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-1cc6708eb565cf8508a83f53e4413049a8116e43c0f9559685868b2b937a08da | Ascertain the following limit without using l'Hospital's rule: \[ L = \lim_{x \to \infty} x \left( \sqrt[p]{x^p + ax^{p-1} + bx^{p-2} + \ldots} + \sqrt[p]{x^p - ax^{p-1} + cx^{p-2} + \ldots} - 2x \right) \] | [
"\\(\\frac{b+c}{p} - \\frac{a^2(p-1)}{p^2}\\)"
] | [
"Ascertain the following limit without using l'Hospital's rule:\n\n\\[ L = \\lim_{x \\to \\infty} x \\left( \\sqrt[p]{x^p + ax^{p-1} + bx^{p-2} + \\ldots} + \\sqrt[p]{x^p - ax^{p-1} + cx^{p-2} + \\ldots} - 2x \\right) \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-99f0919301f28a42b1529a30433320b2c812d630fb6c83efeb352c246adfc3ca | Ascertain the indefinite integral $\int \frac{2x^3+1}{(x^3-1)\sqrt{x^6-2x^3+x^2+1}}\ dx$ on the interval $(1,\infty)$. | [
"\\(\\ln \\frac{\\sqrt{x^6-2x^3+x^2+1}-x}{x^3-1} + C\\)"
] | [
"Ascertain the indefinite integral $\\int \\frac{2x^3+1}{(x^3-1)\\sqrt{x^6-2x^3+x^2+1}}\\ dx$ on the interval $(1,\\infty)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 5
} | 8 | 0 | 0 |
omr-ws-5257f3bc503072ce5a4e814f1b8c9cf23d9f8eda22f59bcc5d1c5c0a6820a551 | Ascertain the indefinite integral $\int \frac{x^2}{(x^2-4)\sin x+4x\cos x}\ dx$ on the interval $(0,\pi)$. | [
"\\ln\\left| (x^2 - 4)\\sin x + 4x \\cos x \\right| + C"
] | [
"Ascertain the indefinite integral $\\int \\frac{x^2}{(x^2-4)\\sin x+4x\\cos x}\\ dx$ on the interval $(0,\\pi)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-11f1f1ba5573783bc119deebcd1ce3b61234bcc54e3c531b66d2fd811dce5c9b | Ascertain the limits of the following sequence \( x_n \), where \( n \in \mathbb{N}^* \): 15. \( x_1 = 1 \), \( e^{1 + x_{n+1}} = 1 + \ln(2 + x_n) \). | [
"-1"
] | [
"Ascertain the limits of the following sequence \\( x_n \\), where \\( n \\in \\mathbb{N}^* \\):\n15. \\( x_1 = 1 \\), \\( e^{1 + x_{n+1}} = 1 + \\ln(2 + x_n) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-798f373dc72502db817fdbc7a25dca5df44c2a9536fc5431b01cb7978dc511e4 | Ascertain the limits of the following sequence \( x_n \), where \( n \in \mathbb{N}^* \): 16. \( x_n \in (0, \pi) \), \( \sin x_n + \cos x_n = 1 \). | [
"\\dfrac{\\pi}{2}"
] | [
"Ascertain the limits of the following sequence \\( x_n \\), where \\( n \\in \\mathbb{N}^* \\):\n16. \\( x_n \\in (0, \\pi) \\), \\( \\sin x_n + \\cos x_n = 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ecef36bb73b649894998880f8f9e3e2027f8ff49881fbb7f8bea5494075d1b10 | Ascertain the limits of the following sequence \( x_n \), where \( n \in \mathbb{N}^* \): 6. \( x_1 > 0 \), \( x_{n+1} = e^{x_n - 1} \). | [
"1"
] | [
"Ascertain the limits of the following sequence \\( x_n \\), where \\( n \\in \\mathbb{N}^* \\):\n6. \\( x_1 > 0 \\), \\( x_{n+1} = e^{x_n - 1} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-afb8fa187a3885a1b1b4f7ae294735c754d8a38e525995d3c2ccb05408ba076e | Ascertain the limits of the following sequences \( x_n \) and \( y_n \), where \( n \in \mathbb{N}^* \): 10. \( x_1 \in (0,1) \), \( x_{n+1} = x_n (1 - x_n)^a \) with \( a > 0 \), \( y_n = n x_n \). | [
"\\dfrac{1}{a}"
] | [
"Ascertain the limits of the following sequences \\( x_n \\) and \\( y_n \\), where \\( n \\in \\mathbb{N}^* \\):\n10. \\( x_1 \\in (0,1) \\), \\( x_{n+1} = x_n (1 - x_n)^a \\) with \\( a > 0 \\), \\( y_n = n x_n \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f087b3040bcc0ffa39342ebd221792d34447c5f752f0a46448ca50db29eab2b0 | Ascertain the limits of the following sequences \( x_n \) and \( y_n \), where \( n \in \mathbb{N}^* \): 11. \( x_1 \in (0,1) \), \( x_{n+1} = x_n - x_n^2 + x_n^3 - x_n^4 \), \( y_n = n x_n \). | [
"1"
] | [
"Ascertain the limits of the following sequences \\( x_n \\) and \\( y_n \\), where \\( n \\in \\mathbb{N}^* \\):\n11. \\( x_1 \\in (0,1) \\), \\( x_{n+1} = x_n - x_n^2 + x_n^3 - x_n^4 \\), \\( y_n = n x_n \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bf4d6bd139bba7ad66789316757f959d5c7b915c38bbffccffd8c875ce0073d0 | Ascertain the limits of the following sequences \( x_n \) and \( y_n \), where \( n \in \mathbb{N}^* \): 12. \( x_1 > 0 \), \( x_{n+1} = |x_n - n| \), \( y_n = \frac{x_n}{n} \). | [
"\\dfrac{1}{2}"
] | [
"Ascertain the limits of the following sequences \\( x_n \\) and \\( y_n \\), where \\( n \\in \\mathbb{N}^* \\):\n12. \\( x_1 > 0 \\), \\( x_{n+1} = |x_n - n| \\), \\( y_n = \\frac{x_n}{n} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4a121b9252c8f9e4fbd0aef1aed21a3085b5edfbe6a3d3886d6e25a70d0a7ef1 | Ascertain the limits of the following sequences \( x_n \) and \( y_n \), where \( n \in \mathbb{N}^* \): 13. \( x_1 \in \left(0, \frac{\pi}{2}\right) \), \( x_{n+1} = x_n \cos x_n \), \( y_n = x_n \sqrt{n} \). | [
"1"
] | [
"Ascertain the limits of the following sequences \\( x_n \\) and \\( y_n \\), where \\( n \\in \\mathbb{N}^* \\):\n13. \\( x_1 \\in \\left(0, \\frac{\\pi}{2}\\right) \\), \\( x_{n+1} = x_n \\cos x_n \\), \\( y_n = x_n \\sqrt{n} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-67bbe1eb6cf4d839ba654f82c15af209587a90bb523e4c65b532447dda158e10 | Ascertain the limits of the following sequences \( x_n \) and \( y_n \), where \( n \in \mathbb{N}^* \): 14. \( x_1 > 0 \), \( x_{n+1} = x_n a^{x_n} \) with \( a \in (0,1) \), \( y_n = n x_n \). | [
"-\\dfrac{1}{\\ln a}"
] | [
"Ascertain the limits of the following sequences \\( x_n \\) and \\( y_n \\), where \\( n \\in \\mathbb{N}^* \\):\n14. \\( x_1 > 0 \\), \\( x_{n+1} = x_n a^{x_n} \\) with \\( a \\in (0,1) \\), \\( y_n = n x_n \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f3c0d971ba94c0b1ca02c3aaa8443c3b199b99ebc4cbb821240189a5420d538 | Ascertain the limits of the following sequences \( x_n \) and \( y_n \), where \( n \in \mathbb{N}^* \): 19. \( x_1 > 1 \), \( x_{n+1} = \sqrt[3]{3 x_n - 2} \), \( y_n = n (x_n - 1) \). | [
"1"
] | [
"Ascertain the limits of the following sequences \\( x_n \\) and \\( y_n \\), where \\( n \\in \\mathbb{N}^* \\):\n19. \\( x_1 > 1 \\), \\( x_{n+1} = \\sqrt[3]{3 x_n - 2} \\), \\( y_n = n (x_n - 1) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6af15759fd868b5e007acba9c0b3aa7cb091c2122df976d1e55f7fc9ed19ba2e | Ascertain the limits of the following sequences \( x_n \) and \( y_n \), where \( n \in \mathbb{N}^* \): 2. \( x_1 > 0 \), \( x_{n+1} = \ln(1 + x_n) \), \( y_n = n x_n \). | [
"2"
] | [
"Ascertain the limits of the following sequences \\( x_n \\) and \\( y_n \\), where \\( n \\in \\mathbb{N}^* \\):\n2. \\( x_1 > 0 \\), \\( x_{n+1} = \\ln(1 + x_n) \\), \\( y_n = n x_n \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-92e4466a6a9b050aada40a9f22dbd4f11bebd7072e6b70d2c23577f51f241ed0 | Ascertain the limits of the following sequences \( x_n \) and \( y_n \), where \( n \in \mathbb{N}^* \): 3. \( x_1 \in (0, \pi) \), \( x_{n+1} = \sin x_n \), \( y_n = x_n \sqrt{n} \). | [
"\\sqrt{3}"
] | [
"Ascertain the limits of the following sequences \\( x_n \\) and \\( y_n \\), where \\( n \\in \\mathbb{N}^* \\):\n3. \\( x_1 \\in (0, \\pi) \\), \\( x_{n+1} = \\sin x_n \\), \\( y_n = x_n \\sqrt{n} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b0c2be6bb487bfdf06b8e4900b17bcf178d791df0e50fe648abcd8429067e8df | Ascertain the limits of the following sequences \( x_n \) and \( y_n \), where \( n \in \mathbb{N}^* \): 4. \( x_1 \in \mathbb{R} \), \( x_{n+1} = \ln(1 + \arctan x_n) \), \( y_n = n x_n \). | [
"2"
] | [
"Ascertain the limits of the following sequences \\( x_n \\) and \\( y_n \\), where \\( n \\in \\mathbb{N}^* \\):\n4. \\( x_1 \\in \\mathbb{R} \\), \\( x_{n+1} = \\ln(1 + \\arctan x_n) \\), \\( y_n = n x_n \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-46ecfe1266e333a33b19ee4c48eb017b4c971be1af5d10abe5738789bc1ecf93 | Ascertain the limits of the following sequences \( x_n \) and \( y_n \), where \( n \in \mathbb{N}^* \): 7. \( x_1 > 0 \), \( x_{n+1} + x_{n+1}^3 = x_n \), \( y_n = x_n \sqrt{n} \). | [
"\\dfrac{\\sqrt{2}}{2}"
] | [
"Ascertain the limits of the following sequences \\( x_n \\) and \\( y_n \\), where \\( n \\in \\mathbb{N}^* \\):\n7. \\( x_1 > 0 \\), \\( x_{n+1} + x_{n+1}^3 = x_n \\), \\( y_n = x_n \\sqrt{n} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bc8e477e9a8996a0354b44cc042526f38d324879ff49d8242721eb2b812bbf6f | Ascertain the limits of the following sequences \( x_n \) and \( y_n \), where \( n \in \mathbb{N}^* \): 8. \( x_1 > 0 \), \( x_{n+1} = \arctan x_n \), \( y_n = x_n \sqrt{n} \). | [
"\\dfrac{\\sqrt{6}}{2}"
] | [
"Ascertain the limits of the following sequences \\( x_n \\) and \\( y_n \\), where \\( n \\in \\mathbb{N}^* \\):\n8. \\( x_1 > 0 \\), \\( x_{n+1} = \\arctan x_n \\), \\( y_n = x_n \\sqrt{n} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e8de9ee53c696778b3c2411d72c9a68760cbc67c6d5805ea7a65c7c951fb9506 | Ascertain the limits of the following sequences \( x_n \) and \( y_n \), where \( n \in \mathbb{N}^* \): 9. \( x_1 \in \mathbb{R} \), \( x_{n+1} = \left(1 + \frac{x_n}{a}\right)^a - 1 \) with \( 0 < a < 1 \), \( y_n = n x_n \). | [
"\\dfrac{2a}{1 - a}"
] | [
"Ascertain the limits of the following sequences \\( x_n \\) and \\( y_n \\), where \\( n \\in \\mathbb{N}^* \\):\n9. \\( x_1 \\in \\mathbb{R} \\), \\( x_{n+1} = \\left(1 + \\frac{x_n}{a}\\right)^a - 1 \\) with \\( 0 < a < 1 \\), \\( y_n = n x_n \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-35b3ffd9913d68f797638d9b31a61563dc076ecb6088267d07cd27e30bf20a29 | Ascertain the limits of the following sequences \( x_n \), \( y_n \), and \( z_n \), where \( n \in \mathbb{N}^* \): 1. \( x_1 \in (0,1) \), \( x_{n+1} = x_n (1 - x_n) \), \( y_n = n x_n \), \( z_n = \sum_{k=1}^n x_k^2 \). | [
"x_1"
] | [
"Ascertain the limits of the following sequences \\( x_n \\), \\( y_n \\), and \\( z_n \\), where \\( n \\in \\mathbb{N}^* \\):\n1. \\( x_1 \\in (0,1) \\), \\( x_{n+1} = x_n (1 - x_n) \\), \\( y_n = n x_n \\), \\( z_n = \\sum_{k=1}^n x_k^2 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-864a7708c999bd72cc51f48682cd37ba14a9640f44bdf25e4ea30cbd5afc0149 | Ascertain the limits of the following sequences \( x_n \), \( y_n \), and \( z_n \), where \( n \in \mathbb{N}^* \): 17. \( x_1 = 1 \), \( x_{n+1} = \sqrt{1 + n x_n} \), \( y_n = \frac{x_n}{n} \), \( z_n = n - x_n \). | [
"2"
] | [
"Ascertain the limits of the following sequences \\( x_n \\), \\( y_n \\), and \\( z_n \\), where \\( n \\in \\mathbb{N}^* \\):\n17. \\( x_1 = 1 \\), \\( x_{n+1} = \\sqrt{1 + n x_n} \\), \\( y_n = \\frac{x_n}{n} \\), \\( z_n = n - x_n \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bbb182f65ddb0efabf72e4937d22e9b6f1eb8c005e73fe285c45c481e359b380 | Ascertain the limits of the following sequences \( x_n \), \( y_n \), and \( z_n \), where \( n \in \mathbb{N}^* \): 18. \( x_1 = 0 \), \( x_{n+1} = \sqrt{n - x_n} \), \( y_n = \frac{x_n}{\sqrt{n}} \), \( z_n = x_n - \sqrt{n} \). | [
"-\\frac{1}{2}"
] | [
"Ascertain the limits of the following sequences \\( x_n \\), \\( y_n \\), and \\( z_n \\), where \\( n \\in \\mathbb{N}^* \\):\n18. \\( x_1 = 0 \\), \\( x_{n+1} = \\sqrt{n - x_n} \\), \\( y_n = \\frac{x_n}{\\sqrt{n}} \\), \\( z_n = x_n - \\sqrt{n} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a86fd3b15d9d9f0f4b77ddb7a68cb26f99a848cac2cce176e794aaf23207ca01 | Ascertain the limits of the following sequences \( x_n \), \( y_n \), and \( z_n \), where \( n \in \mathbb{N}^* \): 20. \( x_1 \in (0,1) \), \( x_{n+1} = \frac{\sqrt{x_n}}{\sqrt{x_n} + \sqrt{1 - x_n}} \), \( y_n = \frac{x_n}{1 - x_n} \). | [
"1"
] | [
"Ascertain the limits of the following sequences \\( x_n \\), \\( y_n \\), and \\( z_n \\), where \\( n \\in \\mathbb{N}^* \\):\n20. \\( x_1 \\in (0,1) \\), \\( x_{n+1} = \\frac{\\sqrt{x_n}}{\\sqrt{x_n} + \\sqrt{1 - x_n}} \\), \\( y_n = \\frac{x_n}{1 - x_n} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-17f884c05bff388ce999a2a47e5c9ea6f26f8e76fddb231fe6d9d7dce47f6a4e | Ascertain the limits of the following sequences \( x_n \), \( y_n \), and \( z_n \), where \( n \in \mathbb{N}^* \): 21. \( x_1 = 1 \), \( x_{n+1} = x_n + \frac{1}{x_n} \), \( y_n = \frac{x_n}{\sqrt{2n}} \), \( z_n = \frac{1}{n \sqrt{n}} \cdot \sum_{k=1}^n x_k \). | [
"\\dfrac{2\\sqrt{2}}{3}"
] | [
"Ascertain the limits of the following sequences \\( x_n \\), \\( y_n \\), and \\( z_n \\), where \\( n \\in \\mathbb{N}^* \\):\n21. \\( x_1 = 1 \\), \\( x_{n+1} = x_n + \\frac{1}{x_n} \\), \\( y_n = \\frac{x_n}{\\sqrt{2n}} \\), \\( z_n = \\frac{1}{n \\sqrt{n}} \\cdot \\sum_{k=1}^n x_k \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6160e7423997efaa0a15888a8ed8c054e5371205db98e5de042027bbdb648fe3 | Ascertain the limits of the following sequences \( x_n \), \( y_n \), and \( z_n \), where \( n \in \mathbb{N}^* \): 5. \( x_1 \neq 0 \), \( x_{n+1} = x_n + e^{-x_n} \), \( y_n = \frac{x_n}{\ln n} \), \( z_n = \frac{x_n}{n} \). | [
"0"
] | [
"Ascertain the limits of the following sequences \\( x_n \\), \\( y_n \\), and \\( z_n \\), where \\( n \\in \\mathbb{N}^* \\):\n5. \\( x_1 \\neq 0 \\), \\( x_{n+1} = x_n + e^{-x_n} \\), \\( y_n = \\frac{x_n}{\\ln n} \\), \\( z_n = \\frac{x_n}{n} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2e96d4c87997764f188f9b32fef03d0e26e4f97c633784147c25c7c8b9c3a0b8 | Ascertain the maximum area of an isosceles trapezoid $ABCD$ which is inscribed in a semicircle with the diameter $[AB]$. | [
"\\dfrac{3\\sqrt{3}}{16} AB^2"
] | [
"Ascertain the maximum area of an isosceles trapezoid $ABCD$ which is inscribed in a semicircle with the diameter $[AB]$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e8a2a92468c2fb8f257cdb4a7bf160c699f25bf65aeb9d12dca2fb4161e1b121 | Ascertain the pairs $(x,y)$ of integer numbers for which $(2x-3)\cdot y^{2}+(3x-1)\cdot y+(x-2)=0$. | [
"\\((2, -5), (4, -2), (2, 0), (1, 1)\\)"
] | [
"Ascertain the pairs $(x,y)$ of integer numbers for which $(2x-3)\\cdot y^{2}+(3x-1)\\cdot y+(x-2)=0$."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-493d7c85075430f8d160c895f8cab9d43c77c1d174e3defebc6d94f2c4745265 | Ascertain the ratio between the radius and the altitude of a cylinder so that its surface area is minimum when its volume is constant (without using calculus of derivatives). | [
"\\(1:2\\)"
] | [
"Ascertain the ratio between the radius and the altitude of a cylinder so that its surface area is minimum when its volume is constant (without using calculus of derivatives)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-48cccee86cbf974d30f64a2860e617db99f9055f0562ac0980ac40a9bd409893 | Ashley has a deck of 45 cards, with 5 cards labeled with each integer from 1 to 9. She randomly draws 20 cards from the deck with replacement and computes the sum of the odd numbers on the cards she drew. The expected value of this sum can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. ... | [
"509"
] | [
"Ashley has a deck of 45 cards, with 5 cards labeled with each integer from 1 to 9. She randomly draws 20 cards from the deck with replacement and computes the sum of the odd numbers on the cards she drew. The expected value of this sum can be written as $\\frac{m}{n}$ for relatively prime positive integers $m$ and... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-85556e09514ffb8915d58c95fe2198c77b60b52cb911ff5b070d208bee4aa90e | Ashley, Betty, Carlos, Dick, and Elgin went shopping with a total of $ \$56$. The absolute differences between their amounts are as follows: $|A - B| = 19$, $|B - C| = 7$, $|C - D| = 5$, $|D - E| = 4$, and $|E - A| = 11$. How much did Elgin have? | [
"10"
] | [
"Ashley, Betty, Carlos, Dick, and Elgin went shopping with a total of $ \\$56$. The absolute differences between their amounts are as follows: $|A - B| = 19$, $|B - C| = 7$, $|C - D| = 5$, $|D - E| = 4$, and $|E - A| = 11$. How much did Elgin have?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 16
} | 39 | 0 | 0 |
omr-ws-171c728599dcf0552aa18406a3a0d0ee2a85c9fd62e0937f89cf158d18d4d9cd | Ashley, Betty, Carlos, Dick, and Elgin went shopping with a total of $56. The absolute differences in their spending amounts are as follows: $|A - B| = 19$, $|B - C| = 7$, $|C - D| = 5$, $|D - E| = 4$, and $|E - A| = 11$. How much did Elgin have? | [
"$10"
] | [
"Ashley, Betty, Carlos, Dick, and Elgin went shopping with a total of $56. The absolute differences in their spending amounts are as follows: $|A - B| = 19$, $|B - C| = 7$, $|C - D| = 5$, $|D - E| = 4$, and $|E - A| = 11$. How much did Elgin have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 18
} | 40 | 0 | 0 |
omr-ws-f483804296c3202c79de715be8f33c32293e75c475ba02c56d2daeb6f35f83fe | Ashok's mother usually picks him up from school at 3 PM. Today, she was not there, so he began walking home at a rate of 3 miles per hour. After 24 minutes of walking, he met his mother who drove him the rest of the way home. They arrived 22 minutes later than usual. What is his mother's average driving speed? | [
"36"
] | [
"Ashok's mother usually picks him up from school at 3 PM. Today, she was not there, so he began walking home at a rate of 3 miles per hour. After 24 minutes of walking, he met his mother who drove him the rest of the way home. They arrived 22 minutes later than usual. What is his mother's average driving speed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-1b348846310f6a11ac2a56c9362363072dae81029e2b6eeae540bd66e64eced3 | Ashwin is biking along the Saratoga Creek Trail one day, as his father walks behind him. He starts at the $0$ mile sign. Once Ashwin bikes to the $0.75$ mile sign, he turns around and bikes back to his father. After reaching his father, Ashwin bikes to the $0.75$ mile sign again, and repeats this process, biking betwee... | [
"9"
] | [
"Ashwin is biking along the Saratoga Creek Trail one day, as his father walks behind him. He starts at the $0$ mile sign. Once Ashwin bikes to the $0.75$ mile sign, he turns around and bikes back to his father. After reaching his father, Ashwin bikes to the $0.75$ mile sign again, and repeats this process, biking b... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c8501a755dba0c8a75df94eb3fdf1e384a006bfda50c79652627dd90e91cf034 | Aside from the values \( w = -4 \), \( x = -8 \), \( y = -1 \), and \( z = 3 \), what are the other values of \( w \), \( x \), \( y \), and \( z \) such that \[ \begin{bmatrix}-4 & -8 \\ -1 & 3\end{bmatrix}^2 = \begin{bmatrix}w & x \\ y & z\end{bmatrix}^2? \] | [
"\\( w = \\frac{44}{9}, x = \\frac{8}{9}, y = \\frac{1}{9}, z = \\frac{37}{9} \\) or \\( w = 4, x = 8, y = 1, z = -3 \\) or \\( w = -\\frac{44}{9}, x = -\\frac{8}{9}, y = -\\frac{1}{9}, z = -\\frac{37}{9} \\)"
] | [
"Aside from the values \\( w = -4 \\), \\( x = -8 \\), \\( y = -1 \\), and \\( z = 3 \\), what are the other values of \\( w \\), \\( x \\), \\( y \\), and \\( z \\) such that\n\\[\n\\begin{bmatrix}-4 & -8 \\\\ -1 & 3\\end{bmatrix}^2 = \\begin{bmatrix}w & x \\\\ y & z\\end{bmatrix}^2?\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 29
} | 40 | 0 | 0 |
omr-ws-d52f4c4e1a681d8835475c88b660ad575304f04fcf253b00d6e6f51e22a05bdd | Asli will distribute 100 candies among her brother and 18 friends of him. Asli splits the friends into several groups and distributes all the candies into these groups such that each child in a group takes the same number of candies, and this number is the largest possible. Asli's brother then takes the remaining candi... | [
"16"
] | [
"Asli will distribute 100 candies among her brother and 18 friends of him. Asli splits the friends into several groups and distributes all the candies into these groups such that each child in a group takes the same number of candies, and this number is the largest possible. Asli's brother then takes the remaining ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 5,
"tir": 4
} | 39 | 0 | 0 |
omr-ws-085f0b3ede72ff4a139ad35e17c46f90de8557c66ab437f7c34d5a91e07749ca | Aslı distributes 1000 balls into 30 boxes. Aslı and Zehra take turns removing one ball from any box, starting with Aslı. The player who removes the last ball from a box wins that box. What is the maximum number of boxes Aslı can guarantee to win regardless of Zehra's moves? | [
"15"
] | [
"Aslı distributes 1000 balls into 30 boxes. Aslı and Zehra take turns removing one ball from any box, starting with Aslı. The player who removes the last ball from a box wins that box. What is the maximum number of boxes Aslı can guarantee to win regardless of Zehra's moves?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-57336186f4693b450001aeea65fd994d18e64b6011a9f7c444b1239ad8035fe6 | Assign 8 distinct numbers from the set $\{0, 1, 2, 3, \ldots, 11, 12\}$ to each vertex of a cube such that the sum of the numbers assigned to the vertices of every edge is even. Calculate the sum of all possible values of the number assigned to a specific vertex, say the vertex labeled $A$. | [
"0"
] | [
"Assign 8 distinct numbers from the set $\\{0, 1, 2, 3, \\ldots, 11, 12\\}$ to each vertex of a cube such that the sum of the numbers assigned to the vertices of every edge is even. Calculate the sum of all possible values of the number assigned to a specific vertex, say the vertex labeled $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-36689d4fa099eb0a6a5770dedc43e211507beb1ead321e43c4a6b3dd423e01ff | Assign a finite value to the series \(\sum_{n=1}^\infty (-1)^n \frac{n}{n+1}\). | [
"\\(\\frac{1}{2} - \\ln 2\\)"
] | [
"Assign a finite value to the series \\(\\sum_{n=1}^\\infty (-1)^n \\frac{n}{n+1}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 26
} | 49 | 0 | 0 |
omr-ws-9e45a4ed2c92dc0483ec330abb0c078b545cb30bfeb4d0def83b50b95c8e6a09 | Assign a real number to each point in the plane such that for every triangle, the number assigned to the incenter is the arithmetic mean of the numbers assigned to the three vertices. Let $A$, $B$, and $C$ be three non-collinear points with assigned numbers $a$, $b$, and $c$ respectively. If $I$ is the incenter of $\tr... | [
"\\dfrac{a + b + c}{3}"
] | [
"Assign a real number to each point in the plane such that for every triangle, the number assigned to the incenter is the arithmetic mean of the numbers assigned to the three vertices. Let $A$, $B$, and $C$ be three non-collinear points with assigned numbers $a$, $b$, and $c$ respectively. If $I$ is the incenter of... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bf36003296bb6be089a8b82992d11f9868afa8449744749b299856f7c7c70d44 | Assign distinct numbers to the vertices of a triangle such that the sum of the numbers on any two adjacent vertices is a square number. | [
"\\(6, 19, 30\\)"
] | [
"Assign distinct numbers to the vertices of a triangle such that the sum of the numbers on any two adjacent vertices is a square number."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 15
} | 15 | 0 | 0 |
omr-ws-bf3255355c096c621c50fa0854d3879305e65e9f7fceda4620f6f4062bc125bc | Assign each positive integer one of four colors: blue, red, purple, yellow. Let \(N\) be the smallest positive integer such that among the integers from 1 to \(N\), there exist four distinct positive integers \(a\), \(b\), \(c\), and \(d\) such that \(ab = cd\) and \(a\), \(b\), \(c\), and \(d\) are all assigned the sa... | [
"243"
] | [
"Assign each positive integer one of four colors: blue, red, purple, yellow. Let \\(N\\) be the smallest positive integer such that among the integers from 1 to \\(N\\), there exist four distinct positive integers \\(a\\), \\(b\\), \\(c\\), and \\(d\\) such that \\(ab = cd\\) and \\(a\\), \\(b\\), \\(c\\), and \\(d... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-42af7557c9d77500e85b06ec30190901650999a224f64d55598a47208debf5e9 | Assign eight numbers which sum to one to the vertices of a cube. Assign to each edge the product of the numbers at its vertices. What is the maximal value the sum of the numbers assigned to the edges can take? | [
"\\(\\frac{1}{4}\\)"
] | [
"Assign eight numbers which sum to one to the vertices of a cube. Assign to each edge the product of the numbers at its vertices. What is the maximal value the sum of the numbers assigned to the edges can take?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 19,
"tir": 9
} | 28 | 0 | 0 |
omr-ws-9f3159fb8a7a1778b39869c6bf87e098d04d91346c2dc1ab52d7c33bdf84e626 | Assign either "+" or "-" to each member of the sequence of consecutive integers from 1 to 2005 and then add them all. What is the smallest possible non-negative value of the sum of all signed members? | [
"1"
] | [
"Assign either \"+\" or \"-\" to each member of the sequence of consecutive integers from 1 to 2005 and then add them all. What is the smallest possible non-negative value of the sum of all signed members?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0233bf4685574aba8f379d70f656349756ecd862e650e7e1e306672638179685 | Assign one of seven distinct colors to each of the numbers $1, 2, 3, \ldots, 2013$ such that all colors are used and for any distinct numbers $a, b, c$ of the same color, $2014 \nmid abc$ and the remainder when $abc$ is divided by $2014$ is of the same color as $a, b, c$. Find the number of valid ways to assign the col... | [
"5040"
] | [
"Assign one of seven distinct colors to each of the numbers $1, 2, 3, \\ldots, 2013$ such that all colors are used and for any distinct numbers $a, b, c$ of the same color, $2014 \\nmid abc$ and the remainder when $abc$ is divided by $2014$ is of the same color as $a, b, c$. Find the number of valid ways to assign ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-94b4ff160096178e2bbcd175c95c15fe90fba97c6235335806d8851e2974d1c4 | Assign positive real numbers to the vertices of a convex $n$-gon such that the distance between any two vertices is strictly less than the difference of their respective numbers. If the maximal assigned number is $M$, find the maximum possible value of the perimeter of the $n$-gon divided by $M$. | [
"2"
] | [
"Assign positive real numbers to the vertices of a convex $n$-gon such that the distance between any two vertices is strictly less than the difference of their respective numbers. If the maximal assigned number is $M$, find the maximum possible value of the perimeter of the $n$-gon divided by $M$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6c3cc2b09758a9382ff51ff0a8b24e113bb61bd8c3c98043b3390b6dabc036b3 | Assign the values 2, 3, -2, -3 to \( u, v, x, \) and \( y \) such that the expression \( uv - xy \) has a maximum value. | [
"31"
] | [
"Assign the values 2, 3, -2, -3 to \\( u, v, x, \\) and \\( y \\) such that the expression \\( uv - xy \\) has a maximum value."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3a05746e02945c70ea21a5a9296b45db72131b42d9c7f806ed59e1aa323af184 | Assign the values 2, 3, -2, -3 to \( u, v, x, \) and \( y \) such that the expression \( uv - xy \) has a minimum value. | [
"-31"
] | [
"Assign the values 2, 3, -2, -3 to \\( u, v, x, \\) and \\( y \\) such that the expression \\( uv - xy \\) has a minimum value."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e312954ed552dfc7c484b4c36b4087e257d9a051ec00db64d3dfc53a9c2e0501 | Assign the values 2, 3, -2, -3 to \( u, v, x, y \) such that the expression \( uv - xy \) has a maximum value. | [
"13"
] | [
"Assign the values 2, 3, -2, -3 to \\( u, v, x, y \\) such that the expression \\( uv - xy \\) has a maximum value."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab964ba7a3b1e9a83b7d14f5d54f04dd06b25a8d515630a491982b1658af0912 | Assign the values 2, 3, -2, -3 to \( u, v, x, y \) such that the expression \( uv - xy \) has a minimum value. | [
"-10"
] | [
"Assign the values 2, 3, -2, -3 to \\( u, v, x, y \\) such that the expression \\( uv - xy \\) has a minimum value."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dec8f7b83c6a13004ddbaf739c43dbfc6fcd3f69608be7ef6b1ac6792fd43179 | Assign to each real number \( x \) a finite set \( F(x) \) (which does not contain \( x \) and is a subset of \( \mathbb{R} \)). A set \( S \) is called independent if \( x \in F(y) \) does not hold for any elements \( x, y \) in \( S \). Let \( S \) be an independent set. Find the maximum possible cardinality of \( S ... | [
"\\mathfrak{c}"
] | [
"Assign to each real number \\( x \\) a finite set \\( F(x) \\) (which does not contain \\( x \\) and is a subset of \\( \\mathbb{R} \\)). A set \\( S \\) is called independent if \\( x \\in F(y) \\) does not hold for any elements \\( x, y \\) in \\( S \\). Let \\( S \\) be an independent set. Find the maximum poss... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-301f2394b80524481c9ea542c84521a4f1039dc99c704e8cd59c091e543b86cf | Assign values to $a$, $b$, and $c$ so that the double integral $$\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{\mathrm dx \mathrm dy}{xy} \exp(-(ax^2+2bxy+cy^2))$$ converges. | [
"a > 0 \\text{ and } ac > b^2"
] | [
"Assign values to $a$, $b$, and $c$ so that the double integral \n$$\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\frac{\\mathrm dx \\mathrm dy}{xy} \\exp(-(ax^2+2bxy+cy^2))$$\nconverges."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-aae4ff7644426c2831f00c91e6462216e1c1e6f8ff80d5f72ccd07afb109876b | Assign variables to the unknown quantities and write a system of inequalities that model the given restrictions for scheduling appointments at a veterinarian's office. Let \( x \) be the number of routine office visits and \( y \) be the number of surgeries. The office is open at least 5 hours a day but will not exceed... | [
"\\( \\frac{1}{3}x + y \\geq 5 \\), \\( \\frac{1}{3}x + y \\leq 8 \\), \\( x \\geq 3y \\), \\( x \\geq 0 \\), \\( y \\geq 0 \\)"
] | [
"Assign variables to the unknown quantities and write a system of inequalities that model the given restrictions for scheduling appointments at a veterinarian's office. Let \\( x \\) be the number of routine office visits and \\( y \\) be the number of surgeries. The office is open at least 5 hours a day but will n... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-a60fb210cfb315557d6766bc84a164a541ad8ed1a951017669281df0c2f3a906 | Assume $1+2+\cdots +n$ is a three-digit number with all digits equal. Then $n$ is divisible by which of the following numbers? | [
"6"
] | [
"Assume $1+2+\\cdots +n$ is a three-digit number with all digits equal. Then $n$ is divisible by which of the following numbers?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-771589c7cb90fa934c8dc3c429448eb71304b3d764d9410ddbd0add83661b6aa | Assume $A = \{a_1, a_2, \ldots, a_{12}\}$ is a set of positive integers such that for each positive integer $n \leq 2500$, there is a subset $S$ of $A$ whose sum of elements is $n$. If $a_1 < a_2 < \ldots < a_{12}$, what is the smallest possible value of $a_1$? | [
"1"
] | [
"Assume $A = \\{a_1, a_2, \\ldots, a_{12}\\}$ is a set of positive integers such that for each positive integer $n \\leq 2500$, there is a subset $S$ of $A$ whose sum of elements is $n$. If $a_1 < a_2 < \\ldots < a_{12}$, what is the smallest possible value of $a_1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-cfc88c1358645275df1f5e5d86290dfb67d1b77228e3631635166bf9104c363b | Assume $A, B, C$ are three collinear points with $B \in [AC]$. Suppose $AA'$ and $BB'$ are parallel lines such that $A'$, $B'$, and $C$ are not collinear. Let $O_1$ be the circumcenter of the circle passing through $A$, $A'$, and $C$, and $O_2$ be the circumcenter of the circle passing through $B$, $B'$, and $C$. If th... | [
"\\(30^\\circ\\) or \\(150^\\circ\\)"
] | [
"Assume $A, B, C$ are three collinear points with $B \\in [AC]$. Suppose $AA'$ and $BB'$ are parallel lines such that $A'$, $B'$, and $C$ are not collinear. Let $O_1$ be the circumcenter of the circle passing through $A$, $A'$, and $C$, and $O_2$ be the circumcenter of the circle passing through $B$, $B'$, and $C$.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 5,
"tir": 5
} | 13 | 0 | 0 |
omr-ws-4460929a1de7bf025e0f4f4051bc2dffccf42f9a37bd19544b2db3de5839b650 | Assume $G$ is a group of order 180 and that it is simple. Consider two distinct $3$-Sylow subgroups $A$ and $B$. Let $g \in A \cap B$ be a non-trivial element. Define $C(g) = \{h \in G : hg = gh\}$. If $|C(g)|$ is not equal to 18, find the smallest possible value of $|C(g)|$. | [
"36"
] | [
"Assume $G$ is a group of order 180 and that it is simple. Consider two distinct $3$-Sylow subgroups $A$ and $B$. Let $g \\in A \\cap B$ be a non-trivial element. Define $C(g) = \\{h \\in G : hg = gh\\}$. If $|C(g)|$ is not equal to 18, find the smallest possible value of $|C(g)|$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-2fc1ee44d1bb17fa0c06cec653b86e28a5ef3d4b5565ff680469054a471aa31f | Assume $PE$ and $PF$ are tangent to circle $O$, and $AOB$ is a diameter. Let $AF$ intersect $EB$ at point $D$. If $\angle PCD = x^\circ$ and $\angle PCE = y^\circ$, calculate the value of $x + y$. | [
"90"
] | [
"Assume $PE$ and $PF$ are tangent to circle $O$, and $AOB$ is a diameter. Let $AF$ intersect $EB$ at point $D$. If $\\angle PCD = x^\\circ$ and $\\angle PCE = y^\\circ$, calculate the value of $x + y$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-73a991eeda375f26319fcf99a6c7cf660e37fb5f904ac03754013078b21b5cdf | Assume $S$ is a compact convex subset of $\mathbb{R}^n$, and $f: S \to \mathbb{R}$ is a continuous function. Let $P$ be the set of all points in $S$ where $f$ attains its maximum value. If $S$ is defined by the intersection of the unit ball in $\mathbb{R}^3$ centered at the origin and the plane $x + y + z = 1$, and $f(... | [
"0"
] | [
"Assume $S$ is a compact convex subset of $\\mathbb{R}^n$, and $f: S \\to \\mathbb{R}$ is a continuous function. Let $P$ be the set of all points in $S$ where $f$ attains its maximum value. If $S$ is defined by the intersection of the unit ball in $\\mathbb{R}^3$ centered at the origin and the plane $x + y + z = 1$... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1c70725b4032849c4a8f86406df52d677c227d7c32fdc5a471ac62495cb828c2 | Assume $S$ is a non-empty set and $G$ is a group. Let $G^S$ denote the set of all mappings from $S$ to $G$. Find an operation on $G^S$ that will yield a group. | [
"\\((f * g)(x) = f(x) \\circ g(x)\\)"
] | [
"Assume $S$ is a non-empty set and $G$ is a group. Let $G^S$ denote the set of all mappings from $S$ to $G$. Find an operation on $G^S$ that will yield a group."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-ced96a8b87712fa1682ffb73557f6461ccb1311cf24cc323968652683edc26e3 | Assume $\{a_n\}$ and $\{b_n\}$ are nonnegative sequences such that $\sum a_n\leq 1$ and $\sum b_n\leq 1$. For each $n\geq 0$, let $p_n:[c,d]\rightarrow [0,1]$ be a polynomial of degree $n$, where $0<c<d<1$. Given that $$\sum_{n=0}^\infty a_n p_n(z)=\sum_{n=0}^\infty b_n p_n(z)\quad\forall z\in[c,d],$$ find the value of... | [
"0"
] | [
"Assume $\\{a_n\\}$ and $\\{b_n\\}$ are nonnegative sequences such that $\\sum a_n\\leq 1$ and $\\sum b_n\\leq 1$. For each $n\\geq 0$, let $p_n:[c,d]\\rightarrow [0,1]$ be a polynomial of degree $n$, where $0<c<d<1$. Given that $$\\sum_{n=0}^\\infty a_n p_n(z)=\\sum_{n=0}^\\infty b_n p_n(z)\\quad\\forall z\\in[c,d... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e93ebfaa51580813f97c2a1f95048a7e6a93883b615b49e5ef22a6be9fe8bb59 | Assume $c$ is a real number. If there exists $x \in [1,2]$ such that $\max\left\{\left |x+\frac{c}{x}\right|, \left |x+\frac{c}{x} + 2\right|\right\} \geq 5$, find the value range of $c$. | [
"\\( c \\in (-\\infty, -6] \\cup [2, \\infty) \\)"
] | [
"Assume $c$ is a real number. If there exists $x \\in [1,2]$ such that $\\max\\left\\{\\left |x+\\frac{c}{x}\\right|, \\left |x+\\frac{c}{x} + 2\\right|\\right\\} \\geq 5$, find the value range of $c$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 8,
"tir": 27
} | 82 | 0 | 0 |
omr-ws-c49bb9909c305baeb2efca8b166de854861469204355ad6974f17d24e05a436f | Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$, $$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$ Calculate $f(k)$ for $0\le k\le12$. | [
"\\( f(k) = k \\) for \\( 0 \\le k \\le 12 \\)"
] | [
"Assume $f:\\mathbb N_0\\to\\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$,\n$$f\\left(m^2+n^2\\right)=f(m)^2+f(n)^2.$$\nCalculate $f(k)$ for $0\\le k\\le12$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 20
} | 20 | 0 | 0 |
omr-ws-e0b4c0e32fcbb487af0ead8862bcb104ae94d7c92098977ce0ec1427be492c7c | Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$, $$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$ Calculate $f(n)$ for any natural number $n$. | [
"\\( f(n) = n \\)"
] | [
"Assume $f:\\mathbb N_0\\to\\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$,\n$$f\\left(m^2+n^2\\right)=f(m)^2+f(n)^2.$$\nCalculate $f(n)$ for any natural number $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-b8ebf48cc7b7fba92f89a0fdf29a21bfbe581717677ff84a3e51cd6ed8f77809 | Assume $n \in \mathbb{N}$ and $t < \frac{1}{2}(n(n+1))$. Evaluate the integral \[ I_n = \int_0^\infty \frac{\sin(x) \sin(2x) \cdots \sin(nx) \sin(tx)}{x^{n+1}} \, dx \] | [
"\\(\\frac{3\\pi}{4}tn!\\)"
] | [
"Assume $n \\in \\mathbb{N}$ and $t < \\frac{1}{2}(n(n+1))$. Evaluate the integral\n\\[ I_n = \\int_0^\\infty \\frac{\\sin(x) \\sin(2x) \\cdots \\sin(nx) \\sin(tx)}{x^{n+1}} \\, dx \\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3ad615e593ff06963b7688668cadb8e27d7768a8d77e54771197b0421f58b632 | Assume $n$ is a positive integer and $z$ is a complex number. If the imaginary part of $z$ is 164 and $\dfrac{z}{z+n}=4i$, then what is the value of $n$? | [
"697"
] | [
"Assume $n$ is a positive integer and $z$ is a complex number. If the imaginary part of $z$ is 164 and $\\dfrac{z}{z+n}=4i$, then what is the value of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-58ecd92b0c31c246e6bb1bf23a951047e056eb92758020c136b295bcd353b72b | Assume $n$ people are standing in a circle, where $n \geq 4$. The rule is that you cannot pass the ball to the two adjacent people. Person 1 starts with the ball and passes it around. What is the probability that after five passes, the ball comes back to Person 1? Assume every person is distinct. | [
"\\(\\frac{1}{n}\\)"
] | [
"Assume $n$ people are standing in a circle, where $n \\geq 4$. The rule is that you cannot pass the ball to the two adjacent people. Person 1 starts with the ball and passes it around. What is the probability that after five passes, the ball comes back to Person 1? Assume every person is distinct."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d5c99aea67598449034b02255352702ddde8eccb2bff0e2edec3066caacca164 | Assume $x$ and $y$ are real numbers. Find the minimum value of $\sqrt{x^{2}+y^{2}}$ given that $x+y=n$, where $n$ is a constant. | [
"\\(\\frac{n\\sqrt{2}}{2}\\)"
] | [
"Assume $x$ and $y$ are real numbers. Find the minimum value of $\\sqrt{x^{2}+y^{2}}$ given that $x+y=n$, where $n$ is a constant."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 25
} | 39 | 0 | 0 |
omr-ws-ef1d3a2254e952ce83d1d2e78b83ad030718c5d0bca89155164c5d17120e8443 | Assume 4 circles such that any 3 of them have a common intersecting point on the surface of a sphere of radius \( R \) and the center \( O \) of the sphere is inside the space blocked by the circles. If \( R_{a} = 2R \), \( R_{b} = R \), and \( R_{c} = R \), find the minimum possible value of \( R_{d} \). | [
"R"
] | [
"Assume 4 circles such that any 3 of them have a common intersecting point on the surface of a sphere of radius \\( R \\) and the center \\( O \\) of the sphere is inside the space blocked by the circles. If \\( R_{a} = 2R \\), \\( R_{b} = R \\), and \\( R_{c} = R \\), find the minimum possible value of \\( R_{d} \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-d6ee723adc09b8e3f005e97236cd6a7933f76fe3369bfc9de820da77c2add153 | Assume ABCD is a rectangle. Two lines PQ and XY are drawn parallel to AB and CD. Both diagonals AC and BD are joined. Find the number of right triangles in the figure so obtained. | [
"44"
] | [
"Assume ABCD is a rectangle. Two lines PQ and XY are drawn parallel to AB and CD. Both diagonals AC and BD are joined. Find the number of right triangles in the figure so obtained."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-013f90ba70a636d1d4286224bd3c78bf332d7c5877dfe24296393c5bcb1667c6 | Assume \( 1 < p < \infty \) and \( U \) is a bounded domain. Let \( u \in W^{1,p}(U) \) be such that \( u = 0 \) on a measurable set \( E \subset U \). If \( |E| > 0 \), calculate the value of the integral \( \int_E |Du|^p \, dx \). | [
"0"
] | [
"Assume \\( 1 < p < \\infty \\) and \\( U \\) is a bounded domain. Let \\( u \\in W^{1,p}(U) \\) be such that \\( u = 0 \\) on a measurable set \\( E \\subset U \\). If \\( |E| > 0 \\), calculate the value of the integral \\( \\int_E |Du|^p \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-886c14bb3e789949fc39b66286982cb595b19ea59b47e3bf48e09f5ec1140fd3 | Assume \( 1 < p < \infty \) and \( U \) is a bounded open set in \( \mathbb{R}^n \). Given \( u \in W^{1,p}(U) \), calculate the value of the integral \( \int_U |\nabla |u||^p \, dx \) in terms of \( u \) and its gradient \( \nabla u \). | [
"\\int_U |\\nabla u|^p \\, dx"
] | [
"Assume \\( 1 < p < \\infty \\) and \\( U \\) is a bounded open set in \\( \\mathbb{R}^n \\). Given \\( u \\in W^{1,p}(U) \\), calculate the value of the integral \\( \\int_U |\\nabla |u||^p \\, dx \\) in terms of \\( u \\) and its gradient \\( \\nabla u \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a98ce38b48ea7cec6268ca6011ec2be4c21e2527c026bd801c2d01d629b5263a | Assume \( A = (a_{ij})_{1 \leq i,j \leq n} \) is an \( n \times n \) matrix with complex entries such that \( A^k = I_n \) for some integer \( k \geq 1 \). Calculate the minimum possible value of \( \sum_{1 \leq i,j \leq n} |a_{ij}|^2 \) for \( n = 5 \). | [
"5"
] | [
"Assume \\( A = (a_{ij})_{1 \\leq i,j \\leq n} \\) is an \\( n \\times n \\) matrix with complex entries such that \\( A^k = I_n \\) for some integer \\( k \\geq 1 \\). Calculate the minimum possible value of \\( \\sum_{1 \\leq i,j \\leq n} |a_{ij}|^2 \\) for \\( n = 5 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ea23761ae15a2f76c4f902defc21149de95e54d6f22c694c34a82865bae2eb95 | Assume \( A \) is a square real matrix of order \( n \) (\( n \geq 2 \)) with rank \( r(A) = n-1 \). Let \( B \) be another square real matrix of order \( n \) such that \( AB = BA = 0 \). If \( B \) is non-zero, find the value of \( \lambda \) such that \( B = \lambda \text{adj}(A)^T \) when \( A \) is given by \[ A =... | [
"1"
] | [
"Assume \\( A \\) is a square real matrix of order \\( n \\) (\\( n \\geq 2 \\)) with rank \\( r(A) = n-1 \\). Let \\( B \\) be another square real matrix of order \\( n \\) such that \\( AB = BA = 0 \\). If \\( B \\) is non-zero, find the value of \\( \\lambda \\) such that \\( B = \\lambda \\text{adj}(A)^T \\) wh... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13b0b986ba258115fbb33fdc8fc5dc5fe9ad216a16934a99201544008aeb391f | Assume \( A \) is an \( n \times n \) matrix with entries \( |a_{ij}| \leq 1 \) for \( 1 \leq i, j \leq n \). Let \( PA = LU \) be the factorization of \( A \), where \( P \) is a permutation matrix, \( L \) is a lower triangular matrix with ones on the diagonal, and \( U \) is an upper triangular matrix. Calculate the... | [
"16"
] | [
"Assume \\( A \\) is an \\( n \\times n \\) matrix with entries \\( |a_{ij}| \\leq 1 \\) for \\( 1 \\leq i, j \\leq n \\). Let \\( PA = LU \\) be the factorization of \\( A \\), where \\( P \\) is a permutation matrix, \\( L \\) is a lower triangular matrix with ones on the diagonal, and \\( U \\) is an upper trian... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d59c7ba7b538cfa4be77322f6a8ee842a3121ccbdbcc4607135cc51bb049416b | Assume \( F \) is a family of subsets of a set \( X \) with 10 elements such that if \( A, B \in F \) and \( B \subset A \), then \( |A - B| \ge 3 \). Calculate the maximum possible number of elements in \( F \). | [
"342"
] | [
"Assume \\( F \\) is a family of subsets of a set \\( X \\) with 10 elements such that if \\( A, B \\in F \\) and \\( B \\subset A \\), then \\( |A - B| \\ge 3 \\). Calculate the maximum possible number of elements in \\( F \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae34718f4486738f1dd8a61859ebf62f431488d100b7964dcf18a3d726630371 | Assume \( G \) acts on \( X \), and let \( K \) be the kernel of the permutation representation of this action. Suppose that \( H \) is a normal subgroup of \( G \) and a subgroup of \( K \) (\( H \triangleleft G \) and \( H < K \)). Define the action of \( G/H \) on \( X \) by \( x(Hg) = xg \) for all \( x \in X \) an... | [
"H"
] | [
"Assume \\( G \\) acts on \\( X \\), and let \\( K \\) be the kernel of the permutation representation of this action. Suppose that \\( H \\) is a normal subgroup of \\( G \\) and a subgroup of \\( K \\) (\\( H \\triangleleft G \\) and \\( H < K \\)). Define the action of \\( G/H \\) on \\( X \\) by \\( x(Hg) = xg ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7d50e8215b1cf3081f63e7607a89bc82592cf36613e22cac2c2e86bb094c254c | Assume \( G \) is a star-shaped domain in \( \mathbb{C} \), and \( f \) and \( g \) are holomorphic functions on \( G \) such that \( f^2 + g^2 = 1 \). Let \( \varphi(z) \) be a holomorphic function on \( G \) such that \( f(z) = \cos \varphi(z) \) and \( g(z) = \sin \varphi(z) \). If \( f(z_0) = \frac{1}{2} \) and \( ... | [
"\\dfrac{\\pi}{3}"
] | [
"Assume \\( G \\) is a star-shaped domain in \\( \\mathbb{C} \\), and \\( f \\) and \\( g \\) are holomorphic functions on \\( G \\) such that \\( f^2 + g^2 = 1 \\). Let \\( \\varphi(z) \\) be a holomorphic function on \\( G \\) such that \\( f(z) = \\cos \\varphi(z) \\) and \\( g(z) = \\sin \\varphi(z) \\). If \\(... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-30afe62f5c212b485fe3e7c12987dc54cff2dc8d46c511731dfa2b3ba56f1f9d | Assume \( P \) is a regular pentagon with a circumcircle radius of 1. Given a circle \( D \) with radius 1.00001, arbitrarily placed in the plane, does there exist a sequence \( S \) such that reflecting \( P \) according to \( S \) results in \( P \) completely residing within \( D \)? | [
"Yes"
] | [
"Assume \\( P \\) is a regular pentagon with a circumcircle radius of 1. Given a circle \\( D \\) with radius 1.00001, arbitrarily placed in the plane, does there exist a sequence \\( S \\) such that reflecting \\( P \\) according to \\( S \\) results in \\( P \\) completely residing within \\( D \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-658c9af18f9193c8ac0cb2668f9928f01955416f7c1baba539a2345548dd3565 | Assume \( R \) is a commutative ring with unity and \( S \) is a multiplicatively closed subset of \( R \). For all \( x, y \in R \), if \( xy \in S \), then \( x \in S \) and \( y \in S \). If there is no prime ideal \( P \) such that \( P \cap S = \emptyset \), find the number of elements in \( R \setminus S \) when ... | [
"0"
] | [
"Assume \\( R \\) is a commutative ring with unity and \\( S \\) is a multiplicatively closed subset of \\( R \\). For all \\( x, y \\in R \\), if \\( xy \\in S \\), then \\( x \\in S \\) and \\( y \\in S \\). If there is no prime ideal \\( P \\) such that \\( P \\cap S = \\emptyset \\), find the number of elements... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e9f5b1aa8a3027129a4b85ab20ac7dd03acb855adb9c0a78d3d2b4e074af511a | Assume \( S = \{a, b, c\} \). List all of the equivalence relations on \( S \), and for each equivalence relation, give the set of equivalence classes. | [
"1. Equivalence classes: \\( \\{\\{a, b, c\\}\\} \\)\n2. Equivalence classes: \\( \\{\\{a, b\\}, \\{c\\}\\} \\)\n3. Equivalence classes: \\( \\{\\{a, c\\}, \\{b\\}\\} \\)\n4. Equivalence classes: \\( \\{\\{b, c\\}, \\{a\\}\\} \\)\n5. Equivalence classes: \\( \\{\\{a\\}, \\{b\\}, \\{c\\}\\} \\)"
] | [
"Assume \\( S = \\{a, b, c\\} \\). List all of the equivalence relations on \\( S \\), and for each equivalence relation, give the set of equivalence classes."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 26
} | 29 | 0 | 0 |
omr-ws-16bbb7835ea9d323f8b7ace3c37afdfce4bbb9ef3a4c989949ee1fbe73a1d4ab | Assume \( T: \mathbb{R}^k \to \mathbb{R}^k \) is a linear map and \( \dim(\text{Im}(T)) < k \). Calculate the Lebesgue measure \( m(\text{Im}(T)) \). | [
"0"
] | [
"Assume \\( T: \\mathbb{R}^k \\to \\mathbb{R}^k \\) is a linear map and \\( \\dim(\\text{Im}(T)) < k \\). Calculate the Lebesgue measure \\( m(\\text{Im}(T)) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-880be3c09673ca4105ad3b658c6699edf28d14289ce1da388e62ad72a67abb38 | Assume \( V \) is a \(\mathbb{R}\)-vector space with \(\operatorname{dim}_{\mathbb{R}}(V) = 10\). Let \( W_{2} \leq V \) with \(\operatorname{dim}_{\mathbb{R}}(W_{2}) = 6\) and \( W_{1} \leq W_{2} \) with \(\operatorname{dim}_{\mathbb{R}}(W_{1}) = 3\). Define \( E \) as the set of all linear transformations \( f: V \ri... | [
"67"
] | [
"Assume \\( V \\) is a \\(\\mathbb{R}\\)-vector space with \\(\\operatorname{dim}_{\\mathbb{R}}(V) = 10\\). Let \\( W_{2} \\leq V \\) with \\(\\operatorname{dim}_{\\mathbb{R}}(W_{2}) = 6\\) and \\( W_{1} \\leq W_{2} \\) with \\(\\operatorname{dim}_{\\mathbb{R}}(W_{1}) = 3\\). Define \\( E \\) as the set of all line... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8d984b397c476b6b4ef91f7be192b75d99f1f145d5aabd62d33493fd18dab701 | Assume \( X \) is a normed vector space over the reals, and \( x_1, \ldots, x_k \in X \) are unit vectors. Additionally, suppose that there exists \( C \in (0, 1/2) \) such that for any constants \( c_i \in \mathbb{R} \), we have \[ \left\|\sum_{i = 1}^k c_i x_i\right\| \le (1 + C) \max_{1 \le i \le k} |c_i|. \] Let \(... | [
"\\sqrt{2} - 1"
] | [
"Assume \\( X \\) is a normed vector space over the reals, and \\( x_1, \\ldots, x_k \\in X \\) are unit vectors. Additionally, suppose that there exists \\( C \\in (0, 1/2) \\) such that for any constants \\( c_i \\in \\mathbb{R} \\), we have\n\\[\n\\left\\|\\sum_{i = 1}^k c_i x_i\\right\\| \\le (1 + C) \\max_{1 \... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-1d5a0fbfa9965d7cc45351e3a4df45d8f93c7673c7666bbddc6e6735a58f7336 | Assume \( [K:F] = 4 \) where \( K = F(\sqrt{a}, \sqrt{b}) \). Find all the different fields \( L \) such that \( F \subset L \subset K \). | [
"\\( F(\\sqrt{a}) \\) and \\( F(\\sqrt{b}) \\)"
] | [
"Assume \\( [K:F] = 4 \\) where \\( K = F(\\sqrt{a}, \\sqrt{b}) \\). Find all the different fields \\( L \\) such that \\( F \\subset L \\subset K \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8c89bd45dd4396788a4e9aa26b0fe12aeda91fd7aa2db588e3345e28f264f5e7 | Assume \( \Delta u = 0 \) in \( \mathbb{R}^n_+ \) and \( u = g \) on \( \partial \mathbb{R}^n_+ \). Using Poisson's formula for the half-space, let \( g \) be bounded and \( g(x) = |x| \) for \( x \in \partial \mathbb{R}^n_+ \) with \( |x| \le 1 \). Calculate the limit \(\lim_{\lambda \to 0^+} \frac{u(\lambda e_n) - u(... | [
"\\infty"
] | [
"Assume \\( \\Delta u = 0 \\) in \\( \\mathbb{R}^n_+ \\) and \\( u = g \\) on \\( \\partial \\mathbb{R}^n_+ \\). Using Poisson's formula for the half-space, let \\( g \\) be bounded and \\( g(x) = |x| \\) for \\( x \\in \\partial \\mathbb{R}^n_+ \\) with \\( |x| \\le 1 \\). Calculate the limit \\(\\lim_{\\lambda \\... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-078a9f8739d3497e7d1425db790532d63335416b38f7821b6ef9f224ee6d31eb | Assume \( \int_{-\infty}^{+\infty} |f(x)|^p \, dx \) exists. Define \( g(\lambda) = \lambda^p \mu\{x : |f(x)| > \lambda\} \). Calculate \( \lim_{\lambda \to +\infty} g(\lambda) \). | [
"0"
] | [
"Assume \\( \\int_{-\\infty}^{+\\infty} |f(x)|^p \\, dx \\) exists. Define \\( g(\\lambda) = \\lambda^p \\mu\\{x : |f(x)| > \\lambda\\} \\). Calculate \\( \\lim_{\\lambda \\to +\\infty} g(\\lambda) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-20e86ad01b31281f0ecc884ed4f2c85a9695ac2d7fce8be1010bf3206c221d32 | Assume \( \lim\limits_{x \to x_0} f(x) = A \) with \( A < 0 \). Let \( \epsilon = 1 \). Find the smallest possible value of \( \delta > 0 \) such that \( 0 < |x - x_0| < \delta \) implies \( |f(x) - A| < 1 \). | [
"0"
] | [
"Assume \\( \\lim\\limits_{x \\to x_0} f(x) = A \\) with \\( A < 0 \\). Let \\( \\epsilon = 1 \\). Find the smallest possible value of \\( \\delta > 0 \\) such that \\( 0 < |x - x_0| < \\delta \\) implies \\( |f(x) - A| < 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-f2bc1201b7fc91e95886c74f1eed396a0a00c351b98fcd76daf10a08ffea47b8 | Assume \( a_i > 0 \) for all \( i \in \mathbb{N}^+ \). Given that the limit \[ \lim_{n \rightarrow \infty} n \left( \frac{a_n}{a_{n+1}} - 1 \right) = L \] exists, calculate the value of \( L \) if it is known that \[ \lim_{n \rightarrow \infty} \frac{\ln \left( \frac{1}{a_n} \right)}{\ln n} = 3. \] | [
"3"
] | [
"Assume \\( a_i > 0 \\) for all \\( i \\in \\mathbb{N}^+ \\). Given that the limit\n\n\\[\n\\lim_{n \\rightarrow \\infty} n \\left( \\frac{a_n}{a_{n+1}} - 1 \\right) = L\n\\]\n\nexists, calculate the value of \\( L \\) if it is known that\n\n\\[\n\\lim_{n \\rightarrow \\infty} \\frac{\\ln \\left( \\frac{1}{a_n} \\r... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a7f522deedba911b1a263d3e521e5c2201f731cb5d10e9faea293cfbdbe487b0 | Assume \( a_i \ge 0 \) for \( i = 1, \ldots, n \) and \( \sum_{i=1}^n a_i = n \). Find the maximum value of \( c \) such that \( \sum_{i=1}^n \frac{1}{n + c a_i^2} \le \frac{n}{n + c} \). | [
"1"
] | [
"Assume \\( a_i \\ge 0 \\) for \\( i = 1, \\ldots, n \\) and \\( \\sum_{i=1}^n a_i = n \\). Find the maximum value of \\( c \\) such that \\( \\sum_{i=1}^n \\frac{1}{n + c a_i^2} \\le \\frac{n}{n + c} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-0d946397e0a2f89f6af45eb7c135f088ac6da7ef31573dcee09ddf418b314708 | Assume \( a_n > 0 \) and \( \lim\limits_{n \to \infty} \frac{a_1 + a_2 + \cdots + a_n}{n} = a < +\infty \). Find \[ \lim\limits_{n \to \infty} \frac{a_1^p + a_2^p + \cdots + a_n^p}{n^p} \] where \( p > 1 \). | [
"0"
] | [
"Assume \\( a_n > 0 \\) and \\( \\lim\\limits_{n \\to \\infty} \\frac{a_1 + a_2 + \\cdots + a_n}{n} = a < +\\infty \\). Find \n\\[ \\lim\\limits_{n \\to \\infty} \\frac{a_1^p + a_2^p + \\cdots + a_n^p}{n^p} \\]\nwhere \\( p > 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6fbd2d30f1887e9b623bd1afcf4714f2078cd3029531cdc73430caba8dc92254 | Assume \( a_{1} \ge a_{2} \ge \dots \ge a_{107} > 0 \) satisfy \( \sum\limits_{k=1}^{107}{a_{k}} = 1070 \) and \( b_{107} \ge b_{106} \ge \dots \ge b_{1} > 0 \) satisfy \( \sum\limits_{k=1}^{107}{b_{k}} = 107 \). Calculate the minimum possible value of the arithmetic mean of the numbers \( \frac{a_{1}}{b_{1}}, \frac{a_... | [
"10"
] | [
"Assume \\( a_{1} \\ge a_{2} \\ge \\dots \\ge a_{107} > 0 \\) satisfy \\( \\sum\\limits_{k=1}^{107}{a_{k}} = 1070 \\) and \\( b_{107} \\ge b_{106} \\ge \\dots \\ge b_{1} > 0 \\) satisfy \\( \\sum\\limits_{k=1}^{107}{b_{k}} = 107 \\). Calculate the minimum possible value of the arithmetic mean of the numbers \\( \\f... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13e1f89be34ce5fb29648400fd0a212d5d267b6964ec6a03ec0a4b49cca5117a | Assume \( f : [0, \infty) \to \mathbb{R} \) is a uniformly continuous function on \([0, \infty)\). Let \( g(x) = |f(x)| \). Find the smallest possible value of \( a + b \) such that \( ax + b \geq g(x) \) for all \( x \in [0, \infty) \), given that \( f(0) = 0 \) and \( f(1) = 1 \). | [
"2"
] | [
"Assume \\( f : [0, \\infty) \\to \\mathbb{R} \\) is a uniformly continuous function on \\([0, \\infty)\\). Let \\( g(x) = |f(x)| \\). Find the smallest possible value of \\( a + b \\) such that \\( ax + b \\geq g(x) \\) for all \\( x \\in [0, \\infty) \\), given that \\( f(0) = 0 \\) and \\( f(1) = 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-7aaa88c01028ed0d7881d89eb4d08b787b4aee6c2b68543165ef1abb81e8fb74 | Assume \( f \) is Riemann integrable with respect to \( \alpha \) on \([a, b]\). For a given \( \epsilon > 0 \), find the smallest integer \( N \) such that there exists a polynomial \( P_N \) satisfying \[ \int_a^b |f - P_N|^2 \, d\alpha < \epsilon. \] Calculate \( N \) when \( \epsilon = 0.01 \), \( f(x) = \sin(x) \)... | [
"2"
] | [
"Assume \\( f \\) is Riemann integrable with respect to \\( \\alpha \\) on \\([a, b]\\). For a given \\( \\epsilon > 0 \\), find the smallest integer \\( N \\) such that there exists a polynomial \\( P_N \\) satisfying\n\n\\[\n\\int_a^b |f - P_N|^2 \\, d\\alpha < \\epsilon.\n\\]\n\nCalculate \\( N \\) when \\( \\ep... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-19510a2f7a7c599093921cf4c660359084dbd4f95eaf616db17c7cc7aa040b10 | Assume \( f \) is a continuous function defined on \([0, 2015]\) such that \( f(0) = f(2015) \). Let \( n \) be the number of pairs \((x, y)\) that satisfy \( f(x) = f(y) \) and \( x - y \) is a positive integer. Find the minimum value of \( n \). | [
"2015"
] | [
"Assume \\( f \\) is a continuous function defined on \\([0, 2015]\\) such that \\( f(0) = f(2015) \\). Let \\( n \\) be the number of pairs \\((x, y)\\) that satisfy \\( f(x) = f(y) \\) and \\( x - y \\) is a positive integer. Find the minimum value of \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-8496fcd221361450159f416cb054f9efd4bd2562c558d496998cc9325b627a9d | Assume \( f \) is a monotone function on the interval \([a, b]\) with \( f(a) \ge a \) and \( f(b) \le b \). Let \( x_0 \) be the smallest value in \([a, b]\) such that \( f(x_0) = x_0 \). If \( a = 1 \) and \( b = 10 \), and \( f(x) = x^2 - 11x + 30 \), find \( x_0 \). | [
"6 - \\sqrt{6}"
] | [
"Assume \\( f \\) is a monotone function on the interval \\([a, b]\\) with \\( f(a) \\ge a \\) and \\( f(b) \\le b \\). Let \\( x_0 \\) be the smallest value in \\([a, b]\\) such that \\( f(x_0) = x_0 \\). If \\( a = 1 \\) and \\( b = 10 \\), and \\( f(x) = x^2 - 11x + 30 \\), find \\( x_0 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab206889c030074304b2dfa75f36a225831088cf76b2e1d63ca60a1858d34f1c | Assume \( f \) is an entire function (analytic in \( \mathbb{C} \)). Assume further that \( |f(z)| \leq |z|^{\frac{5}{2}} \) if \( |z| \geq R \), where \( R \) is a fixed real. If \( f(z) = a_0 + a_1z + a_2z^2 + a_3z^3 + \cdots \), find the sum \( a_3 + a_4 + a_5 + \cdots \). | [
"0"
] | [
"Assume \\( f \\) is an entire function (analytic in \\( \\mathbb{C} \\)). Assume further that \\( |f(z)| \\leq |z|^{\\frac{5}{2}} \\) if \\( |z| \\geq R \\), where \\( R \\) is a fixed real. If \\( f(z) = a_0 + a_1z + a_2z^2 + a_3z^3 + \\cdots \\), find the sum \\( a_3 + a_4 + a_5 + \\cdots \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c8fbe0b712385737275c0763279000a63b09ff442e07c1ff7f64bea95c49f19d | Assume \( f \) is continuous at at least one point \( x' \) and satisfies the functional equation \( f(x+y) = f(x) + f(y) \). If \( f(1) = 3 \), find \( f(10) \). | [
"30"
] | [
"Assume \\( f \\) is continuous at at least one point \\( x' \\) and satisfies the functional equation \\( f(x+y) = f(x) + f(y) \\). If \\( f(1) = 3 \\), find \\( f(10) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2dc1714a27eb22c6814adf7309f634907c17470d0a9a1116f9241be3d75dcc01 | Assume \( f \) is continuous on \( I = [a, b] \) and that \( \int_{a}^{b} f \cdot g = 0 \) for every continuous function \( g \) on \( I \). Let \( h(x) = \int_{a}^{x} f(t) \, dt \). Calculate \( h(b) - h(a) \). | [
"0"
] | [
"Assume \\( f \\) is continuous on \\( I = [a, b] \\) and that \\( \\int_{a}^{b} f \\cdot g = 0 \\) for every continuous function \\( g \\) on \\( I \\). Let \\( h(x) = \\int_{a}^{x} f(t) \\, dt \\). Calculate \\( h(b) - h(a) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e65096cae7f8617c7c38313f4412826e56342bfa0a315522803bc797378c154 | Assume \( f \) is continuous on a rectangle \( Q = [a-h, a+h] \times [b-k, b+k] \) where \( h > 0 \) and \( k > 0 \). Let \( S \) denote the metric subspace of \( C[a-c, a+c] \) consisting of all \( \phi \) such that \( |\phi(x) - b| \leq Mc \) on \( [a-c, a+c] \), where \( |f(x, y)| \leq M \) and \( c = \min\{h, k/M\}... | [
"1"
] | [
"Assume \\( f \\) is continuous on a rectangle \\( Q = [a-h, a+h] \\times [b-k, b+k] \\) where \\( h > 0 \\) and \\( k > 0 \\). Let \\( S \\) denote the metric subspace of \\( C[a-c, a+c] \\) consisting of all \\( \\phi \\) such that \\( |\\phi(x) - b| \\leq Mc \\) on \\( [a-c, a+c] \\), where \\( |f(x, y)| \\leq M... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-39d4e9d2df7364471e12f1f14c80920b24bd7a75e2753fde3f82d6c1f7b808f8 | Assume \( f \) is differentiable on \( (a,b) \) and is continuous on \( [a,b] \) with \( f(a) = f(b) = 0 \). For \( \alpha = 2 \), find the number of distinct values of \( c \) in \( (a,b) \) such that \( f'(c) = 2f(c) \) if \( f(x) = x(x-a)(x-b) \). | [
""
] | [
"Assume \\( f \\) is differentiable on \\( (a,b) \\) and is continuous on \\( [a,b] \\) with \\( f(a) = f(b) = 0 \\). For \\( \\alpha = 2 \\), find the number of distinct values of \\( c \\) in \\( (a,b) \\) such that \\( f'(c) = 2f(c) \\) if \\( f(x) = x(x-a)(x-b) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-b3e53480de4ecf800095e0f872d3755be8339461853937ecee10f49e89265ba1 | Assume \( f \) is increasing on \([0, \infty)\) and \(\lim_{x \to +\infty} f'(x) = L\). Given that \( f(0) = 0 \) and \( f(1) = 1 \), find the value of \( L \) if \( f \) is uniformly continuous on \([0, \infty)\). | [
"0"
] | [
"Assume \\( f \\) is increasing on \\([0, \\infty)\\) and \\(\\lim_{x \\to +\\infty} f'(x) = L\\). Given that \\( f(0) = 0 \\) and \\( f(1) = 1 \\), find the value of \\( L \\) if \\( f \\) is uniformly continuous on \\([0, \\infty)\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
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