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-ba55d23cb60ff39e308a064665a32440bfe82b25da05331eab5fb85227602206 | A function \( f \) is defined for all real numbers and satisfies \( f(2 + x) = f(2 - x) \) and \( f(7 + x) = f(7 - x) \) for all real \( x \). If \( x = 0 \) is a root of \( f(x) = 0 \), what is the least number of roots \( f(x) = 0 \) must have in the interval \( -1000 \leq x \leq 1000 \)? | [
"401"
] | [
"A function \\( f \\) is defined for all real numbers and satisfies \\( f(2 + x) = f(2 - x) \\) and \\( f(7 + x) = f(7 - x) \\) for all real \\( x \\). If \\( x = 0 \\) is a root of \\( f(x) = 0 \\), what is the least number of roots \\( f(x) = 0 \\) must have in the interval \\( -1000 \\leq x \\leq 1000 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c8f44dca322c4587d7fcb7cc6207b0710b24f2468359ddbd0bfbe2f623e990bb | A function \( f \) is defined for all real numbers and satisfies \( f(2+x) = f(2-x) \) and \( f(7+x) = f(7-x) \) for all \( x \). If \( x = 0 \) is a root of \( f(x) = 0 \), what is the least number of roots \( f(x) = 0 \) must have in the interval \( -1000 \le x \le 1000 \)? | [
"201"
] | [
"A function \\( f \\) is defined for all real numbers and satisfies \\( f(2+x) = f(2-x) \\) and \\( f(7+x) = f(7-x) \\) for all \\( x \\). If \\( x = 0 \\) is a root of \\( f(x) = 0 \\), what is the least number of roots \\( f(x) = 0 \\) must have in the interval \\( -1000 \\le x \\le 1000 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1685a8b4f6a97aab89e13693aee9c879e00d5a697c59e16a22ed7260fccc4261 | A function \( f \) is defined on \( \mathbb{R}^* \) (non-zero real numbers) such that \( f^{-1}(x) = \frac{1}{f(x)} \) for all \( x \neq 0 \). If \( f \) is strictly increasing on the interval \( (0, +\infty) \), can we find a general form for this function? What additional conditions must be imposed to determine all s... | [
"f(x) = -\\frac{1}{x^2} \\text{ for } x > 0 \\text{ and } f(x) = \\sqrt{-x} \\text{ for } x < 0"
] | [
"A function \\( f \\) is defined on \\( \\mathbb{R}^* \\) (non-zero real numbers) such that \\( f^{-1}(x) = \\frac{1}{f(x)} \\) for all \\( x \\neq 0 \\). If \\( f \\) is strictly increasing on the interval \\( (0, +\\infty) \\), can we find a general form for this function? What additional conditions must be impos... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-7c0f2e4470bf61bce88582597952888d10f803e5622357e32b01d4b560019239 | A function \( f \) is defined on the complex numbers by \( f(z) = (a + bi)z \), where \( a \) and \( b \) are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that \( |a + bi| = 8 \) and that \( b^2 = \frac{m}{n} \), ... | [
"259"
] | [
"A function \\( f \\) is defined on the complex numbers by \\( f(z) = (a + bi)z \\), where \\( a \\) and \\( b \\) are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that \\( |a + bi| = 8 \\) and that \\( b^2 = ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-76284c01b778c26f2dcb05cdeb5d388f99791604b6e391b83ea47f3f38886a22 | A function \( f \) is defined on the complex numbers by \( f(z) = (a + bi)z \), where \( a \) and \( b \) are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that \( |a + bi| = 8 \), find the value of \( b^2 \). | [
"\\(\\frac{255}{4}\\)"
] | [
"A function \\( f \\) is defined on the complex numbers by \\( f(z) = (a + bi)z \\), where \\( a \\) and \\( b \\) are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that \\( |a + bi| = 8 \\), find the value of ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 58
} | 89 | 0 | 0 |
omr-ws-980fd7d8d39ca2d16fd9ca1c1e3b4614600685ee372e472996591b0c2b6bde0e | A function \( f \) is defined on the complex numbers by \( f(z) = (a + bi)z \), where \( a \) and \( b \) are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. What is the image of each point \( z \) under \( f \)? | [
"\\( (a + bi)z \\)"
] | [
"A function \\( f \\) is defined on the complex numbers by \\( f(z) = (a + bi)z \\), where \\( a \\) and \\( b \\) are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. What is the image of each point \\( z \\) under \\(... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-718c4d8646fd0b41714b54ecbd8bfeca65aa5123b617859c22cc9cfe97635520 | A function \( f \) is defined on the natural numbers \( \mathbb{N} \) and satisfies the following rules: - \( f(1) = 1 \) - \( f(2n) = f(n) \) - \( f(2n+1) = f(2n) + 1 \) for all \( n \in \mathbb{N} \). Calculate the maximum value \( m \) of the set \( \{ f(n) : n \in \mathbb{N}, 1 \le n \le 1989 \} \), and determine t... | [
"\\( m = 10 \\) and the number of natural numbers \\( n \\) with \\( 1 \\leq n \\leq 1989 \\) that satisfy \\( f(n) = 10 \\) is 10."
] | [
"A function \\( f \\) is defined on the natural numbers \\( \\mathbb{N} \\) and satisfies the following rules:\n- \\( f(1) = 1 \\)\n- \\( f(2n) = f(n) \\)\n- \\( f(2n+1) = f(2n) + 1 \\) for all \\( n \\in \\mathbb{N} \\).\n\nCalculate the maximum value \\( m \\) of the set \\( \\{ f(n) : n \\in \\mathbb{N}, 1 \\le ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0c0fa7a6fbd46ec938e9db1a0cb4dc64641f485087f7b080a8e4611407476cbd | A function \( f \) is defined on the set \( D \), where \( D \) is the set of all rational numbers except 0 and 1. The function \( f \) satisfies the equation: \[ f(x) + f\left(1 - \frac{1}{x}\right) = x \] for all \( x \in D \). Determine the function \( f \). | [
"\\( f(x) = \\frac{1}{2} \\left( x + \\frac{1}{x-1} - \\frac{x-1}{x} \\right) \\)"
] | [
"A function \\( f \\) is defined on the set \\( D \\), where \\( D \\) is the set of all rational numbers except 0 and 1. The function \\( f \\) satisfies the equation:\n\n\\[ f(x) + f\\left(1 - \\frac{1}{x}\\right) = x \\]\n\nfor all \\( x \\in D \\). Determine the function \\( f \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 11
} | 25 | 0 | 0 |
omr-ws-9167bcca5f3650ccb531de8830c7999b2930933d54f740f7c9b6d865a238848d | A function \( f \) is defined on the set of pairs of positive integers and has positive integer values. If \( f(f(x,y),z) = f(x,f(y,z)) \) for all positive integers \( x, y, z \), and there exists a constant \( C \) such that \( f(x,y) < C \) for all positive integers \( x, y \), find the smallest positive integer \( x... | [
"1"
] | [
"A function \\( f \\) is defined on the set of pairs of positive integers and has positive integer values. If \\( f(f(x,y),z) = f(x,f(y,z)) \\) for all positive integers \\( x, y, z \\), and there exists a constant \\( C \\) such that \\( f(x,y) < C \\) for all positive integers \\( x, y \\), find the smallest posi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f9bad878e1037cd995aadc4e94acabcd28c3834bb3f6cbd9f9a75454b52cdda | A function \( f \) is defined over all the positive integers and satisfies: \[ f(1) = 1996 \] \[ f(1) + f(2) + f(3) + \cdots + f(n) = n^2 f(n) \] Calculate the exact value of \( f(1996) \). | [
"\\(\\frac{2}{1997}\\)"
] | [
"A function \\( f \\) is defined over all the positive integers and satisfies:\n\\[ f(1) = 1996 \\]\n\\[ f(1) + f(2) + f(3) + \\cdots + f(n) = n^2 f(n) \\]\nCalculate the exact value of \\( f(1996) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-4666605b51a745a4b36c87dd411119065c9cc88e462cd10d35a388cb72747cf3 | A function \( f \) is defined over the set of all positive integers and satisfies \( f(1) = 1996 \) and \( f(1) + f(2) + \cdots + f(n) = n^2 f(n) \) for all \( n > 1 \). Calculate the exact value of \( f(1996) \). | [
"\\(\\frac{2}{1997}\\)"
] | [
"A function \\( f \\) is defined over the set of all positive integers and satisfies \\( f(1) = 1996 \\) and \\( f(1) + f(2) + \\cdots + f(n) = n^2 f(n) \\) for all \\( n > 1 \\). Calculate the exact value of \\( f(1996) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-add567684e21c2f11b4036f59064cb8c302f53fd297452c4c35cd7b40fdbf57e | A function \( f \) is defined over the set of all positive integers and satisfies \( f(1) = 1996 \) and \( f(1) + f(2) + \cdots + f(n-1) + f(n) = n^2 f(n) \) for all \( n > 1 \). Find \( f(2011) \). | [
"\\(\\frac{998}{1011533}\\)"
] | [
"A function \\( f \\) is defined over the set of all positive integers and satisfies \\( f(1) = 1996 \\) and \\( f(1) + f(2) + \\cdots + f(n-1) + f(n) = n^2 f(n) \\) for all \\( n > 1 \\). Find \\( f(2011) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 30
} | 48 | 0 | 0 |
omr-ws-8c1bcd826ebeb9cfdfbedef511b16269b67d06abad494fa23c04ccfad89239bd | A function \( f \) is defined over the set of all positive integers and satisfies \( f(1) = 1996 \) and \[ f(1) + f(2) + \cdots + f(n) = n^2 \cdot f(n) \] for all \( n > 1 \). Calculate the exact value of \( f(1996) \). | [
"\\(\\frac{2}{1997}\\)"
] | [
"A function \\( f \\) is defined over the set of all positive integers and satisfies \\( f(1) = 1996 \\) and \n\\[ f(1) + f(2) + \\cdots + f(n) = n^2 \\cdot f(n) \\]\nfor all \\( n > 1 \\). Calculate the exact value of \\( f(1996) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-749bf52ca32ec3a41572e10384176e36b8da9fdb075eca4cbc240e6754d0df06 | A function \( f \) is defined over the set of all positive integers and satisfies \( f(1) = 1996 \) and \[ f(1) + f(2) + \cdots + f(n) = n^2 f(n) \] for all \( n > 1 \). Calculate the value of \( f(1996) \). | [
"\\(\\frac{2}{1997}\\)"
] | [
"A function \\( f \\) is defined over the set of all positive integers and satisfies \\( f(1) = 1996 \\) and \n\\[ f(1) + f(2) + \\cdots + f(n) = n^2 f(n) \\]\nfor all \\( n > 1 \\). Calculate the value of \\( f(1996) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-858e12446552590dda36beaa7d4b935b6b482f0a0ad43c526a489f8e8177dde3 | A function \( f \) is defined over the set of all positive integers and satisfies \( f(1) = 2022 \) and \[ f(1) + f(2) + \dots + f(n) = n^2 f(n) \] for all \( n > 1 \). Calculate the exact value of \( f(2022) \). | [
"\\(\\frac{2}{2023}\\)"
] | [
"A function \\( f \\) is defined over the set of all positive integers and satisfies \\( f(1) = 2022 \\) and \n\\[ f(1) + f(2) + \\dots + f(n) = n^2 f(n) \\]\nfor all \\( n > 1 \\). Calculate the exact value of \\( f(2022) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-e436a221367ce393c7f23f0b93b180b6d84c230277a34205814450d844cabb34 | A function \( f \) is defined over the set of all positive integers and satisfies \[ \sum_{i=1}^{n} f(i) = n^2 f(n) \] and \( f(1) = 1996 \). What is \( f(1996) \)? | [
"\\(\\frac{2}{1997}\\)"
] | [
"A function \\( f \\) is defined over the set of all positive integers and satisfies\n\\[\n\\sum_{i=1}^{n} f(i) = n^2 f(n)\n\\]\nand \\( f(1) = 1996 \\). What is \\( f(1996) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-b13041258b6ec48429207c9ee6db1f5f9a2c0e2e0f6e9e28318a8da713309663 | A function \( f \) is defined over the set of all positive integers and satisfies: \[ f(1) = 1996 \] \[ f(1) + f(2) + \cdots + f(n) = n^2 f(n) \text{ for all } n > 1 \] Calculate the exact value of \( f(1996) \). | [
"\\(\\frac{2}{1997}\\)"
] | [
"A function \\( f \\) is defined over the set of all positive integers and satisfies:\n\\[ f(1) = 1996 \\]\n\\[ f(1) + f(2) + \\cdots + f(n) = n^2 f(n) \\text{ for all } n > 1 \\]\n\nCalculate the exact value of \\( f(1996) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-6208caf2543c3d03a6a239cf70fcaef1e92a2ac258a3c0e5c8fd6d18a3c1291e | A function \( f \) is defined over the set of all positive integers and satisfies: \[ f(1) = 1996 \] and \[ f(1) + f(2) + f(3) + \cdots + f(n) = n^2 f(n) \text{ for all } n > 1. \] Calculate the exact value of \( f(1996) \). | [
"\\(\\frac{2}{1997}\\)"
] | [
"A function \\( f \\) is defined over the set of all positive integers and satisfies:\n\\[ f(1) = 1996 \\]\nand\n\\[ f(1) + f(2) + f(3) + \\cdots + f(n) = n^2 f(n) \\text{ for all } n > 1. \\]\nCalculate the exact value of \\( f(1996) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-7ba5a0394c01c8dc4c38842cec6fcc3825ece16000b5cdaa0f03840b34c4d69b | A function \( f \) is defined recursively by \( f(1) = f(2) = 1 \) and \( f(n) = f(n - 1) - f(n - 2) + n \) for all integers \( n \geq 3 \). What is \( f(2018) \)? | [
"1"
] | [
"A function \\( f \\) is defined recursively by \\( f(1) = f(2) = 1 \\) and \\( f(n) = f(n - 1) - f(n - 2) + n \\) for all integers \\( n \\geq 3 \\). What is \\( f(2018) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0add4778ce85067755cda42c519d7f72dbe7d0433746c4c05650c93109b01101 | A function \( f \) is defined recursively by \( f(1) = f(2) = 1 \) and \( f(n) = f(n-1) - f(n-2) + n \) for all integers \( n \ge 3 \). What is \( f(2018) \)? | [
"2017"
] | [
"A function \\( f \\) is defined recursively by \\( f(1) = f(2) = 1 \\) and \\( f(n) = f(n-1) - f(n-2) + n \\) for all integers \\( n \\ge 3 \\). What is \\( f(2018) \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d32b49b69cd2bec802ee0edeed181ab81c6bda01c48024ac6bfb4934fc80b54c | A function \( f \) is defined recursively by \( f(1) = f(2) = 1 \) and \( f(n) = f(n-1) - f(n-2) + n \) for all integers \( n \geq 3 \). What is \( f(2018) \)? | [
"2017"
] | [
"A function \\( f \\) is defined recursively by \\( f(1) = f(2) = 1 \\) and \\( f(n) = f(n-1) - f(n-2) + n \\) for all integers \\( n \\geq 3 \\). What is \\( f(2018) \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-4941d980304558b2ebc78dc79402ecb515479d1a1ad4fe43618c61f3e3a5a91f | A function \( f \) satisfies \( f(a+b) = f(a) + f(b) + 2ab \) for all positive integers \( a, b \). If \( f(1) = 4 \), then what is \( f(2012) \)? | [
"4054180"
] | [
"A function \\( f \\) satisfies \\( f(a+b) = f(a) + f(b) + 2ab \\) for all positive integers \\( a, b \\). If \\( f(1) = 4 \\), then what is \\( f(2012) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-1425ee24a1dac4811ed69246eb16a9b242e5f53ff5fcb71923962505b3a7c39a | A function \( f \) satisfies the equation \( 17f(x) + 65f\left(\frac{2}{x}\right) = 257 \). This function has a y-intercept of \( (0, \frac{a}{b}) \), where \( a \) and \( b \) are relatively prime integers and \( b \neq 0 \). What is \( a + b \)? | [
"339"
] | [
"A function \\( f \\) satisfies the equation \\( 17f(x) + 65f\\left(\\frac{2}{x}\\right) = 257 \\). This function has a y-intercept of \\( (0, \\frac{a}{b}) \\), where \\( a \\) and \\( b \\) are relatively prime integers and \\( b \\neq 0 \\). What is \\( a + b \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6eb004488d04ded833f0cf92a324c2b2354454ca8a08ec513b546a7812c15b46 | A function \( f \) satisfies the equation \[ f(x) + f\left(1 - \frac{1}{x}\right) = 1 + x \] for every real number \( x \) except for \( x = 0 \) and \( x = 1 \). Find a closed formula for \( f \). | [
"\\( \\frac{x}{2} + \\frac{1}{2(1-x)} + \\frac{1}{2x} \\)"
] | [
"A function \\( f \\) satisfies the equation\n\\[ f(x) + f\\left(1 - \\frac{1}{x}\\right) = 1 + x \\]\nfor every real number \\( x \\) except for \\( x = 0 \\) and \\( x = 1 \\). Find a closed formula for \\( f \\)."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 10
} | 28 | 0 | 0 |
omr-ws-405b65e7520220328be472d4c002aa8ce3d3429964bc61fc5c4b89e8e82a941c | A function \( f \) satisfies the following conditions: i) For each rational number \( x \), \( f(x) \) is a real number. ii) \( f(1988) \neq f(1987) \). iii) \( f(x+y) = f(x)f(y) - f(xy) + 1 \) holds for all rational numbers \( x \) and \( y \). Calculate the value of \( f\left(\frac{-1987}{1988}\right) \). | [
"\\dfrac{1}{1988}"
] | [
"A function \\( f \\) satisfies the following conditions:\ni) For each rational number \\( x \\), \\( f(x) \\) is a real number.\nii) \\( f(1988) \\neq f(1987) \\).\niii) \\( f(x+y) = f(x)f(y) - f(xy) + 1 \\) holds for all rational numbers \\( x \\) and \\( y \\).\n\nCalculate the value of \\( f\\left(\\frac{-1987}... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-52057416e2a02d115a4db708b36ff991639a1e42b25e28e585eb2ccc41c4c602 | A function \( f \) with domain \( \mathbb{N} \) satisfies: 1. \( f(1) = 1996 \) 2. \( f(1) + f(2) + f(3) + \cdots + f(n) = n^2 f(n) \) Find the exact value of \( f(1996) \). | [
"\\(\\frac{2}{1997}\\)"
] | [
"A function \\( f \\) with domain \\( \\mathbb{N} \\) satisfies:\n1. \\( f(1) = 1996 \\)\n2. \\( f(1) + f(2) + f(3) + \\cdots + f(n) = n^2 f(n) \\)\n\nFind the exact value of \\( f(1996) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-def9c10585c94a57948cfbfd653a8ede5380cc31f921e3b6badc050363407958 | A function \( f \), continuous on the positive real axis, has the property that \[ \int_{1}^{xy} f(t)\ dt = y \int_{1}^{x} f(t)\ dt + x \int_{1}^{y} f(t)\ dt \quad \text{for all} \quad x, y > 0. \] If \( f(1) = 3 \), find \( f(x) \) for each \( x > 0 \). | [
"\\( f(x) = 3 \\ln x + 3 \\)"
] | [
"A function \\( f \\), continuous on the positive real axis, has the property that\n\n\\[\n\\int_{1}^{xy} f(t)\\ dt = y \\int_{1}^{x} f(t)\\ dt + x \\int_{1}^{y} f(t)\\ dt \\quad \\text{for all} \\quad x, y > 0.\n\\]\n\nIf \\( f(1) = 3 \\), find \\( f(x) \\) for each \\( x > 0 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-1a9412d49fe7edce99752a610053c0fc3795d5f11dc3633fe4059a56f58f497f | A function \( f \), continuous on the positive real axis, satisfies the property \[ \int _{ 1 }^{ xy }{ f\left( t \right) \, \text{d}t } = y \int _{ 1 }^{ x }{ f\left( t \right) \, \text{d}t } + x \int _{ 1 }^{ y }{ f\left( t \right) \, \text{d}t } \] for all \( x > 0 \) and \( y > 0 \). Given that \( f(1) = 3 \), comp... | [
"\\( f(x) = 3 + 3 \\ln x \\)"
] | [
"A function \\( f \\), continuous on the positive real axis, satisfies the property\n\\[\n\\int _{ 1 }^{ xy }{ f\\left( t \\right) \\, \\text{d}t } = y \\int _{ 1 }^{ x }{ f\\left( t \\right) \\, \\text{d}t } + x \\int _{ 1 }^{ y }{ f\\left( t \\right) \\, \\text{d}t }\n\\]\nfor all \\( x > 0 \\) and \\( y > 0 \\).... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-4e4ee959856a6c6b3480b203c21c63bfad050aab23d42478d79f58c114143068 | A function \( f \), defined for the set of integers, is such that \( f(x) = x - 10 \) if \( x > 100 \) and \( f(x) = f(f(x + 11)) \) if \( x \leq 100 \). Determine the set of all possible values for \( f \). | [
"\\(\\mathbb{Z} \\cap [91, +\\infty)\\)"
] | [
"A function \\( f \\), defined for the set of integers, is such that \\( f(x) = x - 10 \\) if \\( x > 100 \\) and \\( f(x) = f(f(x + 11)) \\) if \\( x \\leq 100 \\). Determine the set of all possible values for \\( f \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-6089663496cf6d3d9d9d6bd3f29f0628349c559772392f2b0f64943179094cc7 | A function \( f \), with a domain of all positive integers, satisfies the following three properties: 1. \( f(2) = 1 \) 2. \( f(2p) = p \) for all primes \( p \). 3. \( f(ab) = f(a) + f(b) \) for all positive integers \( a \) and \( b \). Find the remainder when \( f(2024^{2024}) \) is divided by 1000. | [
"840"
] | [
"A function \\( f \\), with a domain of all positive integers, satisfies the following three properties:\n1. \\( f(2) = 1 \\)\n2. \\( f(2p) = p \\) for all primes \\( p \\).\n3. \\( f(ab) = f(a) + f(b) \\) for all positive integers \\( a \\) and \\( b \\).\n\nFind the remainder when \\( f(2024^{2024}) \\) is divide... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-3b41202e564e4ebc9b99d3cf5994c2da796d71b375a876086dc0a6a03e57c2de | A function \( f(n) \) has the property for all \( n \geq 1 \) that \( f(n) = f(n-1) + f(n-2) \). If \( f(7) = 120 \), what is \( f(8) \)? | [
"194"
] | [
"A function \\( f(n) \\) has the property for all \\( n \\geq 1 \\) that \\( f(n) = f(n-1) + f(n-2) \\). If \\( f(7) = 120 \\), what is \\( f(8) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-0a86fe3d789816ee1b9b355f73f0a12bbe89d54de50b76c8269be25f7aaaa246 | A function \( f(n) \) is defined as follows: \[ f(n) = 1 \text{ for } n < 0, \] \[ f(n) = 1 - f(n-1)f(n-3)f(n-4) \text{ for } n \geq 0. \] Evaluate \( \sum_{n=0}^{2018} f(n). \) | [
"1441"
] | [
"A function \\( f(n) \\) is defined as follows:\n\\[ f(n) = 1 \\text{ for } n < 0, \\]\n\\[ f(n) = 1 - f(n-1)f(n-3)f(n-4) \\text{ for } n \\geq 0. \\]\nEvaluate \\( \\sum_{n=0}^{2018} f(n). \\)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 10
} | 33 | 0 | 0 |
omr-ws-4389f729cbaeb74d01d53051a00777106b3c8c66c7c7ddfc5e5310f72984258e | A function \( f(n) \) of the positive integer \( n \) has the following properties for all \( n \geq 1 \): \[ f(n+2) = f(n+1) - f(n) \] What is the maximum number of different values of \( f(n) \) as \( n \) ranges over all positive integers? | [
"6"
] | [
"A function \\( f(n) \\) of the positive integer \\( n \\) has the following properties for all \\( n \\geq 1 \\):\n\\[ f(n+2) = f(n+1) - f(n) \\]\nWhat is the maximum number of different values of \\( f(n) \\) as \\( n \\) ranges over all positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-561b86eabe73527d3a25ea0befcbce7654f2fcb0a2aacd060ac4fa13d7062e29 | A function \( f(x) \) defined for \( x \ge 0 \) satisfies the following conditions: i. For \( x, y \ge 0 \), \( f(x)f(y) \le x^2 f(y/2) + y^2 f(x/2) \); ii. There exists a constant \( M > 0 \) such that \( |f(x)| \le M \) when \( 0 \le x \le 1 \). Let \( g(x) = \frac{f(x)}{x^2} \) for \( x > 0 \). Find the maximum poss... | [
"\\dfrac{1}{2}"
] | [
"A function \\( f(x) \\) defined for \\( x \\ge 0 \\) satisfies the following conditions:\ni. For \\( x, y \\ge 0 \\), \\( f(x)f(y) \\le x^2 f(y/2) + y^2 f(x/2) \\);\nii. There exists a constant \\( M > 0 \\) such that \\( |f(x)| \\le M \\) when \\( 0 \\le x \\le 1 \\).\n\nLet \\( g(x) = \\frac{f(x)}{x^2} \\) for \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-b134f9658992450ab393dfc7bd26afa41aa7742d5f5f0ac965ffa16da9417e99 | A function \( f(x) \) defined on \( x > 0 \) satisfies the following conditions: - \( f(1) = 1 \) - \( f'(x) \) is continuous on \( x > 0 \) - \( f'(x) \leq \frac{f(x)}{x} - 1 \) Let \( g(x) = \frac{f(x)}{x} \). Find the value of \( g(1000) \) if \( f(x) \) is such that \( g(x) \) approaches \(-\infty\) as \( x \to \in... | [
"1 - \\ln 1000"
] | [
"A function \\( f(x) \\) defined on \\( x > 0 \\) satisfies the following conditions:\n- \\( f(1) = 1 \\)\n- \\( f'(x) \\) is continuous on \\( x > 0 \\)\n- \\( f'(x) \\leq \\frac{f(x)}{x} - 1 \\)\n\nLet \\( g(x) = \\frac{f(x)}{x} \\). Find the value of \\( g(1000) \\) if \\( f(x) \\) is such that \\( g(x) \\) appr... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d71790d402f73d6b93e8ce3794a759894925ffece6bc8a9b3d64c0717a3d8b73 | A function \( f(x) \) defined on real numbers with real values satisfies \( f(\cot x) = \cos 2x + \sin 2x \) for all \( x \in (0, \pi) \). Find the maximum and minimum values of \( g(x) = f(x)(1 - x) \) for all \( x \in [-1, 1] \). | [
"Maximum value is \\( 0 \\) and minimum value is \\( -2 \\)."
] | [
"A function \\( f(x) \\) defined on real numbers with real values satisfies \\( f(\\cot x) = \\cos 2x + \\sin 2x \\) for all \\( x \\in (0, \\pi) \\). Find the maximum and minimum values of \\( g(x) = f(x)(1 - x) \\) for all \\( x \\in [-1, 1] \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b36ef365be1830f0b003fff1a71cafa7810513db75979bfbae36ec01016a4ff0 | A function \( f(x) \) defined on the positive integers satisfies \( f(1) = 1996 \) and \[ f(1) + f(2) + \cdots + f(n) = n^2 f(n). \] Calculate \( f(1996) \). | [
"\\(\\frac{1996}{3991}\\)"
] | [
"A function \\( f(x) \\) defined on the positive integers satisfies \\( f(1) = 1996 \\) and \n\\[ f(1) + f(2) + \\cdots + f(n) = n^2 f(n). \\]\nCalculate \\( f(1996) \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2dc450324c1a50586d5546fa8286324980fdb9d0feb430f662c2785c826ac6fb | A function \( f(x) \) is defined for all real numbers \( x \). For all non-zero values \( x \), we have \[ 2f(x) + f\left(\frac{1}{x}\right) = 5x + 4. \] Let \( S \) denote the sum of all values of \( x \) for which \( f(x) = 2004 \). Compute the integer nearest to \( S \). | [
"601"
] | [
"A function \\( f(x) \\) is defined for all real numbers \\( x \\). For all non-zero values \\( x \\), we have\n\\[ 2f(x) + f\\left(\\frac{1}{x}\\right) = 5x + 4. \\]\nLet \\( S \\) denote the sum of all values of \\( x \\) for which \\( f(x) = 2004 \\). Compute the integer nearest to \\( S \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3bdf5ba74e247575363a1dfafb1a3a3499d707955a1a2563f7355abaac4bfb6e | A function \( f(x) \) is defined for all real numbers and satisfies the condition \( f(x+2) = f(x) \) for all real \( x \). If \( x = 1 \) is a zero of \( f(x) \), how many zeros does \( f(x) \) have in the interval \([0, 100]\)? | [
"50"
] | [
"A function \\( f(x) \\) is defined for all real numbers and satisfies the condition \\( f(x+2) = f(x) \\) for all real \\( x \\). If \\( x = 1 \\) is a zero of \\( f(x) \\), how many zeros does \\( f(x) \\) have in the interval \\([0, 100]\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-f76c5c2242318518832cae649022fe7811430e4f28d6f45d5d2b5dc75745345e | A function \( f(x) \) is defined on the set of real numbers with the following properties: - \( f(2) = 7 \) - \( f(x+2) \le f(x) + 2 \) - \( f(x+7) \ge f(x) + 7 \) Find \( f(2008) \). | [
"2013"
] | [
"A function \\( f(x) \\) is defined on the set of real numbers with the following properties:\n- \\( f(2) = 7 \\)\n- \\( f(x+2) \\le f(x) + 2 \\)\n- \\( f(x+7) \\ge f(x) + 7 \\)\n\nFind \\( f(2008) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-6975df87ed9534c637e480c456f605205c44ecdf56b7333348e6757046826ab1 | A function \( f(x) \) is defined on the set of real numbers with the following properties: a) \( f(2) = 7 \) b) \( f(x+2) \leq f(x) + 2 \) for all real numbers \( x \) c) \( f(x+7) \geq f(x) + 7 \) for all real numbers \( x \) Find \( f(2008) \). | [
"2013"
] | [
"A function \\( f(x) \\) is defined on the set of real numbers with the following properties:\na) \\( f(2) = 7 \\)\nb) \\( f(x+2) \\leq f(x) + 2 \\) for all real numbers \\( x \\)\nc) \\( f(x+7) \\geq f(x) + 7 \\) for all real numbers \\( x \\)\n\nFind \\( f(2008) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-435d22e3d6511e788f3e9bfcd41476c543cfe70b4e283e1e8a0f3a5a6c7d666b | A function \( f(x) \) is such that \( f(a \cdot b) = f\left(\frac{a}{b}\right) \) if and only if \( a > b \) and \( \frac{a}{b} \) is a positive integer. If \( f(48) = 5 \), what is \( f(27) \)? | [
"5"
] | [
"A function \\( f(x) \\) is such that \\( f(a \\cdot b) = f\\left(\\frac{a}{b}\\right) \\) if and only if \\( a > b \\) and \\( \\frac{a}{b} \\) is a positive integer. If \\( f(48) = 5 \\), what is \\( f(27) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b54429a9c061e02a56aed3ef766429cf0e63cbc96915e579d95bc69ce6dd06c6 | A function \( f(x) \) satisfies \( f(x) = f\left(\frac{c}{x}\right) \) for some real number \( c > 1 \) and all positive numbers \( x \). If \( \int_1^{\sqrt{c}} \frac{f(x)}{x} \, dx = 3 \), evaluate \( \int_1^c \frac{f(x)}{x} \, dx \). | [
"6"
] | [
"A function \\( f(x) \\) satisfies \\( f(x) = f\\left(\\frac{c}{x}\\right) \\) for some real number \\( c > 1 \\) and all positive numbers \\( x \\). If \\( \\int_1^{\\sqrt{c}} \\frac{f(x)}{x} \\, dx = 3 \\), evaluate \\( \\int_1^c \\frac{f(x)}{x} \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-410dbadc2797a6697794e8b3775cd6b698b05c60a4b7fc91a7b1dbc9d4229e19 | A function \( f(x) \) satisfies the differential equation \[ f(x) = -f''(x) - (4x - 2)f'(x) \] with initial conditions \( f(0) = a \) and \( f(1) = b \). Evaluate the integral \[ \int_0^1 f(x)(x^2 - x) \, dx. \] | [
"\\(- \\frac{b+a}{11}\\)"
] | [
"A function \\( f(x) \\) satisfies the differential equation\n\\[ f(x) = -f''(x) - (4x - 2)f'(x) \\]\nwith initial conditions \\( f(0) = a \\) and \\( f(1) = b \\). Evaluate the integral\n\\[ \\int_0^1 f(x)(x^2 - x) \\, dx. \\]"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 13
} | 35 | 0 | 0 |
omr-ws-b724abd82e083eabdb2cdefcc36e90a186e33eb5f4ae657fae4381c550714721 | A function \( f(x) \) satisfies the equation \[ f(x) + f'(x) + f''(x) + \ldots + f^{(n)}(x) = x^n. \] Find the value of \[ f(x) + \frac{f'(x)}{1!} + \frac{f''(x)}{2!} + \ldots + \frac{f^{(n)}(x)}{n!}. \] | [
"\\( x^n \\)"
] | [
"A function \\( f(x) \\) satisfies the equation\n\\[ f(x) + f'(x) + f''(x) + \\ldots + f^{(n)}(x) = x^n. \\]\nFind the value of\n\\[ f(x) + \\frac{f'(x)}{1!} + \\frac{f''(x)}{2!} + \\ldots + \\frac{f^{(n)}(x)}{n!}. \\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e2168793f5916319ca712ec344c31ec18a75b4690458d8fec043f97e5718f90 | A function \( f(x) \) satisfies the following conditions: \[ f(1) = \frac{1}{403} \] \[ f(x)f(2-x) = x^2 f(x-2) \] Compute the value of \( f(2015) \). | [
"\\( 2015^2 \\cdot 403^{1007} \\)"
] | [
"A function \\( f(x) \\) satisfies the following conditions:\n\\[ f(1) = \\frac{1}{403} \\]\n\\[ f(x)f(2-x) = x^2 f(x-2) \\]\nCompute the value of \\( f(2015) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-477991d0688245b5b8fb5f0a906c481a06508871fb60a448bf50e30e5e8fd3c0 | A function \( f(x) \) with a domain of positive integers satisfies \( f(1) = a \) and \[ f(n) = 2^{n-1} f(1) + 2^{n-2} f(2) + \cdots + 2 f(n-1) \] for \( n > 1 \). Given that \( \log_{f(8)} f(17) = 1.3 \) and \( a \) is a positive integer, find the minimum possible value of \( m + n \) where \( a \) can be expressed as... | [
"334"
] | [
"A function \\( f(x) \\) with a domain of positive integers satisfies \\( f(1) = a \\) and\n\\[ f(n) = 2^{n-1} f(1) + 2^{n-2} f(2) + \\cdots + 2 f(n-1) \\]\nfor \\( n > 1 \\). Given that \\( \\log_{f(8)} f(17) = 1.3 \\) and \\( a \\) is a positive integer, find the minimum possible value of \\( m + n \\) where \\( ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5a5d7b007b537ade995ba778e7a03a037806c8293b9f167c31bed62631da358d | A function \( f(x) \), whose domain is the interval \([1, +\infty)\), has the following properties: 1. \( f(1) = \frac{1}{2008} \); 2. \( f(x) + 2007(f(x+1))^2 = f(x+1) \) for all \( x \in [1, +\infty) \). Find the limit \( \lim_{n \to +\infty} \left( \frac{f(2)}{f(1)} + \frac{f(3)}{f(2)} + \cdots + \frac{f(n+1)}{f(n)}... | [
"\\dfrac{2008}{2007}"
] | [
"A function \\( f(x) \\), whose domain is the interval \\([1, +\\infty)\\), has the following properties:\n1. \\( f(1) = \\frac{1}{2008} \\);\n2. \\( f(x) + 2007(f(x+1))^2 = f(x+1) \\) for all \\( x \\in [1, +\\infty) \\).\n\nFind the limit \\( \\lim_{n \\to +\\infty} \\left( \\frac{f(2)}{f(1)} + \\frac{f(3)}{f(2)}... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-0ead2dab18281ef8b125b86bfa9c0d7791adfd46931cbeb794c5838e5134362a | A function \( f(x) \), whose domain is the interval \([1, +\infty)\), has the following properties: i) \( f(1) = \frac{1}{2008} \) ii) \( f(x) + 2007(f(x+1))^2 = f(x+1) \) for all \( x \in [1, +\infty) \) Find the limit \( \lim_{n \to +\infty} \left( \frac{f(2)}{f(1)} + \frac{f(3)}{f(2)} + \cdots + \frac{f(n+1)}{f(n)} ... | [
"2007"
] | [
"A function \\( f(x) \\), whose domain is the interval \\([1, +\\infty)\\), has the following properties:\ni) \\( f(1) = \\frac{1}{2008} \\)\nii) \\( f(x) + 2007(f(x+1))^2 = f(x+1) \\) for all \\( x \\in [1, +\\infty) \\)\n\nFind the limit \\( \\lim_{n \\to +\\infty} \\left( \\frac{f(2)}{f(1)} + \\frac{f(3)}{f(2)} ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-193b9994bf29632253063afaa67d266cf029b085fae61b981cd3c3b1da104d9a | A function \( f: [0,1] \rightarrow \mathbb{R} \) satisfies \( f(x) \geq 0 \) for \( 0 < x < 1 \), \( f(1) = 1 \), and \( f(x+y) \geq f(x) + f(y) \) for \( x, y, x+y \in [0,1] \). Define \( g(x) = 2x - f(x) \). Find the maximum value of \( g(x) \) for \( x \in [0,1] \). | [
"2"
] | [
"A function \\( f: [0,1] \\rightarrow \\mathbb{R} \\) satisfies \\( f(x) \\geq 0 \\) for \\( 0 < x < 1 \\), \\( f(1) = 1 \\), and \\( f(x+y) \\geq f(x) + f(y) \\) for \\( x, y, x+y \\in [0,1] \\). Define \\( g(x) = 2x - f(x) \\). Find the maximum value of \\( g(x) \\) for \\( x \\in [0,1] \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-b19866f92cc8f694415ef598470d8fb210b7d4fb8e2c91d5036fadf0c43142bc | A function \( f: [0,1] \to \mathbb{R} \) which is increasing satisfies the following conditions: (i) \( f(0) = 0 \) (ii) \( f(x/3) = f(x)/2 \) (iii) \( f(1-x) = 1 - f(x) \) Find \( f(18/1991) \). | [
"\\dfrac{5}{128}"
] | [
"A function \\( f: [0,1] \\to \\mathbb{R} \\) which is increasing satisfies the following conditions:\n(i) \\( f(0) = 0 \\)\n(ii) \\( f(x/3) = f(x)/2 \\)\n(iii) \\( f(1-x) = 1 - f(x) \\)\nFind \\( f(18/1991) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-5200fcd728e681d9345749f425c7edd2a79ad91d5e35c08f08fe56b3ec858b50 | A function \( f: \mathbb{N} \longrightarrow \mathbb{N} \) satisfies the following conditions: a) \( f(xy) = f(x) + f(y) - 1 \) for all \( x, y \in \mathbb{N} \) b) There are finite numbers such that \( f(x) = 1 \) c) \( f(30) = 4 \) Find the value of \( f(14400) \). | [
"11"
] | [
"A function \\( f: \\mathbb{N} \\longrightarrow \\mathbb{N} \\) satisfies the following conditions:\na) \\( f(xy) = f(x) + f(y) - 1 \\) for all \\( x, y \\in \\mathbb{N} \\)\nb) There are finite numbers such that \\( f(x) = 1 \\)\nc) \\( f(30) = 4 \\)\n\nFind the value of \\( f(14400) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 1
} | 21 | 0 | 0 |
omr-ws-52b028d88ba6f333918eb364cdb4a500aa763125a9a8823b0933c7b1fe025451 | A function \( f: \mathbb{N} \rightarrow \mathbb{N} \) is called *beautiful* if it satisfies the following conditions for all distinct positive integers \( x \) and \( y \): \[ S(\gcd(f(x), f(y))) = S(\gcd(x, y)) \] \[ S(\text{lcm}(f(x), f(y))) = S(\text{lcm}(x, y)) \] where \( S(x) \) denotes the sum of the digits of t... | [
"\\( f(x) = x \\cdot 10^v \\) for \\( \\gcd(x, 10) > 1 \\) and \\( f(x) = x \\cdot 10^{e_x} \\) for \\( \\gcd(x, 10) = 1 \\) where \\( e_x \\le v \\)."
] | [
"A function \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) is called *beautiful* if it satisfies the following conditions for all distinct positive integers \\( x \\) and \\( y \\):\n\\[ S(\\gcd(f(x), f(y))) = S(\\gcd(x, y)) \\]\n\\[ S(\\text{lcm}(f(x), f(y))) = S(\\text{lcm}(x, y)) \\]\nwhere \\( S(x) \\) denotes... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-44e6be293175726f0973c877a3c4ebf3cd6d38e302e7269cf7ae4a51293e0468 | A function \( f: \mathbb{N} \rightarrow \mathbb{N} \) satisfies the following conditions: (a) \( f(ab) = f(a)f(b) \) for any two coprime positive integers \( a \) and \( b \); (b) \( f(p + q) = f(p) + f(q) \) for any two primes \( p \) and \( q \). Calculate the value of \( f(2) + f(3) + f(1999) \). | [
"2004"
] | [
"A function \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) satisfies the following conditions:\n(a) \\( f(ab) = f(a)f(b) \\) for any two coprime positive integers \\( a \\) and \\( b \\);\n(b) \\( f(p + q) = f(p) + f(q) \\) for any two primes \\( p \\) and \\( q \\).\n\nCalculate the value of \\( f(2) + f(3) + f(1... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b74d0df7727bc86e1ba0cbbfde3225c434d98419ee290d9358fac35d16fe2c0 | A function \( f: \mathbb{N} \rightarrow \mathbb{N} \) satisfies the following properties: - \( f(xy) = f(x) + f(y) - 1 \) - There exist a finite number of \( x \) such that \( f(x) = 1 \) - \( f(30) = 4 \) Determine \( f(14400) \). | [
"11"
] | [
"A function \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) satisfies the following properties:\n- \\( f(xy) = f(x) + f(y) - 1 \\)\n- There exist a finite number of \\( x \\) such that \\( f(x) = 1 \\)\n- \\( f(30) = 4 \\)\n\nDetermine \\( f(14400) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 8,
"tir": 2
} | 26 | 0 | 0 |
omr-ws-4dc3d88c0c0f40a2457b004ad94578fb4ad1b5c7664429111999a0390e51a6f7 | A function \( f: \mathbb{N} \to \mathbb{N} \) is defined such that \( f(x) \) is the number of divisors of \( x \). Find the least value of \( x \) such that \( f(x) = 2016 \). This value can be written as \( a \cdot b \cdot b \), where \( a \) has no square divisors. Determine the value of \( b \). | [
"120"
] | [
"A function \\( f: \\mathbb{N} \\to \\mathbb{N} \\) is defined such that \\( f(x) \\) is the number of divisors of \\( x \\). Find the least value of \\( x \\) such that \\( f(x) = 2016 \\). This value can be written as \\( a \\cdot b \\cdot b \\), where \\( a \\) has no square divisors. Determine the value of \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 8,
"tir": 12
} | 26 | 0 | 0 |
omr-ws-f393423a1dfffce87131fa536ee713e2b51dc95d747412647e0d5d6ad17ca713 | A function \( f: \mathbb{N} \to \mathbb{N} \) is given. If \( a \) and \( b \) are coprime, then \( f(ab) = f(a)f(b) \). Additionally, if \( m \) and \( k \) are primes (not necessarily different), then \( f(m+k-3) = f(m) + f(k) - f(3) \). Find all possible values of \( f(11) \). | [
"1 or 11"
] | [
"A function \\( f: \\mathbb{N} \\to \\mathbb{N} \\) is given. If \\( a \\) and \\( b \\) are coprime, then \\( f(ab) = f(a)f(b) \\). Additionally, if \\( m \\) and \\( k \\) are primes (not necessarily different), then \\( f(m+k-3) = f(m) + f(k) - f(3) \\). Find all possible values of \\( f(11) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 4
} | 5 | 0 | 0 |
omr-ws-f289e9930ac746b0e35d72c920e994b6ebcac556f9dc069acfb1d2f53df5ef7d | A function \( f: \mathbb{N} \to \mathbb{N} \) satisfies the following conditions: 1. \( f(a+b) = f(f(a) + b) \) 2. \( f(a+b) = f(a) + f(b) \) if \( 0 \leq a+b < 10 \) 3. \( f(10) = 1 \) How many three-digit numbers \( n \) are there such that \( f(n) = f(2^{3^{4^5}}) \)? | [
"100"
] | [
"A function \\( f: \\mathbb{N} \\to \\mathbb{N} \\) satisfies the following conditions:\n1. \\( f(a+b) = f(f(a) + b) \\)\n2. \\( f(a+b) = f(a) + f(b) \\) if \\( 0 \\leq a+b < 10 \\)\n3. \\( f(10) = 1 \\)\n\nHow many three-digit numbers \\( n \\) are there such that \\( f(n) = f(2^{3^{4^5}}) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 3
} | 51 | 0 | 0 |
omr-ws-cf8c80374092988a7cb906e76f617167d0966faa27949d672d1276081654b718 | A function \( f: \mathbb{N} \to \mathbb{R} \) is called a MODEL FUNCTION if it satisfies the following conditions: 1. \( f(mn) = f(m)f(n) + f(m) + f(n) \) for all relatively prime positive integers \( m \) and \( n \). 2. \( \sum_{d \mid n} f(d) = n - \tau(n) \) for all \( n \geq 100 \), where \( \tau(n) \) is the numb... | [
"894"
] | [
"A function \\( f: \\mathbb{N} \\to \\mathbb{R} \\) is called a MODEL FUNCTION if it satisfies the following conditions:\n1. \\( f(mn) = f(m)f(n) + f(m) + f(n) \\) for all relatively prime positive integers \\( m \\) and \\( n \\).\n2. \\( \\sum_{d \\mid n} f(d) = n - \\tau(n) \\) for all \\( n \\geq 100 \\), where... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-83aaffd0972ca39bc0b0a116fa90a78ab0ba2f1b6fb2ce68cb46b92414d2e78e | A function \( f: \mathbb{N}_0 \to \mathbb{R} \) satisfies \( f(1) = 3 \) and \( f(m+n) + f(m-n) - m + n - 1 = \frac{f(2m) + f(2n)}{2} \) for any nonnegative integers \( m, n \) with \( m \ge n \). Find all such functions \( f \). | [
"\\( f(x) = x^2 + x + 1 \\)"
] | [
"A function \\( f: \\mathbb{N}_0 \\to \\mathbb{R} \\) satisfies \\( f(1) = 3 \\) and \\( f(m+n) + f(m-n) - m + n - 1 = \\frac{f(2m) + f(2n)}{2} \\) for any nonnegative integers \\( m, n \\) with \\( m \\ge n \\). Find all such functions \\( f \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 17
} | 31 | 0 | 0 |
omr-ws-55bae25803d598cd3617c58f7cbbfb717911aaf6515090804294c53841454b8c | A function \( f: \mathbb{R^+} \rightarrow \mathbb{R^+} \) satisfies the following conditions for all positive real numbers \( x \): \[ f(x^3) = f(x)^3 \] \[ f(2x) = f(x) \] Find all possible values of \( f(\sqrt[2022]{2}) \). | [
"1"
] | [
"A function \\( f: \\mathbb{R^+} \\rightarrow \\mathbb{R^+} \\) satisfies the following conditions for all positive real numbers \\( x \\):\n\\[ f(x^3) = f(x)^3 \\]\n\\[ f(2x) = f(x) \\]\nFind all possible values of \\( f(\\sqrt[2022]{2}) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-95263b6351df50f2ea14bc8bc2ed47116ccaf3f5fe35287429acfee58a680d9f | A function \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfies \( f(0) = 0 \), \( f(1) = 1 \), and \( f\left(x^2 + \frac{1}{x}\right) = f(x)^2 + f\left(\frac{1}{x}\right) \) for all \( x \neq 0 \). Find the smallest positive integer \( n \) such that \( f(n) > 1000 \). | [
"1001"
] | [
"A function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfies \\( f(0) = 0 \\), \\( f(1) = 1 \\), and \\( f\\left(x^2 + \\frac{1}{x}\\right) = f(x)^2 + f\\left(\\frac{1}{x}\\right) \\) for all \\( x \\neq 0 \\). Find the smallest positive integer \\( n \\) such that \\( f(n) > 1000 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fceb78b3d1fe4a3f5fc30f4375d8a724b0982040ac9c45b9d1800bc5927e908c | A function \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfies \( f(x+y) = f(x) f(y) \) for all \( x, y \in \mathbb{R} \) and \( f \) is continuous at \( 0 \). Given that \( f(0) = 1 \), find the value of \( f(2) \) if \( f(1) = 3 \). | [
"9"
] | [
"A function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfies \\( f(x+y) = f(x) f(y) \\) for all \\( x, y \\in \\mathbb{R} \\) and \\( f \\) is continuous at \\( 0 \\). Given that \\( f(0) = 1 \\), find the value of \\( f(2) \\) if \\( f(1) = 3 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-870e3ee7abb63a7d4ee66ff78fe866d603d7b1dd0bdee922b801125570e2d266 | A function \( f: \mathbb{R} \to \mathbb{R} \) satisfies \( f(20 - x) = f(22 + x) \) for all real \( x \). It is known that \( f \) has exactly two roots. What is the sum of these two roots? | [
"42"
] | [
"A function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) satisfies \\( f(20 - x) = f(22 + x) \\) for all real \\( x \\). It is known that \\( f \\) has exactly two roots. What is the sum of these two roots?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-68afd645f32b9671f08cb86dde3fb9b241ab134f4d906e22e4b51df1f932f590 | A function \( f: \mathbb{R} \to \mathbb{R} \) satisfies the following conditions: 1. \( f(x) = 0 \) for negative \( x \). 2. \( f \) is piecewise-constant with a countable discrete set of discontinuities. 3. The Laplace transform \( F(p) = \int_0^{\infty} e^{-pt} f(t) \, dt \) for \( \Re(p) > 0 \) satisfies the formula... | [
"\\dfrac{\\ln 2}{\\ln 6}"
] | [
"A function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) satisfies the following conditions:\n1. \\( f(x) = 0 \\) for negative \\( x \\).\n2. \\( f \\) is piecewise-constant with a countable discrete set of discontinuities.\n3. The Laplace transform \\( F(p) = \\int_0^{\\infty} e^{-pt} f(t) \\, dt \\) for \\( \\Re(p) > ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ed1a9bc52d87e22820f97901e61c37a64df9aaecbcbe40d8719d76292ffe21eb | A function \( f: \mathbb{Z} \rightarrow \mathbb{R} \) satisfies the following conditions for any \( m, n \in \mathbb{Z} \): 1) \( f(mn) = f(m)f(n) \) 2) \( f(m+n) \le f(m) + f(n) \) 3) \( 0 \le f(m) \le 1 \) Let \( g(n) \) be the maximum value of \( f(m+n) \) for all \( m \in \mathbb{Z} \). Calculate \( g(1) \). | [
"1"
] | [
"A function \\( f: \\mathbb{Z} \\rightarrow \\mathbb{R} \\) satisfies the following conditions for any \\( m, n \\in \\mathbb{Z} \\):\n1) \\( f(mn) = f(m)f(n) \\)\n2) \\( f(m+n) \\le f(m) + f(n) \\)\n3) \\( 0 \\le f(m) \\le 1 \\)\n\nLet \\( g(n) \\) be the maximum value of \\( f(m+n) \\) for all \\( m \\in \\mathbb... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c53905f451343e18d5c1bde9bf0ebbb62d58703c4c474127d55f472fb580dcea | A function \( f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+ \) satisfies the following conditions: 1. \( m \ge n \implies f(m) \ge f(n) \) 2. If \(\gcd(m, n) = 1\), then \( f(mn) = f(m) \cdot f(n) \) Let \( S \) be the set of all positive integers \( n \) such that \( f(n) \neq f(1) \). Determine the sum of the first 10 ele... | [
"65"
] | [
"A function \\( f: \\mathbb{Z}^+ \\rightarrow \\mathbb{Z}^+ \\) satisfies the following conditions:\n1. \\( m \\ge n \\implies f(m) \\ge f(n) \\)\n2. If \\(\\gcd(m, n) = 1\\), then \\( f(mn) = f(m) \\cdot f(n) \\)\n\nLet \\( S \\) be the set of all positive integers \\( n \\) such that \\( f(n) \\neq f(1) \\). Dete... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-94cb8930142530ab9798deb9974dea5554dff6d5b2a3e2f23502a5489161325c | A function \( f: \mathbb{Z}^+ \times \mathbb{Z}^+ \to \mathbb{Z}^+ \) is defined by the following properties: - \( f(x, x) = x + 2 \) - \( f(x, y) = f(y, x) \) - \( \frac{f(x, x+y)}{f(x, y)} = \frac{x+y}{y} \) Find a closed expression for \( f(x, y) \) in terms of \( x \) and \( y \). | [
"\\( \\frac{xy \\left( \\gcd(x, y) + 2 \\right)}{\\left( \\gcd(x, y) \\right)^2} \\)"
] | [
"A function \\( f: \\mathbb{Z}^+ \\times \\mathbb{Z}^+ \\to \\mathbb{Z}^+ \\) is defined by the following properties:\n- \\( f(x, x) = x + 2 \\)\n- \\( f(x, y) = f(y, x) \\)\n- \\( \\frac{f(x, x+y)}{f(x, y)} = \\frac{x+y}{y} \\)\n\nFind a closed expression for \\( f(x, y) \\) in terms of \\( x \\) and \\( y \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 33,
"tir": 12
} | 45 | 0 | 0 |
omr-ws-9df0992b485aff3565f7f9d12b13f09237b63843b78a5ea2668019ff93e37d3d | A function \( f: \{ 1,2,3,\cdots ,2016\} \rightarrow \{ 1,2,3,\cdots , 2016\} \) is called good if the function \( g(n) = |f(n) - n| \) is injective. A good function \( f \) is called excellent if there exists another good function \( f' \) such that \( f(n) - f'(n) \) is nonzero for exactly one value of \( n \). Let \... | [
"0"
] | [
"A function \\( f: \\{ 1,2,3,\\cdots ,2016\\} \\rightarrow \\{ 1,2,3,\\cdots , 2016\\} \\) is called good if the function \\( g(n) = |f(n) - n| \\) is injective. A good function \\( f \\) is called excellent if there exists another good function \\( f' \\) such that \\( f(n) - f'(n) \\) is nonzero for exactly one v... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-218c70210ce43f2a0f2d6cc6fd507fe093801d461344d39d9876fa88f0247ea2 | A function \( f_n(x) \) for \( n = 0, 1, 2, 3, \ldots \) satisfies the following conditions: (i) \( f_0(x) = e^{2x} + 1 \). (ii) \( f_n(x) = \int_0^x (n + 2t) f_{n-1}(t) \, dt - \frac{2x^{n+1}}{n+1} \) for \( n = 1, 2, 3, \ldots \). Find \( \sum_{n=1}^{\infty} f_n'\left(\frac{1}{2}\right) \). | [
"\\(4 + 6e\\)"
] | [
"A function \\( f_n(x) \\) for \\( n = 0, 1, 2, 3, \\ldots \\) satisfies the following conditions:\n(i) \\( f_0(x) = e^{2x} + 1 \\).\n(ii) \\( f_n(x) = \\int_0^x (n + 2t) f_{n-1}(t) \\, dt - \\frac{2x^{n+1}}{n+1} \\) for \\( n = 1, 2, 3, \\ldots \\).\n\nFind \\( \\sum_{n=1}^{\\infty} f_n'\\left(\\frac{1}{2}\\right)... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-8343b1298bdbdc40421f1627f1942d40182e9c202dc283620e0af606df2b2ae8 | A function \( f_n(x) \) is defined by \( f_1(x) = x \) and \[ f_n(x) = x + \frac{e}{2} \int_0^1 f_{n-1}(t) e^{x-t} \, dt \quad (n=2, 3, \cdots) \]. Find \( f_n(x) \). | [
"\\( f_n(x) = x + \\left(\\left(\\frac{2}{e}\\right)^{1 - n} - 1\\right)e^{x} \\)"
] | [
"A function \\( f_n(x) \\) is defined by \\( f_1(x) = x \\) and \n\\[ f_n(x) = x + \\frac{e}{2} \\int_0^1 f_{n-1}(t) e^{x-t} \\, dt \\quad (n=2, 3, \\cdots) \\].\nFind \\( f_n(x) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-5e2fb8b7c500818f62c52c4ddb8384e79ed97481285cb2f41426fd085f5beae1 | A function \( g \) defined for all positive integers \( n \) satisfies the following conditions: - \( g(1) = 1 \); - For all \( n \ge 1 \), either \( g(n+1) = g(n) + 1 \) or \( g(n+1) = g(n) - 1 \); - For all \( n \ge 1 \), \( g(3n) = g(n) \); - \( g(k) = 2001 \) for some positive integer \( k \). Find the smallest pos... | [
"2 \\times 3^{1000} - 1"
] | [
"A function \\( g \\) defined for all positive integers \\( n \\) satisfies the following conditions:\n- \\( g(1) = 1 \\);\n- For all \\( n \\ge 1 \\), either \\( g(n+1) = g(n) + 1 \\) or \\( g(n+1) = g(n) - 1 \\);\n- For all \\( n \\ge 1 \\), \\( g(3n) = g(n) \\);\n- \\( g(k) = 2001 \\) for some positive integer \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-bbc3c7666c06de592108c401c82cb4a5e0031d448c2ac9bbc99ab7468b82e19d | A function \( g(x) \) defined for all \( x > 0 \) satisfies \( g(1) = 1 \) and \( g'(x^2) = x^3 \) for all \( x > 0 \). Find \( g(4) \). | [
"\\(\\frac{67}{5}\\)"
] | [
"A function \\( g(x) \\) defined for all \\( x > 0 \\) satisfies \\( g(1) = 1 \\) and \\( g'(x^2) = x^3 \\) for all \\( x > 0 \\). Find \\( g(4) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-ac5df0e121f7c34ab58c4670790accdd5d2cf750cdf846cc9fcd37bac0669aef | A function \( g: [[1,n]] \to [[0,n]] \) is called "good" if for every \( j \in [[1,n]] \), there exists an integer \( i \geq 0 \) such that \( g^{i}(j) = 0 \), where \( g^{i} \) denotes the \( i \)-fold composition of \( g \). How many such "good" functions exist? | [
"\\( (n+1)^{n-1} \\)"
] | [
"A function \\( g: [[1,n]] \\to [[0,n]] \\) is called \"good\" if for every \\( j \\in [[1,n]] \\), there exists an integer \\( i \\geq 0 \\) such that \\( g^{i}(j) = 0 \\), where \\( g^{i} \\) denotes the \\( i \\)-fold composition of \\( g \\). How many such \"good\" functions exist?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 39
} | 39 | 0 | 0 |
omr-ws-2c18572be7c464cc22a4ba363a009da657d64f40b0182692b63412facd21d762 | A function \( g: \mathbb{N} \to \mathbb{N} \) satisfies the following conditions: 1. If \( m \) is a proper divisor of \( n \), then \( g(m) < g(n) \). 2. If \( m \) and \( n \) are relatively prime and greater than 1, then \[ g(mn) = g(m)g(n) + (n+1)g(m) + (m+1)g(n) + m + n. \] Find the least possible value of \( g(20... | [
"6131"
] | [
"A function \\( g: \\mathbb{N} \\to \\mathbb{N} \\) satisfies the following conditions:\n1. If \\( m \\) is a proper divisor of \\( n \\), then \\( g(m) < g(n) \\).\n2. If \\( m \\) and \\( n \\) are relatively prime and greater than 1, then\n\\[ g(mn) = g(m)g(n) + (n+1)g(m) + (m+1)g(n) + m + n. \\]\nFind the least... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9e48a62ce070b4e76c7ae7cb9cab9ded7a00d015463455cd9df66d43c39e0f1 | A function \( g: \mathbb{Z} \to \mathbb{Z} \) is called [i]adjective[/i] if \( g(m) + g(n) > \max(m^2, n^2) \) for any pair of integers \( m \) and \( n \). Let \( f \) be an [i]adjective[/i] function such that the value of \( f(1) + f(2) + f(3) + \cdots + f(30) \) is minimized. Find the smallest possible value of \( f... | [
"498"
] | [
"A function \\( g: \\mathbb{Z} \\to \\mathbb{Z} \\) is called [i]adjective[/i] if \\( g(m) + g(n) > \\max(m^2, n^2) \\) for any pair of integers \\( m \\) and \\( n \\). Let \\( f \\) be an [i]adjective[/i] function such that the value of \\( f(1) + f(2) + f(3) + \\cdots + f(30) \\) is minimized. Find the smallest ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-87394f7431d19e377c8ec1f0a118b2dbb6f66b1c1de1201c5ac8df2f5c1dd369 | A function \( g: \mathbb{Z} \to \mathbb{Z} \) is called adjective if \( g(m) + g(n) > \max(m^2, n^2) \) for any pair of integers \( m \) and \( n \). Let \( f \) be an adjective function such that the value of \( f(1) + f(2) + \dots + f(30) \) is minimized. Find the smallest possible value of \( f(25) \). | [
"498"
] | [
"A function \\( g: \\mathbb{Z} \\to \\mathbb{Z} \\) is called adjective if \\( g(m) + g(n) > \\max(m^2, n^2) \\) for any pair of integers \\( m \\) and \\( n \\). Let \\( f \\) be an adjective function such that the value of \\( f(1) + f(2) + \\dots + f(30) \\) is minimized. Find the smallest possible value of \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e0d9bba4b43d29f0955a19a09cde58eb5c7426bc516464ee125611532916e46 | A function \( y = f(x) \) is defined as follows: 1) \( f(x) = |2x| \) where \( -1 \leq x < 1 \). 2) For all \( x \), \( f(x+2) = f(x) \). Let the intersection of the function \( y = f(x) \) and \( g(x) = \frac{1}{n} \log_2(x) \) be \( a_n \). Find all values of \( n \) such that \( 15 \leq a_n < 255 \). | [
"n \\geq \\dfrac{1}{2} \\log_2 15"
] | [
"A function \\( y = f(x) \\) is defined as follows:\n1) \\( f(x) = |2x| \\) where \\( -1 \\leq x < 1 \\).\n2) For all \\( x \\), \\( f(x+2) = f(x) \\).\n\nLet the intersection of the function \\( y = f(x) \\) and \\( g(x) = \\frac{1}{n} \\log_2(x) \\) be \\( a_n \\). Find all values of \\( n \\) such that \\( 15 \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-b3d84c6a7469447149339b0898c641273604283539f1534d06941e0933ba3e35 | A function \( y(x) \) satisfies the differential equation: \[ 2xy = x^3y^2 + x^2y' + 5y^2 \] with the initial condition \( y(2) = 1 \). What is the greatest integer less than or equal to \( y(1) \)? | [
"-1"
] | [
"A function \\( y(x) \\) satisfies the differential equation: \n\\[ 2xy = x^3y^2 + x^2y' + 5y^2 \\]\nwith the initial condition \\( y(2) = 1 \\). What is the greatest integer less than or equal to \\( y(1) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-51c16fab0b66fa6d93888cb25331ff1eec76b09d5dae63808992bf345fb9a6fc | A function \(f : \mathbb{N}_0 \to \mathbb{R}\) satisfies \(f(1) = 3\) and \[ f(m + n) + f(m - n) - m + n - 1 = \frac{f(2m) + f(2n)}{2}, \] for any nonnegative integers \(m, n\) with \(m \geq n\). Find all such functions \(f\). | [
"f(n) = n^2 + n + 1"
] | [
"A function \\(f : \\mathbb{N}_0 \\to \\mathbb{R}\\) satisfies \\(f(1) = 3\\) and \n\\[ f(m + n) + f(m - n) - m + n - 1 = \\frac{f(2m) + f(2n)}{2}, \\]\nfor any nonnegative integers \\(m, n\\) with \\(m \\geq n\\). Find all such functions \\(f\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d906a51856889e184c46bc08b9b5e9c7daa298a9b5c9458d6ac426cdc4cb2b9b | A function \(f(x)\) is defined for all real numbers \(x\). For all non-zero values \(x\), we have \[ 2f(x) + f\left(\frac{1}{x}\right) = 5x + 4. \] Let \(S\) denote the sum of all values of \(x\) for which \(f(x) = 2004\). Compute the integer nearest to \(S\). | [
"601"
] | [
"A function \\(f(x)\\) is defined for all real numbers \\(x\\). For all non-zero values \\(x\\), we have\n\\[\n2f(x) + f\\left(\\frac{1}{x}\\right) = 5x + 4.\n\\]\nLet \\(S\\) denote the sum of all values of \\(x\\) for which \\(f(x) = 2004\\). Compute the integer nearest to \\(S\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-460a7579c1df851f3bfc1b780da692a04d059273d50cc2ef2f33f0cd2f99d6ce | A function \(g : \mathbb{N} \rightarrow \mathbb{N}\) satisfies the following conditions: (a) If \(m\) is a proper divisor of \(n\), then \(g(m) < g(n)\). (b) If \(m\) and \(n\) are relatively prime and greater than 1, then \(g(mn) = g(m)g(n) + (n+1)g(m) + (m+1)g(n) + m + n\). Find the least possible value of \(g(2016)\... | [
"3053"
] | [
"A function \\(g : \\mathbb{N} \\rightarrow \\mathbb{N}\\) satisfies the following conditions:\n(a) If \\(m\\) is a proper divisor of \\(n\\), then \\(g(m) < g(n)\\).\n(b) If \\(m\\) and \\(n\\) are relatively prime and greater than 1, then \\(g(mn) = g(m)g(n) + (n+1)g(m) + (m+1)g(n) + m + n\\).\n\nFind the least p... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 4
} | 26 | 0 | 0 |
omr-ws-32654de208e3872e8b146c532216b3fb77c116336c56e39304d3770e8e733885 | A function \(h : \mathbb{R} \rightarrow \mathbb{R}\) is differentiable and satisfies \(h(ax) = bh(x)\) for all \(x\), where \(a\) and \(b\) are given positive numbers and \(0 \neq |a| \neq 1\). Suppose that \(h'(0) \neq 0\) and the function \(h'\) is continuous at \(x = 0\). Determine the value of \(a + b\) given that ... | [
"2"
] | [
"A function \\(h : \\mathbb{R} \\rightarrow \\mathbb{R}\\) is differentiable and satisfies \\(h(ax) = bh(x)\\) for all \\(x\\), where \\(a\\) and \\(b\\) are given positive numbers and \\(0 \\neq |a| \\neq 1\\). Suppose that \\(h'(0) \\neq 0\\) and the function \\(h'\\) is continuous at \\(x = 0\\). Determine the v... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d838fff906c2343700eacbbf444ba665044930e7a968ab0b3dd9f3f8886add01 | A function defined over the positive rationals satisfies \( f(a \cdot b) = f(a) + f(b) \) for all positive rationals \( a, b \). Further, \( f(p) = p \) for all primes \( p \). For which of the following numbers \( x \) is \( f(x) < 0 \)? Choices: \[ \textbf{(A) } \text{forgot} \qquad \textbf{(B) } \text{forgot} \qquad... | [
"\\(\\frac{25}{11}\\)"
] | [
"A function defined over the positive rationals satisfies \\( f(a \\cdot b) = f(a) + f(b) \\) for all positive rationals \\( a, b \\). Further, \\( f(p) = p \\) for all primes \\( p \\). For which of the following numbers \\( x \\) is \\( f(x) < 0 \\)?\n\nChoices:\n\\[\n\\textbf{(A) } \\text{forgot} \\qquad \\textb... | [
"c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b12cc88ac09d38392adaba4f6097e483e710e18c7d4f3debbe9b7b317557283f | A function on the whole numbers is defined such that \( f(0) = 1 \) and \[ f(n)f(n-1) = \frac{2n}{(2n+1)} \] for all positive integers \( n \). There exists \( N \) positive integers \( p \) between 1 and 2021 inclusive such that when \( f(p) \) is represented as \( \frac{a}{b} \) where \( a \) and \( b \) are relative... | [
"959"
] | [
"A function on the whole numbers is defined such that \\( f(0) = 1 \\) and \n\\[ f(n)f(n-1) = \\frac{2n}{(2n+1)} \\]\nfor all positive integers \\( n \\). There exists \\( N \\) positive integers \\( p \\) between 1 and 2021 inclusive such that when \\( f(p) \\) is represented as \\( \\frac{a}{b} \\) where \\( a \\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 12
} | 20 | 0 | 0 |
omr-ws-eceb19c094a3cbb317481b8905951585c912a9bcfa949ac94d06f6eb5764f143 | A furniture company has warehouses at Albany, Buffalo, Montreal, and Toronto. Deliveries need to be made to Kingston, Rochester, and Syracuse. Each warehouse has one truck which can visit at most one city in a day. The constraints are: (i) The Albany warehouse does not serve Kingston and Rochester; (ii) The Buffalo war... | [
"2"
] | [
"A furniture company has warehouses at Albany, Buffalo, Montreal, and Toronto. Deliveries need to be made to Kingston, Rochester, and Syracuse. Each warehouse has one truck which can visit at most one city in a day. The constraints are:\n(i) The Albany warehouse does not serve Kingston and Rochester;\n(ii) The Buff... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da012c090df801fcf43739cd82ff0e5bd521940405fb6392ea6fe8565a3683e5 | A furniture company has warehouses at Albany, Buffalo, Montreal, and Toronto. Deliveries need to be made to Kingston, Rochester, and Syracuse. Each warehouse has one truck which can visit at most one city in a day. The constraints are: - Albany does not serve Kingston and Rochester. - Buffalo does not serve Kingston. -... | [
"21"
] | [
"A furniture company has warehouses at Albany, Buffalo, Montreal, and Toronto. Deliveries need to be made to Kingston, Rochester, and Syracuse. Each warehouse has one truck which can visit at most one city in a day. The constraints are:\n- Albany does not serve Kingston and Rochester.\n- Buffalo does not serve King... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac9a52bbc04b4bf6546560ac253e0b864f0060edca751e676cd522579e968d09 | A furniture retailer receives a shipment of sixteen sofas. The retailer inspects a random sample of six items from the shipment and will accept delivery if at most one sampled sofa is defective. Suppose instead that the manufacturer knowingly sends a shipment with four defective sofas. What is the probability that the ... | [
"\\dfrac{93}{182}"
] | [
"A furniture retailer receives a shipment of sixteen sofas. The retailer inspects a random sample of six items from the shipment and will accept delivery if at most one sampled sofa is defective. Suppose instead that the manufacturer knowingly sends a shipment with four defective sofas. What is the probability that... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0af7c770f15a6f9c2cbff83d92f06976dc5b50ffe145cc89589abfbdbd8ada20 | A furniture retailer receives a shipment of sixteen sofas. The retailer inspects a random sample of six items from the shipment and will accept delivery if at most one sampled sofa is defective. Suppose now instead of sending 4 defective sofas, the manufacturer sends 2 defective instead, but the retailer found none to ... | [
"\\(\\frac{1}{3}\\)"
] | [
"A furniture retailer receives a shipment of sixteen sofas. The retailer inspects a random sample of six items from the shipment and will accept delivery if at most one sampled sofa is defective. Suppose now instead of sending 4 defective sofas, the manufacturer sends 2 defective instead, but the retailer found non... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-b987a61ce0d3ba41925ba5973a2548efe21915c77ee0258665924e265f9cdc5f | A furniture retailer receives a shipment of sixteen sofas. The retailer inspects a random sample of six items from the shipment and will accept delivery if at most one sampled sofa is defective. Suppose that the factory has a 22% defective rate when manufacturing sofas. What is the probability that a shipment of sixtee... | [
"0.6063"
] | [
"A furniture retailer receives a shipment of sixteen sofas. The retailer inspects a random sample of six items from the shipment and will accept delivery if at most one sampled sofa is defective. Suppose that the factory has a 22% defective rate when manufacturing sofas. What is the probability that a shipment of s... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-cd9ed1247c95b94caf812cfef7e527589fac7d11196cb88b61fe9f943c738691 | A galaxy contains more than one million stars. The distances between all pairs of stars are recorded. Determine the minimum number of distinct distances that must exist among these stars. | [
"10000"
] | [
"A galaxy contains more than one million stars. The distances between all pairs of stars are recorded. Determine the minimum number of distinct distances that must exist among these stars."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-62505abbed08ad430f0a781758850937d0777ba6802e0474a3e4d516d65a047e | A gambler has $25$ dollars. Each turn, if the gambler has a positive amount of money, a fair coin is flipped: if it is heads, the gambler gains a dollar, and if it is tails, the gambler loses a dollar. If the gambler has no money, he will automatically be given a dollar (which counts as a turn). What is the expected nu... | [
"1875"
] | [
"A gambler has $25$ dollars. Each turn, if the gambler has a positive amount of money, a fair coin is flipped: if it is heads, the gambler gains a dollar, and if it is tails, the gambler loses a dollar. If the gambler has no money, he will automatically be given a dollar (which counts as a turn). What is the expect... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-58ef09ee379697b67e3799edb02055959b061d31ffdd77aa2dfb3a9dcc4f9110 | A gambler has $25$ dollars. Each turn, if the gambler has a positive amount of money, a fair coin is flipped: if it is heads, the gambler gains a dollar, and if it is tails, the gambler loses a dollar. If the gambler has no money, he will automatically be given a dollar (which counts as a turn). What is the expected nu... | [
"1875"
] | [
"A gambler has $25$ dollars. Each turn, if the gambler has a positive amount of money, a fair coin is flipped: if it is heads, the gambler gains a dollar, and if it is tails, the gambler loses a dollar. If the gambler has no money, he will automatically be given a dollar (which counts as a turn). What is the expect... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7c74eb6896c559663eb91d96c2d898fdec2cc084fae4b563313ef574d4f5eec9 | A gambler has a fair coin and a two-headed coin. He selects one at random and flips it twice. If he gets two heads, what is the probability that he selected the fair coin? | [
"\\(\\frac{1}{5}\\)"
] | [
"A gambler has a fair coin and a two-headed coin. He selects one at random and flips it twice. If he gets two heads, what is the probability that he selected the fair coin?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-aaa058f4359f286afb7014a89f0f2438d5cf03511687ca561755c94438fa609c | A gambler offers you a $\$2$ ticket to play a game where you pick a real number $0 \le p \le 1$. You are given a weighted coin with probability $p$ of coming up heads and probability $1 - p$ of coming up tails, and flip this coin twice. The first time the coin comes up heads, you receive $\$1$, and the first time it co... | [
"\\(\\frac{7}{3}\\)"
] | [
"A gambler offers you a $\\$2$ ticket to play a game where you pick a real number $0 \\le p \\le 1$. You are given a weighted coin with probability $p$ of coming up heads and probability $1 - p$ of coming up tails, and flip this coin twice. The first time the coin comes up heads, you receive $\\$1$, and the first t... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-295fbba5c4d1e4adfd037637755c5034c400982ad5fa67257a42f14da59955ab | A gambler plays a game \( n \) times, with the probability of winning each game being \( p \). If it is known that the gambler did not win \( x \) times, what is the expected number of times the gambler won? | [
"n - x"
] | [
"A gambler plays a game \\( n \\) times, with the probability of winning each game being \\( p \\). If it is known that the gambler did not win \\( x \\) times, what is the expected number of times the gambler won?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e666728a094d4c9fe6041bd95b0dfe832d3b5efe5c82670946e19ca31d6574a6 | A gambler plays a game where he commits half of his capital on each game with a probability \( p \) of winning. If he wins, he gains an amount equal to his committed capital; if he loses, he loses the committed capital. Define a "losing position" as having less capital than he started with after \( n \) games. What is ... | [
"\\sum_{k=0}^{\\left\\lfloor n \\log_3 2 \\right\\rfloor} \\binom{n}{k} p^k (1-p)^{n-k}"
] | [
"A gambler plays a game where he commits half of his capital on each game with a probability \\( p \\) of winning. If he wins, he gains an amount equal to his committed capital; if he loses, he loses the committed capital. Define a \"losing position\" as having less capital than he started with after \\( n \\) game... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-070a8ee856b7e06d183f5aa30f99bdd38d8a6ceb5f74d06abec46bdae38c1d57 | A gambler starts with $ r$ dollars and aims to reach $ a$ dollars. At each turn, he chooses an integer $ s$ between $ 1$ and $ \min(r, a-r)$ and tosses a fair coin. If the coin comes up heads, he wins $ s$ dollars; if tails, he loses $ s$ dollars. The game ends if he runs out of money (loses) or reaches $ a$ dollars (w... | [
"\\dfrac{1}{2}"
] | [
"A gambler starts with $ r$ dollars and aims to reach $ a$ dollars. At each turn, he chooses an integer $ s$ between $ 1$ and $ \\min(r, a-r)$ and tosses a fair coin. If the coin comes up heads, he wins $ s$ dollars; if tails, he loses $ s$ dollars. The game ends if he runs out of money (loses) or reaches $ a$ doll... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8d4426d6d013879b71434a2e320f9ba51001902f35a09c64791261ebc788df75 | A gambler starts with $n$ dollars and plays a game for infinitely many rounds. The payoff of the $t^{th}$ round, $X_t$, is an integer between $-1000$ and $1000$. It is known that $\mathbb{E}[X_t|X_{t-1},\cdots,X_1] \geq c$, where $c$ is a positive constant. Let $p(n)$ denote the probability that the gambler goes bankru... | [
"0"
] | [
"A gambler starts with $n$ dollars and plays a game for infinitely many rounds. The payoff of the $t^{th}$ round, $X_t$, is an integer between $-1000$ and $1000$. It is known that $\\mathbb{E}[X_t|X_{t-1},\\cdots,X_1] \\geq c$, where $c$ is a positive constant. Let $p(n)$ denote the probability that the gambler goe... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-70b958331bfde2db3d51658549e41c3bb00a6b17412e3422c07e36979e25ad86 | A gambler starts with $x$ forints, where $0 < x < 2C$. He uses the following strategy in a coin-tossing game: if his capital is $y < C$, he bets all of it; if his capital is $y > C$, he bets $2C - y$. If he has exactly $2C$ forints, he stops playing. Let $f(x)$ be the probability that he reaches $2C$ before going bankr... | [
"\\( \\frac{x}{2C} \\)"
] | [
"A gambler starts with $x$ forints, where $0 < x < 2C$. He uses the following strategy in a coin-tossing game: if his capital is $y < C$, he bets all of it; if his capital is $y > C$, he bets $2C - y$. If he has exactly $2C$ forints, he stops playing. Let $f(x)$ be the probability that he reaches $2C$ before going ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
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