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-169d8c5b6eaa7b9e98b05aae6a1fe4dc2254f2182efbf3344d569ef8cc17bc5f | Calculate the derivatives of the following functions. $f(x)=x\frac{\sqrt {x-2\sqrt {x-1}}}{\sqrt {x-1}-1}$. | [
"$f'(x)= \\left\\{\\begin{array}{rll} -1, & x\\in [1,2),\\\\ 1, & x\\in (2,\\infty ). \\end{array}\\right.$"
] | [
"Calculate the derivatives of the following functions. $f(x)=x\\frac{\\sqrt {x-2\\sqrt {x-1}}}{\\sqrt {x-1}-1}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22c6883e2a023f0d6c234f5228c2e64591b30cabf3ffeeb99cdce6cd19aedf3e | Calculate the determinant \( D_n \) for \( n \in \mathbb{N} \) with \( n \ge 3 \): \[ D_n = \begin{vmatrix} x_2 x_3 \cdots x_n & x_1 x_3 \cdots x_n & \ldots & x_1 x_2 \cdots x_{n-1} \\ 1 & 1 & \ldots & 1 \\ x_1 & x_2 & \ldots & x_n \\ x_1^2 & x_2^2 & \ldots & x_n^2 \\ \vdots & \vdots & \ddots & \vdots \\ x_1^{n-2} & x_... | [
"\\prod_{1 \\leq i < j \\leq n} (x_j - x_i)"
] | [
"Calculate the determinant \\( D_n \\) for \\( n \\in \\mathbb{N} \\) with \\( n \\ge 3 \\):\n\n\\[\nD_n = \\begin{vmatrix}\nx_2 x_3 \\cdots x_n & x_1 x_3 \\cdots x_n & \\ldots & x_1 x_2 \\cdots x_{n-1} \\\\\n1 & 1 & \\ldots & 1 \\\\\nx_1 & x_2 & \\ldots & x_n \\\\\nx_1^2 & x_2^2 & \\ldots & x_n^2 \\\\\n\\vdots & \... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01c273496d91163adbb2080e92c76bcba85b9084abd95fcdf1bb73bb5eba661c | Calculate the determinant \( D_n \) of the \( n \times n \) matrix defined as: \[ D_n = \left| \begin{array}{ccccc} a & b \\ c & a & b \\ & c & a & \ddots \\ & & \ddots & \ddots & b \\ & & & c & a \end{array} \right| \] | [
" \\frac{ \\left( a + \\sqrt{a^2 - 4bc} \\right)^{n+1} - \\left( a - \\sqrt{a^2 - 4bc} \\right)^{n+1} }{ 2^{n+1} \\sqrt{a^2 - 4bc} } "
] | [
"Calculate the determinant \\( D_n \\) of the \\( n \\times n \\) matrix defined as:\n\\[ D_n = \\left| \\begin{array}{ccccc}\na & b \\\\\nc & a & b \\\\\n& c & a & \\ddots \\\\\n& & \\ddots & \\ddots & b \\\\\n& & & c & a\n\\end{array} \\right| \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-a733cab9b3a3cf8c263e16b20b8a173f562fca5d43b5f48d11f0455abb969f9b | Calculate the determinant \( D_n \) of the \( n \times n \) matrix given by: \[ D_n = \begin{vmatrix} 2\cos a & 1 & 0 & 0 & \cdots & 0 & 0 \\ 1 & 2\cos a & 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 2\cos a & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 2\cos a &... | [
"\\( D_n = \\frac{\\sin (n+1)a}{\\sin a} \\)"
] | [
"Calculate the determinant \\( D_n \\) of the \\( n \\times n \\) matrix given by:\n\\[ D_n = \\begin{vmatrix}\n2\\cos a & 1 & 0 & 0 & \\cdots & 0 & 0 \\\\\n1 & 2\\cos a & 1 & 0 & \\cdots & 0 & 0 \\\\\n0 & 1 & 2\\cos a & 1 & \\cdots & 0 & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-3cd015348e1339e4fc377a107efb2d6b3f82367ce1f22334be010453584ae17f | Calculate the determinant \( d_n \) of the matrix: \[ d_n = \left| \begin{array}{ccccc} 0 & x & x^2 & \ldots & x^{n-1} \\ \frac{1}{x} & 0 & x & \ldots & x^{n-2} \\ \frac{1}{x^2} & \frac{1}{x} & 0 & \ldots & x^{n-3} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ \frac{1}{x^{n-1}} & \frac{1}{x^{n-2}} & \frac{1}{x^{n-3}... | [
"(-1)^{n-1} (n-1)"
] | [
"Calculate the determinant \\( d_n \\) of the matrix:\n\\[ d_n = \\left| \\begin{array}{ccccc} 0 & x & x^2 & \\ldots & x^{n-1} \\\\ \\frac{1}{x} & 0 & x & \\ldots & x^{n-2} \\\\ \\frac{1}{x^2} & \\frac{1}{x} & 0 & \\ldots & x^{n-3} \\\\ \\ldots & \\ldots & \\ldots & \\ldots & \\ldots \\\\ \\frac{1}{x^{n-1}} & \\fra... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-d1a98838e0e3434c08a887b22ca31742e298a2deeb87ea76b6dda73a3692239e | Calculate the determinant \( d_n \) of the matrix: \[ d_n = \left| \begin{array}{ccccc} 1 & 1 & 1 & \ldots & 1 \\ 1 & 2 & 2 & \ldots & 2 \\ 1 & 2 & 3 & \ldots & 3 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 1 & 2 & 3 & \ldots & n \end{array} \right| \] | [
"1"
] | [
"Calculate the determinant \\( d_n \\) of the matrix:\n\\[ d_n = \\left| \\begin{array}{ccccc} 1 & 1 & 1 & \\ldots & 1 \\\\ 1 & 2 & 2 & \\ldots & 2 \\\\ 1 & 2 & 3 & \\ldots & 3 \\\\ \\ldots & \\ldots & \\ldots & \\ldots & \\ldots \\\\ 1 & 2 & 3 & \\ldots & n \\end{array} \\right| \\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-544f7913632fbe96131198c5bd0f2a8473bf1f90326b563bcf57c9e07d182a2f | Calculate the determinant \( d_n \) of the matrix: \[ d_n = \left| \begin{array}{ccccc} a_1 & x & x & \ldots & x \\ x & a_2 & x & \ldots & x \\ x & x & a_3 & \ldots & x \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ x & x & x & \ldots & a_n \end{array} \right| \] | [
" \\prod_{i=1}^{n} (a_i - x) + x \\sum_{i=1}^{n} \\prod_{\\substack{j=1 \\\\ j \\neq i}}^{n} (a_j - x) "
] | [
"Calculate the determinant \\( d_n \\) of the matrix:\n\\[ d_n = \\left| \\begin{array}{ccccc} a_1 & x & x & \\ldots & x \\\\ x & a_2 & x & \\ldots & x \\\\ x & x & a_3 & \\ldots & x \\\\ \\ldots & \\ldots & \\ldots & \\ldots & \\ldots \\\\ x & x & x & \\ldots & a_n \\end{array} \\right| \\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-eefe1995071fc33a244b79f40975ce97a60df8093a745a42c8b955d33349a908 | Calculate the determinant \( x_n \) of the \( n \times n \) matrix given by: \[ x_n = \begin{vmatrix} \cos a & 1 & 0 & 0 & \cdots & 0 & 0 \\ 1 & 2\cos a & 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 2\cos a & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 2\cos a & ... | [
"\\(\\cos na\\)"
] | [
"Calculate the determinant \\( x_n \\) of the \\( n \\times n \\) matrix given by:\n\\[ x_n = \\begin{vmatrix}\n\\cos a & 1 & 0 & 0 & \\cdots & 0 & 0 \\\\\n1 & 2\\cos a & 1 & 0 & \\cdots & 0 & 0 \\\\\n0 & 1 & 2\\cos a & 1 & \\cdots & 0 & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-4dab61e8c4f4c150e32eb0a55f53e5225eaab2feff284c4946e3a01f52c56782 | Calculate the determinant of an \( n \times n \) matrix where all elements are \( a \) except for the diagonal elements, which are given by \( a_{ii} = a + \frac{1}{i} \). Also, find the inverse of the matrix if it exists. | [
"\\(\\frac{1}{n!}\\left(1+\\frac{n(n+1)}{2}a\\right)\\)"
] | [
"Calculate the determinant of an \\( n \\times n \\) matrix where all elements are \\( a \\) except for the diagonal elements, which are given by \\( a_{ii} = a + \\frac{1}{i} \\). Also, find the inverse of the matrix if it exists."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 7
} | 13 | 0 | 0 |
omr-ws-30c29f87890d07e6234fc1f59a60c74908e0b0d269f7599ba511afda54acd488 | Calculate the determinant of the $ n \times n $ complex matrix $ \left(a_j^i\right)_{1\le j\le n}^{1\le i\le n} $ defined by $$ a_j^i=\begin{cases} 1+x^2, & \text{if } i=j \\ x, & \text{if } |i-j|=1 \\ 0, & \text{if } |i-j|\ge 2 \end{cases}, $$ where $ n $ is a natural number greater than $ 2. $ | [
"\\(\\frac{x^{2n+2} - 1}{x^2 - 1}\\) for \\( |x| \\neq 1 \\) and \\( n + 1 \\) for \\( |x| = 1 \\)."
] | [
"Calculate the determinant of the $ n \\times n $ complex matrix $ \\left(a_j^i\\right)_{1\\le j\\le n}^{1\\le i\\le n} $ defined by\n$$ a_j^i=\\begin{cases} 1+x^2, & \\text{if } i=j \\\\ x, & \\text{if } |i-j|=1 \\\\ 0, & \\text{if } |i-j|\\ge 2 \\end{cases}, $$\nwhere $ n $ is a natural number greater than $ 2. $... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-4a54a6ea1fc2f51d0687657617a6ebf4a52b4a8dba8f1738ba3aa84e7953d740 | Calculate the determinant of the \( n \times n \) matrix \( A \) given by: \[ A = \begin{pmatrix} 1 & 2 & 3 & \cdots & n \\ x & 1 & 2 & \cdots & n-1 \\ x & x & 1 & \cdots & n-2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x & x & x & \cdots & 1 \end{pmatrix} \] | [
"\\(-(-x)^n + (1 - x)^n\\)"
] | [
"Calculate the determinant of the \\( n \\times n \\) matrix \\( A \\) given by:\n\\[ A = \\begin{pmatrix} \n1 & 2 & 3 & \\cdots & n \\\\ \nx & 1 & 2 & \\cdots & n-1 \\\\ \nx & x & 1 & \\cdots & n-2 \\\\ \n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ \nx & x & x & \\cdots & 1 \n\\end{pmatrix} \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-4137471116eb0afb8867b40fed548a94c523bb2095f3e7df110c1d81aa34f48c | Calculate the determinant of the \( n \times n \) matrix: \[ \begin{vmatrix} 5 & 7 & 7 & \cdots & 7\\ 7 & 5 & 7 & \cdots & 7\\ 7 & 7 & 5 & \cdots & 7\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 7 & 7 & 7 & \cdots & 5 \end{vmatrix} \] | [
"\\((-2)^{n-1}(7n-2)\\)"
] | [
"Calculate the determinant of the \\( n \\times n \\) matrix:\n\\[\n\\begin{vmatrix}\n5 & 7 & 7 & \\cdots & 7\\\\ \n7 & 5 & 7 & \\cdots & 7\\\\ \n7 & 7 & 5 & \\cdots & 7\\\\ \n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots\\\\ \n7 & 7 & 7 & \\cdots & 5\n\\end{vmatrix}\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-58827e7eb9f04b0682a94ffc4cb625d65e571762cd3cd10e23e31fafdb9b6235 | Calculate the determinant of the distance matrix $\det_{1\le i,j \le 5} ({\mathrm {dist}}(i,j))$ for both the star graph $S_5$ and the path graph $P_5$, and find the sum of these two determinants. | [
"64"
] | [
"Calculate the determinant of the distance matrix $\\det_{1\\le i,j \\le 5} ({\\mathrm {dist}}(i,j))$ for both the star graph $S_5$ and the path graph $P_5$, and find the sum of these two determinants."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-73c34b6d48585f0946d7f7a5799153b2df7eb294105dafd3d0b0d8f2cb195e12 | Calculate the determinant of the following $2n \times 2n$ matrix: $$ \begin{vmatrix} 0 & 0 & \cdots & 0 & 5 & 3 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \cdots & 5 & 0 & 0 & 3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \\ 0 & 5 & \cdots & 0 & 0 & 0 & 0 & \cdots & 3 & 0 \... | [
"\\(16^n\\)"
] | [
"Calculate the determinant of the following $2n \\times 2n$ matrix:\n$$ \\begin{vmatrix} \n0 & 0 & \\cdots & 0 & 5 & 3 & 0 & \\cdots & 0 & 0 \\\\ \n0 & 0 & \\cdots & 5 & 0 & 0 & 3 & \\cdots & 0 & 0 \\\\ \n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\\\ \n0 & 5 & \\cdot... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 6,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-dcca4527042120da931b8d8b2dc5027a78213bca187e40daa1ce0643a4f38a01 | Calculate the determinant of the following $n \times n$ matrix: \[ \begin{vmatrix} a & b & \cdots & b & b\\ -b & a & \cdots & b & b\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ -b & -b & \cdots & a & b\\ -b & -b & \cdots & -b & a \end{vmatrix} \] | [
"\\dfrac{(a + b)^n + (a - b)^n}{2}"
] | [
"Calculate the determinant of the following $n \\times n$ matrix:\n\\[\n\\begin{vmatrix}\na & b & \\cdots & b & b\\\\ \n-b & a & \\cdots & b & b\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots\\\\\n-b & -b & \\cdots & a & b\\\\\n-b & -b & \\cdots & -b & a\n\\end{vmatrix}\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-d14a5b653dbafb91854aaac32f0213a61f7e6a2912ee7dbfa68101ec73a85aff | Calculate the determinant of the following $n \times n$ matrix: \[ \begin{vmatrix} a+b & a & \cdots & a\\ a & a + b & \cdots & a\\ \vdots & \vdots & \ddots & \vdots\\ a & a & \cdots & a + b \end{vmatrix} \] | [
"\\( b^{n-1}(an + b) \\)"
] | [
"Calculate the determinant of the following $n \\times n$ matrix:\n\\[\n\\begin{vmatrix}\na+b & a & \\cdots & a\\\\\na & a + b & \\cdots & a\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\na & a & \\cdots & a + b\n\\end{vmatrix}\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-03c8cb310746a4077f4de278012524a89d0538b524494f6a4bff70887c82b327 | Calculate the determinant of the following \( n \times n \) matrix: \[ \Delta_n = \left|\begin{array}{cccccccc} 1 & \sin t & 0 & 0 & 0 & \cdots & 0 & 0 \\ \sin t & 2 & \sin t & 0 & 0 & \cdots & 0 & 0 \\ 0 & \sin t & 2 & \sin t & 0 & \cdots & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots... | [
"\\(\\frac{(1+\\cos t)^{n} + (1-\\cos t)^{n}}{2}\\)"
] | [
"Calculate the determinant of the following \\( n \\times n \\) matrix:\n\n\\[\n\\Delta_n = \\left|\\begin{array}{cccccccc}\n1 & \\sin t & 0 & 0 & 0 & \\cdots & 0 & 0 \\\\\n\\sin t & 2 & \\sin t & 0 & 0 & \\cdots & 0 & 0 \\\\\n0 & \\sin t & 2 & \\sin t & 0 & \\cdots & 0 & 0 \\\\\n\\cdots & \\cdots & \\cdots & \\cd... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-95effd9d21b7bc4a572f3d3e2ae9bfaa1c8b12d087996541b58d5ba7779423c7 | Calculate the determinant of the following \( n \times n \) matrix: \[ \begin{vmatrix} 1 & 2 & 3 & \cdots & n\\ 2 & 3 & 4 & \cdots & 1\\ 3 & 4 & 5 & \cdots & 2\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ n & 1 & 2 & \cdots & n-1 \end{vmatrix} \] | [
"\\(\\frac{n^{n+1}(n+1)}{2}(-1)^{\\frac{(n-1)n}{2}}\\)"
] | [
"Calculate the determinant of the following \\( n \\times n \\) matrix:\n\\[\n\\begin{vmatrix}\n1 & 2 & 3 & \\cdots & n\\\\ \n2 & 3 & 4 & \\cdots & 1\\\\ \n3 & 4 & 5 & \\cdots & 2\\\\ \n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ \nn & 1 & 2 & \\cdots & n-1\n\\end{vmatrix}\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 5
} | 13 | 0 | 0 |
omr-ws-fc1d382b3799d4451eb362cb996b8f1cc0238398dc199b5cd225ba02f93cba44 | Calculate the determinant of the following matrix for \( n \geq 2 \) and \( n \in \mathbb{N} \): \[ \begin{vmatrix} 1 & 0 & 0 & 0 & \ldots & 1 \\ 1 & 1! & 0 & 0 & \ldots & x \\ 1 & 2 & 2! & 0 & \ldots & x^2 \\ 1 & 3 & 3 \cdot 2 & 3! & \ldots & x^3 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & n & n(n-1)... | [
"\\( 1! \\cdot 2! \\cdot \\cdots \\cdot (n-2)! \\cdot (n-1)! \\cdot (x-1)^n \\)"
] | [
"Calculate the determinant of the following matrix for \\( n \\geq 2 \\) and \\( n \\in \\mathbb{N} \\):\n\n\\[\n\\begin{vmatrix}\n1 & 0 & 0 & 0 & \\ldots & 1 \\\\ \n1 & 1! & 0 & 0 & \\ldots & x \\\\ \n1 & 2 & 2! & 0 & \\ldots & x^2 \\\\ \n1 & 3 & 3 \\cdot 2 & 3! & \\ldots & x^3 \\\\ \n\\vdots & \\vdots & \\vdots... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 4
} | 17 | 0 | 0 |
omr-ws-a44a9e28f3014e3d5dbe4a7af03f5293a8ed6366c6e8570852ba529da3f4367d | Calculate the determinant of the following matrix: $$ \begin{bmatrix} 1 & 2 & 8 & 9 & 81 \\ 3 & 2 & 4 & 4 & 36 \\ 213 & 342 & 2 & 5 & 45 \\ 32 & 245 & 42 & 1 & 9 \\ 344 & 53 & 23 & 2 & 18 \end{bmatrix} $$ | [
"0"
] | [
"Calculate the determinant of the following matrix:\n\n$$\n\\begin{bmatrix}\n1 & 2 & 8 & 9 & 81 \\\\\n3 & 2 & 4 & 4 & 36 \\\\\n213 & 342 & 2 & 5 & 45 \\\\\n32 & 245 & 42 & 1 & 9 \\\\\n344 & 53 & 23 & 2 & 18\n\\end{bmatrix}\n$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 21
} | 44 | 0 | 0 |
omr-ws-fce5a3cb0f6727f09307887e577118429edd7d94ca66b71c6f02c1628351df2e | Calculate the determinant of the following matrix: $$ \begin{vmatrix} -1 & 1 & 1 & \cdots & 1 & 1 & 1 \\ -2 & -1 & 0 & \cdots & 0 & 0 & 1 \\ -2 & 0 & -1 & \cdots & 0 & 0 & 1 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ -2 & 0 & 0 & \cdots & -1 & 0 & 1 \\ -2 & 0 & 0 & \cdots & 0 & -1 & 1 \\ -2 & -2... | [
"\\((-1)^n(2n-1)\\)"
] | [
"Calculate the determinant of the following matrix:\n$$ \\begin{vmatrix} -1 & 1 & 1 & \\cdots & 1 & 1 & 1 \\\\ -2 & -1 & 0 & \\cdots & 0 & 0 & 1 \\\\ -2 & 0 & -1 & \\cdots & 0 & 0 & 1 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ -2 & 0 & 0 & \\cdots & -1 & 0 & 1 \\\\ -2 & 0 & 0 & \\... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 24
} | 36 | 0 | 0 |
omr-ws-a3cd612689f43d729b69340ae23b8e320cf6b494daf45c56f53c39233a8183dc | Calculate the determinant of the following matrix: $$ \begin{vmatrix} -5 & 1 & 0 & 0 & \cdots & 0 & 0 & 0 \\ 4 & -5 & 1 & 0 & \cdots & 0 & 0 & 0 \\ 4 & 4 & -5 & 1 & \cdots & 0 & 0 & 0 \\ 4 & 4 & 4 & -5 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 4 & 4 & 4 & 4 & \cdot... | [
"\\(\\Delta_n = \\frac{5 + 2n}{3} \\cdot (-3)^n\\)"
] | [
"Calculate the determinant of the following matrix:\n$$ \\begin{vmatrix} -5 & 1 & 0 & 0 & \\cdots & 0 & 0 & 0 \\\\ 4 & -5 & 1 & 0 & \\cdots & 0 & 0 & 0 \\\\ 4 & 4 & -5 & 1 & \\cdots & 0 & 0 & 0 \\\\ 4 & 4 & 4 & -5 & \\cdots & 0 & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdo... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 4
} | 10 | 0 | 0 |
omr-ws-b6bd640806b5089d71084cf6b9574b0c297ecc92dbb993da9ab4502b695e84c6 | Calculate the determinant of the following matrix: $$\begin{pmatrix} \alpha +\beta&\alpha\beta&0&\cdots&0&0\\ 1&\alpha + \beta &\alpha \beta&\cdots&0& 0\\ 0&1&\alpha + \beta & \cdots &0&0\\ .&.&.&.&.&.\\ .&.&.&.&.&.\\ .&.&.&.&.&.\\ 0&0&0&0&1&\alpha + \beta\end{pmatrix}$$ | [
"\\(\\frac{\\alpha^{n+1}-\\beta^{n+1}}{\\alpha-\\beta}\\)"
] | [
"Calculate the determinant of the following matrix:\n$$\\begin{pmatrix} \\alpha +\\beta&\\alpha\\beta&0&\\cdots&0&0\\\\\n1&\\alpha + \\beta &\\alpha \\beta&\\cdots&0& 0\\\\\n0&1&\\alpha + \\beta & \\cdots &0&0\\\\\n.&.&.&.&.&.\\\\\n.&.&.&.&.&.\\\\\n.&.&.&.&.&.\\\\\n0&0&0&0&1&\\alpha + \\beta\\end{pmatrix}$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 6
} | 20 | 0 | 0 |
omr-ws-109219b6d45e34c02e166be5698a7d97020b4e932d36c680d25ade23dd04ca2f | Calculate the determinant of the following matrix: $$\begin{vmatrix} \cos\theta&\cos 2\theta&\cos 3\theta&\cdots &\cos n\theta\\ \cos n\theta&\cos\theta&\cos 2\theta&\cdots &\cos (n-1)\theta\\ \cos (n-1)\theta&\cos n\theta&\cos\theta&\cdots&\cos(n-2)\theta\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \cos 2\theta&\cos 3\thet... | [
"0"
] | [
"Calculate the determinant of the following matrix:\n$$\\begin{vmatrix}\n\\cos\\theta&\\cos 2\\theta&\\cos 3\\theta&\\cdots &\\cos n\\theta\\\\\n\\cos n\\theta&\\cos\\theta&\\cos 2\\theta&\\cdots &\\cos (n-1)\\theta\\\\\n\\cos (n-1)\\theta&\\cos n\\theta&\\cos\\theta&\\cdots&\\cos(n-2)\\theta\\\\\n\\vdots&\\vdots&\... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-04071a60dc12dda3c0df8f3be333d66f08d97f520fabe1c325e8d32691bceead | Calculate the determinant of the following matrix: \[ \begin{vmatrix} (a_0+b_0)^n & (a_0+b_1)^n & (a_0+b_2)^n & \cdots & (a_0+b_n)^n \\ (a_1+b_0)^n & (a_1+b_1)^n & (a_1+b_2)^n & \cdots & (a_1+b_n)^n \\ (a_2+b_0)^n & (a_2+b_1)^n & (a_2+b_2)^n & \cdots & (a_2+b_n)^n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ (a_n+b... | [
" (-1)^{\\frac{n(n+1)}{2}} \\left( \\prod_{k=0}^{n} \\binom{n}{k} \\right) \\prod_{0 \\leq i < j \\leq n} (a_j - a_i)(b_j - b_i) "
] | [
"Calculate the determinant of the following matrix:\n\\[\n\\begin{vmatrix}\n(a_0+b_0)^n & (a_0+b_1)^n & (a_0+b_2)^n & \\cdots & (a_0+b_n)^n \\\\\n(a_1+b_0)^n & (a_1+b_1)^n & (a_1+b_2)^n & \\cdots & (a_1+b_n)^n \\\\\n(a_2+b_0)^n & (a_2+b_1)^n & (a_2+b_2)^n & \\cdots & (a_2+b_n)^n \\\\\n\\vdots & \\vdots & \\vdots & ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-d376b57efa38e37c66b55d0222dad01e9fa033b936e9430625d471fe5c5654ab | Calculate the determinant of the following matrix: \[ \begin{vmatrix} 0 & \ldots & 0 & 0 & 1 \\ 0 & \ldots & 0 & 2 & 0 \\ 0 & \ldots & 3 & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ n & \ldots & 0 & 0 & 0 \end{vmatrix} \] | [
"\\((-1)^{\\frac{(n-1)n}{2}} \\cdot n!\\)"
] | [
"Calculate the determinant of the following matrix:\n\\[\n\\begin{vmatrix}\n0 & \\ldots & 0 & 0 & 1 \\\\ \n0 & \\ldots & 0 & 2 & 0 \\\\ \n0 & \\ldots & 3 & 0 & 0 \\\\ \n\\ldots & \\ldots & \\ldots & \\ldots & \\ldots \\\\ \nn & \\ldots & 0 & 0 & 0 \n\\end{vmatrix}\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-231a665b0eb10de12fa9f3c4c2279beeb32900e12eaef922b9d004bf00e8727d | Calculate the determinant of the following matrix: \[ \begin{vmatrix} 1 & 1 & 1 & \ldots & 1 \\ 1 & \epsilon & \epsilon^2 & \ldots & \epsilon^{n-1}\\ 1 & \epsilon^2 & \epsilon^4 & \ldots & \epsilon^{2n-2} \\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & \epsilon^{n-1} & \epsilon^{2(n-1)} & \ldots & \epsilon^{(n-1)^2... | [
"\\(n\\)"
] | [
"Calculate the determinant of the following matrix:\n\n\\[\n\\begin{vmatrix}\n1 & 1 & 1 & \\ldots & 1 \\\\ \n1 & \\epsilon & \\epsilon^2 & \\ldots & \\epsilon^{n-1}\\\\ \n1 & \\epsilon^2 & \\epsilon^4 & \\ldots & \\epsilon^{2n-2} \\\\ \n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots\\\\ \n1 & \\epsilon^{n-1} & \... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bd1d7a342c5b1cc349c77d6a1f65b5f5e0273635a9e149192c9b488190c87d97 | Calculate the determinant of the following matrix: \[ \begin{vmatrix} 1 & 2 & 3 & \cdots & n-1 & n \\ 1 & 3 & 4 & \cdots & n & n+1 \\ 0 & 3 & 5 & \cdots & n+1 & n+2 \\ 0 & 0 & 5 & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & \cdots & \cdots \\ \vdots & \vdots & \vdots & \ddots & \cdots & \cdots \\ 0 & 0 & \cdots & ... | [
"(-1)^{n(n-1)/2}"
] | [
"Calculate the determinant of the following matrix:\n\\[\n\\begin{vmatrix}\n1 & 2 & 3 & \\cdots & n-1 & n \\\\\n1 & 3 & 4 & \\cdots & n & n+1 \\\\\n0 & 3 & 5 & \\cdots & n+1 & n+2 \\\\\n0 & 0 & 5 & \\cdots & \\cdots & \\cdots \\\\\n0 & 0 & 0 & \\cdots & \\cdots & \\cdots \\\\\n\\vdots & \\vdots & \\vdots & \\ddots ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-08477ff4870c08514cdb327f4d72d2b27fecca70f9cf27b2757898a6ea6dee00 | Calculate the determinant of the following matrix: \[ \begin{vmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n} \\ \frac{1}{n} & 1 & \frac{1}{2} & \cdots & \frac{1}{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \cdots & 1 \end{vmatrix} \] | [
"0"
] | [
"Calculate the determinant of the following matrix:\n\\[\n\\begin{vmatrix}\n1 & \\frac{1}{2} & \\frac{1}{3} & \\cdots & \\frac{1}{n} \\\\\n\\frac{1}{n} & 1 & \\frac{1}{2} & \\cdots & \\frac{1}{n-1} \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\frac{1}{2} & \\frac{1}{3} & \\frac{1}{4} & \\cdots & 1\n... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 2
} | 17 | 0 | 0 |
omr-ws-3b3911ec2dd4177434d6d3b7ea28652c35fea0d2a451b569397b07ae3ede34b5 | Calculate the determinant of the following matrix: \[ \begin{vmatrix} s_0 & s_1 & s_2 & \ldots & s_{n-1} & 1 \\ s_1 & s_2 & s_3 & \ldots & s_{n} & x \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ s_{n-1} & s_{n} & s_{n+1} & \ldots & s_{2n-2} & x^{n-1} \\ s_{n} & s_{n+1} & s_{n+2} & \ldots & s_{2n-1} & x^{n} ... | [
"(x - x_1)(x - x_2) \\cdots (x - x_n) \\prod_{1 \\leq i < j \\leq n} (x_i - x_j)^2"
] | [
"Calculate the determinant of the following matrix:\n\\[ \\begin{vmatrix}\ns_0 & s_1 & s_2 & \\ldots & s_{n-1} & 1 \\\\\ns_1 & s_2 & s_3 & \\ldots & s_{n} & x \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\ns_{n-1} & s_{n} & s_{n+1} & \\ldots & s_{2n-2} & x^{n-1} \\\\\ns_{n} & s_{n+1} & s_{n+2... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-9daa17da1ecf7cdb2c66909966e9fbcb79e79379a00e08a5effda9a2ef54a5e4 | Calculate the determinant of the following matrix: \[ \left|\begin{array}{ccccc} 1 & x_1 & \cdots & x_1^{n-2} & \left(x_2+x_3+\cdots+x_n\right)^{n-1} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & x_n & \cdots & x_n^{n-2} & \left(x_1+x_2+\cdots+x_{n-1}\right)^{n-1} \end{array}\right| \] Express your answer in ter... | [
"(-1)^{n-1}"
] | [
"Calculate the determinant of the following matrix:\n\\[\n\\left|\\begin{array}{ccccc}\n1 & x_1 & \\cdots & x_1^{n-2} & \\left(x_2+x_3+\\cdots+x_n\\right)^{n-1} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n1 & x_n & \\cdots & x_n^{n-2} & \\left(x_1+x_2+\\cdots+x_{n-1}\\right)^{n-1}\n\\end{array}\\rig... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4e712b8f28e84bfb4c03315f642cc63d71e37123b75f2cf7f4679f762cc7bc0 | Calculate the determinant of the linear transformation \( T \) on the vector space \( V \) of all real \( n \times n \) matrices, where \( T(X) = \frac{1}{2}(AX + XA) \) and \( A \) is a diagonal matrix with positive entries \( \lambda_1, \lambda_2, \ldots, \lambda_n \). | [
"\\(\\prod_{r,s=1}^{n} \\frac{\\lambda_r + \\lambda_s}{2}\\)"
] | [
"Calculate the determinant of the linear transformation \\( T \\) on the vector space \\( V \\) of all real \\( n \\times n \\) matrices, where \\( T(X) = \\frac{1}{2}(AX + XA) \\) and \\( A \\) is a diagonal matrix with positive entries \\( \\lambda_1, \\lambda_2, \\ldots, \\lambda_n \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-b7d7e9c1a86dc764cf6995705f2b803ab61c7a26b90f2f52eb9ebad9d3a0be05 | Calculate the determinant of the matrix $ \left|\begin{array}{c} \frac{1}{(i + j - 1)!} \end{array}\right|_{1 \leq i,j \leq n}$. | [
" \\dfrac{(-1)^{\\frac{n(n-1)}{2}}}{\\prod_{k=n}^{2n-1} k!} "
] | [
"Calculate the determinant of the matrix $ \\left|\\begin{array}{c} \\frac{1}{(i + j - 1)!} \\end{array}\\right|_{1 \\leq i,j \\leq n}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-cda7149d56254e953ea0d630a728a884ed5a698b212fc261e84e6097504abae4 | Calculate the determinant of the matrix $ \left|\begin{array}{c} \frac{1}{(i + j - 2)!} \end{array}\right|_{1 \leq i,j \leq n}$. | [
" \\dfrac{(-1)^{\\frac{n(n-1)}{2}}}{\\prod_{k=0}^{n-1} (k! \\cdot (k+1)!)} "
] | [
"Calculate the determinant of the matrix $ \\left|\\begin{array}{c} \\frac{1}{(i + j - 2)!} \\end{array}\\right|_{1 \\leq i,j \\leq n}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-83f591d09c1e0c39a89e5bbec4394509624d5e23be20456721b24887ab4e4011 | Calculate the determinant of the matrix $$\begin{vmatrix} \cos x & \sin x & \cos x & \sin x \\ \cos 2x & \sin 2x & 2\cos 2x & 2\sin 2x \\ \cos 3x & \sin 3x & 3\cos 3x & 3\sin 3x \\ \cos 4x & \sin 4x & 4\cos 4x & 4\sin 4x \end{vmatrix}$$ | [
"\\(4\\sin^4 x\\)"
] | [
"Calculate the determinant of the matrix\n$$\\begin{vmatrix}\n\\cos x & \\sin x & \\cos x & \\sin x \\\\ \n\\cos 2x & \\sin 2x & 2\\cos 2x & 2\\sin 2x \\\\ \n\\cos 3x & \\sin 3x & 3\\cos 3x & 3\\sin 3x \\\\ \n\\cos 4x & \\sin 4x & 4\\cos 4x & 4\\sin 4x \n\\end{vmatrix}$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 14
} | 17 | 0 | 0 |
omr-ws-5ceeebea23fa51c9c59f8c7ec874c7e38d2d7c7f3d71aa657d5c5002636d8ac3 | Calculate the determinant of the matrix $$\begin{vmatrix} a+b & ab & a^2+b^2\\ b+c & bc & b^2+c^2\\ c+a & ca & c^2+a^2 \end{vmatrix}.$$ | [
"\\((a-b)(b-c)(c-a)(ab+bc+ca)\\)"
] | [
"Calculate the determinant of the matrix\n$$\\begin{vmatrix}\na+b & ab & a^2+b^2\\\\\nb+c & bc & b^2+c^2\\\\\nc+a & ca & c^2+a^2\n\\end{vmatrix}.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 42,
"tir": 26
} | 68 | 0 | 0 |
omr-ws-6f5295a9b1df7c12da50a73a76fc4b51638efe0e7ef2ad6fad9821a9b9dbaac1 | Calculate the determinant of the matrix $\left|{\begin{array}{ccc}b^2+c^2 & ab & ca \\ ab & c^2+a^2 & bc \\ ca & bc & a^2+b^2 \end{array}}\right|$. | [
"\\(4a^2b^2c^2\\)"
] | [
"Calculate the determinant of the matrix $\\left|{\\begin{array}{ccc}b^2+c^2 & ab & ca \\\\ ab & c^2+a^2 & bc \\\\ ca & bc & a^2+b^2 \\end{array}}\\right|$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-ec12155fbe59f338583b5c3d1ac44f50d0fcc1b17a78e68cbe27881c49bf4cbc | Calculate the determinant of the matrix \( A - I_n + (a_1, a_2, \cdots, a_n)'(a_1, a_2, \cdots, a_n) \), where all elements of \( A \) are 1. | [
"(-1)^{n-1} \\left( \\left( \\sum_{i=1}^n a_i \\right)^2 - (n - 1) \\sum_{i=1}^n a_i^2 + (n - 1) \\right)"
] | [
"Calculate the determinant of the matrix \\( A - I_n + (a_1, a_2, \\cdots, a_n)'(a_1, a_2, \\cdots, a_n) \\), where all elements of \\( A \\) are 1."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cda9c559e647ef1b74c4104b17877f75b6b7424ce501a5a97d205b5deb9ff829 | Calculate the determinant of the matrix \( A = [a_{ij}] \) where \( a_{ij} = ij \) if \( i \neq j \) and \( a_{ij} = 1 + ij \) if \( i = j \). | [
"\\( 1 + \\frac{n(n+1)(2n+1)}{6} \\)"
] | [
"Calculate the determinant of the matrix \\( A = [a_{ij}] \\) where \\( a_{ij} = ij \\) if \\( i \\neq j \\) and \\( a_{ij} = 1 + ij \\) if \\( i = j \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-b0e54e02ede841765c854029278e29153f312b9b9a93c6b01288f5e2ef9c506a | Calculate the determinant of the matrix \( A = [a_{ij}]_{i, j = 1}^{n} \), where the entries are defined as follows: - \( a_{i,i} = (-1)^{i+1} \) for \( 1 \leq i \leq n \), - \( a_{i,i-1} = (-1)^{i} \) for \( 2 \leq i \leq n \), - \( a_{1,i} = (-1)^{i+1} \) for \( 1 \leq i \leq n \), - All other entries are 0. | [
"\\(\\begin{cases}(-1)^{k+1} & \\text{if } n = 2k - 1 \\\\ (-1)^{k+1} + (-1)^{3k} & \\text{if } n = 2k \\end{cases}\\)"
] | [
"Calculate the determinant of the matrix \\( A = [a_{ij}]_{i, j = 1}^{n} \\), where the entries are defined as follows:\n- \\( a_{i,i} = (-1)^{i+1} \\) for \\( 1 \\leq i \\leq n \\),\n- \\( a_{i,i-1} = (-1)^{i} \\) for \\( 2 \\leq i \\leq n \\),\n- \\( a_{1,i} = (-1)^{i+1} \\) for \\( 1 \\leq i \\leq n \\),\n- All ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 7
} | 17 | 0 | 0 |
omr-ws-b08658dd5f551a65a8fc1b8a4e8d329834dee6e2ea8b3c9fc4da2a153f109139 | Calculate the determinant of the matrix \( A = \begin{pmatrix} 30 & 20 & 15 & 12 \\ 20 & 15 & 12 & 15 \\ 15 & 12 & 15 & 20 \\ 12 & 15 & 20 & 30 \end{pmatrix} \). | [
"-2639"
] | [
"Calculate the determinant of the matrix \\( A = \\begin{pmatrix} 30 & 20 & 15 & 12 \\\\ 20 & 15 & 12 & 15 \\\\ 15 & 12 & 15 & 20 \\\\ 12 & 15 & 20 & 30 \\end{pmatrix} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a943b94b13fe11037961473f65e9c5d9b6d0fe7904379413a3675795658d2e51 | Calculate the determinant of the matrix \( A = \begin{vmatrix} 5 & 4 & 3 & 2 & 1 \\ 0 & a & b & c & d \\ 0 & a^2 & b^2 & c^2 & d^2 \\ 0 & a^3 & b^3 & c^3 & d^3 \\ 0 & a^4 & b^4 & c^4 & d^4 \end{vmatrix} \). | [
"\\( 5abcd(b-a)(c-a)(d-a)(c-b)(d-b)(d-c) \\)"
] | [
"Calculate the determinant of the matrix \\( A = \\begin{vmatrix} 5 & 4 & 3 & 2 & 1 \\\\ 0 & a & b & c & d \\\\ 0 & a^2 & b^2 & c^2 & d^2 \\\\ 0 & a^3 & b^3 & c^3 & d^3 \\\\ 0 & a^4 & b^4 & c^4 & d^4 \\end{vmatrix} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-f7d3b0d6957b2c5c6a1f60d518a75fc4f8bc0b93bb969ab13de0608b93a654d6 | Calculate the determinant of the matrix \( A \) and diagonalize it using its characteristic polynomial. \[ A = \begin{pmatrix} a^2 - x & ab & ab & b^2 \\ ab & a^2 - x & b^2 & ab \\ ab & b^2 & a^2 - x & ab \\ b^2 & ab & ab & a^2 - x \end{pmatrix} \] The characteristic polynomial is given by: \[ p_A(x) = [(a+b)^2 - x][(a... | [
"\\((a^2 - b^2)^4\\)"
] | [
"Calculate the determinant of the matrix \\( A \\) and diagonalize it using its characteristic polynomial.\n\n\\[ A = \\begin{pmatrix} a^2 - x & ab & ab & b^2 \\\\ ab & a^2 - x & b^2 & ab \\\\ ab & b^2 & a^2 - x & ab \\\\ b^2 & ab & ab & a^2 - x \\end{pmatrix} \\]\n\nThe characteristic polynomial is given by:\n\n\\... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 8
} | 13 | 0 | 0 |
omr-ws-eb176b13aff2fdafe0d266076d3f3ba813a37aa5841c62c889723f414d513e25 | Calculate the determinant of the matrix \( A \) given by \[ A = \begin{bmatrix} a_1 & x_1 & x_1^2 & \cdots & x_1^{n-1} \\ a_2 & x_2 & x_2^2 & \cdots & x_2^{n-1} \\ a_3 & x_3 & x_3^2 & \cdots & x_3^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_n & x_n & x_n^2 & \cdots & x_n^{n-1} \end{bmatrix}. \] | [
"\\(\\sum_{i=1}^{n} (-1)^{i+1} a_i \\prod_{\\substack{1 \\leq l \\leq n \\\\ l \\neq i}} x_l \\prod_{\\substack{1 \\leq j < k \\leq n \\\\ j, k \\neq i}} (x_k - x_j)\\)"
] | [
"Calculate the determinant of the matrix \\( A \\) given by\n\\[ A = \\begin{bmatrix}\na_1 & x_1 & x_1^2 & \\cdots & x_1^{n-1} \\\\\na_2 & x_2 & x_2^2 & \\cdots & x_2^{n-1} \\\\\na_3 & x_3 & x_3^2 & \\cdots & x_3^{n-1} \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\na_n & x_n & x_n^2 & \\cdots & x_n^{n-... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 3
} | 5 | 0 | 0 |
omr-ws-11eb98a024f408dc40746298f2530d1a7e1f3e01d0de9b858a2edeb363fd2eab | Calculate the determinant of the matrix \( A \) given by \[ A = \begin{vmatrix} a_1 & x & x & \cdots & x \\ x & a_2 & x & \cdots & x \\ x & x & a_3 & \cdots & x \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x & x & x & \cdots & a_n \end{vmatrix} \] | [
"\\(\\left(1+\\sum_{j=1}^n\\frac{x}{a_j-x}\\right)\\prod_{k=1}^n (a_k-x)\\)"
] | [
"Calculate the determinant of the matrix \\( A \\) given by\n\\[ A = \\begin{vmatrix}\na_1 & x & x & \\cdots & x \\\\ \nx & a_2 & x & \\cdots & x \\\\ \nx & x & a_3 & \\cdots & x \\\\ \n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ \nx & x & x & \\cdots & a_n\n\\end{vmatrix} \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 18
} | 32 | 0 | 0 |
omr-ws-01b86963862c78b8e91ef172e973545cc8bb6d77ad8c5fc95a4829178e94b299 | Calculate the determinant of the matrix \( A \) given by \[ \det{A} = \left|\begin{array}{cccccc} 1 & 2 & 3 & \cdots & n-1 & n \\ 2 & 3 & 4 & \cdots & n & 1 \\ 3 & 4 & 5 & \cdots & 1 & 2 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ n-1 & n & 1 & \cdots & n-3 & n-2 \\ n & 1 & 2 & \cdots & n-2 & n-1 \end{arr... | [
"\\(\\frac{n^{n-1}(n+1)}{2} (-1)^{\\frac{n^2 - n + 4}{2}}\\)"
] | [
"Calculate the determinant of the matrix \\( A \\) given by\n\\[\n\\det{A} = \\left|\\begin{array}{cccccc}\n1 & 2 & 3 & \\cdots & n-1 & n \\\\\n2 & 3 & 4 & \\cdots & n & 1 \\\\\n3 & 4 & 5 & \\cdots & 1 & 2 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\nn-1 & n & 1 & \\cdots & n-3 & n-2 \\\\\n... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 4,
"tir": 3
} | 11 | 0 | 0 |
omr-ws-9c29c6cca40a3fa550c1fd8b86c9e058bd663b3af8b5c02f66dc5647a8d38cd7 | Calculate the determinant of the matrix \( A \) given by: \[ A = \begin{pmatrix} a_1 & x & x & \cdots & x \\ x & a_2 & x & \cdots & x \\ x & x & a_3 & \cdots & x \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x & x & x & \cdots & a_n \end{pmatrix} \] | [
"\\(\\left[a_n + x(a_n - x) \\sum_{i = 1}^{n-1} \\frac{1}{a_i - x}\\right] \\cdot \\prod_{i = 1}^{n-1}(a_i - x)\\)"
] | [
"Calculate the determinant of the matrix \\( A \\) given by:\n\\[ A = \\begin{pmatrix}\na_1 & x & x & \\cdots & x \\\\\nx & a_2 & x & \\cdots & x \\\\\nx & x & a_3 & \\cdots & x \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\nx & x & x & \\cdots & a_n\n\\end{pmatrix} \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 17
} | 38 | 0 | 0 |
omr-ws-b8e3dbf54c9f40c088399c04155511c8a553da7e40dafc5e775b0125f24eae49 | Calculate the determinant of the matrix \( A \): \[ A = \begin{vmatrix} -3 & 1 & 1 & 1 & 1 \\ 1 & a & b & c & d \\ 1 & a^2 & b^2 & c^2 & d^2 \\ 1 & a^3 & b^3 & c^3 & d^3 \\ 1 & a^4 & b^4 & c^4 & d^4 \end{vmatrix} \] | [
"\\(\\left[(d-1)(c-1)(b-1)(a-1) - 4abcd\\right] (d-c)(d-b)(d-a)(c-b)(c-a)(b-a)\\)"
] | [
"Calculate the determinant of the matrix \\( A \\):\n\n\\[ A = \\begin{vmatrix} -3 & 1 & 1 & 1 & 1 \\\\ 1 & a & b & c & d \\\\ 1 & a^2 & b^2 & c^2 & d^2 \\\\ 1 & a^3 & b^3 & c^3 & d^3 \\\\ 1 & a^4 & b^4 & c^4 & d^4 \\end{vmatrix} \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-481daef816f0f2f73384bef003bb939514932bb84b87bfaae91ca2232d7fd4a6 | Calculate the determinant of the matrix \( A \): \[ A = \begin{vmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & a & b & c & d \\ 1 & a^2 & b^2 & c^2 & d^2 \\ 1 & a^3 & b^3 & c^3 & d^3 \\ 1 & a^4 & b^4 & c^4 & d^4 \end{vmatrix} \] | [
"\\((a-1)(b-1)(c-1)(d-1)(b-a)(c-a)(d-a)(c-b)(d-b)(d-c)\\)"
] | [
"Calculate the determinant of the matrix \\( A \\):\n\n\\[ A = \\begin{vmatrix} \n1 & 1 & 1 & 1 & 1 \\\\ \n1 & a & b & c & d \\\\ \n1 & a^2 & b^2 & c^2 & d^2 \\\\ \n1 & a^3 & b^3 & c^3 & d^3 \\\\ \n1 & a^4 & b^4 & c^4 & d^4 \n\\end{vmatrix} \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-45e2cf51922e12ea264c42b92bf540d1fd5c126f3ef3b211248d6cfd5b604172 | Calculate the determinant of the matrix \( A \): \[ A = \left | \begin{array}{ccccc} 2 & 3 & 4 & \cdots & n \\ 3 & 6 & 10 & \cdots & \frac{n(n+1)}{2!} \\ 4 & 10 & 20 & \cdots & \frac{n(n+1)(n+2)}{3!} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ n & \frac{n(n+1)}{2!} & \frac{n(n+1)(n+2)}{3!} & \cdots & \frac{n(n+1)(... | [
"\\( n \\)"
] | [
"Calculate the determinant of the matrix \\( A \\):\n\n\\[ A = \\left | \\begin{array}{ccccc} \n2 & 3 & 4 & \\cdots & n \\\\\n3 & 6 & 10 & \\cdots & \\frac{n(n+1)}{2!} \\\\\n4 & 10 & 20 & \\cdots & \\frac{n(n+1)(n+2)}{3!} \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\nn & \\frac{n(n+1)}{2!} & \\frac{n(... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-f4a161d642a6964daa9d49d5bd69d8cb137dd794d2346ccf00e6c08f8fa5ad57 | Calculate the determinant of the matrix \( A \in M_n(\mathbb{Z}) \) defined as: \[ A = \left[\begin{array}{ccccc} 1^{n-1} & 2^{n-1} & 3^{n-1} & \cdots & n^{n-1} \\ 2^{n-1} & 3^{n-1} & 4^{n-1} & \cdots & (n+1)^{n-1} \\ 3^{n-1} & 4^{n-1} & 5^{n-1} & \cdots & (n+2)^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ n^... | [
"(-1)^{\\frac{n(n-1)}{2}} \\cdot \\left( (n-1)! \\right)^n"
] | [
"Calculate the determinant of the matrix \\( A \\in M_n(\\mathbb{Z}) \\) defined as:\n\\[ A = \\left[\\begin{array}{ccccc}\n1^{n-1} & 2^{n-1} & 3^{n-1} & \\cdots & n^{n-1} \\\\\n2^{n-1} & 3^{n-1} & 4^{n-1} & \\cdots & (n+1)^{n-1} \\\\\n3^{n-1} & 4^{n-1} & 5^{n-1} & \\cdots & (n+2)^{n-1} \\\\\n\\vdots & \\vdots & \\... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1afb1734e0f996486e68bf0c98fdff22994b0375c8071b7c1936546ec310f87c | Calculate the determinant of the matrix \( A_{n \times n} \) where \( x \in \mathbb{R} \) and \( n \ge 2 \): \[ \det A_{n \times n} = \det \begin{pmatrix} 1 & 1 & 1 & 1 & \cdots & 1 \\ 2 & 1 & x & x & \cdots & x \\ 3 & 2 & 1 & x & \cdots & x \\ 4 & 3 & 2 & 1 & \cdots & x \\ \vdots & \vdots & \vdots & \vdots & \ddots & ... | [
"\\((-1)^{n-1} x^{n-2}\\)"
] | [
"Calculate the determinant of the matrix \\( A_{n \\times n} \\) where \\( x \\in \\mathbb{R} \\) and \\( n \\ge 2 \\):\n\\[ \\det A_{n \\times n} = \\det \\begin{pmatrix} \n1 & 1 & 1 & 1 & \\cdots & 1 \\\\ \n2 & 1 & x & x & \\cdots & x \\\\ \n3 & 2 & 1 & x & \\cdots & x \\\\ \n4 & 3 & 2 & 1 & \\cdots & x \\\\ \n\\... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 9
} | 23 | 0 | 0 |
omr-ws-32c1b5ebd85a0ecabbc712fc5bba1ae0587f4e243147eb199cac9d8df96f1306 | Calculate the determinant of the matrix \( B = \begin{pmatrix} x & a & a & a \\ -a & x & a & a \\ -a & -a & x & a \\ -a & -a & -a & x \end{pmatrix} \). | [
"x^4 + 6a^2x^2 + a^4"
] | [
"Calculate the determinant of the matrix \\( B = \\begin{pmatrix} x & a & a & a \\\\ -a & x & a & a \\\\ -a & -a & x & a \\\\ -a & -a & -a & x \\end{pmatrix} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-52d2e551fffb53898fecf8726d3d3c7a19e2bc96d07595a24898946766da5288 | Calculate the determinant of the matrix \( B \) where \[ B = \begin{pmatrix} A & J & J & \cdots & J\\ J & A & J & \cdots & J\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ J & J & J & \cdots & A \end{pmatrix} \] Here, \( A \) is an \( n \times n \) matrix, and \( J \) is an \( n \times n \) matrix with all elements equ... | [
"\\det(A - J)^{k-1} \\cdot \\det(A + (k-1)J)"
] | [
"Calculate the determinant of the matrix \\( B \\) where\n\\[ B = \\begin{pmatrix}\nA & J & J & \\cdots & J\\\\ \nJ & A & J & \\cdots & J\\\\ \n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots\\\\ \nJ & J & J & \\cdots & A\n\\end{pmatrix} \\]\nHere, \\( A \\) is an \\( n \\times n \\) matrix, and \\( J \\) is an \\(... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-734b518696ab9ca8468a09dba37acf4ddeccded30b7a86ced8c2f82e384e0f99 | Calculate the determinant of the matrix \( B \in M_n(\mathbb{Z}) \) defined as: \[ b_{ij} = \begin{cases} 1 & \text{if } |i - j| = 2 \\ 0 & \text{otherwise} \end{cases} \] | [
"\\(\\det(B) = \\begin{cases}\n0 & \\text{if } n \\text{ is odd} \\\\\n(-1)^{n/2} & \\text{if } n \\text{ is even}\n\\end{cases}\\)"
] | [
"Calculate the determinant of the matrix \\( B \\in M_n(\\mathbb{Z}) \\) defined as:\n\\[ b_{ij} = \\begin{cases}\n1 & \\text{if } |i - j| = 2 \\\\\n0 & \\text{otherwise}\n\\end{cases} \\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1f0c384d0194c730502e36ce6387d2c13b0c6228e877877564f10ed9bb07bcda | Calculate the determinant of the matrix \( E \) given by \[ E = \left|\begin{matrix} a_1 & 0 &\cdots & 0 & b_1 & 0 &\cdots & 0 \\ 0 & a_2 &\cdots & 0 & 0 & b_2 &\cdots & 0 \\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ 0 & 0 &\cdots & a_n & 0 & 0 &\cdots & b_n \\ c_1 & 0 &\cdots & 0 & d_1 & 0 &\cdots & 0 ... | [
"\\(\\prod_{i=1}^{n} (a_i d_i - b_i c_i)\\)"
] | [
"Calculate the determinant of the matrix \\( E \\) given by\n\\[ E = \\left|\\begin{matrix}\na_1 & 0 &\\cdots & 0 & b_1 & 0 &\\cdots & 0 \\\\\n0 & a_2 &\\cdots & 0 & 0 & b_2 &\\cdots & 0 \\\\\n\\cdots&\\cdots&\\cdots&\\cdots&\\cdots&\\cdots&\\cdots&\\cdots\\\\\n0 & 0 &\\cdots & a_n & 0 & 0 &\\cdots & b_n \\\\\nc_1... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-95eceef1935c1a71937ccd9037892c3f4a6ac8b7b8d0e3ea746130f9e160fc39 | Calculate the determinant of the matrix \( E \): \[ E = \begin{vmatrix} 1+x_1 & 1+x_1^2 & 1+x_1^3 & \cdots & 1+x_1^n \\ 1+x_2 & 1+x_2^2 & 1+x_2^3 & \cdots & 1+x_2^n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1+x_n & 1+x_n^2 & 1+x_n^3 & \cdots & 1+x_n^n \end{vmatrix} \] | [
"\\([2x_{1}x_{2}...x_{n}-(x_{1}-1)(x_{2}-1)...(x_{n}-1)]\\prod_{1\\leq{k}<i\\leq{n}}(x_{i}-x_{k})\\)"
] | [
"Calculate the determinant of the matrix \\( E \\):\n\n\\[ E = \\begin{vmatrix} \n1+x_1 & 1+x_1^2 & 1+x_1^3 & \\cdots & 1+x_1^n \\\\ \n1+x_2 & 1+x_2^2 & 1+x_2^3 & \\cdots & 1+x_2^n \\\\ \n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ \n1+x_n & 1+x_n^2 & 1+x_n^3 & \\cdots & 1+x_n^n \n\\end{vmatrix} \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-d53519d7ccd82b193f2cb70dec0bd77c7184e2bd6a8cbcee40400227e7ea34ce | Calculate the determinant of the matrix \( H \) given by \[ H = \begin{vmatrix} 1 & \frac{1}{2} & \cdots & \frac{1}{n} \\ \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n+1} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{1}{n} & \frac{1}{n+1} & \cdots & \frac{1}{2n-1} \end{vmatrix}. \] | [
"\\(\\frac{c_n^4}{c_{2n}}\\) where \\( c_n = \\prod_{i=1}^{n-1} i! \\)"
] | [
"Calculate the determinant of the matrix \\( H \\) given by\n\\[ H = \\begin{vmatrix}\n1 & \\frac{1}{2} & \\cdots & \\frac{1}{n} \\\\\n\\frac{1}{2} & \\frac{1}{3} & \\cdots & \\frac{1}{n+1} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\frac{1}{n} & \\frac{1}{n+1} & \\cdots & \\frac{1}{2n-1}\n\\end{vmatrix}. \... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-c2143fb0ac957bcf329397225f468909e45647cf86831cb15f8f1448f42f65f5 | Calculate the determinant of the matrix \(D_n\): \[D_n = \left| \begin{array}{ccccc} 1 & 2 & \cdots & n-1 & n \\ n & 1 & \cdots & n-2 & n-1 \\ n-1 & n & \cdots & n-3 & n-2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 2 & 3 & \cdots & n & 1 \end{array} \right|\] | [
"\\(\\frac{n^{n-1}(n+1)(-1)^{n-1}}{2}\\)"
] | [
"Calculate the determinant of the matrix \\(D_n\\):\n\\[D_n = \\left| \\begin{array}{ccccc}\n1 & 2 & \\cdots & n-1 & n \\\\\nn & 1 & \\cdots & n-2 & n-1 \\\\\nn-1 & n & \\cdots & n-3 & n-2 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n2 & 3 & \\cdots & n & 1\n\\end{array} \\right|\\]"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-736cbb06e471643183973d237853b559414f3c69453e33460e7a8cf276aa65bc | Calculate the determinant of the matrix \(\begin{pmatrix}0&1&2&3&...&n\\ 1&0&1&2&...&n-1\\ 2&1&0&1&...&n-2\\ 3&2&1&0&..&n-3\\ ...&...&...&...&...&...\\ n&n-1&n-2&n-3&...&0\end{pmatrix}\). | [
"\\((-1)^n \\cdot n \\cdot 2^{n-1}\\)"
] | [
"Calculate the determinant of the matrix \\(\\begin{pmatrix}0&1&2&3&...&n\\\\ 1&0&1&2&...&n-1\\\\ 2&1&0&1&...&n-2\\\\ 3&2&1&0&..&n-3\\\\ ...&...&...&...&...&...\\\\ n&n-1&n-2&n-3&...&0\\end{pmatrix}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 20
} | 43 | 0 | 0 |
omr-ws-bf8dbbe4af47988ac02b9869d20ea9a5cbe415bdfd5a302b9d5382f838997ee4 | Calculate the determinant of the matrix \(\begin{vmatrix} x_1 & x_2 & x_3 & x_4 \\ x_2 & x_3 & x_4 & x_1 \\ x_3 & x_4 & x_1 & x_2 \\ x_4 & x_1 & x_2 & x_3 \end{vmatrix}\), where \(x_1, x_2, x_3,\) and \(x_4\) are the roots of the equation \(x^4 + px^2 + qx + r = 0\). | [
"0"
] | [
"Calculate the determinant of the matrix \\(\\begin{vmatrix} x_1 & x_2 & x_3 & x_4 \\\\ x_2 & x_3 & x_4 & x_1 \\\\ x_3 & x_4 & x_1 & x_2 \\\\ x_4 & x_1 & x_2 & x_3 \\end{vmatrix}\\), where \\(x_1, x_2, x_3,\\) and \\(x_4\\) are the roots of the equation \\(x^4 + px^2 + qx + r = 0\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4df46f487deed5185a990ccc4b9a181a57eedab0fda1f83a0c2cfce74566f633 | Calculate the determinant of the matrix \(\begin{vmatrix}-bc & b^2+bc & c^2+bc \\ a^2+ac & -ac & c^2+ac \\ a^2+ab & b^2+ab & -ab\end{vmatrix}\) and express your answer in terms of \(a\), \(b\), and \(c\). | [
"(ab + bc + ca)^3"
] | [
"Calculate the determinant of the matrix \\(\\begin{vmatrix}-bc & b^2+bc & c^2+bc \\\\ a^2+ac & -ac & c^2+ac \\\\ a^2+ab & b^2+ab & -ab\\end{vmatrix}\\) and express your answer in terms of \\(a\\), \\(b\\), and \\(c\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e48a2925cf07e7712c7264f179ba96b98ca1ea087ab82eee9a8eb448e34ef3cf | Calculate the determinant of the matrix \(\delta = \begin{vmatrix} 0 & 3 & 3 & 3 & \cdots & 3 \\ 3 & 0 & a_1 + a_2 & a_1 + a_3 & \cdots & a_1 + a_n \\ 3 & a_2 + a_1 & 0 & a_2 + a_3 & \cdots & a_2 + a_n \\ 3 & a_3 + a_1 & a_3 + a_2 & 0 & \cdots & a_3 + a_n \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 3 & a_... | [
"(-1)^n \\cdot 9 \\cdot 2^{n-1} \\cdot \\sum_{1 \\leq i_1 < i_2 < \\cdots < i_{n-1} \\leq n} a_{i_1} a_{i_2} \\cdots a_{i_{n-1}}"
] | [
"Calculate the determinant of the matrix \\(\\delta = \\begin{vmatrix} 0 & 3 & 3 & 3 & \\cdots & 3 \\\\ 3 & 0 & a_1 + a_2 & a_1 + a_3 & \\cdots & a_1 + a_n \\\\ 3 & a_2 + a_1 & 0 & a_2 + a_3 & \\cdots & a_2 + a_n \\\\ 3 & a_3 + a_1 & a_3 + a_2 & 0 & \\cdots & a_3 + a_n \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-584b4f35b4b57fc789caa751c22925dbf4ff1d62c4151b8ac47a43d7fba19416 | Calculate the determinant of the matrix \(\|(\sum_{k=0}^{n} a_k x_k^{m+i-j})\|_{i,j=1}^{n}\), where \(a_1, a_2, \ldots, a_n, x_1, x_2, \ldots, x_n\) are non-zero real numbers and \(m \geq n\) are positive integers. | [
"(-1)^{\\frac{n(n-1)}{2}} \\left( \\prod_{k=1}^{n} a_k \\right) \\left( \\prod_{k=1}^{n} x_k^{m - n + 1} \\right) \\left( \\prod_{1 \\leq i < j \\leq n} (x_j - x_i)^2 \\right)"
] | [
"Calculate the determinant of the matrix \\(\\|(\\sum_{k=0}^{n} a_k x_k^{m+i-j})\\|_{i,j=1}^{n}\\), where \\(a_1, a_2, \\ldots, a_n, x_1, x_2, \\ldots, x_n\\) are non-zero real numbers and \\(m \\geq n\\) are positive integers."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-2efea93ab9f15c18a31ce62269ec8235c7748c91449dd1e99e9c81fe657d0d0b | Calculate the determinant of the matrix \[ D= \left| \begin{array}{ccc} (a+b)^2 & c^2 & c^2 \\ a^2 & ( b+c)^2 & a^2 \\ b^2 & b^2 & (c+a)^2 \\ \end{array} \right| \] and express your answer in terms of \(a\), \(b\), and \(c\). | [
"2abc(a + b + c)^3"
] | [
"Calculate the determinant of the matrix \n\\[ D= \\left| \\begin{array}{ccc}\n(a+b)^2 & c^2 & c^2 \\\\\na^2 & ( b+c)^2 & a^2 \\\\\nb^2 & b^2 & (c+a)^2 \\\\\n\\end{array} \\right| \\]\nand express your answer in terms of \\(a\\), \\(b\\), and \\(c\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c93603c977ca936fbcdebd16ab1ad4e808dfc8ca9312a52fbeb66236ad2ec853 | Calculate the determinant of the matrix \[ \begin{vmatrix} 2bc - a^2 & c^2 & b^2 \\ c^2 & 2ac - b^2 & a^2 \\ b^2 & a^2 & 2ab - c^2 \end{vmatrix} \] and express your answer in terms of \(a\), \(b\), and \(c\). | [
"(a^3 + b^3 + c^3 - 3abc)^2"
] | [
"Calculate the determinant of the matrix\n\n\\[\n\\begin{vmatrix}\n2bc - a^2 & c^2 & b^2 \\\\\nc^2 & 2ac - b^2 & a^2 \\\\\nb^2 & a^2 & 2ab - c^2\n\\end{vmatrix}\n\\]\n\nand express your answer in terms of \\(a\\), \\(b\\), and \\(c\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-0cef558ebc4118c1b6e849553c891186493fe48e7dc0553fd749f2171d1e0141 | Calculate the determinant of the matrix \[ \begin{vmatrix} x_1 + y_1 + 1 & x_1 + y_2 & \cdots & x_1 + y_n\\ x_2 + y_1 & x_2 + y_2 + 1 & \cdots & x_2 + y_n\\ \vdots & \vdots & \ddots & \vdots \\ x_n + y_1 & x_n + y_2 & \cdots & x_n + y_n + 1 \end{vmatrix} \] where \( n \in \mathbb{N}^* \) and \( x_i, y_i \in \mathbb{R} ... | [
"\\((1 + x_1 + x_2 + \\cdots + x_n)(1 + y_1 + y_2 + \\cdots + y_n) - n(x_1 y_1 + x_2 y_2 + \\cdots + x_n y_n)\\)"
] | [
"Calculate the determinant of the matrix\n\\[\n\\begin{vmatrix}\nx_1 + y_1 + 1 & x_1 + y_2 & \\cdots & x_1 + y_n\\\\ \nx_2 + y_1 & x_2 + y_2 + 1 & \\cdots & x_2 + y_n\\\\ \n\\vdots & \\vdots & \\ddots & \\vdots \\\\ \nx_n + y_1 & x_n + y_2 & \\cdots & x_n + y_n + 1\n\\end{vmatrix}\n\\]\nwhere \\( n \\in \\mathbb{N}... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-b58ccd9a273cfbcd4a583cd756f71cc073f58a412f5b060efc89dbe8f8ca86ca | Calculate the determinant of the matrix \[ \det \begin{pmatrix} x & y & z & v \\ y & x & v & z \\ z & v & x & y \\ v & z & y & x \end{pmatrix} \] | [
"\\((x+y+z+v)(x+y-z-v)(x-y+z-v)(x-y-z+v)\\)"
] | [
"Calculate the determinant of the matrix\n\\[ \\det \\begin{pmatrix}\nx & y & z & v \\\\\ny & x & v & z \\\\\nz & v & x & y \\\\\nv & z & y & x\n\\end{pmatrix} \\]"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 7,
"tir": 23
} | 66 | 0 | 0 |
omr-ws-9f9fd41ffe63d8e5996b2d7cd6cd26f598df26c4529f8f2b747bd49399ee9ba4 | Calculate the determinant of the matrix \[ \left| \begin{array}{ccc} -2a & a+b & a+c \\ b+a & -2b & b+c \\ c+a & c+b & -2c \end{array} \right| \] and express your answer in the form $k(b+c)(c+a)(a+b)$, where $k$ is an integer. Find the value of $k$. | [
"4"
] | [
"Calculate the determinant of the matrix\n\\[\n\\left|\n\\begin{array}{ccc}\n-2a & a+b & a+c \\\\\nb+a & -2b & b+c \\\\ \nc+a & c+b & -2c\n\\end{array}\n\\right|\n\\]\nand express your answer in the form $k(b+c)(c+a)(a+b)$, where $k$ is an integer. Find the value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a775a7aa7be1f434c450c350ef09e73e9583945c4fbc22556cb2736261644e9f | Calculate the determinant of the matrix defined by: \[ \begin{vmatrix} a & b & b & \cdots & b \\ -b & a & b & \cdots & b \\ -b & -b & a & \cdots & b \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -b & -b & -b & \cdots & a \end{vmatrix} \] or equivalently, where the entries are given by: \[ a_{ij} = \begin{cases} b & ... | [
"\\(\\frac{1}{2}[(a-b)^n + (a+b)^n]\\)"
] | [
"Calculate the determinant of the matrix defined by:\n\\[ \\begin{vmatrix} a & b & b & \\cdots & b \\\\ -b & a & b & \\cdots & b \\\\ -b & -b & a & \\cdots & b \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ -b & -b & -b & \\cdots & a \\end{vmatrix} \\]\nor equivalently, where the entries are given by:\n\... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 8
} | 21 | 0 | 0 |
omr-ws-03088300f8739b8c3e46a2ce07cee5f9bf858e40a98247372bbe205ecbfea2cf | Calculate the determinant of the matrix with the following entries: - \( a_{i,i+2} = -2 \) for \( i \in \{1, \ldots, n-2\} \) - \( a_{i,i+1} = -1 \) for \( i \in \{1, \ldots, n-1\} \) - \( a_{i,i} = 2 \) for \( i \in \{1, \ldots, n\} \) - \( a_{i+1,i} = 1 \) for \( i \in \{1, \ldots, n-1\} \) All other entries are \( 0... | [
"\\( \\frac{2^{n+3} + (-1)^n - 3}{6} \\)"
] | [
"Calculate the determinant of the matrix with the following entries:\n- \\( a_{i,i+2} = -2 \\) for \\( i \\in \\{1, \\ldots, n-2\\} \\)\n- \\( a_{i,i+1} = -1 \\) for \\( i \\in \\{1, \\ldots, n-1\\} \\)\n- \\( a_{i,i} = 2 \\) for \\( i \\in \\{1, \\ldots, n\\} \\)\n- \\( a_{i+1,i} = 1 \\) for \\( i \\in \\{1, \\ldo... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 12
} | 26 | 0 | 0 |
omr-ws-de000aeacbf77e123d64c6528356d4ed53a5156a168c0f4db69fbe67ea2871d9 | Calculate the determinant of the matrix: \[ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \] | [
"0"
] | [
"Calculate the determinant of the matrix:\n\\[\n\\begin{pmatrix}\n1 & 2 & 3 \\\\\n4 & 5 & 6 \\\\\n7 & 8 & 9\n\\end{pmatrix}\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3e5295094a5c9632bfcf6aa409787c9e53656bd147938752c836aa909c1b8e7d | Calculate the determinant of the matrix: \[ \begin{vmatrix} s-a_1 & s-a_2 & \ldots & s-a_n \\ s-a_n & s-a_1 & \ldots & s-a_{n-1} \\ \vdots & \vdots & \ddots & \vdots \\ s-a_2 & s-a_3 & \ldots & s-a_1 \end{vmatrix} \] where \( s = a_1 + a_2 + \cdots + a_n \). | [
"(-1)^{n-1} (n - 1) \\det(C)"
] | [
"Calculate the determinant of the matrix:\n\\[ \\begin{vmatrix}\ns-a_1 & s-a_2 & \\ldots & s-a_n \\\\\ns-a_n & s-a_1 & \\ldots & s-a_{n-1} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\ns-a_2 & s-a_3 & \\ldots & s-a_1\n\\end{vmatrix} \\]\nwhere \\( s = a_1 + a_2 + \\cdots + a_n \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-51234897d2bca65cb99d537c2bbd538af86ee3c268e07c75ac3eb25ba44ac077 | Calculate the determinant of the matrix: \[ \begin{vmatrix} t^{n-1} & \binom{n}{1} t^{n-2} & \binom{n}{2} t^{n-3} & \ldots & \binom{n}{n-2} t & \binom{n}{n-1} \\ \binom{n}{n-1} & t^{n-1} & \binom{n}{1} t^{n-2} & \ldots & \binom{n}{n-3} t^2 & \binom{n}{n-2} t \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ \bi... | [
"t^{n(n-1)} - (-n)^n"
] | [
"Calculate the determinant of the matrix:\n\\[ \\begin{vmatrix}\nt^{n-1} & \\binom{n}{1} t^{n-2} & \\binom{n}{2} t^{n-3} & \\ldots & \\binom{n}{n-2} t & \\binom{n}{n-1} \\\\\n\\binom{n}{n-1} & t^{n-1} & \\binom{n}{1} t^{n-2} & \\ldots & \\binom{n}{n-3} t^2 & \\binom{n}{n-2} t \\\\\n\\vdots & \\vdots & \\vdots & \\d... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb6cf83770e92949b38bce42810078863b6af6dff039f44550ff2dec1d0efe60 | Calculate the determinant of the matrix: \[ \begin{vmatrix}\binom{k+m}{m}& \binom{k+m+1}{m}& \ldots & \binom{k+2m}{m} \\ \binom{k+m+1}{m}& \binom{k+m+2}{m}& \ldots & \binom{k+2m+1}{m}\\ \vdots &\vdots &\ddots &\vdots\\ \binom{k+2m}{m}& \binom{k+2m+1}{m}& \ldots & \binom{k+3m}{m}\end{vmatrix} \] | [
"$(-1)^{m+1}$"
] | [
"Calculate the determinant of the matrix:\n\\[ \\begin{vmatrix}\\binom{k+m}{m}& \\binom{k+m+1}{m}& \\ldots & \\binom{k+2m}{m} \\\\ \\binom{k+m+1}{m}& \\binom{k+m+2}{m}& \\ldots & \\binom{k+2m+1}{m}\\\\ \\vdots &\\vdots &\\ddots &\\vdots\\\\ \\binom{k+2m}{m}& \\binom{k+2m+1}{m}& \\ldots & \\binom{k+3m}{m}\\end{vmatr... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eadf5bbac65e55b1fd87b4f156079d2c7fbd9cc5052e22127493165420932901 | Calculate the determinant of the matrix: \[ \begin{vmatrix}\binom{m}{k}& \binom{m}{k+1}& \ldots & \binom{m}{k+n} \\ \binom{m+1}{k}& \binom{m+1}{k+1}& \ldots & \binom{m+1}{k+n}\\ \vdots &\vdots &\ddots &\vdots\\ \binom{m+n}{k}& \binom{m+n}{k+1}& \ldots & \binom{m+n}{k+n}\end{vmatrix} \] | [
"1"
] | [
"Calculate the determinant of the matrix:\n\\[ \\begin{vmatrix}\\binom{m}{k}& \\binom{m}{k+1}& \\ldots & \\binom{m}{k+n} \\\\ \\binom{m+1}{k}& \\binom{m+1}{k+1}& \\ldots & \\binom{m+1}{k+n}\\\\ \\vdots &\\vdots &\\ddots &\\vdots\\\\ \\binom{m+n}{k}& \\binom{m+n}{k+1}& \\ldots & \\binom{m+n}{k+n}\\end{vmatrix} \\]"
... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-716832f73625282133f149f5f8c9d573a3b35183acd87133a01af0cf0eb3765f | Calculate the determinant of the matrix: \[ \left|\begin{array}{ccccccc}F(0)& F(1)& F(2) &\cdots&F(n)\\F(1)& F(2)& F(3) &\cdots&F(n+1)\\\vdots&\vdots&\vdots&\ddots&\vdots\\F(n)& F(n+1)& F(n+2) &\cdots&F(2n)\end{array}\right| \] where \( F(x) = x(x-1)(x-2)\cdots(x-n+1) \). | [
"\\((-1)^{\\frac{n(n+1)}{2}} (n!)^{n+1}\\)"
] | [
"Calculate the determinant of the matrix:\n\\[ \\left|\\begin{array}{ccccccc}F(0)& F(1)& F(2) &\\cdots&F(n)\\\\F(1)& F(2)& F(3) &\\cdots&F(n+1)\\\\\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\F(n)& F(n+1)& F(n+2) &\\cdots&F(2n)\\end{array}\\right| \\]\nwhere \\( F(x) = x(x-1)(x-2)\\cdots(x-n+1) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 3,
"tir": 17
} | 35 | 0 | 0 |
omr-ws-e13d4b51a44b3c66b28c72bb1d45518fb1adb7b5117b799da2a1c3ea24c0e5ec | Calculate the determinant of the matrix: \[ |A| = \begin{vmatrix} a & 0 & 0 & \cdots & 0 & 0 & b \\ 0 & a & 0 & \cdots & 0 & b & 0 \\ 0 & 0 & a & \cdots & b & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & \cdots & a & b & 0 & 0 \\ 0 & 0 & \cdots & b & a & 0 & 0 \\ \vdots & \vdots & \v... | [
"(a^2 - b^2)^{n/2}"
] | [
"Calculate the determinant of the matrix:\n\\[ |A| = \\begin{vmatrix}\na & 0 & 0 & \\cdots & 0 & 0 & b \\\\\n0 & a & 0 & \\cdots & 0 & b & 0 \\\\\n0 & 0 & a & \\cdots & b & 0 & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots \\\\\n0 & 0 & \\cdots & a & b & 0 & 0 \\\\\n0 & 0 & \\cdots & b... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-54a26ddf366130b50fd75d1b50561493f167a79472616a7fc00106fcf62e645b | Calculate the determinant of the real matrix \( A = \begin{pmatrix} a_{1} & x & x & \dotsc & x\\ x & a_{2} & x & \dotsc & x\\ x & x & a_{3} & \dotsc & x\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x & x & x & \dotsc & a_{n} \end{pmatrix} \). | [
"\\( P(x) - xP'(x) \\) where \\( P(x) = \\prod_{i=1}^n (a_i - x) \\)"
] | [
"Calculate the determinant of the real matrix \\( A = \\begin{pmatrix}\na_{1} & x & x & \\dotsc & x\\\\\nx & a_{2} & x & \\dotsc & x\\\\\nx & x & a_{3} & \\dotsc & x\\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\nx & x & x & \\dotsc & a_{n}\n\\end{pmatrix} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 4,
"tir": 18
} | 31 | 0 | 0 |
omr-ws-4dc1e095a8422c8e9ee79e4d6079eb8e05457ad10e423c47993fe2deae81f3b4 | Calculate the difference $D$ between the number of ways to rearrange "baby hamster" and "babyhamster". Given that baby spamster writes "SpamsterHamster" 50 times and then writes "Spams" until she runs out of room, find $\dfrac{D}{p}$ where $p$ is the probability of randomly selecting the letter A from all the character... | [
"548856000"
] | [
"Calculate the difference $D$ between the number of ways to rearrange \"baby hamster\" and \"babyhamster\". Given that baby spamster writes \"SpamsterHamster\" 50 times and then writes \"Spams\" until she runs out of room, find $\\dfrac{D}{p}$ where $p$ is the probability of randomly selecting the letter A from all... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-4185bcdb8916144b2a095b1363adb14fe3b7fdee8d5fcbbad33c78918d2f6f66 | Calculate the difference between the area of a unit circle computed using degrees and the area computed using radians. Express your answer in terms of $\pi$. | [
"180 - \\pi"
] | [
"Calculate the difference between the area of a unit circle computed using degrees and the area computed using radians. Express your answer in terms of $\\pi$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2876f74be52a503e58280580cdb5a643df221de3f5d46d208ccdb8c6d92ea5d8 | Calculate the difference between the lengths of the largest and smallest diagonal of a regular 9-sided convex polygon with each side of length 1. | [
"1"
] | [
"Calculate the difference between the lengths of the largest and smallest diagonal of a regular 9-sided convex polygon with each side of length 1."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-cf263635da6a01a1a7a93e6da759c6481f353bb3b864f59c973cf1f3e698d5b3 | Calculate the digit in the hundreds place of the expansion of $7^{707}$. | [
"5"
] | [
"Calculate the digit in the hundreds place of the expansion of $7^{707}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-1057a7c2ee0631b0e5cde54034ff924bf15e9d2c1b06581aae6e8fd9e97d00cd | Calculate the dimension of the group \({\rm PSL}(n + 1,\mathbb{C})/{\rm PSU}(n,\mathbb{C})\). | [
"n^2 + 4n + 1"
] | [
"Calculate the dimension of the group \\({\\rm PSL}(n + 1,\\mathbb{C})/{\\rm PSU}(n,\\mathbb{C})\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-108901938d4b5caed129920eab18ae6de35c7854226be285eeb25c0993fcab7e | Calculate the directional derivative of the scalar function \( f(x_1, x_2, \ldots, x_n) = x_1^2 + x_2^2 + \ldots + x_n^2 \) at the point \( (1, 2, 3, \ldots, n) \) in the direction of the unit vector \( (1, 1, \ldots, 1) \). | [
"\\(\\sqrt{n}(n+1)\\)"
] | [
"Calculate the directional derivative of the scalar function \\( f(x_1, x_2, \\ldots, x_n) = x_1^2 + x_2^2 + \\ldots + x_n^2 \\) at the point \\( (1, 2, 3, \\ldots, n) \\) in the direction of the unit vector \\( (1, 1, \\ldots, 1) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 11
} | 33 | 0 | 0 |
omr-ws-b982afdb6780fb836e42ccd01bc6a79e1e92c8e64a353a15087029e05e777926 | Calculate the distance $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$, where $p$ runs over polynomials of degree less than $n$ with real coefficients and $f$ runs over functions $f(x)= \sum_{k=n}^{\infty} c_k x^k$ defined on the closed interval $[0,1]$, where $c_k \ge 0$ and $\sum_{k=n}^{\infty} c_k=1$. | [
"$\\frac{1}{2^{2n-1}}$"
] | [
"Calculate the distance $\\inf_{p,f}\\, \\max_{0\\le x\\le 1} |f(x)-p(x)|$, where $p$ runs over polynomials of degree less than $n$ with real coefficients and $f$ runs over functions $f(x)= \\sum_{k=n}^{\\infty} c_k x^k$ defined on the closed interval $[0,1]$, where $c_k \\ge 0$ and $\\sum_{k=n}^{\\infty} c_k=1$."
... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 15
} | 29 | 0 | 0 |
omr-ws-0ba3e710398df6e0b4f5dfb12279e54696aa90362d2e3dbbd55322c87617c63e | Calculate the distance between line $EG$ and a line that passes through $B$ and is skew to $EG$ in a cube $ABCD.EFGH$ with an edge length of 4 cm. | [
"4"
] | [
"Calculate the distance between line $EG$ and a line that passes through $B$ and is skew to $EG$ in a cube $ABCD.EFGH$ with an edge length of 4 cm."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f5e1c98d5ed1e6226aa9da4946a03caed101759c567ba216f5398488fb3953e0 | Calculate the distance between line $FH$ and plane $\alpha$ that contains $AC$ and is parallel to $FH$ in a cube $ABCD.EFGH$ with an edge length of 4 cm. | [
"4 cm"
] | [
"Calculate the distance between line $FH$ and plane $\\alpha$ that contains $AC$ and is parallel to $FH$ in a cube $ABCD.EFGH$ with an edge length of 4 cm."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-66817988161266f95a0f9539c001534f6aaf3f6b0966e01386e7f16b46690482 | Calculate the distance between planes $BEG$ and $ACH$ in a cube $ABCD.EFGH$ with an edge length of 4 cm. | [
"4 cm"
] | [
"Calculate the distance between planes $BEG$ and $ACH$ in a cube $ABCD.EFGH$ with an edge length of 4 cm."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-efe7000ef15f34c399960a77230e1b23fe7ac078b040900fa9b062d6a42f09c4 | Calculate the distance between the bus stop and the traffic light for a cyclist who travels at a constant speed of $3 \text{ m/s}$ for $20$ seconds and then decelerates at a constant rate of $0.2 \text{ m/s}^2$ until coming to rest. | [
"82.5 meters"
] | [
"Calculate the distance between the bus stop and the traffic light for a cyclist who travels at a constant speed of $3 \\text{ m/s}$ for $20$ seconds and then decelerates at a constant rate of $0.2 \\text{ m/s}^2$ until coming to rest."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-1312407a2e0f82c03df29c92a1e5a4f003cd1836bf614a38583c64422deb33f9 | Calculate the distance between the lines $BF'$ and $ND$ in the regular hexagonal prism $ABCDEFA'B C'D'E'F'$ with base edge length $12$ and height $12 \sqrt{3}$, where $N$ is the midpoint of the edge $CC'$. | [
"12"
] | [
"Calculate the distance between the lines $BF'$ and $ND$ in the regular hexagonal prism $ABCDEFA'B C'D'E'F'$ with base edge length $12$ and height $12 \\sqrt{3}$, where $N$ is the midpoint of the edge $CC'$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-12957d50bd83e79a90c0ab231e2bd5ae305a9cbfdb612e5bb50b95ee2cb48310 | Calculate the distance for an airplane to reduce its speed from $50$ $m/s$ to $20$ $m/s$ given that its acceleration is $a = - v^{2} /250$. | [
"229.07 m"
] | [
"Calculate the distance for an airplane to reduce its speed from $50$ $m/s$ to $20$ $m/s$ given that its acceleration is $a = - v^{2} /250$."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 4,
"tir": 30
} | 38 | 0 | 0 |
omr-ws-9648b0252cd47e40de58233c44aa9fce466830894018d8848ab31edf7ad531a6 | Calculate the distance in the normed space $(C[0,1],||\cdot||)$ endowed with the supremum norm \[||f||=\sup_{t\in[0,1]}|f(t)|\] from the function $f(t)=t^2$ to the linear hull of the functions $g(t)=t$ and $h(t)=\sin(t)$. | [
"\\dfrac{1}{8}"
] | [
"Calculate the distance in the normed space $(C[0,1],||\\cdot||)$ endowed with the supremum norm \\[||f||=\\sup_{t\\in[0,1]}|f(t)|\\] from the function $f(t)=t^2$ to the linear hull of the functions $g(t)=t$ and $h(t)=\\sin(t)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-9265ca0e77751067a9e408f1b6485603dec9835918625b087f9b2969424a089b | Calculate the distance of point $P$ from either base of the isosceles trapezoid with bases $a$ and $c$ and altitude $h$. | [
"\\(\\frac{h + \\sqrt{h^2 - ac}}{2}\\) and \\(\\frac{h - \\sqrt{h^2 - ac}}{2}\\)"
] | [
"Calculate the distance of point $P$ from either base of the isosceles trapezoid with bases $a$ and $c$ and altitude $h$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-64f43127d9526d7689324bab02e5b7f2f84ad078139946a4bd1d40570f56c6e0 | Calculate the distance: (a) from the point $(3, 1, -1)$ to the plane $22x+4y-20z-45=0$; (b) from the point $(4, 3, -2)$ to the plane $3x-y+5z+1=0$; (c) from the point $ (2, 0, -1/2)$ to the plane $4x-4y+2z+17=0$. | [
"(a) $\\frac{3}{2}$; (b) 0; (c) 4"
] | [
"Calculate the distance: (a) from the point $(3, 1, -1)$ to the plane $22x+4y-20z-45=0$; (b) from the point $(4, 3, -2)$ to the plane $3x-y+5z+1=0$; (c) from the point $ (2, 0, -1/2)$ to the plane $4x-4y+2z+17=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bd90cc4400f7d879be48faa83914dcb707e60b48d6aabe2fd2fde8c8f3827d41 | Calculate the distances for the following routes: - From A to X at 5 km/h and then from X to B at 2 km/h, which takes 16 minutes and 59 seconds. - From A to Y at 5 km/h and then from Y to B at 2 km/h, which takes 19 minutes and 25 seconds. Given: - The angle between A-> X and A-> Y is 95°. - The angles between A-> X an... | [
"0.4 \\, \\text{km}"
] | [
"Calculate the distances for the following routes:\n- From A to X at 5 km/h and then from X to B at 2 km/h, which takes 16 minutes and 59 seconds.\n- From A to Y at 5 km/h and then from Y to B at 2 km/h, which takes 19 minutes and 25 seconds.\n\nGiven:\n- The angle between A-> X and A-> Y is 95°.\n- The angles betw... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-19c75782720b039d94cde898a8d40f4b874f8c4b12dea6096377fc2fce5e4602 | Calculate the distribution of the random variable \[ \int\limits_{0}^{t}W_{s}ds \] where \( W_{t} \) is Brownian motion. | [
"\\mathcal{N}\\left(0, \\frac{t^3}{3}\\right)"
] | [
"Calculate the distribution of the random variable\n\n\\[\n\\int\\limits_{0}^{t}W_{s}ds\n\\]\n\nwhere \\( W_{t} \\) is Brownian motion."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-da5ad3eb8c7e30507672f8b3f85ad803d9da47f523d0f80690e4cd0581ab7d06 | Calculate the divergence of the vector field $\vec{F} = \left(\frac{x}{(x^2 + y^2 + z^2)^{3/2}}, \frac{y}{(x^2 + y^2 + z^2)^{3/2}}, \frac{z}{(x^2 + y^2 + z^2)^{3/2}}\right)$ in Cartesian coordinates and express your answer in terms of $x$, $y$, and $z$. Then, find the value of the divergence at the point $(1, 1, 1)$. | [
"0"
] | [
"Calculate the divergence of the vector field $\\vec{F} = \\left(\\frac{x}{(x^2 + y^2 + z^2)^{3/2}}, \\frac{y}{(x^2 + y^2 + z^2)^{3/2}}, \\frac{z}{(x^2 + y^2 + z^2)^{3/2}}\\right)$ in Cartesian coordinates and express your answer in terms of $x$, $y$, and $z$. Then, find the value of the divergence at the point $(1... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b8cf4d7272fd0ac44363e6f557468eaef6347d69cf591b738c651ca1f9182093 | Calculate the double integral $\iint\limits_R {e^{x + 2y} \, dy \, dx}$, where $R$ is the region defined by $\left| x \right| + \left| y \right| = 1$. | [
"\\(\\frac{2}{3}(e^2+e^{-2})-\\frac{1}{6}(e+7e^{-1})\\)"
] | [
"Calculate the double integral $\\iint\\limits_R {e^{x + 2y} \\, dy \\, dx}$, where $R$ is the region defined by $\\left| x \\right| + \\left| y \\right| = 1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 8
} | 13 | 0 | 0 |
omr-ws-ad132c142a6d66ee7fc57ed3fe9768ee964860f3b4915b6500a325ed2fce4f19 | Calculate the double integral $\int\int\frac{dxdy}{1-xy(1-x)}$ where $0\le x\le 1$ and $0\leq y \leq 1$. | [
"\\(\\frac{\\pi^2}{9}\\)"
] | [
"Calculate the double integral $\\int\\int\\frac{dxdy}{1-xy(1-x)}$ where $0\\le x\\le 1$ and $0\\leq y \\leq 1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
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