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-f7b72a39ca364fcdc8e9a8e1de66af399729e2534fbf38b8f902a37f81b85792 | A sequence \( b_1, b_2, b_3, \ldots \) of nonzero real numbers has the property that \( b_{n+2} = \frac{b_{n+1}^2 - 1}{b_n} \) for all positive integers \( n \). Suppose that \( b_1 = 1 \) and \( b_2 = k \) where \( 1 < k < 2 \). Determine the smallest integer \( n \) such that \( b_n > 2020 \) for \( k = \frac{3}{2} \... | [
"2024"
] | [
"A sequence \\( b_1, b_2, b_3, \\ldots \\) of nonzero real numbers has the property that \\( b_{n+2} = \\frac{b_{n+1}^2 - 1}{b_n} \\) for all positive integers \\( n \\). Suppose that \\( b_1 = 1 \\) and \\( b_2 = k \\) where \\( 1 < k < 2 \\). Determine the smallest integer \\( n \\) such that \\( b_n > 2020 \\) f... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6dc8728bc235cf772f296244622a556488af16be050770ce69ef880155e545a8 | A sequence \( b_1, b_2, b_3, \ldots \) of nonzero real numbers has the property that \[ b_{n+2} = \frac{b_{n+1}^2 - 1}{b_n} \] for all positive integers \( n \). Suppose that \( b_1 = 1 \) and \( b_2 = k \) where \( 1 < k < 2 \). Determine the maximum value of \( |b_{10}| \) for \( k = \frac{3}{2} \). | [
"\\dfrac{627}{512}"
] | [
"A sequence \\( b_1, b_2, b_3, \\ldots \\) of nonzero real numbers has the property that\n\\[ b_{n+2} = \\frac{b_{n+1}^2 - 1}{b_n} \\]\nfor all positive integers \\( n \\). Suppose that \\( b_1 = 1 \\) and \\( b_2 = k \\) where \\( 1 < k < 2 \\). Determine the maximum value of \\( |b_{10}| \\) for \\( k = \\frac{3}... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf45aafa9236f6f13f2615801a70a0ecdf7673c550ef23a0e64812a0d1aa3401 | A sequence \( b_1, b_2, b_3, \ldots \) of nonzero real numbers has the property that \[ b_{n+2} = \frac{b_{n+1}^2 - 1}{b_n} \] for all positive integers \( n \). Suppose that \( b_1 = 1 \) and \( b_2 = k \) where \( 1 < k < 2 \). Find the smallest integer \( n \) such that \( b_n > 2020 \) for \( k = \frac{3}{2} \). | [
"no_valid_answer_found"
] | [
"A sequence \\( b_1, b_2, b_3, \\ldots \\) of nonzero real numbers has the property that\n\\[ b_{n+2} = \\frac{b_{n+1}^2 - 1}{b_n} \\]\nfor all positive integers \\( n \\). Suppose that \\( b_1 = 1 \\) and \\( b_2 = k \\) where \\( 1 < k < 2 \\). Find the smallest integer \\( n \\) such that \\( b_n > 2020 \\) for ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ec63f8f41954142d900a4c38760c8f85dc9bb29d81e2bede6d3ff6424a74a7be | A sequence \( t_1, t_2, t_3, \ldots, t_n \) is defined as follows: \( t_1 = 1 \), \( t_2 = 4 \), and \( t_k = t_{k-1} + t_{k-2} \) for \( k = 3, 4, \ldots, n \). Let \( T \) be the set of all terms in this sequence; that is, \( T = \{ t_1, t_2, t_3, \ldots, t_n \} \). How many positive integers can be expressed as the ... | [
"\\(\\binom{n}{3}\\)"
] | [
"A sequence \\( t_1, t_2, t_3, \\ldots, t_n \\) is defined as follows: \\( t_1 = 1 \\), \\( t_2 = 4 \\), and \\( t_k = t_{k-1} + t_{k-2} \\) for \\( k = 3, 4, \\ldots, n \\). Let \\( T \\) be the set of all terms in this sequence; that is, \\( T = \\{ t_1, t_2, t_3, \\ldots, t_n \\} \\). How many positive integers ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df28ba390dac25b7912b4ae7c6bf51a2fbf596f0a57f0cf284360dac39989192 | A sequence \( t_{1}, t_{2}, \ldots, t_{n}, \ldots \) is defined as follows: \[ t_{1} = 14 \] \[ t_{k} = 24 - 5t_{k-1} \text{ for each } k \ge 2. \] For every positive integer \( n \), \( t_{n} \) can be expressed as \( p \cdot q^n + r \), where \( p, q, \) and \( r \) are constants. Find \( p + q + r \). | [
"-3"
] | [
"A sequence \\( t_{1}, t_{2}, \\ldots, t_{n}, \\ldots \\) is defined as follows:\n\\[ t_{1} = 14 \\]\n\\[ t_{k} = 24 - 5t_{k-1} \\text{ for each } k \\ge 2. \\]\nFor every positive integer \\( n \\), \\( t_{n} \\) can be expressed as \\( p \\cdot q^n + r \\), where \\( p, q, \\) and \\( r \\) are constants. Find \\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-5801a1d3175988412eb2bbf52a97312240da73bc9dbcd1c94e5aa33e84856857 | A sequence \( x_1, x_2, \cdots \) is defined by \( x_1 = 10 \) and \[ x_n = 3n \cdot x_{n-1} + n! - 3^n \cdot (n^2 - 2) \] for integers \( n \ge 2 \). Find a closed form for \( x_n \) that does not include summations. | [
"\\( x_n = \\frac{1}{2}3^n n! - \\frac{1}{2}n! + (n+2)3^n \\)"
] | [
"A sequence \\( x_1, x_2, \\cdots \\) is defined by \\( x_1 = 10 \\) and \n\\[ x_n = 3n \\cdot x_{n-1} + n! - 3^n \\cdot (n^2 - 2) \\]\nfor integers \\( n \\ge 2 \\). Find a closed form for \\( x_n \\) that does not include summations."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 43,
"tir": 13
} | 56 | 0 | 0 |
omr-ws-ec86ec772a76c963f5ad917ae18369df586687ba4446c9e4052bc253961c9890 | A sequence \( x_1, x_2, \ldots, x_n, \ldots \) consists of an initial block of \( p \) positive distinct integers that then repeat periodically. Given that \( 1 \le p \le 10 \), find the least \( n \) such that there is a strategy which allows to find \( p \) by revealing at most \( n \) terms of the sequence. | [
"4"
] | [
"A sequence \\( x_1, x_2, \\ldots, x_n, \\ldots \\) consists of an initial block of \\( p \\) positive distinct integers that then repeat periodically. Given that \\( 1 \\le p \\le 10 \\), find the least \\( n \\) such that there is a strategy which allows to find \\( p \\) by revealing at most \\( n \\) terms of t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b38ab222334c6d8045bbba4530381a18fd61b55585c09fc9b611b3ffd76cd65d | A sequence \( x_1, x_2, \ldots, x_n, \ldots \) consists of an initial block of \( p \) positive distinct integers that then repeat periodically. Given that \( p \) is one of the first \( k \) prime numbers, find for which values of \( k \) there exists a strategy that allows to find \( p \) by revealing at most 5 terms... | [
"\\( k \\le 52 \\)"
] | [
"A sequence \\( x_1, x_2, \\ldots, x_n, \\ldots \\) consists of an initial block of \\( p \\) positive distinct integers that then repeat periodically. Given that \\( p \\) is one of the first \\( k \\) prime numbers, find for which values of \\( k \\) there exists a strategy that allows to find \\( p \\) by reveal... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dee9f11eb715aa5e7e4a324a0a4e42a3ed35f65a14ff7c8dfe9429efbbc6f643 | A sequence \( x_1, x_2, \ldots, x_n, \ldots \) is defined by \( x_1 = 2 \) and \( x_{n+1} = \left(1 + \frac{1}{n}\right) x_n + \frac{2}{n} \) for every natural number \( n \). The value of \( x_{2013} \) is... | [
"4026"
] | [
"A sequence \\( x_1, x_2, \\ldots, x_n, \\ldots \\) is defined by \\( x_1 = 2 \\) and \\( x_{n+1} = \\left(1 + \\frac{1}{n}\\right) x_n + \\frac{2}{n} \\) for every natural number \\( n \\). The value of \\( x_{2013} \\) is..."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-51e93b4b70992a0e6da528706e0ae26440d96ed828e001ad78723ee46ffa40ee | A sequence \( x_n \) for \( n \geq 0 \) satisfies \( x_{n+1} = x_n + \frac{1}{\sqrt{x_n}} \). Find \( \lim_{n \to \infty} x_n \) if \( x_0 > 0 \). | [
"\\infty"
] | [
"A sequence \\( x_n \\) for \\( n \\geq 0 \\) satisfies \\( x_{n+1} = x_n + \\frac{1}{\\sqrt{x_n}} \\). Find \\( \\lim_{n \\to \\infty} x_n \\) if \\( x_0 > 0 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-80a0f70b55eb13202b3adffd8153e925cac99baa80169f19a98d15d6fb2234c1 | A sequence \( x_n \) is defined by \( x_1 = a \), \( x_2 = b \), and \( x_n = x_{n-1} + x_{n-2} \) for \( n \ge 3 \), where \( a \) and \( b \) are integers and \( 1 \le a, b \le 9 \). Compute the number of possible values of \( a \) and \( b \) for which the sequence \( x_n \) contains the term 10. | [
"15"
] | [
"A sequence \\( x_n \\) is defined by \\( x_1 = a \\), \\( x_2 = b \\), and \\( x_n = x_{n-1} + x_{n-2} \\) for \\( n \\ge 3 \\), where \\( a \\) and \\( b \\) are integers and \\( 1 \\le a, b \\le 9 \\). Compute the number of possible values of \\( a \\) and \\( b \\) for which the sequence \\( x_n \\) contains th... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2ce56f1395164d64c606a809f13bab20faef4cc16a107ae7aadb42b2e93f8f3f | A sequence \( x_n \) is defined by the rules: \( x_1 = 2 \) and \( nx_n = 2(2n - 1)x_{n-1} \) for \( n = 2, 3, \ldots \). Calculate the value of \( x_5 \). | [
"252"
] | [
"A sequence \\( x_n \\) is defined by the rules: \\( x_1 = 2 \\) and \\( nx_n = 2(2n - 1)x_{n-1} \\) for \\( n = 2, 3, \\ldots \\). Calculate the value of \\( x_5 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ab1895d7ee333873a64964729288229cbbb3c674bfcc4941558d5bd0de2946b | A sequence \( x_n \) of natural numbers satisfies \( x_{n+1} = x_n + x_{n-1} + 1 \) for all \( n \geq 2 \) with initial conditions \( x_1 = x_2 = 1 \). Find all pairs of natural numbers \( (n, m) \) such that \( x_n = 2^m - 1 \). | [
"(1, 1), (2, 1), (6, 4)"
] | [
"A sequence \\( x_n \\) of natural numbers satisfies \\( x_{n+1} = x_n + x_{n-1} + 1 \\) for all \\( n \\geq 2 \\) with initial conditions \\( x_1 = x_2 = 1 \\). Find all pairs of natural numbers \\( (n, m) \\) such that \\( x_n = 2^m - 1 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4864bc2a76ac2f729a27998f8178afc5ca54ca86b0ca477d2e51e90c61219e17 | A sequence \((a_i)_{1\leq i\leq n}\) of positive integers is called beautiful if \(a_k\) is the number of multiples of \(k\) found in \((a_i)_{1\leq i\leq n}\), for all \(1\leq k\leq n\). Find all beautiful sequences. | [
"[2, 2]"
] | [
"A sequence \\((a_i)_{1\\leq i\\leq n}\\) of positive integers is called beautiful if \\(a_k\\) is the number of multiples of \\(k\\) found in \\((a_i)_{1\\leq i\\leq n}\\), for all \\(1\\leq k\\leq n\\). Find all beautiful sequences."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-313232e903c4e503e279008ad62edf7a850faeb50943c6ba2f7354012831595e | A sequence \((a_n)\) is defined with \(a_1 = 1\) and the recursion: In the decimal expansion of \(a_n\) (without trailing zeros), let \(k\) be the smallest digit, then \(a_{n+1} = a_n + 2^k\). How many digits does \(a_{9 \cdot 10^{2010}}\) have in its decimal expansion? | [
"2011"
] | [
"A sequence \\((a_n)\\) is defined with \\(a_1 = 1\\) and the recursion: In the decimal expansion of \\(a_n\\) (without trailing zeros), let \\(k\\) be the smallest digit, then \\(a_{n+1} = a_n + 2^k\\). How many digits does \\(a_{9 \\cdot 10^{2010}}\\) have in its decimal expansion?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 20,
"genselect": 3
} | 23 | 0 | 0 |
omr-ws-1279887ae51f9a33b08cca61e318fe005743a591e18ace10cc36f99867e16182 | A sequence \((a_n)_{n \ge 1}\) of integers is defined by the recurrence relation \[a_{n+1} + (-1)^n a_n = 2n - 1.\] Find the value of the sum \[\sum_{n=1}^{60} a_n.\] | [
"1830"
] | [
"A sequence \\((a_n)_{n \\ge 1}\\) of integers is defined by the recurrence relation\n\\[a_{n+1} + (-1)^n a_n = 2n - 1.\\]\nFind the value of the sum\n\\[\\sum_{n=1}^{60} a_n.\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-7977168c9dda68298fed607c1f398b5401d90026a44b9da8c798b152c288f5f3 | A sequence \((c_1, c_2, \ldots, c_n)\) of positive integers is called admissible if each integer \(k\) with \(1 \leq k \leq 2c_1 + \cdots + 2c_n\) can be represented in the form \(k = a_1c_1 + \cdots + a_nc_n\) where each \(a_i \in \{-2, -1, 0, 1, 2\}\). For each \(n\), find the maximum value of \(c_1 + \cdots + c_n\). | [
"\\dfrac{5^n - 1}{4}"
] | [
"A sequence \\((c_1, c_2, \\ldots, c_n)\\) of positive integers is called admissible if each integer \\(k\\) with \\(1 \\leq k \\leq 2c_1 + \\cdots + 2c_n\\) can be represented in the form \\(k = a_1c_1 + \\cdots + a_nc_n\\) where each \\(a_i \\in \\{-2, -1, 0, 1, 2\\}\\). For each \\(n\\), find the maximum value o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-7c7c1135e1fdd2271967c8ba9374f2dea764023f88ece3a1dd67fa1ef672168e | A sequence \((p_n)\) of natural numbers satisfies: 1. The numbers \( p_1 \) and \( p_2 \) are primes. 2. For every \( n \geq 3 \), the number \( p_n \) is the greatest prime divisor of the number \( p_{n-1} + p_{n-2} + 2000 \). Let \( M \) be the maximum value of \( p_n \) for \( n \geq 1 \). Calculate the smallest pos... | [
"no_valid_answer_found"
] | [
"A sequence \\((p_n)\\) of natural numbers satisfies:\n1. The numbers \\( p_1 \\) and \\( p_2 \\) are primes.\n2. For every \\( n \\geq 3 \\), the number \\( p_n \\) is the greatest prime divisor of the number \\( p_{n-1} + p_{n-2} + 2000 \\).\nLet \\( M \\) be the maximum value of \\( p_n \\) for \\( n \\geq 1 \\)... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4029e348d775302b0c4c5da866a9a3694f05aa39c0bd0de98d9f98ffab448382 | A sequence \((u_n)\) is defined by \( u_0 = 2 \), \( u_1 = \frac{5}{2} \), and \( u_{n+1} = u_n(u_{n-1}^2 - 2) - u_1 \) for \( n \geq 1 \). Let \( f(n) = [u_n] \), where \([x]\) denotes the greatest integer less than or equal to \( x \). Find the value of \( f(5) \). | [
"2048"
] | [
"A sequence \\((u_n)\\) is defined by \\( u_0 = 2 \\), \\( u_1 = \\frac{5}{2} \\), and \\( u_{n+1} = u_n(u_{n-1}^2 - 2) - u_1 \\) for \\( n \\geq 1 \\). Let \\( f(n) = [u_n] \\), where \\([x]\\) denotes the greatest integer less than or equal to \\( x \\). Find the value of \\( f(5) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-119161c328b4a586f0c4179531cf8cbdf34dea07f8ff6b2d60a48dec1939ec20 | A sequence \((u_n)\) is defined by \( u_0 = 2 \), \( u_1 = \frac{5}{2} \), and \( u_{n+1} = u_n(u_{n-1}^2 - 2) - u_1 \) for \( n \geq 1 \). Let \( f(n) = [u_n] \), where \([x]\) denotes the smallest integer less than or equal to \( x \). Find the value of \( f(5) \). | [
"2048"
] | [
"A sequence \\((u_n)\\) is defined by \\( u_0 = 2 \\), \\( u_1 = \\frac{5}{2} \\), and \\( u_{n+1} = u_n(u_{n-1}^2 - 2) - u_1 \\) for \\( n \\geq 1 \\). Let \\( f(n) = [u_n] \\), where \\([x]\\) denotes the smallest integer less than or equal to \\( x \\). Find the value of \\( f(5) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a80e555423fb0fa1d104fb6d4d424ec1483ad7190131d468f41f5d413a23e777 | A sequence \((u_n)\) is defined by \(u_1 = 1\) and \(u_n = 5u_{n-1} + \sqrt{24u_{n-1}^2 - 8}\) for \(n \geq 2\). Find \(u_n\). | [
"\\( u_n = \\left(\\frac{3-\\sqrt{6}}{6}\\right)(5+2\\sqrt{6})^n + \\left(\\frac{3+\\sqrt{6}}{6}\\right)(5-2\\sqrt{6})^n \\)"
] | [
"A sequence \\((u_n)\\) is defined by \\(u_1 = 1\\) and \\(u_n = 5u_{n-1} + \\sqrt{24u_{n-1}^2 - 8}\\) for \\(n \\geq 2\\). Find \\(u_n\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 4,
"tir": 16
} | 35 | 0 | 0 |
omr-ws-3fcc329a4d433fa3eade61cedc99b9047e437714d092428c8a0f356ff5059e1d | A sequence \((u_n)\) is defined by \(u_1 = 2019\) and \(u_{n+1} = \frac{-2019}{n}(u_1 + u_2 + \cdots + u_n)\). Find \(S = u_1 + u_2 + \cdots + u_{2018}\). | [
"-4074342"
] | [
"A sequence \\((u_n)\\) is defined by \\(u_1 = 2019\\) and \\(u_{n+1} = \\frac{-2019}{n}(u_1 + u_2 + \\cdots + u_n)\\). Find \\(S = u_1 + u_2 + \\cdots + u_{2018}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-b6ace744a3e6d45ba97112f9b35f4cb2547123c3f8013841700e9b2aa69d68d3 | A sequence \((u_n)\) is defined by the relation \(u_{m+n} + u_{m-n} = \frac{1}{2}(u_{2m} + u_{2n})\) for \(m \geq n \geq 0\). Find the general term \(u_n\). | [
"\\( u_n = c n^2 \\) where \\( c = u_1 \\in \\mathbb{R} \\)."
] | [
"A sequence \\((u_n)\\) is defined by the relation \\(u_{m+n} + u_{m-n} = \\frac{1}{2}(u_{2m} + u_{2n})\\) for \\(m \\geq n \\geq 0\\). Find the general term \\(u_n\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-766f9a4c96064d22f5b08afdb10c3265634c9f17556f6c87cf0d1485e977e9c1 | A sequence \(\{a_n\}\) is defined as follows: \(a_1 = 1\), \(a_2 = \frac{1}{3}\), and for all \(n \geq 1\), \(\frac{(1+a_n)(1+a_{n+2})}{(1+a_{n+1})^2} = \frac{a_n a_{n+2}}{a_{n+1}^2}\). Calculate the smallest integer \(N\) such that \(a_1 + a_2 + \cdots + a_N \geq \frac{34}{21}\). | [
"7"
] | [
"A sequence \\(\\{a_n\\}\\) is defined as follows: \\(a_1 = 1\\), \\(a_2 = \\frac{1}{3}\\), and for all \\(n \\geq 1\\), \\(\\frac{(1+a_n)(1+a_{n+2})}{(1+a_{n+1})^2} = \\frac{a_n a_{n+2}}{a_{n+1}^2}\\). Calculate the smallest integer \\(N\\) such that \\(a_1 + a_2 + \\cdots + a_N \\geq \\frac{34}{21}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-2ae0f84a1c40026180e93136a6b2711bb415a1f8dd56c93dc9d9491bb3481515 | A sequence \(\{a_n\}\) of positive integers is defined by \(a_1 = 2\) and for integers \(n > 1\), \[ \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_{n-1}} + \frac{n}{a_n} = 1. \] Determine the value of \(\sum_{k=1}^{\infty} \frac{3^k(k+3)}{4^k a_k}.\) | [
"\\(\\frac{21}{8}\\)"
] | [
"A sequence \\(\\{a_n\\}\\) of positive integers is defined by \\(a_1 = 2\\) and for integers \\(n > 1\\),\n\\[\n\\frac{1}{a_1} + \\frac{1}{a_2} + \\cdots + \\frac{1}{a_{n-1}} + \\frac{n}{a_n} = 1.\n\\]\nDetermine the value of \\(\\sum_{k=1}^{\\infty} \\frac{3^k(k+3)}{4^k a_k}.\\)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-10e395503349ccef4c7d8c3659553a87e716ff664c06bc385027443380704714 | A sequence \(\{a_n\}\) satisfies \(a_{n+1} = \frac{a_n}{1 + \sqrt{a_n}}\) for \(n \in \mathbb{N}_+\) with \(a_1 = 1\). Find \(a_n\). | [
"\\dfrac{1}{\\left( \\sqrt{2 + \\sqrt{2 + \\sqrt{2 + \\cdots + \\sqrt{2}}}} \\right)^2}"
] | [
"A sequence \\(\\{a_n\\}\\) satisfies \\(a_{n+1} = \\frac{a_n}{1 + \\sqrt{a_n}}\\) for \\(n \\in \\mathbb{N}_+\\) with \\(a_1 = 1\\). Find \\(a_n\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-5bdec8b3f9948e17678517e0f66e15b74504907e839577c2650fad869ceb372b | A sequence \(\{a_n\}\) satisfies \(a_{n+1} = a_n + \frac{a_n^2}{n^3}\) for \(n \in \mathbb{N}_+\). Find the largest \(a_1 \in \mathbb{R}\) such that the sequence \(\{a_n\}\) is bounded. | [
"2"
] | [
"A sequence \\(\\{a_n\\}\\) satisfies \\(a_{n+1} = a_n + \\frac{a_n^2}{n^3}\\) for \\(n \\in \\mathbb{N}_+\\). Find the largest \\(a_1 \\in \\mathbb{R}\\) such that the sequence \\(\\{a_n\\}\\) is bounded."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 22,
"genselect": 4
} | 26 | 0 | 0 |
omr-ws-64e36ce927644c0734a81b57b030d3d4e43638bf79f188ec3a117d9fa0db0760 | A sequence \(\{u_n\}\) of real numbers is given by \(u_1\) and, for \(n \geq 1\), by the recurrence relation \(4u_{n+1} = (64u_n + 15)^{1/3}\). Describe the behavior of \(u_n\) as \(n \to \infty\). | [
"\\(u_n \\to -\\frac{1}{4}, \\frac{1 + \\sqrt{61}}{8}, \\text{ or } \\frac{1 - \\sqrt{61}}{8}\\) depending on the initial value \\(u_1\\)."
] | [
"A sequence \\(\\{u_n\\}\\) of real numbers is given by \\(u_1\\) and, for \\(n \\geq 1\\), by the recurrence relation \\(4u_{n+1} = (64u_n + 15)^{1/3}\\). Describe the behavior of \\(u_n\\) as \\(n \\to \\infty\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 6,
"tir": 26
} | 33 | 0 | 0 |
omr-ws-31ec9e2cf4161d516c2fb1386b9fe68c34f9652738c17303afd1f85f1ee24b18 | A sequence \(a_0, a_1, \dots, a_n, \dots\) of positive integers is constructed as follows: - If the last digit of \(a_n\) is less than or equal to 5, then this digit is deleted and \(a_{n+1}\) is the number consisting of the remaining digits. (If \(a_{n+1}\) contains no digits, the process stops.) - Otherwise, \(a_{n+1... | [
"No"
] | [
"A sequence \\(a_0, a_1, \\dots, a_n, \\dots\\) of positive integers is constructed as follows:\n- If the last digit of \\(a_n\\) is less than or equal to 5, then this digit is deleted and \\(a_{n+1}\\) is the number consisting of the remaining digits. (If \\(a_{n+1}\\) contains no digits, the process stops.)\n- Ot... | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-fa412a5b8a85695ce851cf375dcdca66cd7fce85bf347d07a106de847ca90750 | A sequence \(a_1, a_2, \cdots, a_{100}\) is called a "good" sequence if each of the integers \(1901, 1902, \cdots, 2000\) appears exactly once in it and none of the sums \(a_1, a_1 + a_2, \cdots, a_1 + a_2 + \cdots + a_{100}\) is divisible by 3. Determine the number of "good" sequences. | [
"\\(34! \\cdot 33! \\cdot 67^{33}\\)"
] | [
"A sequence \\(a_1, a_2, \\cdots, a_{100}\\) is called a \"good\" sequence if each of the integers \\(1901, 1902, \\cdots, 2000\\) appears exactly once in it and none of the sums \\(a_1, a_1 + a_2, \\cdots, a_1 + a_2 + \\cdots + a_{100}\\) is divisible by 3. Determine the number of \"good\" sequences."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cbf5bb012d5f380df419dd7967b4c038ce8446dc04ed1a7e01beafbb640a7a60 | A sequence \(a_1, a_2, \dots, a_n\) is called meanly if \(a_i \in \mathbb{N}\) for all \(i\), \(a_i = a_j\) if and only if \(i = j\), and \(\frac{1}{k}\sum_{j=1}^k a_j \in \mathbb{N}\) for all \(k \leq n\). Find the minimum sum of a meanly sequence of integers of length \(n\). | [
""
] | [
"A sequence \\(a_1, a_2, \\dots, a_n\\) is called meanly if \\(a_i \\in \\mathbb{N}\\) for all \\(i\\), \\(a_i = a_j\\) if and only if \\(i = j\\), and \\(\\frac{1}{k}\\sum_{j=1}^k a_j \\in \\mathbb{N}\\) for all \\(k \\leq n\\). Find the minimum sum of a meanly sequence of integers of length \\(n\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-0126357e3c91bdaf6a27f38cb6414d8bba3e7430f4c5b495fcfb972d92a93faa | A sequence \(a_1, a_2, \dots, a_n\) of positive integers is called *alagoana* if for every positive integer \(n\), the following two conditions hold: I. \(a_{n!} = a_1 \cdot a_2 \cdot a_3 \cdots a_n\) II. The number \(a_n\) is the \(n\)-th power of a positive integer. Find all such *alagoana* sequences. | [
"\\( a_n = 1 \\) for all \\( n \\)."
] | [
"A sequence \\(a_1, a_2, \\dots, a_n\\) of positive integers is called *alagoana* if for every positive integer \\(n\\), the following two conditions hold:\nI. \\(a_{n!} = a_1 \\cdot a_2 \\cdot a_3 \\cdots a_n\\)\nII. The number \\(a_n\\) is the \\(n\\)-th power of a positive integer.\nFind all such *alagoana* sequ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-356cbbd1e8c18463ce04aacebbbc4ea4cb7debff10d8a27e3c664c593442eacd | A sequence \(a_1, a_2, \dots\) is defined as \(a_n = \int_0^{\pi} \cot\left(\frac{x}{2}\right) \sin(nx) \, \mathrm{d}x\). If \(S = \sum_{n=1}^{2013} a_n\), find the value of \(\cos(S)\). | [
"-1"
] | [
"A sequence \\(a_1, a_2, \\dots\\) is defined as \\(a_n = \\int_0^{\\pi} \\cot\\left(\\frac{x}{2}\\right) \\sin(nx) \\, \\mathrm{d}x\\). If \\(S = \\sum_{n=1}^{2013} a_n\\), find the value of \\(\\cos(S)\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-9e8f65c4e41bf8764f7cc528684482ce19b74632fb0d9038f83833b8c4e9addb | A sequence \(a_1, a_2, \dots\) of non-negative integers is defined by the rule \(a_{n+2} = |a_{n+1} - a_n|\) for \(n \geq 1\). If \(a_1 = 999\), \(a_2 < 999\), and \(a_{2006} = 1\), how many different values of \(a_2\) are possible? | [
"648"
] | [
"A sequence \\(a_1, a_2, \\dots\\) of non-negative integers is defined by the rule \\(a_{n+2} = |a_{n+1} - a_n|\\) for \\(n \\geq 1\\). If \\(a_1 = 999\\), \\(a_2 < 999\\), and \\(a_{2006} = 1\\), how many different values of \\(a_2\\) are possible?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-9db87ae4eab4f43322b8c681aaeb2730f52ad94ceb0f905362b6065e6111071c | A sequence \(a_1, a_2, \dots\) satisfies \[ \sum_{i=1}^n a_{\lfloor \frac{n}{i} \rfloor} = n^{10}, \] for every \(n \in \mathbb{N}\). Let \(c\) be a positive integer. Find the smallest positive integer \(n\) such that \[ \frac{c^{a_n} - c^{a_{n-1}}}{n} \] is an integer for all \(c \geq 2\). | [
"2"
] | [
"A sequence \\(a_1, a_2, \\dots\\) satisfies \n\\[\n\\sum_{i=1}^n a_{\\lfloor \\frac{n}{i} \\rfloor} = n^{10},\n\\]\nfor every \\(n \\in \\mathbb{N}\\). Let \\(c\\) be a positive integer. Find the smallest positive integer \\(n\\) such that \n\\[\n\\frac{c^{a_n} - c^{a_{n-1}}}{n}\n\\]\nis an integer for all \\(c \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-1ff975d888c0a3db73f271689820e0c9f08a1646b6852985e30535ecf7f9eba7 | A sequence \(a_1, a_2, \ldots\) of positive integers satisfies \[ a_n = \sqrt{(n+1)a_{n-1} + 1} \quad \text{for all } n \geq 2. \] What are the possible values of \(a_1\)? | [
"1"
] | [
"A sequence \\(a_1, a_2, \\ldots\\) of positive integers satisfies\n\\[ a_n = \\sqrt{(n+1)a_{n-1} + 1} \\quad \\text{for all } n \\geq 2. \\]\nWhat are the possible values of \\(a_1\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb8facb9e5f071b3a7902676ef1887e92d64876f920aa53ac9e20538ab5dcf2f | A sequence \(a_1, a_2, \ldots\) of positive integers satisfies the following properties: - \(a_1 = 1\) - \(a_{3n+1} = 2a_n + 1\) - \(a_{n+1} \ge a_n\) - \(a_{2001} = 200\) Find the value of \(a_{1000}\). | [
"105"
] | [
"A sequence \\(a_1, a_2, \\ldots\\) of positive integers satisfies the following properties:\n- \\(a_1 = 1\\)\n- \\(a_{3n+1} = 2a_n + 1\\)\n- \\(a_{n+1} \\ge a_n\\)\n- \\(a_{2001} = 200\\)\n\nFind the value of \\(a_{1000}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b88845c0087e061c10adf040e7bb5c460410aec2198a899dfed41461766e7fed | A sequence \(a_1, a_2, \ldots\) satisfies \(a_1 = \frac{5}{2}\) and \(a_{n+1} = a_n^2 - 2\) for all \(n \ge 1\). Let \(M\) be the integer which is closest to \(a_{2023}\). What is the last digit of \(M\)? | [
"6"
] | [
"A sequence \\(a_1, a_2, \\ldots\\) satisfies \\(a_1 = \\frac{5}{2}\\) and \\(a_{n+1} = a_n^2 - 2\\) for all \\(n \\ge 1\\). Let \\(M\\) be the integer which is closest to \\(a_{2023}\\). What is the last digit of \\(M\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-fd3f8abe5f7013ce727126fe828e88e5fc0b0ecdb6e0890fe3418e5c264393a0 | A sequence \(a_1, a_2, a_3, \ldots, a_{2n-1}\) is called rugged if the largest element in the set \(\{a_{2k-1}, a_{2k}, a_{2k+1}\}\) is equal to \(a_{2k}\) for all integers \(k \le n\). Let \(N\) be the number of unique rugged permutations of \(\{1, 2, 3, \ldots, 9\}\). Find the remainder when \(N\) is divided by 1000. | [
"936"
] | [
"A sequence \\(a_1, a_2, a_3, \\ldots, a_{2n-1}\\) is called rugged if the largest element in the set \\(\\{a_{2k-1}, a_{2k}, a_{2k+1}\\}\\) is equal to \\(a_{2k}\\) for all integers \\(k \\le n\\). Let \\(N\\) be the number of unique rugged permutations of \\(\\{1, 2, 3, \\ldots, 9\\}\\). Find the remainder when \... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 5,
"tir": 3
} | 11 | 0 | 0 |
omr-ws-3c4afbf35c6ac04af778f3dec4df80a427e30e588e123caea7f5bbf2f94e9e3e | A sequence \(a_1, a_2, a_3, \ldots\) is defined as follows: \(a_1 = 2007\), and \(a_n = a_{n-1} + n \pmod{k}\), where \(0 \leq a_n < k\). For how many values of \(k\), where \(2007 < k < 10^{12}\), does the sequence assume all \(k\) possible values (modulo \(k\) residues)? | [
"3.619 \\times 10^{10}"
] | [
"A sequence \\(a_1, a_2, a_3, \\ldots\\) is defined as follows: \\(a_1 = 2007\\), and \\(a_n = a_{n-1} + n \\pmod{k}\\), where \\(0 \\leq a_n < k\\). For how many values of \\(k\\), where \\(2007 < k < 10^{12}\\), does the sequence assume all \\(k\\) possible values (modulo \\(k\\) residues)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-860fc3daec436907eb66901270401e9d97303b78243e09c3c124ed28ab21bb47 | A sequence \(a_1, a_2, a_3, \ldots\) is defined recursively by \(a_1 = 1\), \(a_2 = 1\), and for \(k \geq 3\), \(a_k = \frac{1}{3}a_{k-1} + \frac{1}{4}a_{k-2}\). Evaluate the sum \(a_1 + a_2 + a_3 + \ldots\). | [
"4"
] | [
"A sequence \\(a_1, a_2, a_3, \\ldots\\) is defined recursively by \\(a_1 = 1\\), \\(a_2 = 1\\), and for \\(k \\geq 3\\), \\(a_k = \\frac{1}{3}a_{k-1} + \\frac{1}{4}a_{k-2}\\). Evaluate the sum \\(a_1 + a_2 + a_3 + \\ldots\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-602ccfd45a864ff6b58940593064f813a934983cb5ef7ace72fc124ddfe83dfb | A sequence \(a_1, a_2, a_3, \ldots\) of positive integers satisfies \(a_1 > 5\) and \(a_{n+1} = 5 + 6 + \cdots + a_n\) for all positive integers \(n\). Determine all prime numbers \(p\) such that, regardless of the value of \(a_1\), this sequence must contain a multiple of \(p\). | [
"2"
] | [
"A sequence \\(a_1, a_2, a_3, \\ldots\\) of positive integers satisfies \\(a_1 > 5\\) and \\(a_{n+1} = 5 + 6 + \\cdots + a_n\\) for all positive integers \\(n\\). Determine all prime numbers \\(p\\) such that, regardless of the value of \\(a_1\\), this sequence must contain a multiple of \\(p\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-d20c982864fd9a8eb7e29f40aef9efa965104428c9caa3355b93bdab42eadaf2 | A sequence \(a_1, a_2, a_3, \ldots\) satisfies the recurrence relation \(a_{n+3} = 2a_{n+2} - 2a_{n+1} + a_n\) for \(n = 1, 2, 3, \ldots\). Given \(a_1 = 222\), \(a_2 = 222\), and \(a_3 = 7777\), find the value of \(a_{2016}\). | [
"7777"
] | [
"A sequence \\(a_1, a_2, a_3, \\ldots\\) satisfies the recurrence relation \\(a_{n+3} = 2a_{n+2} - 2a_{n+1} + a_n\\) for \\(n = 1, 2, 3, \\ldots\\). Given \\(a_1 = 222\\), \\(a_2 = 222\\), and \\(a_3 = 7777\\), find the value of \\(a_{2016}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c81864e06763cbaad6d63734e7cabd469aa231cba01afc174665dd576b967367 | A sequence \(a_n\) is defined by \[a_0 = 0,\] \[a_1 = 3,\] \[a_n = 8a_{n-1} + 9a_{n-2} + 16 \text{ for } n \ge 2.\] Find the least positive integer \(h\) such that \(a_{n+h} - a_n\) is divisible by 1999 for all \(n \ge 0\). | [
"1998"
] | [
"A sequence \\(a_n\\) is defined by\n\\[a_0 = 0,\\]\n\\[a_1 = 3,\\]\n\\[a_n = 8a_{n-1} + 9a_{n-2} + 16 \\text{ for } n \\ge 2.\\]\nFind the least positive integer \\(h\\) such that \\(a_{n+h} - a_n\\) is divisible by 1999 for all \\(n \\ge 0\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-39c4424f0e77308c43a3eebc83dc8e198368377160fc6d34922d8b7b6cfc0e63 | A sequence \(a_n\) is recursively defined where \(a_n = 3(a_{n-1} - 1000)\) for \(n > 0\). Compute the smallest integer \(x\) such that when \(a_0 = x\), \(a_n > a_0\) for all \(n > 0\). | [
"1501"
] | [
"A sequence \\(a_n\\) is recursively defined where \\(a_n = 3(a_{n-1} - 1000)\\) for \\(n > 0\\). Compute the smallest integer \\(x\\) such that when \\(a_0 = x\\), \\(a_n > a_0\\) for all \\(n > 0\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2683068d773a65e88a84238fd5e78f1ae2613ba246c179b706c2650b84db0139 | A sequence \(a_n\) satisfies the property that \(a_{n+1}\) is the smallest triangular number that is relatively prime to each of \(a_1, a_2, \ldots, a_n\). If \(a_1 = 1\), find the remainder when \(a_7\) is divided by 1000. | [
"711"
] | [
"A sequence \\(a_n\\) satisfies the property that \\(a_{n+1}\\) is the smallest triangular number that is relatively prime to each of \\(a_1, a_2, \\ldots, a_n\\). If \\(a_1 = 1\\), find the remainder when \\(a_7\\) is divided by 1000."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 9
} | 32 | 0 | 0 |
omr-ws-efb83a78623afc2d8acb39d8efe1dadb80125226050a82b48f817352343a5379 | A sequence \(a_{1}, a_{2}, a_{3}, \cdots\) is defined by \[a_{n} = 2a_{n}a_{n+1} + 3a_{n+1}\] for all \(n = 1, 2, 3, \cdots\). If \(b_{n} = 1 + \frac{1}{a_{n}}\) for all \(n = 1, 2, \cdots\), find the largest integer \(m\) such that \[\sum_{k=1}^{n} \frac{1}{n + \log_{3}{b_{k}}} > \frac{m}{24}\] for all positive intege... | [
"16"
] | [
"A sequence \\(a_{1}, a_{2}, a_{3}, \\cdots\\) is defined by\n\\[a_{n} = 2a_{n}a_{n+1} + 3a_{n+1}\\]\nfor all \\(n = 1, 2, 3, \\cdots\\). If \\(b_{n} = 1 + \\frac{1}{a_{n}}\\) for all \\(n = 1, 2, \\cdots\\), find the largest integer \\(m\\) such that\n\\[\\sum_{k=1}^{n} \\frac{1}{n + \\log_{3}{b_{k}}} > \\frac{m}{... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-240a95d305bec46fae969abb08b6556153a0b5d8e8e32212c3d8469867a306e6 | A sequence begins with a number $n$. Each term of the sequence is found by squaring the sum of the digits of the previous number. If the sequence begins with the number 32, find the 1337th term. | [
"169"
] | [
"A sequence begins with a number $n$. Each term of the sequence is found by squaring the sum of the digits of the previous number. If the sequence begins with the number 32, find the 1337th term."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-08903f4da098d8ca716d4b8a25b8e004ce5862018db34dde73f1cd6a4a8a80b3 | A sequence begins with two unidentified integers. If the third and subsequent elements of the sequence can be computed by summing the previous two elements, and the $11$th and $12$th elements are $107$ and $173$ respectively, then what are the first two elements of the sequence? | [
"8 and -3"
] | [
"A sequence begins with two unidentified integers. If the third and subsequent elements of the sequence can be computed by summing the previous two elements, and the $11$th and $12$th elements are $107$ and $173$ respectively, then what are the first two elements of the sequence?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 27
} | 30 | 0 | 0 |
omr-ws-b83d75ace006b980e19f3108f8fef0bd3be216a237134c47aed52770a9f705b9 | A sequence consisting of X's and O's is considered basic if every subset of consecutive symbols containing the first symbol satisfies the following conditions: 1. The number of X's isn't three more than the number of O's. 2. The number of O's isn't two more than the number of X's. For example, this sequence would be ba... | [
"17711"
] | [
"A sequence consisting of X's and O's is considered basic if every subset of consecutive symbols containing the first symbol satisfies the following conditions:\n1. The number of X's isn't three more than the number of O's.\n2. The number of O's isn't two more than the number of X's.\nFor example, this sequence wou... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-221b5618e012d28b702d6ac9c0319414f1dfde3b2c3c308a12a4880cc92e3b2b | A sequence consists of $2010$ terms. Each term after the first is 1 larger than the previous term. The sum of the $2010$ terms is $5307$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum? | [
"2151"
] | [
"A sequence consists of $2010$ terms. Each term after the first is 1 larger than the previous term. The sum of the $2010$ terms is $5307$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-017ec2a2755a1a8db2e6c8d5da03d3d03ba065e99b8eebace2b23dfe94d9f6f7 | A sequence consists of \(4n\) complex numbers \(a_1, a_2, \ldots, a_{4n}\) which satisfy the following properties: - For each \(j\), \(a_j = 1, -1, i,\) or \(-i\), where \(i = \sqrt{-1}\). - For each \(k = 1, \ldots, 4n\), the real and imaginary parts of the sum \(a_1 + a_2 + \cdots + a_k\) fall in the interval \([0, 3... | [
"4"
] | [
"A sequence consists of \\(4n\\) complex numbers \\(a_1, a_2, \\ldots, a_{4n}\\) which satisfy the following properties:\n- For each \\(j\\), \\(a_j = 1, -1, i,\\) or \\(-i\\), where \\(i = \\sqrt{-1}\\).\n- For each \\(k = 1, \\ldots, 4n\\), the real and imaginary parts of the sum \\(a_1 + a_2 + \\cdots + a_k\\) f... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-dc1c36a9c1901b7875bff928a51499798651aba874d4524a21a6ce413f96ceb7 | A sequence follows the pattern: $0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, \ldots$, where $0$ divides the sequence into groups with $n$ $1$'s in the $n$-th group. Find the sum of the first 500 terms. | [
"469"
] | [
"A sequence follows the pattern: $0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, \\ldots$, where $0$ divides the sequence into groups with $n$ $1$'s in the $n$-th group. Find the sum of the first 500 terms."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-b2bc526c5976a1676badbadf0fbc14ce38d43362f9cbd3f41cbfaa0de126498e | A sequence has 15 terms, each H or T. There are 14 pairs of adjacent terms: 2 are HH, 3 are HT, 4 are TH, and 5 are TT. How many sequences meet these criteria? | [
"560"
] | [
"A sequence has 15 terms, each H or T. There are 14 pairs of adjacent terms: 2 are HH, 3 are HT, 4 are TH, and 5 are TT. How many sequences meet these criteria?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-29f0a502bdc37300e63eaaf87e395e3fee2048ae8b7289ec69af7ed4cf4c3b27 | A sequence has its first term as $2007$. Each subsequent term is the sum of the squares of the digits of the preceding term. What is the total sum of the sequence up to the $2013$th term? | [
"131400"
] | [
"A sequence has its first term as $2007$. Each subsequent term is the sum of the squares of the digits of the preceding term. What is the total sum of the sequence up to the $2013$th term?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cc937a102b2a8899b717b995da167ef51d19e2ee0b370b6ca1e395cdc7aede21 | A sequence has its first term as 2007, and each subsequent term is the sum of the squares of the digits of the previous term. Find the sum of the first 2013 terms of this sequence. | [
"105288"
] | [
"A sequence has its first term as 2007, and each subsequent term is the sum of the squares of the digits of the previous term. Find the sum of the first 2013 terms of this sequence."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eda5d23f62af6f5cfd5a9749cecc666434a4e5c3c2ea3cd6aa550228bfb72dcc | A sequence has its first term positive, and each subsequent term is the area of an equilateral triangle whose perimeter is the preceding term. If the first three terms form an arithmetic progression, determine all possible values of the first term. | [
"\\( x = 12\\sqrt{3} \\) or \\( x = 6\\sqrt{15} - 6\\sqrt{3} \\)"
] | [
"A sequence has its first term positive, and each subsequent term is the area of an equilateral triangle whose perimeter is the preceding term. If the first three terms form an arithmetic progression, determine all possible values of the first term."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 20
} | 20 | 0 | 0 |
omr-ws-420864739c0a0ce1908f0db5c17a140ac6455fcb784f765639bd6ef89b5e0c99 | A sequence is called an arithmetic progression of the first order if the differences of the successive terms are constant. It is called an arithmetic progression of the second order if the differences of the successive terms form an arithmetic progression of the first order. In general, for $k \geq 2$, a sequence is ca... | [
"The least possible order is 3, and the formula for the \\( n \\)-th term is \\( a_n = \\frac{1}{3}n^3 + \\frac{1}{2}n^2 - \\frac{11}{6}n + 5 \\)."
] | [
"A sequence is called an arithmetic progression of the first order if the differences of the successive terms are constant. It is called an arithmetic progression of the second order if the differences of the successive terms form an arithmetic progression of the first order. In general, for $k \\geq 2$, a sequence... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-6d46396a96b17faed68b1a34404bab1988a6603a7136d311cf2577ea21eed2de | A sequence is defined as $a_0 = e$, $a_1 = \pi$, and $a_{n+2} = \frac{1 + a_{n+1}}{a_n}$ for $n \ge 0$. Find $a_{2016}$. | [
"\\(\\pi\\)"
] | [
"A sequence is defined as $a_0 = e$, $a_1 = \\pi$, and $a_{n+2} = \\frac{1 + a_{n+1}}{a_n}$ for $n \\ge 0$. Find $a_{2016}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f47bdd1e9302adb5076d5a35506708b30486bb49f254d201823c3b1543a5f633 | A sequence is defined as \( a_n = 13a_{n-1} - 30a_{n-2} \). Let \( a_0 = 10 \) and \( a_1 = 58 \). The product of the units digits of \( a_{2023} \), \( a_{2024} \), and \( a_{2025} \) can be expressed as \( n \). Find \( n \). | [
"96"
] | [
"A sequence is defined as \\( a_n = 13a_{n-1} - 30a_{n-2} \\). Let \\( a_0 = 10 \\) and \\( a_1 = 58 \\). The product of the units digits of \\( a_{2023} \\), \\( a_{2024} \\), and \\( a_{2025} \\) can be expressed as \\( n \\). Find \\( n \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 17
} | 65 | 0 | 0 |
omr-ws-05269f71c50ae2633f9fe8bb404839d8555b451c38f97986d5153ddd9f757384 | A sequence is defined as follows $a_1=a_2=a_3=1,$ and, for all positive integers $n, a_{n+3}=a_{n+2}+a_{n+1}+a_n.$ Given that $a_{28}=6090307, a_{29}=11201821,$ and $a_{30}=20603361,$ find the remainder when $\sum^{28}_{k=1} a_k$ is divided by 1000. | [
"834"
] | [
"A sequence is defined as follows $a_1=a_2=a_3=1,$ and, for all positive integers $n, a_{n+3}=a_{n+2}+a_{n+1}+a_n.$ Given that $a_{28}=6090307, a_{29}=11201821,$ and $a_{30}=20603361,$ find the remainder when $\\sum^{28}_{k=1} a_k$ is divided by 1000."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-da5f3d2d61665f7b8cbf0a8e743762d9fa4e733c5eb8d9d3f1384e6b624f0331 | A sequence is defined as follows: $a_1=a_2=a_3=1$, and, for all positive integers $n$, $a_{n+3}=a_{n+2}+a_{n+1}+a_n$. Given that $a_{28}= 6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\displaystyle \sum_{k=1}^{28}a_k$ is divided by 1000. | [
"834"
] | [
"A sequence is defined as follows: $a_1=a_2=a_3=1$, and, for all positive integers $n$, $a_{n+3}=a_{n+2}+a_{n+1}+a_n$. Given that $a_{28}= 6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\\displaystyle \\sum_{k=1}^{28}a_k$ is divided by 1000."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7a426a5995a48a69b43470761417babee19633282dd53eff0a9df01b185aeae0 | A sequence is defined as follows: - The first sequence is $1, 1$. - The $(n+1)$-th sequence is obtained by inserting $0$ between each pair of adjacent terms of the $n$-th sequence and then adding the first term of the $n$-th sequence to each term of the resulting sequence. For example, the second sequence is $2, 1, 2$ ... | [
"Number of terms: \\( 2^{n-1} + 1 \\), Sum of terms: \\( 2^{n-1} + 2 \\frac{4^{n-1} - 1}{3} + 1 \\)"
] | [
"A sequence is defined as follows:\n- The first sequence is $1, 1$.\n- The $(n+1)$-th sequence is obtained by inserting $0$ between each pair of adjacent terms of the $n$-th sequence and then adding the first term of the $n$-th sequence to each term of the resulting sequence.\nFor example, the second sequence is $2... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 14
} | 14 | 0 | 0 |
omr-ws-e4a6d24486af89b2d409c891b8ddb76ba6c5715469721efd8e3909c989d0b4ee | A sequence is defined as follows: \(a_1 = a_2 = a_3 = 1\), and for all positive integers \(n\), \(a_{n+3} = a_{n+2} + a_{n+1} + a_n\). Given that \(a_{28} = 6090307\), \(a_{29} = 11201821\), and \(a_{30} = 20603361\), find the remainder when \(\sum_{k=1}^{28} a_k\) is divided by 1000. | [
"834"
] | [
"A sequence is defined as follows: \\(a_1 = a_2 = a_3 = 1\\), and for all positive integers \\(n\\), \\(a_{n+3} = a_{n+2} + a_{n+1} + a_n\\). Given that \\(a_{28} = 6090307\\), \\(a_{29} = 11201821\\), and \\(a_{30} = 20603361\\), find the remainder when \\(\\sum_{k=1}^{28} a_k\\) is divided by 1000."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-24e6e8d899612ec0d93269778dfb8edf23fe609fbcc1ba5c17bf95c6cb7e43b2 | A sequence is defined as follows: \[ t_1 = 3^8 \] \[ t_2 = 1000 \] \[ t_n = \frac{1 + t_{n-1}}{t_{n-2}} \] What is the sum of the numerator and the denominator of \( t_{2008} \)? | [
"7562"
] | [
"A sequence is defined as follows:\n\\[ t_1 = 3^8 \\]\n\\[ t_2 = 1000 \\]\n\\[ t_n = \\frac{1 + t_{n-1}}{t_{n-2}} \\]\n\nWhat is the sum of the numerator and the denominator of \\( t_{2008} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 10
} | 32 | 0 | 0 |
omr-ws-400ff6c8bb4d38bdc32480f7c2f933c815cc23e347f428b89690b6801243a5a4 | A sequence is defined by $a_0 = \frac{1}{2}$ and $a_n = 1 + (a_{n - 1} - 1)^2.$ Compute \[a_0 a_1 a_2 \dotsm.\] | [
"\\frac{2}{3}"
] | [
"A sequence is defined by $a_0 = \\frac{1}{2}$ and $a_n = 1 + (a_{n - 1} - 1)^2.$ Compute\n\\[a_0 a_1 a_2 \\dotsm.\\]"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-327e8409b617d62e3f3a34ec86e3f23e283664254f3707f5657193508a05a56d | A sequence is defined by $a_1 = 8$, $a_2 = 18$, and $a_{n + 2} = a_n \cdot a_{n + 1}$ for $n \ge 1$. Determine the values of $n$ for which $a_n$ is a perfect square. | [
"3k"
] | [
"A sequence is defined by $a_1 = 8$, $a_2 = 18$, and $a_{n + 2} = a_n \\cdot a_{n + 1}$ for $n \\ge 1$. Determine the values of $n$ for which $a_n$ is a perfect square."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6db292bd7c9235ecd728fb21482f1733322c9ec949cf1994c35d61f1023998ea | A sequence is defined by $a_{n+1} = \frac{a_n}{2} + \frac{1}{a_n}$ with $a_0 = 2$. Find the closed form of the $k^{\text{th}}$ term of the sequence. | [
"\\( a_k = \\frac{\\sqrt{2}((1 + \\sqrt{2})^{2^k} + (\\sqrt{2} - 1)^{2^k})}{(1 + \\sqrt{2})^{2^k} - (\\sqrt{2} - 1)^{2^k}} \\)"
] | [
"A sequence is defined by $a_{n+1} = \\frac{a_n}{2} + \\frac{1}{a_n}$ with $a_0 = 2$. Find the closed form of the $k^{\\text{th}}$ term of the sequence."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 32,
"genselect": 7,
"tir": 2
} | 41 | 0 | 0 |
omr-ws-cefd31fc65d982390485554b3ca4395f49aa3e7cddb20ee8b35654dd0cf0b50a | A sequence is defined by $x_1=1$, $x_2=4$, and $x_{n+2}=4x_{n+1}-x_n$ for $n \ge 1$. Find all natural numbers $m$ such that the number $2x_n^2+m$ is a perfect square for all natural numbers $n$. | [
"2"
] | [
"A sequence is defined by $x_1=1$, $x_2=4$, and $x_{n+2}=4x_{n+1}-x_n$ for $n \\ge 1$. Find all natural numbers $m$ such that the number $2x_n^2+m$ is a perfect square for all natural numbers $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-6ae762c8c5f295038371e7c88b06e86da78976b7ab0f5ea696a2a9e99cb55c99 | A sequence is defined by \( a_0 = 1 \) and \( a_{n+1} = (a_n)^2 + 1 \) for each \( n \geq 0 \). Find the greatest common divisor of \( a_{999} \) and \( a_{2004} \). | [
"677"
] | [
"A sequence is defined by \\( a_0 = 1 \\) and \\( a_{n+1} = (a_n)^2 + 1 \\) for each \\( n \\geq 0 \\). Find the greatest common divisor of \\( a_{999} \\) and \\( a_{2004} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-97acbf3c92240549ac87b80b705c2c79efe64a5d5cdc4d325adb6dc9c4575cf4 | A sequence is defined by \( x_1 = 0 \) and the recurrence relation \( x_{n+1} = x_n + \frac{n}{23} \) for all natural numbers \( n \). How many of the first 1000 terms of this sequence are integers? | [
"8"
] | [
"A sequence is defined by \\( x_1 = 0 \\) and the recurrence relation \\( x_{n+1} = x_n + \\frac{n}{23} \\) for all natural numbers \\( n \\). How many of the first 1000 terms of this sequence are integers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1bcb14c81a85a7a617c5884c6792778f6eccaa51da7655e7c677a43d57d32f9 | A sequence is defined by \( x_1 = 1 \), \( x_2 = 4 \), and \( x_{n+2} = 4x_{n+1} - x_n \) for \( n \geq 1 \). Find all natural numbers \( m \) such that \( 3x_n^2 + m \) is a perfect square for all natural numbers \( n \). | [
"1"
] | [
"A sequence is defined by \\( x_1 = 1 \\), \\( x_2 = 4 \\), and \\( x_{n+2} = 4x_{n+1} - x_n \\) for \\( n \\geq 1 \\). Find all natural numbers \\( m \\) such that \\( 3x_n^2 + m \\) is a perfect square for all natural numbers \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7b8a4912c22a952c3f77e6c28d945c560315919ab8e24e3c7590a450b6cb015e | A sequence is defined by \( x_1 = 1 \), \( x_2 = 4 \), and \( x_{n+2} = 4x_{n+1} - x_n \) for \( n \geq 1 \). Find all natural numbers \( m \) such that the number \( 3x_n^2 + m \) is a perfect square for all natural numbers \( n \). | [
"1"
] | [
"A sequence is defined by \\( x_1 = 1 \\), \\( x_2 = 4 \\), and \\( x_{n+2} = 4x_{n+1} - x_n \\) for \\( n \\geq 1 \\). Find all natural numbers \\( m \\) such that the number \\( 3x_n^2 + m \\) is a perfect square for all natural numbers \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-6ee9b721c029bed8e79321d3f6c272b7f3de7caa11423aa2be3a78450ffe21e5 | A sequence is defined by \(a_1 = 1995\) and \(a_{n+1} = \frac{a_n}{a_n + 1}\) for all \(n \geq 1\). If \(a_{1995} = \frac{m}{n}\), where \(m\) and \(n\) are relatively prime integers, find the remainder when \(m + n\) is divided by 1000. | [
"26"
] | [
"A sequence is defined by \\(a_1 = 1995\\) and \\(a_{n+1} = \\frac{a_n}{a_n + 1}\\) for all \\(n \\geq 1\\). If \\(a_{1995} = \\frac{m}{n}\\), where \\(m\\) and \\(n\\) are relatively prime integers, find the remainder when \\(m + n\\) is divided by 1000."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-1b93bf11287c84887c3a39db82ea0895725503623c1a2540c58f66ea6133d608 | A sequence is defined by \(a_1 = 1\) and \[ a_{n+1} = \frac{a_n}{1 + n a_n}. \] Find the 2013th term of the sequence. | [
"\\(\\frac{2}{4050158}\\)"
] | [
"A sequence is defined by \\(a_1 = 1\\) and \n\\[ a_{n+1} = \\frac{a_n}{1 + n a_n}. \\]\nFind the 2013th term of the sequence."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 30
} | 78 | 0 | 0 |
omr-ws-57c3b11549bf0e17d0152267cd14beab266b25057c878e15b30f1cdbf68270cd | A sequence is defined by \(a_1 = 3\) and the following rules: - \(a_{n+1} = \frac{a_n}{2}\) if \(a_n\) is even, - \(a_{n+1} = \frac{a_n + 1983}{2}\) if \(a_n\) is odd. Let \(P\) be the minimum period of the sequence. Calculate the value of \(P\). | [
"660"
] | [
"A sequence is defined by \\(a_1 = 3\\) and the following rules:\n- \\(a_{n+1} = \\frac{a_n}{2}\\) if \\(a_n\\) is even,\n- \\(a_{n+1} = \\frac{a_n + 1983}{2}\\) if \\(a_n\\) is odd.\n\nLet \\(P\\) be the minimum period of the sequence. Calculate the value of \\(P\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b3f296caf66f7e671bc622ee5b307010ad720a9a701280403af19503814161a | A sequence is defined by the formula $$a_n=\frac{2021a_{n-2}-2020a_{n-1}}{2021a_{n-1}-2020a_{n-2}}.$$ Given that $a_1=1$ and $a_2=10$, find $a_{2021}$. | [
"1"
] | [
"A sequence is defined by the formula $$a_n=\\frac{2021a_{n-2}-2020a_{n-1}}{2021a_{n-1}-2020a_{n-2}}.$$ Given that $a_1=1$ and $a_2=10$, find $a_{2021}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-5814b15fae128df4551bf26c3ea1937f122c1ebed95cca0e863171dcdde15fc2 | A sequence is defined by the recursion \(a_1 = t\) and \(a_n = \text{ranint}[1, a_{n-1}]\), where \(t\) is a positive integer and \(\text{ranint}[x, y]\) denotes a randomly chosen integer from the interval \([x, y]\). Calculate the expected number of terms in the sequence that are not equal to 1. | [
"\\( 1 + \\sum_{i=1}^{t-1} \\frac{1}{i} \\) or for \\( t = 2011 \\), \\( 1 + \\sum_{i=1}^{2010} \\frac{1}{i} \\)"
] | [
"A sequence is defined by the recursion \\(a_1 = t\\) and \\(a_n = \\text{ranint}[1, a_{n-1}]\\), where \\(t\\) is a positive integer and \\(\\text{ranint}[x, y]\\) denotes a randomly chosen integer from the interval \\([x, y]\\). Calculate the expected number of terms in the sequence that are not equal to 1."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 23,
"tir": 16
} | 39 | 0 | 0 |
omr-ws-71d604a0fc3e39a0ad694201d4f85ebb8030e968fe279b35334acd4e30e998cd | A sequence is defined for positive integer values of \( n \) by \( u_{n+3} = u_{n+2} + u_{n+1} - u_{n} \), with initial conditions \( u_{1} = 0 \), \( u_{2} = 1 \), and \( u_{3} = 1 \). What is the sum of the first 100 terms of the sequence? | [
"2500"
] | [
"A sequence is defined for positive integer values of \\( n \\) by \\( u_{n+3} = u_{n+2} + u_{n+1} - u_{n} \\), with initial conditions \\( u_{1} = 0 \\), \\( u_{2} = 1 \\), and \\( u_{3} = 1 \\). What is the sum of the first 100 terms of the sequence?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-c163029099f5630169a5a4eba00d505e75bbe1a9ba1a21008993fd6a2f01e84e | A sequence is defined over non-negative integral indexes as follows: \(a_0 = a_1 = 1\) and \(a_{n+1}a_{n-1} = a_n^2 + 2007\). Find the invariant for this sequence. | [
"2007"
] | [
"A sequence is defined over non-negative integral indexes as follows: \\(a_0 = a_1 = 1\\) and \\(a_{n+1}a_{n-1} = a_n^2 + 2007\\). Find the invariant for this sequence."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-aa35a1577a6fe6c087f842aa1c917c5345f1afa28921fdb58500200a485be207 | A sequence is defined over non-negative integral indexes in the following way: \( a_0 = a_1 = 3 \), \( a_{n+1} a_{n-1} = a_n^2 + 2007 \). Find the greatest integer that does not exceed \( \frac{a_{2006}^2 + a_{2007}^2}{a_{2006} a_{2007}} \). | [
"225"
] | [
"A sequence is defined over non-negative integral indexes in the following way: \\( a_0 = a_1 = 3 \\), \\( a_{n+1} a_{n-1} = a_n^2 + 2007 \\). Find the greatest integer that does not exceed \\( \\frac{a_{2006}^2 + a_{2007}^2}{a_{2006} a_{2007}} \\)."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-aaf622ccb41fe075adbf51a1a666058088dada379f70037306fab49e39e9a528 | A sequence is defined recursively by \( a_1 = 2 \) and \[ a_{n+1} = \left(1 + \frac{1}{n}\right) \cdot a_n + \frac{1}{n}. \] What is \( a_{100} \)? | [
"299"
] | [
"A sequence is defined recursively by \\( a_1 = 2 \\) and \n\\[ a_{n+1} = \\left(1 + \\frac{1}{n}\\right) \\cdot a_n + \\frac{1}{n}. \\]\nWhat is \\( a_{100} \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-de338511c98170cf8ead9555d440c207bb974b35822aa228c4af815886410b14 | A sequence is defined recursively by \(a_n - a_{n-1} \in \{-1, 0, 1\}\) for all integers \(n\) chosen from \(\{1, 2, 3, \dots, 9\}\). Find the remainder when the number of sequences with \(a_1 = a_9 = 0\) is divided by 1000. | [
"1107"
] | [
"A sequence is defined recursively by \\(a_n - a_{n-1} \\in \\{-1, 0, 1\\}\\) for all integers \\(n\\) chosen from \\(\\{1, 2, 3, \\dots, 9\\}\\). Find the remainder when the number of sequences with \\(a_1 = a_9 = 0\\) is divided by 1000."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 18
} | 41 | 0 | 0 |
omr-ws-1e5688ce990dcfd0104fbd111420ffe755f309372fc718c59ae9ffbc48dbecf5 | A sequence is defined recursively with $F_0=F_1=F_2=1,$ and $F_n=F_{n-1}+F_{n-2}+F_{n-3},$ for $n \ge 4.$ (a) How many of the first $2023$ terms are even? How many are divisible by $3$? (b) Find a closed form for this sequence. | [
"622"
] | [
"A sequence is defined recursively with $F_0=F_1=F_2=1,$ and $F_n=F_{n-1}+F_{n-2}+F_{n-3},$ for $n \\ge 4.$\n\n(a) How many of the first $2023$ terms are even? How many are divisible by $3$?\n\n(b) Find a closed form for this sequence."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-4d75d826608aae0b328f6a75c5b7589561a72a1f82865e8e11b3f423c2c227f8 | A sequence is defined such that $a_1=1$, $a_2=2$, and $a_{n+2}=a_{n+1}^{a_n}$ for $n\ge 1$. What is the 9th term of this sequence? | [
"\\(2^{2^{280}}\\)"
] | [
"A sequence is defined such that $a_1=1$, $a_2=2$, and $a_{n+2}=a_{n+1}^{a_n}$ for $n\\ge 1$. What is the 9th term of this sequence?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-caf889893d82d2fe3fb66132a0c501c7d94644b8b7cf80f360ec2bc894860286 | A sequence is defined such that \(a_1=1\), \(a_2=2\), and \(a_{n+2}=a_{n+1}^{a_n}\) for \(n \ge 1\). What is the 9th term of this sequence? | [
"\\(2^{2^{280}}\\)"
] | [
"A sequence is defined such that \\(a_1=1\\), \\(a_2=2\\), and \\(a_{n+2}=a_{n+1}^{a_n}\\) for \\(n \\ge 1\\). What is the 9th term of this sequence?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-eb4e597eb4eb76a74004f163a5450c020e8dd5bfaafac8c0125f2d21ab12ec96 | A sequence is formed by writing the nth term n times: 1, 2, 2, 3, 3, 3, and so on. What is the sum of the first 2007 terms in this sequence? | [
"84452"
] | [
"A sequence is formed by writing the nth term n times: 1, 2, 2, 3, 3, 3, and so on. What is the sum of the first 2007 terms in this sequence?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19ae9a3edf9049d72d881c8897b4f613e4fc85a1a1d7de374900c9df293a4a0f | A sequence is formed such that $a_n = 2a_{n-1} + 4$, where $n$ is a positive integer and $a_1 = 1$. Given this, what is $a_1 + a_2 + \ldots + a_8$? | [
"1243"
] | [
"A sequence is formed such that $a_n = 2a_{n-1} + 4$, where $n$ is a positive integer and $a_1 = 1$. Given this, what is $a_1 + a_2 + \\ldots + a_8$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3cf5f6e92f54a8af1e9f7c938f029d45c854ec03b589198cad27950872fb9812 | A sequence is generated as follows: if the \( n \)-th term is even, then the \( (n+1) \)-th term is half the \( n \)-th term; otherwise it is two more than twice the \( n \)-th term. If the first term is 10, what is the 2008-th term? | [
"4"
] | [
"A sequence is generated as follows: if the \\( n \\)-th term is even, then the \\( (n+1) \\)-th term is half the \\( n \\)-th term; otherwise it is two more than twice the \\( n \\)-th term. If the first term is 10, what is the 2008-th term?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-37a2b76a49811cd107e684c354899296511c15cbb7c13c914efda61705267c3d | A sequence made of letters A and B is called "[color=#00f]JSB[/color]" if it can be composed by merging two smaller "[color=#00f]JSB[/color]" sequences. Every sequence of length 1 or 2 is "[color=#00f]JSB[/color]", while none of the sequences of length 3 are "[color=#00f]JSB[/color]". How many sequences of a given leng... | [
"0"
] | [
"A sequence made of letters A and B is called \"[color=#00f]JSB[/color]\" if it can be composed by merging two smaller \"[color=#00f]JSB[/color]\" sequences. Every sequence of length 1 or 2 is \"[color=#00f]JSB[/color]\", while none of the sequences of length 3 are \"[color=#00f]JSB[/color]\". How many sequences of... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-c18eb6382ee961a177c92f5394910587f5db55e5dc8359a1d0471465e84e6ded | A sequence of $2n + 1$ non-negative integers $a_1, a_2, \ldots, a_{2n + 1}$ is given. There's also a sequence of $2n + 1$ consecutive cells enumerated from $1$ to $2n + 1$ from left to right, such that initially the number $a_i$ is written on the $i$-th cell, for $i = 1, 2, \ldots, 2n + 1$. Starting from this initial p... | [
"\\( \\left( \\frac{1}{n+1} \\binom{2n}{n} \\right)^2 \\)"
] | [
"A sequence of $2n + 1$ non-negative integers $a_1, a_2, \\ldots, a_{2n + 1}$ is given. There's also a sequence of $2n + 1$ consecutive cells enumerated from $1$ to $2n + 1$ from left to right, such that initially the number $a_i$ is written on the $i$-th cell, for $i = 1, 2, \\ldots, 2n + 1$. Starting from this in... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cfa5b8218b71e25a713ee486528dec5aa63c02fc27874d0a1a9c581d851e5106 | A sequence of $A$s and $B$s is called antipalindromic if writing it backwards and then turning all the $A$s into $B$s and vice versa produces the original sequence. For example, $ABBAAB$ is antipalindromic. The cost of a sequence is defined as the product of the positions of the $A$s. For example, the string $ABBAAB$ h... | [
"\\(2021^{1010}\\)"
] | [
"A sequence of $A$s and $B$s is called antipalindromic if writing it backwards and then turning all the $A$s into $B$s and vice versa produces the original sequence. For example, $ABBAAB$ is antipalindromic. The cost of a sequence is defined as the product of the positions of the $A$s. For example, the string $ABBA... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 11
} | 59 | 0 | 0 |
omr-ws-c322430129813592b803a209c721f76dd3fd0eb26d473eedd25c5c9315301e3a | A sequence of $X$s and $O$s is given, such that no three consecutive characters in the sequence are all the same, and let $N$ be the number of characters in this sequence. Maia may swap two consecutive characters in the sequence. After each swap, any consecutive block of three or more of the same character will be eras... | [
"5"
] | [
"A sequence of $X$s and $O$s is given, such that no three consecutive characters in the sequence are all the same, and let $N$ be the number of characters in this sequence. Maia may swap two consecutive characters in the sequence. After each swap, any consecutive block of three or more of the same character will be... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-acf2367a244c17d5a9cb7d522104786aeb12d2e2c784ae77e6268ba739d1898e | A sequence of $n$ positive integers ($a_1, a_2, ..., a_n$) is defined as $k$-beautiful if for each $1 \le i \le n-1$, the number $\overline{a_ia_{i+1}}$ is a multiple of $k$ and $1 \le a_i \le 9$ for each $i$. Find the greatest possible value of the sum $a_1 + a_2 + \cdots + a_{2020}$ for a $13$-beautiful sequence of l... | [
"4377"
] | [
"A sequence of $n$ positive integers ($a_1, a_2, ..., a_n$) is defined as $k$-beautiful if for each $1 \\le i \\le n-1$, the number $\\overline{a_ia_{i+1}}$ is a multiple of $k$ and $1 \\le a_i \\le 9$ for each $i$. Find the greatest possible value of the sum $a_1 + a_2 + \\cdots + a_{2020}$ for a $13$-beautiful se... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7c62690e9fffb67bbdbbdea91f91b1f05d243f27a2bb2eee7dd90c9d17f2d82a | A sequence of 0s and 1s of length $n$ is randomly generated, with each digit having an equal probability of being 0 or 1. Given that $m$ of the digits are 0s, what is the probability that the sequence contains no consecutive 0s? | [
"\\(\\frac{C(n-m+1, n-2m+1)}{C(n, m)}\\) if \\( n \\geq 2m-1 \\), otherwise 0."
] | [
"A sequence of 0s and 1s of length $n$ is randomly generated, with each digit having an equal probability of being 0 or 1. Given that $m$ of the digits are 0s, what is the probability that the sequence contains no consecutive 0s?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-c3e31b89f83f02ebcc4a4446e9a56679b01b1950c9ef1553d0faae6636582d5f | A sequence of 10 letters is made according to the following rules: - The letter P can only be followed by Q or R. - The letter Q can only be followed by R or S. - The letter R can only be followed by S or T. - The letter S can only be followed by T or P. - The letter T can only be followed by P or Q. How many possible ... | [
"45"
] | [
"A sequence of 10 letters is made according to the following rules:\n- The letter P can only be followed by Q or R.\n- The letter Q can only be followed by R or S.\n- The letter R can only be followed by S or T.\n- The letter S can only be followed by T or P.\n- The letter T can only be followed by P or Q.\nHow man... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 27
} | 48 | 0 | 0 |
omr-ws-82528f6422386c00a085feb3d5a564c1df07aa76c42fbe0e1e62717b464d4d8d | A sequence of 15 random draws, one at a time with replacement, is made from the set $\{A, B, C, \dots, X, Y, Z\}$ of the English alphabet. What is the probability that the string "UNIVERSITY" occurs as a block in the sequence? | [
"\\( 6 \\times \\left(\\frac{1}{26}\\right)^{10} \\)"
] | [
"A sequence of 15 random draws, one at a time with replacement, is made from the set $\\{A, B, C, \\dots, X, Y, Z\\}$ of the English alphabet. What is the probability that the string \"UNIVERSITY\" occurs as a block in the sequence?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-7b2dd8b95e48600ecc83286cb54a92023ba2718b1b7e60922c471342b43bde8f | A sequence of 2005 digits has the property that each pair of consecutive digits forms a two-digit number which is either divisible by 17 or 23. If the first digit is 6 and the final digit is not 4, then what is the final digit? | [
"7"
] | [
"A sequence of 2005 digits has the property that each pair of consecutive digits forms a two-digit number which is either divisible by 17 or 23. If the first digit is 6 and the final digit is not 4, then what is the final digit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-45d1942f7f826bcaac600cf71961eba3b793cf03a5a7c75b787bb74c82e1cd7a | A sequence of 4 digits, when considered as a number in base 10, is four times the number it represents in base 6. What is the sum of the digits of the sequence? | [
"8"
] | [
"A sequence of 4 digits, when considered as a number in base 10, is four times the number it represents in base 6. What is the sum of the digits of the sequence?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4a8843ff44e0bde6f7833bf5a146743ac5dc6b7b7b4d3ba446aeffb811901753 | A sequence of As and Bs is called antipalindromic if writing it backwards and then turning all the As into Bs and vice versa produces the original sequence. For example, ABBAAB is antipalindromic. The cost of a sequence is defined as the product of the positions of the As. Find the sum of the costs of all antipalindrom... | [
"\\( 2021^{1010} \\)"
] | [
"A sequence of As and Bs is called antipalindromic if writing it backwards and then turning all the As into Bs and vice versa produces the original sequence. For example, ABBAAB is antipalindromic. The cost of a sequence is defined as the product of the positions of the As. Find the sum of the costs of all antipali... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.