y' = x + y - \\cos t \\)\n(c) \\( x' = ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-29634979402297840645015a6d8d04eda523a3e93db97f48a10d782869548f31 | Write either $1$ or $-1$ in each of the cells of a $(2n) \times (2n)$-table, such that there are exactly $2n^2$ entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be $M$. Determine the largest possible value of $M$. | [
"2"
] | [
"Write either $1$ or $-1$ in each of the cells of a $(2n) \\times (2n)$-table, such that there are exactly $2n^2$ entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be $M$. Determine the largest possible value of $M$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-e35c67aa8e1943ff50c0c0d5a107d60810e737039f953e0cd5a23233d47432f1 | Write either \( 1 \) or \( -1 \) in each of the cells of a \( (2n) \times (2n) \)-table, in such a way that there are exactly \( 2n^2 \) entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be \( M \). Determine the largest possible value of \( M \). | [
"2"
] | [
"Write either \\( 1 \\) or \\( -1 \\) in each of the cells of a \\( (2n) \\times (2n) \\)-table, in such a way that there are exactly \\( 2n^2 \\) entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be \\( M \\). Determine the largest possible value of \\( M \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e04beb716a4516e8447a18b14f6c19125eeb7378420492e454b74c609fa2365e | Write equations whose roots are equal to the numbers $\cot^2 \frac{\pi }{2n+1},\ \cot^2 \frac{2\pi }{2n+1},\ \cot^2 \frac{3\pi }{2n+1},\ldots ,\ \cot^2 \frac{n\pi }{2n+1}$. | [
"Make use of the formulas of Problem 321(b)"
] | [
"Write equations whose roots are equal to the numbers\n$\\cot^2 \\frac{\\pi }{2n+1},\\ \\cot^2 \\frac{2\\pi }{2n+1},\\ \\cot^2 \\frac{3\\pi }{2n+1},\\ldots ,\\ \\cot^2 \\frac{n\\pi }{2n+1}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1c71c7df0b6420016a0bb53e3d603b96bce5ce2d115d39b7f7c217d720746998 | Write every even number greater than 46 as a sum of two abundant numbers. | [
"\\text{Every even number greater than 46 can be written as the sum of two abundant numbers.}"
] | [
"Write every even number greater than 46 as a sum of two abundant numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-b58505ae3dd7e7cb60a50a868ec3f3043c1b76674189c11c7470486f34f8957c | Write every number between 1 and 2500 using exactly four 4s and the operations +, -, *, /, !, \(\sqrt{\cdot}\), and \(d(\cdot)\), where \(d(n)\) is the number of divisors of \(n\). | [
"2500"
] | [
"Write every number between 1 and 2500 using exactly four 4s and the operations +, -, *, /, !, \\(\\sqrt{\\cdot}\\), and \\(d(\\cdot)\\), where \\(d(n)\\) is the number of divisors of \\(n\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-13c0dd62fafa2ca8448e67b91e50cba10dde6f1fb3177c6104eea4e7156ddd7a | Write explicit polynomials that vanish on the set of $2 \times 3$ matrices with entries in $k$ that have rank $\le 1$. | [
"\\(bf - ce\\), \\(-af + cd\\), \\(ae - bd\\)"
] | [
"Write explicit polynomials that vanish on the set of $2 \\times 3$ matrices with entries in $k$ that have rank $\\le 1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 11
} | 17 | 0 | 0 |
omr-ws-d9e0c7af83f64f4be17c443ac96290c1a57b606422c86dd0db9952b786aedfe9 | Write five other ways to express the iterated integral \[ \int_{0}^{1}\int_{y}^{1}\int_{0}^{y}f(x,y,z)\, {\rm d}z{\rm d}x{\rm d}y. \] | [
"\\[\n\\int_{0}^{1}\\int_{0}^{y}\\int_{y}^{1}f(x,y,z)\\, {\\rm d}x{\\rm d}z{\\rm d}y, \\quad \\int_{0}^{1}\\int_{0}^{x}\\int_{z}^{x}f(x,y,z)\\, {\\rm d}y{\\rm d}z{\\rm d}x, \\quad \\int_{0}^{1}\\int_{0}^{x}\\int_{z}^{x}f(x,y,z)\\, {\\rm d}y{\\rm d}z{\\rm d}x, \\quad \\int_{0}^{1}\\int_{z}^{1}\\int_{0}^{x}f(x,y,z)\\... | [
"Write five other ways to express the iterated integral \n\\[\n\\int_{0}^{1}\\int_{y}^{1}\\int_{0}^{y}f(x,y,z)\\, {\\rm d}z{\\rm d}x{\\rm d}y.\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-7985ee7f61dc3dba158f06e530901a92f352c1187f249dbb01f592a7b79e81b2 | Write in a trigonometric way the expression $\frac{(2-2i)^{\frac{1}{94}}}{(-\sqrt{3}+i)^\frac{1}{108}}$. | [
"2^{\\frac{17}{2538}} \\left( \\cos \\frac{83\\pi}{7614} + i \\sin \\frac{83\\pi}{7614} \\right)"
] | [
"Write in a trigonometric way the expression $\\frac{(2-2i)^{\\frac{1}{94}}}{(-\\sqrt{3}+i)^\\frac{1}{108}}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-f05d7f8577b696f2b770414b42473ddd7b0d9d889afe95724e6b7df21378fb99 | Write in algebraic form the product of two odd numbers and one even number, using only one variable. | [
"\\( (2x-1)(2x+1)(2x) \\)"
] | [
"Write in algebraic form the product of two odd numbers and one even number, using only one variable."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 4
} | 26 | 0 | 0 |
omr-ws-a46aadb83d66c19e9129b0ae49f147548e632798097a4cc545e41998d113d2a1 | Write in base 2 the result of $(101010)_2 - (3)_{10}$. | [
"\\(100111_2\\)"
] | [
"Write in base 2 the result of $(101010)_2 - (3)_{10}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-50eae4442b4a2d65da8f000ab3b4ccb57d4511470961e535e638b4ef9cf55532 | Write numbers from $0$ to $9$ in the cells of a $10 \times 10$ square array so that each number appears exactly $10$ times. Let $S$ be the maximum number of different numbers in any row or column. Find the minimum possible value of $S$. | [
"5"
] | [
"Write numbers from $0$ to $9$ in the cells of a $10 \\times 10$ square array so that each number appears exactly $10$ times. Let $S$ be the maximum number of different numbers in any row or column. Find the minimum possible value of $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-76ef8e05dccee97b1229b6dd0f9e08814c4817a49ccb68933a934639542dbd86 | Write numbers from $1$ to $15$, placing $k$ numbers in the $k$-th column for $k=1,2,3,4,5$. Let $M_k$ be the maximum number in the $k$-th column. What is the probability that $M_1 < M_2 < M_3 < M_4 < M_5$? | [
"\\(\\frac{2}{45}\\)"
] | [
"Write numbers from $1$ to $15$, placing $k$ numbers in the $k$-th column for $k=1,2,3,4,5$. Let $M_k$ be the maximum number in the $k$-th column. What is the probability that $M_1 < M_2 < M_3 < M_4 < M_5$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 6,
"tir": 16
} | 38 | 0 | 0 |
omr-ws-2592b57ef70b9014f5d6d393f17ab9335c07a0122ff9e4b7a4fe2f7d3d733fec | Write odd integers on the 8 vertices of a cube. The number on each face of the cube is formed by the product of the 4 corner vertices. Calculate the sum of all numbers on the vertices and on the faces, and find the remainder when this sum is divided by 2. | [
"0"
] | [
"Write odd integers on the 8 vertices of a cube. The number on each face of the cube is formed by the product of the 4 corner vertices. Calculate the sum of all numbers on the vertices and on the faces, and find the remainder when this sum is divided by 2."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d723d57681563a5d2518977fb0ebc383441e74c0ae3a96c1f3c93778ef43f03a | Write out all left ideals in the algebra \( A \) over a field \( K \), where \( A = \left\{ \left[\begin{array}{cc}a & b \\ 0 & a\end{array}\right] : a, b \in K \right\} \). | [
"\\(\\left\\{ \\left[\\begin{array}{cc}0 & 0 \\\\ 0 & 0\\end{array}\\right] \\right\\}, \\left\\{ \\left[\\begin{array}{cc}a & 0 \\\\ 0 & a\\end{array}\\right] : a \\in K \\right\\}, \\left\\{ \\left[\\begin{array}{cc}a & b \\\\ 0 & a\\end{array}\\right] : a, b \\in K \\right\\}\\)"
] | [
"Write out all left ideals in the algebra \\( A \\) over a field \\( K \\), where \\( A = \\left\\{ \\left[\\begin{array}{cc}a & b \\\\ 0 & a\\end{array}\\right] : a, b \\in K \\right\\} \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-69d87eb8b91b35749985f65a18cdf0eb3f045781d0bd151aa149477d17d8229d | Write parametric equations for two curves on the ellipsoid \(E: x^2 + y^2 + 2z^2 = 4\) passing through the point \((1, 1, 1)\). | [
"\\[\n\\text{Curve 1: } x = \\sqrt{2} \\cos(t), \\quad y = \\sqrt{2} \\sin(t), \\quad z = 1\n\\]\n\\[\n\\text{Curve 2: } x = \\sqrt{3} \\cos(u), \\quad y = 1, \\quad z = \\sqrt{\\frac{3}{2}} \\sin(u)\n\\]"
] | [
"Write parametric equations for two curves on the ellipsoid \\(E: x^2 + y^2 + 2z^2 = 4\\) passing through the point \\((1, 1, 1)\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-62784b4bae5e09b69cceae936569a65bcca2daaee04408c02a436b4ffd9586d1 | Write the Euler relation for the function \( f(x) = \langle x, x \rangle^k = \|x\|^{2k} \). | [
"\\( kf(x) = \\sum_{i=1}^{n} x_i \\frac{\\partial f}{\\partial x_i}(x) \\)"
] | [
"Write the Euler relation for the function \\( f(x) = \\langle x, x \\rangle^k = \\|x\\|^{2k} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 11
} | 31 | 0 | 0 |
omr-ws-601bb0a8754070e61c88a5c524a40b138eb017ff043e5b190127427002563330 | Write the Laurent series expansion for the function \( f(z) = \frac{z}{5z+1} \) centered at \( z_0 = 1 \). State the possible domains. | [
"\\frac{1}{5} + \\sum_{n=1}^{\\infty} \\frac{(-1)^n 6^{n-1}}{5^{n+1}} (z - 1)^{-n}"
] | [
"Write the Laurent series expansion for the function \\( f(z) = \\frac{z}{5z+1} \\) centered at \\( z_0 = 1 \\). State the possible domains."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-d030f3280e46867c9dfb143cac4b798e6d6ab07b04ec8d7f53d0e96cf3fdc9c1 | Write the Laurent series expansion for the function \( g(z) = \frac{z}{z^2 - 3iz - 2} \) centered at \( z_0 = 0 \). State the possible domains. | [
"The Laurent series expansion for \\( g(z) = \\frac{z}{z^2 - 3iz - 2} \\) centered at \\( z_0 = 0 \\) depends on the region:\n- For \\( |z| < 1 \\):\n\\[ g(z) = 2 \\left( \\frac{-1}{2i} \\left(1 + \\frac{z}{2i} + \\left(\\frac{z}{2i}\\right)^2 + \\cdots \\right) \\right) - \\left( \\frac{-1}{i} \\left(1 + \\frac{z}... | [
"Write the Laurent series expansion for the function \\( g(z) = \\frac{z}{z^2 - 3iz - 2} \\) centered at \\( z_0 = 0 \\). State the possible domains."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-50c480f285553db6445bc92de7be7ff1675465e1565dae9bec0f8a32a7ae75ee | Write the Maclaurin formula for the function \( f(x) = \frac{1 + x + x^2}{1 - x + x^2} \) up to terms with \( x^4 \). | [
"1 + 2x + 2x^2 - 2x^4"
] | [
"Write the Maclaurin formula for the function \\( f(x) = \\frac{1 + x + x^2}{1 - x + x^2} \\) up to terms with \\( x^4 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1baeeec1dd793f2baf36a3a0d07dcb4d77d8b7c280b1e8c581c087c32f0f1efc | Write the Maclaurin formula for the function \( g(x) = e^{2x - x^2} \) up to terms with \( x^5 \). | [
"1 + 2x + x^2 - \\dfrac{2}{3}x^3 - \\dfrac{5}{6}x^4 - \\dfrac{1}{15}x^5"
] | [
"Write the Maclaurin formula for the function \\( g(x) = e^{2x - x^2} \\) up to terms with \\( x^5 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-cdad2911a2b8f078bc3fd41df8f38477728ac31be5ea61eefeea78408ba042f2 | Write the Maclaurin formula for the function \( h(x) = \ln (\cos x) \) up to terms with \( x^6 \). | [
"-\\frac{x^2}{2} - \\frac{x^4}{12} - \\frac{x^6}{45}"
] | [
"Write the Maclaurin formula for the function \\( h(x) = \\ln (\\cos x) \\) up to terms with \\( x^6 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed8ecab7e089fcd3d574b63a0813bba041e4d77a57b0656e59e9db8f498110d6 | Write the Maclaurin formula for the function \( k(x) = \tan x \) up to terms with \( x^5 \). | [
"x + \\frac{1}{3}x^3 + \\frac{2}{15}x^5"
] | [
"Write the Maclaurin formula for the function \\( k(x) = \\tan x \\) up to terms with \\( x^5 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ddde6bda00c4f4745b7f02460ba2930963d6a81d125f2c837dbdd73cd0370b64 | Write the Maclaurin series for \(\cos(x^2)\). By antidifferentiating its first three (nonzero) terms, obtain an approximation for the integral of \(\cos(x^2)\) from 0.6 to 1. Is this approximation larger than or smaller than the actual value of the integral of \(\cos(x^2)\) from 0.6 to 1? How does the series itself tel... | [
"0.31241028"
] | [
"Write the Maclaurin series for \\(\\cos(x^2)\\). By antidifferentiating its first three (nonzero) terms, obtain an approximation for the integral of \\(\\cos(x^2)\\) from 0.6 to 1. Is this approximation larger than or smaller than the actual value of the integral of \\(\\cos(x^2)\\) from 0.6 to 1? How does the ser... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 2,
"tir": 5
} | 15 | 0 | 0 |
omr-ws-974cc9b99a529acd38d92da94bb517472f1b904bbc23e1b81202d5ae3e1b0f53 | Write the SOS (Sum of Squares) form for the expression $$(n-2)\sum_{i<j} a_i a_j (a_i + a_j) - 6\sum_{i<j<k} a_i a_j a_k.$$ | [
"\\sum_{1 \\leq i < j < k \\leq n} \\left( (a_i - a_j)^2 a_k + (a_j - a_k)^2 a_i + (a_k - a_i)^2 a_j \\right)"
] | [
"Write the SOS (Sum of Squares) form for the expression $$(n-2)\\sum_{i<j} a_i a_j (a_i + a_j) - 6\\sum_{i<j<k} a_i a_j a_k.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-3347a1a5c09df7786808239d9eed27b30251cb8372c370fdc73a5f22fde025de | Write the Taylor polynomial of degree $3$, $P_3(x)$ for $f(x)=\ln(\sin x)$ expanded about $x=\frac{\pi}{2}$. Find $P_3\left(\frac{\pi}{3}\right)$. What is the general form of the actual remainder term $R_3(x)$ (not an upper bound)? What is an upper bound on $R_3\left(\frac{\pi}{3}\right)$? What are the bounds for the a... | [
"-\\dfrac{\\pi^2}{72}"
] | [
"Write the Taylor polynomial of degree $3$, $P_3(x)$ for $f(x)=\\ln(\\sin x)$ expanded about $x=\\frac{\\pi}{2}$. Find $P_3\\left(\\frac{\\pi}{3}\\right)$. What is the general form of the actual remainder term $R_3(x)$ (not an upper bound)? What is an upper bound on $R_3\\left(\\frac{\\pi}{3}\\right)$? What are the... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-8ec9d6027654f5967a804e1f4cf9009a78c8c043f74bbd5b255b54c7310f5894 | Write the Taylor series for $e^x$ up to the term in $x^n$ together with the remainder term. Determine the smallest integer $n$ to ensure that the estimated value of $e$ is correct to 4 decimal places and find the estimated value. | [
"\\( n = 8 \\), estimated value of \\( e \\approx 2.7183 \\)"
] | [
"Write the Taylor series for $e^x$ up to the term in $x^n$ together with the remainder term. Determine the smallest integer $n$ to ensure that the estimated value of $e$ is correct to 4 decimal places and find the estimated value."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-9144f25a959f5ffa34617ab77aae17d63e2ad763f2e53f86a25732bc45bc326f | Write the Taylor series for $e^{x}$ up to the term in $x^{n}$ together with the remainder term. Determine the smallest integer $n$ to ensure that the estimated value of $e$ is correct to four decimal places and find the estimated value. | [
"2.7183"
] | [
"Write the Taylor series for $e^{x}$ up to the term in $x^{n}$ together with the remainder term. Determine the smallest integer $n$ to ensure that the estimated value of $e$ is correct to four decimal places and find the estimated value."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-576160a151ef6b2aa2f7a12f75021b2fc2f1a86d3a046e8f2ee18c9ec075efc0 | Write the Taylor series of \( \sin x \) at the point \( a = \frac{\pi}{4} \) using \( \sum \) notation. | [
"\\( \\sum_{n=0}^{\\infty} (-1)^{\\frac{n(n-1)}{2}} \\frac{\\sqrt{2}}{2} \\frac{(x - \\pi/4)^n}{n!} \\)"
] | [
"Write the Taylor series of \\( \\sin x \\) at the point \\( a = \\frac{\\pi}{4} \\) using \\( \\sum \\) notation."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 3,
"tir": 23
} | 31 | 0 | 0 |
omr-ws-6d22bbeb2bc8348b6cdac60d5d3aae6b13eee28ad9b1575710e2f6ec582c826a | Write the Taylor series of the function \( f(x) = \frac{1}{1-x} + e^x \) at \( a = 0 \) and determine its convergence domain. | [
"\\( \\sum_{k=0}^{\\infty} \\left(1 + \\frac{1}{k!}\\right) x^k \\) for \\( x \\in (-1, 1) \\)"
] | [
"Write the Taylor series of the function \\( f(x) = \\frac{1}{1-x} + e^x \\) at \\( a = 0 \\) and determine its convergence domain."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-3ef91a1ed5f32c50e53991b3b45e06b6ed845f8a06bcfcbbfbefd358e533bfb3 | Write the Taylor series of the function \( f(x) = \ln(1-x^2) \) at \( a = 0 \) and determine its convergence domain. | [
"\\(-\\sum_{k=1}^{\\infty} \\frac{x^{2k}}{k}\\) for \\( x \\in (-1, 1) \\)"
] | [
"Write the Taylor series of the function \\( f(x) = \\ln(1-x^2) \\) at \\( a = 0 \\) and determine its convergence domain."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 2,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-50a53f69b2af122819beeb33616b9f214d7b6b266c911bf7bfdde7d7f0d9796f | Write the barycentric coordinates of the midpoint $M$ of segment $AB$. | [
"\\left( \\frac{1}{2}, \\frac{1}{2}, 0 \\right)"
] | [
"Write the barycentric coordinates of the midpoint $M$ of segment $AB$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-ec7bd578568e9b86c2e9a9c5f947e86b33472cd2c147ee19adb1674249d759ce | Write the complex expression $Z = \sin(\phi) + 2i \sin^2(\phi/2)$ in trigonometric form. | [
"\\( Z = 2 \\sin(\\phi/2) e^{i \\phi/2} \\)"
] | [
"Write the complex expression $Z = \\sin(\\phi) + 2i \\sin^2(\\phi/2)$ in trigonometric form."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-4c6601c2112c8d37262b04a0a6c20b0523686f30b2581dfe98e5ce09e0c455dc | Write the complex number \( z = \frac{-\sqrt{3} + i}{1 - i} \cdot \left( \cos\frac{\pi}{12} - i \sin\frac{\pi}{12} \right) \) in the exponential form. | [
"\\( \\sqrt{2} e^{\\pi i} \\)"
] | [
"Write the complex number \\( z = \\frac{-\\sqrt{3} + i}{1 - i} \\cdot \\left( \\cos\\frac{\\pi}{12} - i \\sin\\frac{\\pi}{12} \\right) \\) in the exponential form."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-dce37b7bc004f49e1c97a072c281e4a40542fda32ec6b3390b4805b75fc7368a | Write the composite rule for the quadrature rule \[ \int_0^{3h} f(x) \, \mathrm{d}x \approx \alpha_1 f(0) + \alpha_2 f(2h) + \alpha_3 f(3h) \] for the integral $\int_a^b f(x) \, \mathrm{d}x$ with $h = \frac{b - a}{3n}$ and $x_i = a + ih$, $i = 0, 1, \ldots, 3n$. | [
"\\(\\int_a^b f(x) \\, \\mathrm{d}x \\approx \\frac{b-a}{4n} \\left( f(x_0) + 3f(x_2) + f(x_3) + 3f(x_5) + \\cdots + f(x_{3n-3}) + 3f(x_{3n-1}) \\right)\\)"
] | [
"Write the composite rule for the quadrature rule\n\\[ \\int_0^{3h} f(x) \\, \\mathrm{d}x \\approx \\alpha_1 f(0) + \\alpha_2 f(2h) + \\alpha_3 f(3h) \\]\nfor the integral $\\int_a^b f(x) \\, \\mathrm{d}x$ with $h = \\frac{b - a}{3n}$ and $x_i = a + ih$, $i = 0, 1, \\ldots, 3n$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-29d05c338b344d513504aa372d4eca875574531a790921fae1f11c6fbfe69767 | Write the condition \( n \equiv 1 \pmod{4} \) and \( n \geq 9 \) in binary form. | [
"n \\text{ in binary ends with } 01 \\text{ and has at least four digits}"
] | [
"Write the condition \\( n \\equiv 1 \\pmod{4} \\) and \\( n \\geq 9 \\) in binary form."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66c2cccd757fa716ac1bad2d4d3ba2184ebf3241fb7cbd6d93fb92961d10806a | Write the condition \( n \equiv 2 \pmod{4} \) and \( n \geq 34 \) in binary form. | [
"n_2 \\geq 100010_2 \\text{ and } n_2 \\equiv 10_2 \\pmod{100_2}"
] | [
"Write the condition \\( n \\equiv 2 \\pmod{4} \\) and \\( n \\geq 34 \\) in binary form."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e33ef72b5121a5c4bc0a6a15260642a7c7eab87f4cadd4893a0a361e4cb4167c | Write the condition \( n \equiv 3 \pmod{4} \) and \( n \geq 27 \) in binary form. | [
"n \\equiv 11_2 \\pmod{100_2} \\text{ and } n \\geq 11011_2"
] | [
"Write the condition \\( n \\equiv 3 \\pmod{4} \\) and \\( n \\geq 27 \\) in binary form."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-76446ffed1bdf64a6548a5dc405c5d4fd0dfc1a58a81c6f0acec6453b21d0ca7 | Write the constraint for the availability of 500 dozen blank sweatshirts and T-shirts each in stock, ready for production. Here, the variables are: - \( sf \): number of units of Sweatshirt – Front to produce - \( sb \): number of units of Sweatshirt – Back to produce - \( tf \): number of units of T-Shirt – Front to p... | [
"\\( sf + sb \\leq 6000 \\) and \\( tf + tb \\leq 6000 \\)"
] | [
"Write the constraint for the availability of 500 dozen blank sweatshirts and T-shirts each in stock, ready for production. \n\nHere, the variables are:\n- \\( sf \\): number of units of Sweatshirt – Front to produce\n- \\( sb \\): number of units of Sweatshirt – Back to produce\n- \\( tf \\): number of units of T-... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-f15ef729867f0714dd026716f0a00aadbc0432d86a48e9ffba52fa82d5eaf674 | Write the constraint for the box capacity given that a standard-size box holds 12 T-shirts and a box of 12 sweatshirts is three times the size of a standard box. The truck has enough capacity to accommodate 1,200 standard-size boxes. | [
"\\( X_{tf} + X_{tb} + 3X_{sf} + 3X_{sb} \\leq 14400 \\)"
] | [
"Write the constraint for the box capacity given that a standard-size box holds 12 T-shirts and a box of 12 sweatshirts is three times the size of a standard box. The truck has enough capacity to accommodate 1,200 standard-size boxes."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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