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-9f7653663924b2b9ca301923348bd3ef8d12717c777c835a344c44eef8296202 | A sequence $(a_n)$ has the property that every term is equal to $1$ or $2$, and $a_n$ is the number of $2$s between the $n$th and the $(n+1)$st $1$ in the sequence. The first few terms are as follows: \[1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, \dots.\] Find the number of $1$s in the first 1000 terms of the sequence. | [
"610"
] | [
"A sequence $(a_n)$ has the property that every term is equal to $1$ or $2$, and $a_n$ is the number of $2$s between the $n$th and the $(n+1)$st $1$ in the sequence. The first few terms are as follows: \\[1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, \\dots.\\] Find the number of $1$s in the first 1000 terms of the sequence."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e465665e6e3b02bd74904b7e50736193b5a8a6c5b816d3fa5b28b706dbf9f141 | A sequence $(a_n)$ is defined as follows: $a_1 = 1,$ $a_2 = \frac{1}{2},$ and \[a_n = \frac{1 - a_{n - 1}}{2a_{n - 2}}\]for all $n \ge 3.$ Find $a_{120}.$ | [
"\\frac{20}{41}"
] | [
"A sequence $(a_n)$ is defined as follows: $a_1 = 1,$ $a_2 = \\frac{1}{2},$ and\n\\[a_n = \\frac{1 - a_{n - 1}}{2a_{n - 2}}\\]for all $n \\ge 3.$ Find $a_{120}.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-39e12b902afb1251b19a327fa928c05796cfa77c9027df3394d43f7eb0db6b04 | A sequence $(a_n)$ is defined as follows: $a_1 = p$ is a prime number with 300 nonzero digits, and for each $n \geq 1$, $a_{n+1}$ is twice the decimal period of $\frac{1}{a_n}$. Find $a_{2003}$. | [
"0"
] | [
"A sequence $(a_n)$ is defined as follows: $a_1 = p$ is a prime number with 300 nonzero digits, and for each $n \\geq 1$, $a_{n+1}$ is twice the decimal period of $\\frac{1}{a_n}$. Find $a_{2003}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-8435b4c58ccd180619cff3f4432a6a31ae1de3bbc69c84226f9ee1e2f62ca1cd | A sequence $(a_n)$ is defined as follows: $a_1 = p$ is a prime number with exactly $300$ nonzero digits, and for each $n \geq 1$, $a_{n+1}$ is the decimal period of $1/a_n$ multiplied by $2$. Determine $a_{2003}$. | [
"\\(10p\\)"
] | [
"A sequence $(a_n)$ is defined as follows: $a_1 = p$ is a prime number with exactly $300$ nonzero digits, and for each $n \\geq 1$, $a_{n+1}$ is the decimal period of $1/a_n$ multiplied by $2$. Determine $a_{2003}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bdcecf3d1eb0f5a1292e9c4d7d509aa67e9a970bb516a36fa3cf7d9c27aabf94 | A sequence $(a_n)$ is defined by $a_0 = -1$, $a_1 = 0$, and $a_{n+1} = a_n^2 - (n+1)^2 a_{n-1} - 1$ for all positive integers $n$. Find $a_{100}$. | [
"9999"
] | [
"A sequence $(a_n)$ is defined by $a_0 = -1$, $a_1 = 0$, and $a_{n+1} = a_n^2 - (n+1)^2 a_{n-1} - 1$ for all positive integers $n$. Find $a_{100}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4962ca8a90f5dddad8b1a9ac4e2ee2be300a6b32cea5080af671e38f5887366c | A sequence $(a_n)$ is defined by $a_0 = \frac{350}{27}$, $a_1 = 200$, and $a_n = 36 a_{n-1} - 243 a_{n-2}$ for $n \geq 2$. Find the value of $\sqrt[3]{a_0 + \sqrt[3]{a_1 + \sqrt[3]{a_2 + \cdots}}}$. | [
"\\(\\frac{8}{3}\\)",
"\\dfrac{8}{3}"
] | [
"A sequence $(a_n)$ is defined by $a_0 = \\frac{350}{27}$, $a_1 = 200$, and $a_n = 36 a_{n-1} - 243 a_{n-2}$ for $n \\geq 2$. Find the value of $\\sqrt[3]{a_0 + \\sqrt[3]{a_1 + \\sqrt[3]{a_2 + \\cdots}}}$."
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 49,
"tir": 24
} | 73 | 0 | 0 |
omr-ws-b6a544f86a1cf9922a262e88563852adf311e06299f66464dacdecfffe9f1f01 | A sequence $(a_n)$ is defined by $a_0=\dfrac{350}{27}$, $a_1=200$, and $a_n=36a_{n-1}-243a_{n-2}$ for $n\ge 2$. Then, $\sqrt[3]{a_0 + \sqrt[3]{a_1 + \sqrt[3]{a_2 + \cdots}}}$ can be expressed in the form $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"11"
] | [
"A sequence $(a_n)$ is defined by $a_0=\\dfrac{350}{27}$, $a_1=200$, and $a_n=36a_{n-1}-243a_{n-2}$ for $n\\ge 2$. Then, $\\sqrt[3]{a_0 + \\sqrt[3]{a_1 + \\sqrt[3]{a_2 + \\cdots}}}$ can be expressed in the form $\\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-1bf19aa0669d7cfe1220255a19142c0c7ff60f6f212dabf272972fa60cb3e9c0 | A sequence $(a_n)$ is defined by $a_1 = 1$ and $a_n = \frac{n+1}{n-1}(a_1 + a_2 + \cdots + a_{n-1})$. Determine $a_{1998}$. | [
"\\(2^{1996} \\cdot 1999\\)."
] | [
"A sequence $(a_n)$ is defined by $a_1 = 1$ and $a_n = \\frac{n+1}{n-1}(a_1 + a_2 + \\cdots + a_{n-1})$. Determine $a_{1998}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-d5178dad7d6f86d4917a7fec734c7c3175a1ad48a009a69793a1a1f9e4485888 | A sequence $(a_n)$ is defined by the recursion \[ a_1 = 1 \] \[ a_{n+1} = \frac{1 + 4a_n + \sqrt{1 + 24a_n}}{16} \] Find an explicit formula for $a_n$. | [
"\\dfrac{(2^n + 1)(2^n + 2)}{3 \\cdot 4^n}"
] | [
"A sequence $(a_n)$ is defined by the recursion\n\\[ a_1 = 1 \\]\n\\[ a_{n+1} = \\frac{1 + 4a_n + \\sqrt{1 + 24a_n}}{16} \\]\nFind an explicit formula for $a_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-443e14a77db5e5e158ac686f8c9617e73baa6792d7a949c5834c449a0d98a079 | A sequence $(a_n)$ is defined by the recursion \[a_1 = 1, \quad a_{n+1} = \frac{1 + 4a_n + \sqrt{1 + 24a_n}}{16}.\] Find an explicit formula for $a_n.$ | [
"\\(\\frac{(3 + 2^{2-n})^2 - 1}{24}\\)"
] | [
"A sequence $(a_n)$ is defined by the recursion\n\\[a_1 = 1, \\quad a_{n+1} = \\frac{1 + 4a_n + \\sqrt{1 + 24a_n}}{16}.\\]\nFind an explicit formula for $a_n.$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-6d2e39bbdd5a3493ff95c94fef86319b40c0c2fa4838d76b9fcdf37c1c9670bd | A sequence $(a_n)$ is defined by the recursive form: $a_1=2$ and \[ a_{n+1}=a_n\left(\dfrac{1}{2}-\dfrac{1}{4n}\right)+\left(\dfrac{3}{4}-\dfrac{1}{4n}\right), \quad \forall n=1,2,3,\ldots \] Find the formula for this sequence. | [
"no_valid_answer_found"
] | [
"A sequence $(a_n)$ is defined by the recursive form: $a_1=2$ and \n\\[ a_{n+1}=a_n\\left(\\dfrac{1}{2}-\\dfrac{1}{4n}\\right)+\\left(\\dfrac{3}{4}-\\dfrac{1}{4n}\\right), \\quad \\forall n=1,2,3,\\ldots \\]\nFind the formula for this sequence."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d19a81c45e56a8eb4ad3eb72733c87085b6a41bf34f48c16b246adb9664c23a0 | A sequence $(a_n)$ is defined recursively by $$a_1=1, \quad a_2=4, \quad \text{and} \quad a_{n}=2a_{n-1}-a_{n-2}+2 \quad \text{for all } n \geq 3.$$ Find a closed form for $(a_n)$. | [
"\\( a_n = n^2 \\)"
] | [
"A sequence $(a_n)$ is defined recursively by $$a_1=1, \\quad a_2=4, \\quad \\text{and} \\quad a_{n}=2a_{n-1}-a_{n-2}+2 \\quad \\text{for all } n \\geq 3.$$ Find a closed form for $(a_n)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-4f0f47fe3970af3b01aeb66197261a3da48b9271f3ee1724bf0293898abc4b42 | A sequence $(a_n)$ is defined recursively by $a_1=0$, $a_2=1$, and for $n \geq 3$, \[a_n = \frac{1}{2} n a_{n-1} + \frac{1}{2} n (n-1) a_{n-2} + (-1)^n \left(1 - \frac{n}{2}\right).\] Find a closed-form expression for \[f_n = a_n + 2 \binom{n}{1} a_{n-1} + 3 \binom{n}{2} a_{n-2} + \ldots + (n-1) \binom{n}{n-2} a_2 + n ... | [
"\\( 2n! - n - 1 \\)"
] | [
"A sequence $(a_n)$ is defined recursively by $a_1=0$, $a_2=1$, and for $n \\geq 3$,\n\\[a_n = \\frac{1}{2} n a_{n-1} + \\frac{1}{2} n (n-1) a_{n-2} + (-1)^n \\left(1 - \\frac{n}{2}\\right).\\]\nFind a closed-form expression for\n\\[f_n = a_n + 2 \\binom{n}{1} a_{n-1} + 3 \\binom{n}{2} a_{n-2} + \\ldots + (n-1) \\b... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 26,
"tir": 17
} | 43 | 0 | 0 |
omr-ws-9228a6b36e195c76327c2d357c5aa88a65f9344144010ac5d1a871070b89c28f | A sequence $(a_n)$ is defined recursively for all $n \ge 1$ by the relation $a_{n+1}^2 = a_{n+2} - a_n + \frac{1}{n^2}$, with initial conditions $a_1 = 1$ and $a_2 = \frac{1}{2}$. Find the closed form of $(a_n)$. | [
"\\frac{1}{n}"
] | [
"A sequence $(a_n)$ is defined recursively for all $n \\ge 1$ by the relation $a_{n+1}^2 = a_{n+2} - a_n + \\frac{1}{n^2}$, with initial conditions $a_1 = 1$ and $a_2 = \\frac{1}{2}$. Find the closed form of $(a_n)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-e5450671d72a1d98d5dbef49fae02cf5c62992d66edc04f8d661ddc5695dcdc4 | A sequence $(a_n)$ of positive integers is defined by $a_0 = m$ and $a_{n+1} = a_n^5 + 487$ for all $n \ge 0$. Find all positive integers $m$ such that the sequence contains the maximum possible number of perfect squares. | [
"\\(m = 9\\)"
] | [
"A sequence $(a_n)$ of positive integers is defined by $a_0 = m$ and $a_{n+1} = a_n^5 + 487$ for all $n \\ge 0$. Find all positive integers $m$ such that the sequence contains the maximum possible number of perfect squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-a927bdb6b049598da5045545e50ab29ece1dc60bcac51bc0586dc32d83821ebb | A sequence $(a_n)$ of positive reals is defined by $a_{n+2}=a_{n-1}+\frac{1}{a_na_{n+1}}$, where $a_0=\frac{3}{4}$ and $a_1=\frac{3}{7}$. Find the limit of this sequence. | [
"\\(\\infty\\)"
] | [
"A sequence $(a_n)$ of positive reals is defined by $a_{n+2}=a_{n-1}+\\frac{1}{a_na_{n+1}}$, where $a_0=\\frac{3}{4}$ and $a_1=\\frac{3}{7}$. Find the limit of this sequence."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-74df9efebadd7a44dbc72156a8fbbfbb8f022b2e2f7b29cce196c67c0a282218 | A sequence $(a_n)_{-\infty}^{-\infty}$ of zeros and ones is given. It is known that $a_n = 0$ if and only if $a_{n-6} + a_{n-5} + \ldots + a_{n-1}$ is a multiple of $3$, and not all terms of the sequence are zero. Determine the maximum possible number of zeros among $a_0, a_1, \ldots, a_{97}$. | [
"65"
] | [
"A sequence $(a_n)_{-\\infty}^{-\\infty}$ of zeros and ones is given. It is known that $a_n = 0$ if and only if $a_{n-6} + a_{n-5} + \\ldots + a_{n-1}$ is a multiple of $3$, and not all terms of the sequence are zero. Determine the maximum possible number of zeros among $a_0, a_1, \\ldots, a_{97}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-bdce9eeef9c9d4370b7d73930847189ef9085951eb8875af26051cbc275a06e2 | A sequence $(a_n)_{n\geqslant0}$ is defined as follows: \[ a_0 = 1 \] \[ \frac{a_{n-1}}{a_n} = n+1+\frac{1}{2n-1}, \quad n > 0 \] An integrable function $f$ is defined by: \[ \int_0^x f(t) \, \mathrm{d}t = \sum_{n=0}^{\infty} a_n x^{2n+1} \] Find $f(x)$. | [
"\\( e^{x^2} \\)"
] | [
"A sequence $(a_n)_{n\\geqslant0}$ is defined as follows:\n\\[ a_0 = 1 \\]\n\\[ \\frac{a_{n-1}}{a_n} = n+1+\\frac{1}{2n-1}, \\quad n > 0 \\]\n\nAn integrable function $f$ is defined by:\n\\[ \\int_0^x f(t) \\, \\mathrm{d}t = \\sum_{n=0}^{\\infty} a_n x^{2n+1} \\]\n\nFind $f(x)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 10
} | 33 | 0 | 0 |
omr-ws-fe77f225c83ab7de2a062b5e118ac1864195d32534c70edf813ed80041a6c4e2 | A sequence $(a_{n})$ is defined by $a_{1}= 1$ and $a_{n}= a_{1}a_{2}...a_{n-1}+1$ for $n \geq 2$. Find the smallest real number $M$ such that $\sum_{n=1}^{m}\frac{1}{a_{n}}<M$ for all $m \in \mathbb{N}$. | [
"2"
] | [
"A sequence $(a_{n})$ is defined by $a_{1}= 1$ and $a_{n}= a_{1}a_{2}...a_{n-1}+1$ for $n \\geq 2$. Find the smallest real number $M$ such that $\\sum_{n=1}^{m}\\frac{1}{a_{n}}<M$ for all $m \\in \\mathbb{N}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a4196293c6ec7c394e8f420dbc624aa1a2fa4368b251adb5716ca0abba71abde | A sequence $(a_{n})$ of natural numbers is defined by $a_{1} = 1$, $a_{2} = 3$, and $a_{n} = (n + 1)a_{n-1} - na_{n-2}$ for $n \geq 3$. Find all values of $n$ such that $11 \mid a_{n}$. | [
"\\( n = 4, 8, \\text{ and } n \\ge 10 \\)"
] | [
"A sequence $(a_{n})$ of natural numbers is defined by $a_{1} = 1$, $a_{2} = 3$, and $a_{n} = (n + 1)a_{n-1} - na_{n-2}$ for $n \\geq 3$. Find all values of $n$ such that $11 \\mid a_{n}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 29
} | 34 | 0 | 0 |
omr-ws-86a7a8b7f8ec406f84cce6ceca306e96f84b838221f70074844b7298055633ba | A sequence $(a_{n})$ of positive numbers is defined by \[ a_{n} = \frac{1}{16} \left( 1 + 4 a_{n-1} + \sqrt{1 + 24 a_{n-1}} \right) \] with $a_{0} = 1$. Find a general formula for $a_{n}$. | [
"\\( a_{n} = \\frac{(2^{n+1}+1)(2^{n+1}+2)}{3 \\cdot 4^{n+1}} \\)"
] | [
"A sequence $(a_{n})$ of positive numbers is defined by\n\n\\[ a_{n} = \\frac{1}{16} \\left( 1 + 4 a_{n-1} + \\sqrt{1 + 24 a_{n-1}} \\right) \\]\n\nwith $a_{0} = 1$. Find a general formula for $a_{n}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 2
} | 7 | 0 | 0 |
omr-ws-c6113a896860c6e5938e53f57420e69aa78e6be6a9b749eda1efb7e4951cf812 | A sequence $(a_{n})$ of positive numbers is defined by \[ a_{n} = \frac{1}{16} \left(1 + 4a_{n-1} + \sqrt{1 + 24a_{n-1}}\right) \] with $a_{0} = 1$. Find a general formula for $a_{n}$. | [
"\\( a_{n} = \\frac{1}{24} \\{(2^{1-n} + 3)^{2} - 1\\} \\)"
] | [
"A sequence $(a_{n})$ of positive numbers is defined by \n\\[ a_{n} = \\frac{1}{16} \\left(1 + 4a_{n-1} + \\sqrt{1 + 24a_{n-1}}\\right) \\]\nwith $a_{0} = 1$. Find a general formula for $a_{n}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 4
} | 14 | 0 | 0 |
omr-ws-17e76e0a8eafce04d92214df40fd74222d3ed2abec3fc9109ec63a1272b18fd3 | A sequence $(a_{n})$ satisfies $a_{1} = \frac{1}{2}$ and $a_{1} + a_{2} + \ldots + a_{n} = n^{2} a_{n}$. Determine $a_{n}$. | [
"\\( \\frac{1}{n(n+1)} \\)"
] | [
"A sequence $(a_{n})$ satisfies $a_{1} = \\frac{1}{2}$ and $a_{1} + a_{2} + \\ldots + a_{n} = n^{2} a_{n}$. Determine $a_{n}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-d3dd0aea91a1e3d101d16d82a57a2dc94a99ae37edef73b79a1eda8f4ae15f61 | A sequence $(a_{n})$ satisfies the recurrence relation \[ \left(\frac{a_{n}}{a_{n-1}}\right)^{2} = \frac{a_{n-1}}{a_{n-2}} \] with initial conditions $a_{0} = \frac{1}{4}$ and $a_{1} = 1$. Solve the recurrence relation. | [
"\\( a_n = 4^{1 - \\left(\\frac{1}{2}\\right)^{n-1}} \\)"
] | [
"A sequence $(a_{n})$ satisfies the recurrence relation \n\\[\n\\left(\\frac{a_{n}}{a_{n-1}}\\right)^{2} = \\frac{a_{n-1}}{a_{n-2}}\n\\]\nwith initial conditions $a_{0} = \\frac{1}{4}$ and $a_{1} = 1$. Solve the recurrence relation."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 20
} | 32 | 0 | 0 |
omr-ws-f012ff6014f3ecabca12c589a851f6b1f25b0b03520e3ebb5dc47c1aa7392f03 | A sequence $(t_{n})$ is defined by $t_{n} = \left(\tan\frac{a}{2^n}\right)^{2^{n+1}}$ for $n=0,1,2,\ldots$ and $0 < a < \frac{\pi}{2}$. Compute the product $T_{n} = t_{0} \cdot t_{1} \cdot t_{2} \cdots t_{n}$. | [
"\\( 2^{2^{n+2}-2} \\cdot \\frac{\\left(\\sin\\left(\\frac{a}{2^n}\\right)\\right)^{2^{n+2}}}{(\\sin(2a))^2} \\)"
] | [
"A sequence $(t_{n})$ is defined by $t_{n} = \\left(\\tan\\frac{a}{2^n}\\right)^{2^{n+1}}$ for $n=0,1,2,\\ldots$ and $0 < a < \\frac{\\pi}{2}$. Compute the product $T_{n} = t_{0} \\cdot t_{1} \\cdot t_{2} \\cdots t_{n}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d350f0fd669180808b9217053588af259967359bf8e7d1b789058434cf512974 | A sequence $(u_n)$ is defined by $u_0 = 2.9$ and $u_{n+1} = \sqrt{3} + \frac{u_n}{\sqrt{u_n^2 - 1}}$ for every $n \in \mathbb{N}$. Find $\alpha$ such that $u_{2k} < \alpha < u_{2k-1}$ for every $k \in \mathbb{N}$. | [
"\\(\\alpha = \\frac{\\sqrt{3} + \\sqrt{15}}{2}\\)"
] | [
"A sequence $(u_n)$ is defined by $u_0 = 2.9$ and $u_{n+1} = \\sqrt{3} + \\frac{u_n}{\\sqrt{u_n^2 - 1}}$ for every $n \\in \\mathbb{N}$. Find $\\alpha$ such that $u_{2k} < \\alpha < u_{2k-1}$ for every $k \\in \\mathbb{N}$."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 30
} | 52 | 0 | 0 |
omr-ws-41d00faa5b9fce8c759c8dfd3a21cfd20b8b372ee32673324532e9c0bb71852c | A sequence $(u_n)$ is defined by $u_0 = 3$ and $u_{n+1} = (u_n - 1)(u_n + \sqrt{4u_n - 3}) + 1$ for $n = 0, 1, 2, 3, \ldots$. Compute $u_n$ and $P_n = u_0 \cdot u_1 \cdot u_2 \cdots u_n$. | [
"\\( u_n = 2^{2^{n+1}} - 2^{2^n + 1} + 1 \\)"
] | [
"A sequence $(u_n)$ is defined by $u_0 = 3$ and $u_{n+1} = (u_n - 1)(u_n + \\sqrt{4u_n - 3}) + 1$ for $n = 0, 1, 2, 3, \\ldots$. Compute $u_n$ and $P_n = u_0 \\cdot u_1 \\cdot u_2 \\cdots u_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1dd749729a950aefc3a87b6f8279a6d5032971dde8f8b45ea53f68159a2dca64 | A sequence $(u_n)$ is defined by $u_0 = u_1 = 0$, $u_2 = 9$, and $u_{n+3} = \frac{u_{n+2} + u_{n+1} + u_n}{3}$ for every positive integer $n$. Find the limit of the sequence. | [
"\\(\\frac{9}{2}\\)"
] | [
"A sequence $(u_n)$ is defined by $u_0 = u_1 = 0$, $u_2 = 9$, and $u_{n+3} = \\frac{u_{n+2} + u_{n+1} + u_n}{3}$ for every positive integer $n$. Find the limit of the sequence."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-36da4e38618ffbf28c760babf8e699066def7f9f69eed4e1f4a8082b280aac9d | A sequence $(u_n)$ is defined by $u_0 \in [0, 2)$ and $u_n^2 - (n + 1)u_{n + 1} = 1$ for every $n \in \mathbb{N}$. Find the limit of the sequence $u_n$. | [
"0"
] | [
"A sequence $(u_n)$ is defined by $u_0 \\in [0, 2)$ and $u_n^2 - (n + 1)u_{n + 1} = 1$ for every $n \\in \\mathbb{N}$. Find the limit of the sequence $u_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fe6375d78b99378234ddf6aa4ab84318e475baa72c391b4281d8d847316ce827 | A sequence $(u_n)$ of integers is defined for $n \geq 0$ by $u_0 = 0$, $u_1 = 1$, and $u_n - 2u_{n-1} + (1-c)u_{n-2} = 0$ for $n \geq 2$, where $c$ is a fixed integer independent of $n$. Find the least value of $c$ for which both of the following statements are true: (i) If $p$ is a prime less than or equal to $P$, the... | [
"1"
] | [
"A sequence $(u_n)$ of integers is defined for $n \\geq 0$ by $u_0 = 0$, $u_1 = 1$, and $u_n - 2u_{n-1} + (1-c)u_{n-2} = 0$ for $n \\geq 2$, where $c$ is a fixed integer independent of $n$. Find the least value of $c$ for which both of the following statements are true:\n(i) If $p$ is a prime less than or equal to ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-521f1b8c5e0f53287ce4c310a056c927ccc72d0eacd05e9339ab98bcb6840ed4 | A sequence $(u_n)_{n\ge 1}$ is defined by $u_1=2$ and $u_{n+1}=u_n^2-u_n+1$ for $n\ge 1$. Let $a_1,a_2,...,a_n$ be positive integers and $r$ be a rational number such that $r>\max(a_1,a_2,...,a_n)$ and \[\frac{1}{r}+\sum_{i=1}^{n}{\frac{1}{a_i}}=1\]. Calculate the maximum possible value of $\frac{a_1a_2...a_n(r+1)}{u_1... | [
"1"
] | [
"A sequence $(u_n)_{n\\ge 1}$ is defined by $u_1=2$ and $u_{n+1}=u_n^2-u_n+1$ for $n\\ge 1$. Let $a_1,a_2,...,a_n$ be positive integers and $r$ be a rational number such that $r>\\max(a_1,a_2,...,a_n)$ and \\[\\frac{1}{r}+\\sum_{i=1}^{n}{\\frac{1}{a_i}}=1\\]. Calculate the maximum possible value of $\\frac{a_1a_2..... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7f7b00c4b83a805db09944dfffd6282fcf22564172b642306b3fd7a458d11de3 | A sequence $(u_{n})$ is defined by $u_{1}=1$ and $u_{n+1}=\left(1+\frac{1}{n}\right) u_{n}+\frac{n}{n!}$. Find $u_{n}$ for all $n=1,2,3,\ldots$. | [
"\\( u_n = 2n - \\frac{1}{(n-1)!} \\)"
] | [
"A sequence $(u_{n})$ is defined by $u_{1}=1$ and $u_{n+1}=\\left(1+\\frac{1}{n}\\right) u_{n}+\\frac{n}{n!}$. Find $u_{n}$ for all $n=1,2,3,\\ldots$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 21
} | 67 | 0 | 0 |
omr-ws-45cf7e6d90d1a206391dc9422c3a4770a127324070caefbbf59bf235c0701ec7 | A sequence $(x_n)$ is given as follows: $x_0$ and $x_1$ are arbitrary positive real numbers, and $x_{n+2} = \frac{1 + x_{n+1}}{x_n}$ for $n \ge 0$. Find $x_{1998}$. | [
"\\(\\frac{x_0 + x_1 + 1}{x_0 x_1}\\)"
] | [
"A sequence $(x_n)$ is given as follows: $x_0$ and $x_1$ are arbitrary positive real numbers, and $x_{n+2} = \\frac{1 + x_{n+1}}{x_n}$ for $n \\ge 0$. Find $x_{1998}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-6e6a54f8c5dbe9f5cc79e184bfbeee320ab17ce7310426ceebefb9aca7fdaf9d | A sequence $(x_n)$ is given by $x_1 = 2$ and $nx_n = 2(2n-1)x_{n-1}$ for $n > 1$. Calculate the value of $x_{10}$. | [
"184756"
] | [
"A sequence $(x_n)$ is given by $x_1 = 2$ and $nx_n = 2(2n-1)x_{n-1}$ for $n > 1$. Calculate the value of $x_{10}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c334648f19194ee434a8c0840441043f63813c158ad49d87db76acaf446be7f4 | A sequence $(x_n)_{n\ge 0}$ is defined as follows: $x_0=a$, $x_1=2$, and $x_n=2x_{n-1}x_{n-2}-x_{n-1}-x_{n-2}+1$ for all $n>1$. Find all integers $a$ such that $2x_{3n}-1$ is a perfect square for all $n\ge 1$. | [
"\\( a = \\frac{k^2 + 1}{2} \\) for odd integers \\( k \\)."
] | [
"A sequence $(x_n)_{n\\ge 0}$ is defined as follows: $x_0=a$, $x_1=2$, and $x_n=2x_{n-1}x_{n-2}-x_{n-1}-x_{n-2}+1$ for all $n>1$. Find all integers $a$ such that $2x_{3n}-1$ is a perfect square for all $n\\ge 1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 4
} | 7 | 0 | 0 |
omr-ws-101324f809f902686b866ea5f81e0a5cae762727c12b767e839bd7f84e631a7f | A sequence $(x_n)_{n\ge 1}$ is defined by $x_1=\frac{1}{2}$ and $2x_n=1+x_{n-1}^2$ for $n\ge 2$. Find $x_n$. | [
"\\frac{1}{2^{2^{0}}}+\\frac{1}{2^{2^{1}}}+\\frac{1}{2^{2^{2}}}+\\cdots+\\frac{1}{2^{2^{n-1}}}"
] | [
"A sequence $(x_n)_{n\\ge 1}$ is defined by $x_1=\\frac{1}{2}$ and $2x_n=1+x_{n-1}^2$ for $n\\ge 2$. Find $x_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-81f0d4569998f08f8c0c577b62e202c7e07f54d40ac361afea0cf8cdba823d08 | A sequence $(x_{n})$ is defined by $$ \left\{\begin{matrix}x_{1}=1 \\ x_{n+1}=\dfrac{6x_{n}}{x_{n}^{2}+3};\forall n\geq 1 \end{matrix}\right. $$ for every $n$, set $a_{n}=\frac{x_{n}^{2}x_{n+1}}{2(x_{n+1}-x_{n})}$. Find the smallest positive integer $k$ such that $a_{k}$ has at least $10$ distinct prime divisors. | [
"11"
] | [
"A sequence $(x_{n})$ is defined by\n$$ \\left\\{\\begin{matrix}x_{1}=1 \\\\ x_{n+1}=\\dfrac{6x_{n}}{x_{n}^{2}+3};\\forall n\\geq 1\n\\end{matrix}\\right. $$\nfor every $n$, set $a_{n}=\\frac{x_{n}^{2}x_{n+1}}{2(x_{n+1}-x_{n})}$.\nFind the smallest positive integer $k$ such that $a_{k}$ has at least $10$ distinct p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-fb76bbbd68cdf484a2ef3216f1b5b64995800c8c3d9f7ad81fabda1611d51443 | A sequence $(x_{n})$ is defined by $$ \left\{\begin{matrix}x_{1}=1 \\ x_{n+1}=\dfrac{6x_{n}}{x_{n}^{2}+3};\forall n\geq 1 \end{matrix}\right. $$ for every $n$, set $a_{n}=\frac{x_{n}^{2}x_{n+1}}{2(x_{n+1}-x_{n})}$. Let $L_{n}=\sum _{k=1}^{n}\frac{a_{k}}{(4a_{k}+3)(4a_{k}+9)}$. Calculate the value of $\lim_{n\to\infty} ... | [
"\\dfrac{1}{60}"
] | [
"A sequence $(x_{n})$ is defined by\n$$ \\left\\{\\begin{matrix}x_{1}=1 \\\\ x_{n+1}=\\dfrac{6x_{n}}{x_{n}^{2}+3};\\forall n\\geq 1\n\\end{matrix}\\right. $$\nfor every $n$, set $a_{n}=\\frac{x_{n}^{2}x_{n+1}}{2(x_{n+1}-x_{n})}$.\nLet $L_{n}=\\sum _{k=1}^{n}\\frac{a_{k}}{(4a_{k}+3)(4a_{k}+9)}$.\nCalculate the value... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e0120a32947f4c22dbc0479598838233f6aa3b82a5a7f609425269f5678fa610 | A sequence $\{ a_i \}$ is defined as follows: $a_{i+1}=\frac{1}{1-a_i}$ for $i\geq 1$. If $a_3=a_1$, compute $(a_9)^9$. | [
"1"
] | [
"A sequence $\\{ a_i \\}$ is defined as follows: $a_{i+1}=\\frac{1}{1-a_i}$ for $i\\geq 1$. If $a_3=a_1$, compute $(a_9)^9$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d669e90d0feb17227c5f8aa2d3225c97a35e42e3d23f3c2a124cd90bf634ecfc | A sequence $\{ a_n \}$ has the property that $\sum_{k=1}^{n} a_k = \left(a_n+\frac{1}{4}\right)^2$ for all natural numbers $n$. Find the explicit formula for $a_n$. | [
"\\( a_n = \\frac{2n-1}{4} \\)"
] | [
"A sequence $\\{ a_n \\}$ has the property that $\\sum_{k=1}^{n} a_k = \\left(a_n+\\frac{1}{4}\\right)^2$ for all natural numbers $n$. Find the explicit formula for $a_n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-7006b0eda8adf8024ce2f447c1ebdaf4f14d35b76019caf124c3b7402b1b944a | A sequence $\{ a_n\}$ is defined by $a_{n+1}=2(a_n)^2$, where $n$ is a natural number and $a_1=1$. Find the maximum possible value of $n$ such that $a_n < 10^{60}$. | [
"8"
] | [
"A sequence $\\{ a_n\\}$ is defined by $a_{n+1}=2(a_n)^2$, where $n$ is a natural number and $a_1=1$. Find the maximum possible value of $n$ such that $a_n < 10^{60}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b187fa0d08d6f44592e8c089afa8aef7cc9b8327f4be2ca530e440506bc22443 | A sequence $\{ r_n \}$ is defined recursively as follows: $r_0 = r_1 = 1$ and $r_{n} = 3r_{n-2}^2 + 2r_{n-1}$ for all $n \geq 2$. Find the remainder obtained on dividing $r_{2012}$ by $1000$. | [
"285"
] | [
"A sequence $\\{ r_n \\}$ is defined recursively as follows:\n$r_0 = r_1 = 1$ and $r_{n} = 3r_{n-2}^2 + 2r_{n-1}$ for all $n \\geq 2$.\nFind the remainder obtained on dividing $r_{2012}$ by $1000$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-0096729a9e0d43e68550a3af0c6c46361eff56e682354e9bf5dbdd7d20255230 | A sequence $\{ u_n \}$ is defined as $u_1=1$ and $$u_{n+1} = u_n + \dfrac{1}{2u_n}, \quad \forall n=1,2,\ldots$$ Find $\lfloor u_{48} \rfloor$. | [
"7"
] | [
"A sequence $\\{ u_n \\}$ is defined as $u_1=1$ and\n$$u_{n+1} = u_n + \\dfrac{1}{2u_n}, \\quad \\forall n=1,2,\\ldots$$\nFind $\\lfloor u_{48} \\rfloor$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-7946c902f675b7aaba3373fbec44cc2e14894f84e03961fd84fe4bf5c5e614ee | A sequence $\{ x_n \}$ is defined by $x_n x_{n+1} = 2n$ and $x_1 = 1$. Evaluate the limit $\lim_{n \to \infty} \frac{|x_{n+1} - x_n|}{\sqrt{n}}$. | [
"\\(\\frac{4 - \\pi}{\\sqrt{2\\pi}}\\)"
] | [
"A sequence $\\{ x_n \\}$ is defined by $x_n x_{n+1} = 2n$ and $x_1 = 1$. Evaluate the limit $\\lim_{n \\to \\infty} \\frac{|x_{n+1} - x_n|}{\\sqrt{n}}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 4
} | 10 | 0 | 0 |
omr-ws-42acfdef0e1814ad681857eefb2107a803638fea5490b9f354980a37079a2d79 | A sequence $\{R_n\}_{n \ge 0}$ obeys the recurrence $7R_n = 64 - 2R_{n-1} + 9R_{n-2}$ for any integers $n \ge 2$. Additionally, $R_0 = 10$ and $R_1 = -2$. Let $$S = \sum^{\infty}_{i=0} \frac{R_i}{2^i}.$$ $S$ can be expressed as $\frac{m}{n}$ for two relatively prime positive integers $m$ and $n$. Determine the value of... | [
"443"
] | [
"A sequence $\\{R_n\\}_{n \\ge 0}$ obeys the recurrence $7R_n = 64 - 2R_{n-1} + 9R_{n-2}$ for any integers $n \\ge 2$. Additionally, $R_0 = 10$ and $R_1 = -2$. Let $$S = \\sum^{\\infty}_{i=0} \\frac{R_i}{2^i}.$$ $S$ can be expressed as $\\frac{m}{n}$ for two relatively prime positive integers $m$ and $n$. Determine... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4d5dba56dc5b4fd4fae25c0bc3f2eef183ed4c907d4e2d85944a1b38fd6eb4d5 | A sequence $\{a_1, a_2, \ldots, a_{10000}\}$ is defined as follows. Let $a_1$ be a positive integer with 10000 digits, each digit being 0, 1, or 2. For a positive integer $i$ with $1 \le i < 10000$, if $a_i$ has digits $b_1b_2 \ldots b_p$, then \[ a_{i+1} = \sum_{k=1}^{p-1} 10^{k-1} \cdot (2b_k + 2b_{k+1} \mod 3). \] S... | [
"1"
] | [
"A sequence $\\{a_1, a_2, \\ldots, a_{10000}\\}$ is defined as follows. Let $a_1$ be a positive integer with 10000 digits, each digit being 0, 1, or 2. For a positive integer $i$ with $1 \\le i < 10000$, if $a_i$ has digits $b_1b_2 \\ldots b_p$, then\n\\[ a_{i+1} = \\sum_{k=1}^{p-1} 10^{k-1} \\cdot (2b_k + 2b_{k+1}... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dbb6cc588341dbfb41499ed196b4a7db8e2436f4850e2cf0dbddea96b89f1848 | A sequence $\{a_i\}$ is defined as $a_{i+1} = \frac{1}{1-a_i}$ for $i \ge 1$. If $a_3 = a_1$, compute $(a_9)^9$. | [
"-1"
] | [
"A sequence $\\{a_i\\}$ is defined as $a_{i+1} = \\frac{1}{1-a_i}$ for $i \\ge 1$. If $a_3 = a_1$, compute $(a_9)^9$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-1cea323dca7a4110962c7f8315c9ca59fa749486c695213347beabb49b808d3b | A sequence $\{a_n\} = a_1, a_2, a_3, \ldots$ of positive integers is called *Fib-o’nacci* if it satisfies $a_n = a_{n-1} + a_{n-2}$ for all $n \ge 3$. Suppose that $m$ is the largest even positive integer such that exactly one *Fib-o’nacci* sequence satisfies $a_5 = m$, and suppose that $n$ is the largest odd positive ... | [
"132"
] | [
"A sequence $\\{a_n\\} = a_1, a_2, a_3, \\ldots$ of positive integers is called *Fib-o’nacci* if it satisfies $a_n = a_{n-1} + a_{n-2}$ for all $n \\ge 3$. Suppose that $m$ is the largest even positive integer such that exactly one *Fib-o’nacci* sequence satisfies $a_5 = m$, and suppose that $n$ is the largest odd ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b74816177f24aaf532375be90fbe605f7e5cca4dce1aa8b2888b389bd2c259b | A sequence $\{a_n\}$ for $n \geq 1$ 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 $\sum^{\infty}_{k=1} \frac{(3^k)(k+3)}{4^k a_k}$. | [
"\\dfrac{21}{8}"
] | [
"A sequence $\\{a_n\\}$ for $n \\geq 1$ of positive integers is defined by $a_1 = 2$ and for integers $n > 1$, \n\\[ \\frac{1}{a_1} + \\frac{1}{a_2} + \\cdots + \\frac{1}{a_{n-1}} + \\frac{n}{a_n} = 1. \\]\nDetermine the value of $\\sum^{\\infty}_{k=1} \\frac{(3^k)(k+3)}{4^k a_k}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f7bc1a842c43891763ea6479b43a5b51a7c05ff920f5c8dd7e873d8451e7607a | A sequence $\{a_n\}$ is defined as follows: $a_1 = 1$ and $a_{n+1} = 420a_n - 69$ for all positive integers $n$. Determine the number of squares in this sequence. If there are infinitely many square numbers in the sequence, enter 999. | [
"1"
] | [
"A sequence $\\{a_n\\}$ is defined as follows: $a_1 = 1$ and $a_{n+1} = 420a_n - 69$ for all positive integers $n$. Determine the number of squares in this sequence. If there are infinitely many square numbers in the sequence, enter 999."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4ae4469b86a1060bc86fb6b7a84b70bccf94ed583cbeb71c7fce3818f9a1524d | A sequence $\{a_n\}$ is defined as follows: $a_n$ is the $n$th digit after the decimal point in the infinite decimal representation of the rational number $\frac{1}{n}$. For example, $a_1=0$ since $\frac{1}{1} = 1.00000...$, and $a_6 = 6$ since $\frac{1}{6} = .1666666...$, and $a_8 = 0$ since $\frac{1}{8} = .125000000.... | [
"8"
] | [
"A sequence $\\{a_n\\}$ is defined as follows: $a_n$ is the $n$th digit after the decimal point in the infinite decimal representation of the rational number $\\frac{1}{n}$. For example, $a_1=0$ since $\\frac{1}{1} = 1.00000...$, and $a_6 = 6$ since $\\frac{1}{6} = .1666666...$, and $a_8 = 0$ since $\\frac{1}{8} = ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 2
} | 24 | 0 | 0 |
omr-ws-23a497894923989d377dd440b045c3658f50dfef58bf4397017eda740c0a1953 | A sequence $\{a_n\}$ is defined as follows: \[a_1 = 1\] For $n \geq 1$, \[a_{n+1} = \frac{a_n}{1 + n \cdot a_n}\]. Find an explicit formula for $a_n$, if it exists. Hence, find the value of $a_{2012}$. | [
"\\(\\frac{1}{2023067}\\)"
] | [
"A sequence $\\{a_n\\}$ is defined as follows:\n\\[a_1 = 1\\]\nFor $n \\geq 1$,\n\\[a_{n+1} = \\frac{a_n}{1 + n \\cdot a_n}\\].\nFind an explicit formula for $a_n$, if it exists. Hence, find the value of $a_{2012}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-b203b82fbe7eaaf6af87b747d4bdc96f37fbdeb371244a69413756518ec8cc64 | A sequence $\{a_n\}$ is defined by $a_1=1$, and for $n \geq 2$, $a_n$ is the $x$-coordinate of the midpoint of the points of intersection between the line $y=a_{n-1}x$ and the curve $y=|x^2-1|$. Find $\lim_{n\to\infty} a_n$. | [
"\\(\\frac{2}{3}\\sqrt{3}\\)"
] | [
"A sequence $\\{a_n\\}$ is defined by $a_1=1$, and for $n \\geq 2$, $a_n$ is the $x$-coordinate of the midpoint of the points of intersection between the line $y=a_{n-1}x$ and the curve $y=|x^2-1|$. Find $\\lim_{n\\to\\infty} a_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 16
} | 64 | 0 | 0 |
omr-ws-0de0bf28232befd791ce3edb0cafc01830e793920da13f4cd3eb643e309fe6e0 | A sequence $\{a_n\}$ is defined by $a_1=a_2=1$ and $a_{n+2}=a_{n+1}+a_n$. Find all integers $n$ such that 13 divides $a_n$. | [
"\\(7 \\mid n\\)"
] | [
"A sequence $\\{a_n\\}$ is defined by $a_1=a_2=1$ and $a_{n+2}=a_{n+1}+a_n$. Find all integers $n$ such that 13 divides $a_n$."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-d7a1bdd83d535a807cc4e7b14872cf7e1092f127f7605e15ab1545c81b1182d6 | A sequence $\{a_n\}$ is defined by $a_n = \int_0^1 x^3(1-x)^n \, dx$ for $n = 1, 2, 3, \ldots$. Find the constant number $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n - a_{n+1}) = \frac{1}{3}$. | [
"5"
] | [
"A sequence $\\{a_n\\}$ is defined by $a_n = \\int_0^1 x^3(1-x)^n \\, dx$ for $n = 1, 2, 3, \\ldots$. Find the constant number $c$ such that $\\sum_{n=1}^{\\infty} (n+c)(a_n - a_{n+1}) = \\frac{1}{3}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-5bde64a471ea60c802c6ae0258a8e5590f43e0721fc51837037b8b8651d2c1b8 | A sequence $\{a_n\}$ is defined by $a_n=\int_0^1 x^3(1-x)^n dx$ for $n=1,2,3,\ldots$. Find the constant number $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}$. | [
"5"
] | [
"A sequence $\\{a_n\\}$ is defined by $a_n=\\int_0^1 x^3(1-x)^n dx$ for $n=1,2,3,\\ldots$. Find the constant number $c$ such that $\\sum_{n=1}^{\\infty} (n+c)(a_n-a_{n+1})=\\frac{1}{3}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dbc3f7e6ee21a6d60b3d2224eaff3bbe4451e8faf7dd47fa70b0410e312551e3 | A sequence $\{a_n\}$ is defined by: $a_1 = 1, a_{n+1} = \frac{a_n^2 + 1}{n}$. Calculate the value of $\lim_{n \to \infty} \frac{a_1 + a_2 + \cdots + a_n}{\ln n}$. | [
"1"
] | [
"A sequence $\\{a_n\\}$ is defined by: $a_1 = 1, a_{n+1} = \\frac{a_n^2 + 1}{n}$. Calculate the value of $\\lim_{n \\to \\infty} \\frac{a_1 + a_2 + \\cdots + a_n}{\\ln n}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3694ee4acb4ce983cd5ec370c971d67461106023e189dca4df4589e50f6403dd | A sequence $\{a_n\}$ is defined by: $a_1 = 1, a_{n+1} = a_n + \dfrac{1}{\sqrt{a_n}}$ for $n = 1, 2, 3, \ldots$. Find all real numbers $q$ such that the sequence $\{u_n\}$ defined by $u_n = \frac{a_n^q}{n}$, $n = 1, 2, 3, \ldots$ has a nonzero finite limit as $n$ goes to infinity. | [
"\\dfrac{3}{2}"
] | [
"A sequence $\\{a_n\\}$ is defined by: $a_1 = 1, a_{n+1} = a_n + \\dfrac{1}{\\sqrt{a_n}}$ for $n = 1, 2, 3, \\ldots$. Find all real numbers $q$ such that the sequence $\\{u_n\\}$ defined by $u_n = \\frac{a_n^q}{n}$, $n = 1, 2, 3, \\ldots$ has a nonzero finite limit as $n$ goes to infinity."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b5a4da18b14fae823d11081ebc93cee6bf72bd34923fcfbe4612e881811fc98a | A sequence $\{a_n\}$ is defined by: $a_1 = 1, a_{n+1} = a_n + \dfrac{1}{\sqrt{a_n}}$ for $n = 1, 2, 3, \ldots$. Find all real numbers $q$ such that the sequence $\{u_n\}$ defined by $u_n = a_n^q$, $n = 1, 2, 3, \ldots$ has a nonzero finite limit as $n$ goes to infinity. | [
"\\( q = \\frac{3}{2} \\)"
] | [
"A sequence $\\{a_n\\}$ is defined by: $a_1 = 1, a_{n+1} = a_n + \\dfrac{1}{\\sqrt{a_n}}$ for $n = 1, 2, 3, \\ldots$. Find all real numbers $q$ such that the sequence $\\{u_n\\}$ defined by $u_n = a_n^q$, $n = 1, 2, 3, \\ldots$ has a nonzero finite limit as $n$ goes to infinity."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf170bc2a85d1ffdce7d8d22bf14ed8b9649cfcaba628da9c7a7ba81bb71c3b8 | A sequence $\{a_n\}$ is defined by: $a_1 = 2, a_2 = 3, a_{n+1} = \frac{n+1}{2n}a_n + \frac{n-2}{3n}a_{n-1} + \frac{1}{6}$ for all $n \ge 2$. Calculate the value of $\lim_{n \to \infty} a_n$. | [
"1"
] | [
"A sequence $\\{a_n\\}$ is defined by: $a_1 = 2, a_2 = 3, a_{n+1} = \\frac{n+1}{2n}a_n + \\frac{n-2}{3n}a_{n-1} + \\frac{1}{6}$ for all $n \\ge 2$. Calculate the value of $\\lim_{n \\to \\infty} a_n$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04905fa287b6179cb859d766ac33718d55e10ea357415d0dfa26358e2dcf8d46 | A sequence $\{a_n\}$ is defined such that $a_i=i$ for $i=1,2,3,\ldots,2020$ and for $i>2020$, $a_i$ is the average of the previous $2020$ terms. What is the largest integer less than or equal to $\lim_{n\to\infty}a_n$? | [
"1347"
] | [
"A sequence $\\{a_n\\}$ is defined such that $a_i=i$ for $i=1,2,3,\\ldots,2020$ and for $i>2020$, $a_i$ is the average of the previous $2020$ terms. What is the largest integer less than or equal to $\\lim_{n\\to\\infty}a_n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 4
} | 21 | 0 | 0 |
omr-ws-516aae9ec31524ec71908af0db2292668398e4e37aa8ecae3d62716f19b900f4 | A sequence $\{a_n\}$ is given by $a_1 = 1$ and $a_{n+1} = \frac{3a_n - 1}{4a_n - 1}$. Find the general term $a_n$. | [
"\\( a_n = \\frac{2n - 1}{2n + 1} \\)"
] | [
"A sequence $\\{a_n\\}$ is given by $a_1 = 1$ and $a_{n+1} = \\frac{3a_n - 1}{4a_n - 1}$. Find the general term $a_n$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-08691d7ac481d63cf37078d595bcbff931e4ccf3cf1dfc7ef9ce5617906b8542 | A sequence $\{a_n\}$ of real numbers is defined by $a_1 = 1$ and for all integers $n \geq 1$, $$a_{n+1} = \frac{a_n \sqrt{n^2+n}}{\sqrt{n^2+n+2a_n^2}}.$$ Compute the sum of all positive integers $n < 1000$ for which $a_n$ is a rational number. | [
"131"
] | [
"A sequence $\\{a_n\\}$ of real numbers is defined by $a_1 = 1$ and for all integers $n \\geq 1$,\n$$a_{n+1} = \\frac{a_n \\sqrt{n^2+n}}{\\sqrt{n^2+n+2a_n^2}}.$$\nCompute the sum of all positive integers $n < 1000$ for which $a_n$ is a rational number."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4d404563908c83f6f2eed25bb2f5aa1561316961365908883c951589b7504f39 | A sequence $\{a_n\}$ satisfies $a_n=a_{n-1}+a_{n-3}$ for all $n \ge 4$. If $a_1 =3$ and $a_6 =30$, find $a_8$. | [
"63"
] | [
"A sequence $\\{a_n\\}$ satisfies $a_n=a_{n-1}+a_{n-3}$ for all $n \\ge 4$. If $a_1 =3$ and $a_6 =30$, find $a_8$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a7b6c1915507c2b9aa45c80161073c1309fd6a6d8e61164aab78b5203130c97c | A sequence $\{a_n\}$ satisfies the following conditions: \[ a_0 = 0, \quad a_1 = a_2 = 1, \quad a_{3n} = a_n, \quad a_{3n+1} = a_{3n+2} = a_n + 1 \] for all $n \geq 1$. Compute $a_{2021}$. | [
"6"
] | [
"A sequence $\\{a_n\\}$ satisfies the following conditions:\n\\[ a_0 = 0, \\quad a_1 = a_2 = 1, \\quad a_{3n} = a_n, \\quad a_{3n+1} = a_{3n+2} = a_n + 1 \\]\nfor all $n \\geq 1$. Compute $a_{2021}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-c4a66868da62927b8838a4da3185d35306b452635f14f55925e4d623070d3301 | A sequence $\{a_n\}$ satisfies the recurrence relation $(3 - a_{n+1})(6 + a_n) = 18$ with the initial condition $a_0 = 3$. Find the sum $\sum_{n=0}^{\infty} a_n$. | [
"6"
] | [
"A sequence $\\{a_n\\}$ satisfies the recurrence relation $(3 - a_{n+1})(6 + a_n) = 18$ with the initial condition $a_0 = 3$. Find the sum $\\sum_{n=0}^{\\infty} a_n$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9de4848b6f076d0e8adab423377bf54b5fce42c3fd165a356b82e17d40ab5da5 | A sequence $\{a_n\}_{n\geq 1}$ is defined by the recurrence relation $$a_1 = 1,\quad a_{n+1} = \log \frac{e^{a_n}-1}{a_n}.$$ Another sequence $\{b_n\}_{n\geq 1}$ is defined as $b_n = \prod\limits_{i=1}^n a_i$. Evaluate the infinite series $\sum\limits_{n=1}^\infty b_n$. | [
"e - 1"
] | [
"A sequence $\\{a_n\\}_{n\\geq 1}$ is defined by the recurrence relation\n$$a_1 = 1,\\quad a_{n+1} = \\log \\frac{e^{a_n}-1}{a_n}.$$\nAnother sequence $\\{b_n\\}_{n\\geq 1}$ is defined as $b_n = \\prod\\limits_{i=1}^n a_i$. Evaluate the infinite series $\\sum\\limits_{n=1}^\\infty b_n$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dec838ccf7b8af143840bf41ae84ea4202ec002b3db762a77c887e5d6d270f88 | A sequence $\{a_n\}_{n\geq 1}$ 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 $\\{a_n\\}_{n\\geq 1}$ 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",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-1084f86f1dd4278c29d55768ea25c8f83047b484e276b9940d501e6fb7690f4c | A sequence $\{a_{n}\}$ with $n = 2,3,...$ of positive integers satisfies the following conditions: 1. For each natural number $m \geq 1$: $a_{2^{m}}= \frac{1}{m}$. 2. For each natural number $n \geq 2$: $a_{2n-1}a_{2n}= a_{n}$. 3. For all natural numbers $n,m$ with $2^{m}> n \geq 1$: $a_{2n}a_{2n+1}= a_{2^{m}+n}$. Dete... | [
"\\(\\frac{2}{3}\\)"
] | [
"A sequence $\\{a_{n}\\}$ with $n = 2,3,...$ of positive integers satisfies the following conditions:\n1. For each natural number $m \\geq 1$: $a_{2^{m}}= \\frac{1}{m}$.\n2. For each natural number $n \\geq 2$: $a_{2n-1}a_{2n}= a_{n}$.\n3. For all natural numbers $n,m$ with $2^{m}> n \\geq 1$: $a_{2n}a_{2n+1}= a_{2... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f960171a3500a09ac220a443c7fc0207726b9b961d106d9be3aecccde0531c3b | A sequence $\{a_{n}\}_{n=0}$ is defined as: $a_{0}=1$ and $a_{n} = \sum_{k=1}^{[\sqrt{n}]} a_{n-k^2}$ for $n \in \mathbb{Z}_{+}$. Find all $n \in \mathbb{N}$ such that $a_{2n} \equiv 1 \pmod{2}$. | [
"n = k^2"
] | [
"A sequence $\\{a_{n}\\}_{n=0}$ is defined as: $a_{0}=1$ and $a_{n} = \\sum_{k=1}^{[\\sqrt{n}]} a_{n-k^2}$ for $n \\in \\mathbb{Z}_{+}$. Find all $n \\in \\mathbb{N}$ such that $a_{2n} \\equiv 1 \\pmod{2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5da2f6fae4c04e8176981f812dc8400de46ce7654ff4d1d03adeb248644744db | A sequence $\{x_n\}$ is defined by $x_0 = a$ where $1 \le a \le 2019$ and $a \in \mathbb{R}$. The sequence follows the rule: \[ x_{n+1} = \begin{cases} 1 + 1009x_n & \text{if } x_n \le 2 \\ 2021 - x_n & \text{if } 2 < x_n \le 1010 \\ 3031 - 2x_n & \text{if } 1010 < x_n \le 1011 \\ 2020 - x_n & \text{if } x_n > 1011 \en... | [
"All even integers \\( k \\) and all odd integers \\( k \\geq 2019 \\)."
] | [
"A sequence $\\{x_n\\}$ is defined by $x_0 = a$ where $1 \\le a \\le 2019$ and $a \\in \\mathbb{R}$. The sequence follows the rule:\n\\[ x_{n+1} = \\begin{cases} \n1 + 1009x_n & \\text{if } x_n \\le 2 \\\\\n2021 - x_n & \\text{if } 2 < x_n \\le 1010 \\\\\n3031 - 2x_n & \\text{if } 1010 < x_n \\le 1011 \\\\\n2020 - ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d33e00669f4497fd4ecf3fd93ff0959effb70f8fa0480bcbd32511f56ef3a3e2 | A sequence $\{z_n\}$ satisfies that for any positive integer $i,$ $z_i\in\{0,1,\cdots,9\}$ and $z_i\equiv i-1 \pmod {10}.$ Suppose there are $2021$ non-negative reals $x_1,x_2,\cdots,x_{2021}$ such that for $k=1,2,\cdots,2021,$ $$\sum_{i=1}^kx_i\geq\sum_{i=1}^kz_i,\quad \sum_{i=1}^kx_i\leq\sum_{i=1}^kz_i+\sum_{j=1}^{10... | [
"46399.4"
] | [
"A sequence $\\{z_n\\}$ satisfies that for any positive integer $i,$ $z_i\\in\\{0,1,\\cdots,9\\}$ and $z_i\\equiv i-1 \\pmod {10}.$ Suppose there are $2021$ non-negative reals $x_1,x_2,\\cdots,x_{2021}$ such that for $k=1,2,\\cdots,2021,$\n$$\\sum_{i=1}^kx_i\\geq\\sum_{i=1}^kz_i,\\quad \\sum_{i=1}^kx_i\\leq\\sum_{i... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-0ab25a124a6f5e9e4d20ee48aff3e789fae3bab2c62b675934efb8f3b428c299 | A sequence $a_1$, $a_2$, $\ldots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\geq1$. If $a_1=999$, $a_2<999$, and $a_{2006}=1$, how many different values of $a_2$ are possible? | [
"324"
] | [
"A sequence $a_1$, $a_2$, $\\ldots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\\geq1$. If $a_1=999$, $a_2<999$, and $a_{2006}=1$, how many different values of $a_2$ are possible?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-a68b03249308599a06cdac41d4ab6ec00b3c333643b8bb7212bf1cedfb50cbb8 | 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 $a_{1000}$. | [
"63"
] | [
"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- $a_{2001} = 200$\nFind the value of $a_{1000}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-c449d6d9ae6278a9aa73982a4b1592ecbe5a065342d100cab7347d75d1959d69 | A sequence $a_1, a_2, \ldots, a_n$ is called "rioplatense" if it satisfies the following conditions: $a_1 = 1$ and for all $k \geq 2$, $a_k = a_{k-1} + 2k - 3$. Find the value of $a_{10}$. | [
"82"
] | [
"A sequence $a_1, a_2, \\ldots, a_n$ is called \"rioplatense\" if it satisfies the following conditions: $a_1 = 1$ and for all $k \\geq 2$, $a_k = a_{k-1} + 2k - 3$. Find the value of $a_{10}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c40f50c096f8827181d7123e8c7ed77b0fbc12646feae01e39a856301e659e8 | A sequence $a_1,$ $a_2,$ $a_3,$ $\dots,$ is defined recursively by $a_1 = 1,$ $a_2 = 1,$ and for $k \ge 3,$ \[a_k = \frac{1}{3} a_{k - 1} + \frac{1}{4} a_{k - 2}.\]Evaluate $a_1 + a_2 + a_3 + \dotsb.$ | [
"4"
] | [
"A sequence $a_1,$ $a_2,$ $a_3,$ $\\dots,$ is defined recursively by $a_1 = 1,$ $a_2 = 1,$ and for $k \\ge 3,$\n\\[a_k = \\frac{1}{3} a_{k - 1} + \\frac{1}{4} a_{k - 2}.\\]Evaluate $a_1 + a_2 + a_3 + \\dotsb.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6bb85a8f1964577071da8fbbdc7a34856e6e62d282a503de9a463fe48e6bceb8 | A sequence \( \{R_n\}_{n \ge 0} \) obeys the recurrence \( 7R_n = 64 - 2R_{n-1} + 9R_{n-2} \) for any integers \( n \ge 2 \). Additionally, \( R_0 = 10 \) and \( R_1 = -2 \). Let \( S = \sum_{i=0}^{\infty} \frac{R_i}{2^i} \). \( S \) can be expressed as \( \frac{m}{n} \) for two relatively prime positive integers \( m ... | [
"443"
] | [
"A sequence \\( \\{R_n\\}_{n \\ge 0} \\) obeys the recurrence \\( 7R_n = 64 - 2R_{n-1} + 9R_{n-2} \\) for any integers \\( n \\ge 2 \\). Additionally, \\( R_0 = 10 \\) and \\( R_1 = -2 \\). Let \\( S = \\sum_{i=0}^{\\infty} \\frac{R_i}{2^i} \\). \\( S \\) can be expressed as \\( \\frac{m}{n} \\) for two relatively ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b5e79c17d78796c48430615dc0383ad8f43026f9963b77993e21aa02e52030b | A sequence \( \{a_i\} \) is defined as follows: \( a_{i+1} = \frac{1}{1 - a_i} \) for \( i \geq 1 \). If \( a_3 = a_1 \), then what is the value of \( a_9^9 \)? | [
"-1"
] | [
"A sequence \\( \\{a_i\\} \\) is defined as follows: \\( a_{i+1} = \\frac{1}{1 - a_i} \\) for \\( i \\geq 1 \\). If \\( a_3 = a_1 \\), then what is the value of \\( a_9^9 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5d80f3f4fb6541c920fac096561a7bc1abc9f6b61058b690edbeb8894e8d035a | A sequence \( \{a_n\} \) is defined such that \( a_1 = 56 \) and \( a_{n+1} = a_n - \frac{1}{a_n} \) for \( n \geq 1 \). Find the smallest integer \( k \) with \( 1 \leq k \leq 2002 \) such that \( a_k < 0 \). | [
"1570"
] | [
"A sequence \\( \\{a_n\\} \\) is defined such that \\( a_1 = 56 \\) and \\( a_{n+1} = a_n - \\frac{1}{a_n} \\) for \\( n \\geq 1 \\). Find the smallest integer \\( k \\) with \\( 1 \\leq k \\leq 2002 \\) such that \\( a_k < 0 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-4388ee642ca9a0460f835c1cf4897eb3ecb40e3da38bf0c0dd508df237a5fa30 | A sequence \( \{a_n\} \) of integers is defined by \( a_1 = 1 \), \( a_2 = 2 \), and \( a_n = 3a_{n-1} + 5a_{n-2} \) for \( n \geq 3 \). Does there exist an integer \( k \geq 2 \) for which \( a_k \) divides \( a_{k+1}a_{k+2} \)? | [
"No such \\( k \\) exists."
] | [
"A sequence \\( \\{a_n\\} \\) of integers is defined by \\( a_1 = 1 \\), \\( a_2 = 2 \\), and \\( a_n = 3a_{n-1} + 5a_{n-2} \\) for \\( n \\geq 3 \\). Does there exist an integer \\( k \\geq 2 \\) for which \\( a_k \\) divides \\( a_{k+1}a_{k+2} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d95a56bc7e12091fa75bf241ec6c5d16b06ee9a00c66aa09cb1c32052a50fa85 | A sequence \( \{a_n\} \) of positive real numbers is defined recursively as follows: \( a_1 = 1 \) and \( a_2 = 1 \). For \( n \geq 1 \), \( a_{n+2} \) is the length of the altitude to the hypotenuse of a right triangle with legs of lengths \( a_n \) and \( a_{n+1} \). Compute the value of \( a_{12} \). | [
"\\(\\frac{1}{12}\\)"
] | [
"A sequence \\( \\{a_n\\} \\) of positive real numbers is defined recursively as follows: \\( a_1 = 1 \\) and \\( a_2 = 1 \\). For \\( n \\geq 1 \\), \\( a_{n+2} \\) is the length of the altitude to the hypotenuse of a right triangle with legs of lengths \\( a_n \\) and \\( a_{n+1} \\). Compute the value of \\( a_{... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 29
} | 43 | 0 | 0 |
omr-ws-1be5fea11bd1793026184fa7e8abfba4c840b0ae8f677bbb84fd89d8575ccc3f | A sequence \( \{u_k\} \) satisfies the recurrence relations: \[ u_1 = 1 \] \[ u_n = (n-1)u_{n-1} + 1 \] Find all \( n \) for which \( n \mid u_n \). | [
"10"
] | [
"A sequence \\( \\{u_k\\} \\) satisfies the recurrence relations:\n\\[ u_1 = 1 \\]\n\\[ u_n = (n-1)u_{n-1} + 1 \\]\nFind all \\( n \\) for which \\( n \\mid u_n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-bd8ead9cc8aed6053f7d885977f9f2fff7bd6732fd29e2157e3270c1f2d74018 | A sequence \( a \) is defined as: \[ a_1 = 1, \quad a_2 = 2, \quad a_n = \frac{(a_{n-1}+1)^2}{a_{n-2}} \ (n \geq 3) \] Find an explicit formula for \( a_n \). Then, calculate the value of \( \sqrt{a_{10}} + \sqrt{\frac{a_{10}}{2}} \). | [
"985(1+\\sqrt{2})"
] | [
"A sequence \\( a \\) is defined as:\n\\[\na_1 = 1, \\quad a_2 = 2, \\quad a_n = \\frac{(a_{n-1}+1)^2}{a_{n-2}} \\ (n \\geq 3)\n\\]\nFind an explicit formula for \\( a_n \\). Then, calculate the value of \\( \\sqrt{a_{10}} + \\sqrt{\\frac{a_{10}}{2}} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-ef1bb9ed014bc5ae302e7864a19797ce01be7536b5b8164a743b210ce2c79c05 | A sequence \( a_0, a_1, \ldots \) of natural numbers is defined such that \( a_0 = a \) and \( a_{i+1} = a_i + L(a_i) \), where \( L(a_i) \) is the last digit of \( a_i \). Suppose that \( a_n < a_{n+1} \) for all \( n \in \mathbb{N} \). Let \( S \) be the set of all \( m \in \mathbb{N} \) such that \( 3 \mid a_m \). F... | [
"3"
] | [
"A sequence \\( a_0, a_1, \\ldots \\) of natural numbers is defined such that \\( a_0 = a \\) and \\( a_{i+1} = a_i + L(a_i) \\), where \\( L(a_i) \\) is the last digit of \\( a_i \\). Suppose that \\( a_n < a_{n+1} \\) for all \\( n \\in \\mathbb{N} \\). Let \\( S \\) be the set of all \\( m \\in \\mathbb{N} \\) s... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5348c926d863986176d21aeb2ebda994d870f27cd080c85937b3cf9075982533 | A sequence \( a_1, a_2, \ldots \) of non-negative integers is defined by the rule \( a_{n+2} = |a_{n+1} - a_n| \) for \( n \ge 1 \). If \( a_1 = 999 \), \( a_2 < 999 \), and \( a_{2006} = 1 \), how many different values of \( a_2 \) are possible? | [
"324"
] | [
"A sequence \\( a_1, a_2, \\ldots \\) of non-negative integers is defined by the rule \\( a_{n+2} = |a_{n+1} - a_n| \\) for \\( n \\ge 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",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 8
} | 18 | 0 | 0 |
omr-ws-e983459ae553969323bd70a01cba2ae3f440103869a297a292dedff55f633c06 | A sequence \( a_1, a_2, \ldots, a_n \) of real numbers is such that \( a_1 = 0 \), \( |a_2| = |a_1 - 2| \), \( |a_3| = |a_2 - 2| \), \(\ldots\), \( |a_n| = |a_{n-1} - 2| \). Find the maximum value of the arithmetic mean of these numbers. | [
"\\( 1 \\) if \\( n \\) is even, or \\( 1 - \\frac{1}{n} \\) if \\( n \\) is odd."
] | [
"A sequence \\( a_1, a_2, \\ldots, a_n \\) of real numbers is such that \\( a_1 = 0 \\), \\( |a_2| = |a_1 - 2| \\), \\( |a_3| = |a_2 - 2| \\), \\(\\ldots\\), \\( |a_n| = |a_{n-1} - 2| \\). Find the maximum value of the arithmetic mean of these numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 3,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-f70b21370852f75eb7aa7dad3001a6184bf87077fd992df304369481d9b99458 | A sequence \( a_1, a_2, a_3 \) is formed by selecting \( a_1 \) at random from the set \([1, 5]\), \( a_2 \) at random from the set \([11, 15]\), and \( a_3 \) at random from the set \([21, 25]\). What is the probability that the sequence \( a_1, a_2, a_3 \) is either an arithmetic sequence or a geometric sequence? | [
"0"
] | [
"A sequence \\( a_1, a_2, a_3 \\) is formed by selecting \\( a_1 \\) at random from the set \\([1, 5]\\), \\( a_2 \\) at random from the set \\([11, 15]\\), and \\( a_3 \\) at random from the set \\([21, 25]\\). What is the probability that the sequence \\( a_1, a_2, a_3 \\) is either an arithmetic sequence or a ge... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fb991c5cfc8089ffe6cd96cccab8898d0d908caafa08ead279f79e4787c3686f | A sequence \( a_1, a_2, a_3, \cdots \) is defined as follows. Let \( a_1 \) (where \( a_1 \neq 0 \) and \( a_1 \neq 2 \)) be a given real first term. For each \( n \), take a point \( P \) with \( x \)-coordinate \( a_n \) on the curve \( y = \frac{1}{x} - 1 \). Let \( Q(a_{n+1}, 0) \) be the intersection point of the ... | [
"1"
] | [
"A sequence \\( a_1, a_2, a_3, \\cdots \\) is defined as follows. Let \\( a_1 \\) (where \\( a_1 \\neq 0 \\) and \\( a_1 \\neq 2 \\)) be a given real first term. For each \\( n \\), take a point \\( P \\) with \\( x \\)-coordinate \\( a_n \\) on the curve \\( y = \\frac{1}{x} - 1 \\). Let \\( Q(a_{n+1}, 0) \\) be t... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-58b8dbb198c87a421ebe61b4e8881e22f8774d52a8e200a32af7be1dfce0b53d | A sequence \( a_i \) is defined by the recurrence relation \( 2a_{n-1}a_n + 2 = 4a_{n-1} + (a_n - 1)a_{n-1}^2 \). Given that \( a_{100} = 2 \) and \( 0 < a_0 < 1 \), if \( N \) is the number of possible values for \( a_0 \), compute the last 3 digits of \( N \). | [
"344"
] | [
"A sequence \\( a_i \\) is defined by the recurrence relation \\( 2a_{n-1}a_n + 2 = 4a_{n-1} + (a_n - 1)a_{n-1}^2 \\). Given that \\( a_{100} = 2 \\) and \\( 0 < a_0 < 1 \\), if \\( N \\) is the number of possible values for \\( a_0 \\), compute the last 3 digits of \\( N \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 7,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-fdd2f834702dcb41e271eaf0c33dce504f8ed2ab5f4cccbd4b008af05beea17f | A sequence \( a_n \) is defined as follows: \( a_0 = 1 \) and \( a_{n+1} = \frac{1 + 4a_n + \sqrt{1 + 24a_n}}{16} \) for \( n \ge 0 \). Find an explicit formula for \( a_n \). | [
"\\( a_n = \\frac{(2^n + 1)(2^{n+1} + 1)}{3 \\cdot 2^{2n+1}} = \\frac{1}{3} \\left(1 + \\frac{1}{2^n}\\right) \\left(1 + \\frac{1}{2^{n+1}}\\right) \\)"
] | [
"A sequence \\( a_n \\) is defined as follows: \\( a_0 = 1 \\) and \\( a_{n+1} = \\frac{1 + 4a_n + \\sqrt{1 + 24a_n}}{16} \\) for \\( n \\ge 0 \\). Find an explicit formula for \\( a_n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 15
} | 29 | 0 | 0 |
omr-ws-4062ba81e7c2727fca8baa178650eeb0a8b8a554bf3c12d7e8753cfb27d299ab | A sequence \( a_n \) is defined as follows: \( a_1 = 32 \), \( a_n \) is the smallest natural number such that \( a_n > a_{n-1} \) and \( \frac{a_n}{\varphi(a_n)} > \frac{a_{n-1}}{\varphi(a_{n-1})} \). For \( n \geq 1 \), \( \varphi(n) \) denotes the number of positive integers less than or equal to \( n \) that are re... | [
"04"
] | [
"A sequence \\( a_n \\) is defined as follows: \\( a_1 = 32 \\), \\( a_n \\) is the smallest natural number such that \\( a_n > a_{n-1} \\) and \\( \\frac{a_n}{\\varphi(a_n)} > \\frac{a_{n-1}}{\\varphi(a_{n-1})} \\). For \\( n \\geq 1 \\), \\( \\varphi(n) \\) denotes the number of positive integers less than or equ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-359ee48a1b47da53462e657d6afaabca9c937841f596acaff3a45f9ca3043a64 | A sequence \( a_n \) is defined as follows: \( a_1 = p \) is a prime number with exactly 300 nonzero digits, and for each \( n \geq 1 \), \( a_{n+1} \) is the decimal period of \( \frac{1}{a_n} \) multiplied by 2. Determine \( a_{2003} \). | [
"2"
] | [
"A sequence \\( a_n \\) is defined as follows: \\( a_1 = p \\) is a prime number with exactly 300 nonzero digits, and for each \\( n \\geq 1 \\), \\( a_{n+1} \\) is the decimal period of \\( \\frac{1}{a_n} \\) multiplied by 2. Determine \\( a_{2003} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-1910a23000e0f421e08a699ac92b5493169e2eac90891fb28024915b4bf21c9d | A sequence \( a_n \) is defined by the integral equation \(\int_{a_n}^{a_{n+1}} (1+|\sin x|) \, dx = (n+1)^2\) for \( n = 1, 2, \cdots \) with \( a_1 = 0 \). Find \(\lim_{n \to \infty} \frac{a_n}{n^3}\). | [
"\\(\\frac{\\pi}{6+3\\pi}\\)"
] | [
"A sequence \\( a_n \\) is defined by the integral equation \\(\\int_{a_n}^{a_{n+1}} (1+|\\sin x|) \\, dx = (n+1)^2\\) for \\( n = 1, 2, \\cdots \\) with \\( a_1 = 0 \\). Find \\(\\lim_{n \\to \\infty} \\frac{a_n}{n^3}\\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-ed1c67f5908570379e36a0d79ccfd63607a0e48c99cc933a779d285693e3584a | A sequence \( a_n \) is defined by the recurrence relation: \[ a_0 = 1 \] \[ a_{n+1} = 10a_n + 4 \] Find all pairs \((m, n)\) such that \( m < n \) and \( a_m a_n \) is a perfect square. | [
"\\((0, 2), (0, 3), (2, 3)\\)"
] | [
"A sequence \\( a_n \\) is defined by the recurrence relation:\n\\[ a_0 = 1 \\]\n\\[ a_{n+1} = 10a_n + 4 \\]\nFind all pairs \\((m, n)\\) such that \\( m < n \\) and \\( a_m a_n \\) is a perfect square."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-a95f9010bd97ab26b2b0f3970234af10d6fe48358410d41bb4e090b2538a5873 | A sequence \( a_n \) is defined for \( n \ge 1 \) by \( a_n = \frac{1}{2} \left( a_{n-1} - \frac{1}{3^n} \right) \) with \( a_1 = \frac{5}{6} \). Determine \( \sum_{n=1}^{\infty} a_n \). | [
"\\(\\frac{3}{2}\\)"
] | [
"A sequence \\( a_n \\) is defined for \\( n \\ge 1 \\) by \\( a_n = \\frac{1}{2} \\left( a_{n-1} - \\frac{1}{3^n} \\right) \\) with \\( a_1 = \\frac{5}{6} \\). Determine \\( \\sum_{n=1}^{\\infty} a_n \\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-27ed7621940cbfcdb63b50f7869c0682f16e9a4cbf81d868e30baded9b30d508 | A sequence \( a_n \) of positive integers is given by \( a_0 = m \) and \( a_{n+1} = a_n^5 + 487 \). Find all values of \( m \) for which this sequence contains the maximum possible number of squares. | [
"9"
] | [
"A sequence \\( a_n \\) of positive integers is given by \\( a_0 = m \\) and \\( a_{n+1} = a_n^5 + 487 \\). Find all values of \\( m \\) for which this sequence contains the maximum possible number of squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-c15feabfed089c529d3a30c669f42182b389a06689db50d269d20030015f9473 | A sequence \( a_n \) satisfies \( a_1 = 2 \) and \( a_{n+1} = \frac{2(n+2)}{n+1} \). Find \( \frac{a_{2014}}{a_1 + a_2 + \cdots + a_{2013}} \). | [
"\\(\\frac{2015}{2013}\\)"
] | [
"A sequence \\( a_n \\) satisfies \\( a_1 = 2 \\) and \\( a_{n+1} = \\frac{2(n+2)}{n+1} \\). Find \\( \\frac{a_{2014}}{a_1 + a_2 + \\cdots + a_{2013}} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-352267b43f9ba729139cc7e087a0923af8bde6f4982a4852e16f5633682ffe3b | A sequence \( a_n \) satisfies \( a_1 = \frac{\pi}{6} \) and \( a_{n+1} = \arctan(\sec a_n) \). If \( \sin a_1 \cdot \sin a_2 \cdots \sin a_m = \frac{1}{100} \), find \( m \). | [
"3333"
] | [
"A sequence \\( a_n \\) satisfies \\( a_1 = \\frac{\\pi}{6} \\) and \\( a_{n+1} = \\arctan(\\sec a_n) \\). If \\( \\sin a_1 \\cdot \\sin a_2 \\cdots \\sin a_m = \\frac{1}{100} \\), find \\( m \\)."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-9b7b38f4c8dc8f199f071c51b43e001212f5be884b13ee8e1d8486ca3148999f | A sequence \( a_n \) satisfies the following for natural integers \( n \): \[ a_{2n} = \frac{2}{n} (a_n)^2 \] \[ a_{2n+1} = \frac{a_{2n}}{2} + \frac{1}{2^{2n+1}} \] Find the least natural integer \( m \) such that \( m \cdot a_{102} \) is an integer, and determine \( m \mod 1000 \). | [
"752"
] | [
"A sequence \\( a_n \\) satisfies the following for natural integers \\( n \\):\n\\[ a_{2n} = \\frac{2}{n} (a_n)^2 \\]\n\\[ a_{2n+1} = \\frac{a_{2n}}{2} + \\frac{1}{2^{2n+1}} \\]\nFind the least natural integer \\( m \\) such that \\( m \\cdot a_{102} \\) is an integer, and determine \\( m \\mod 1000 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 4,
"tir": 2
} | 21 | 0 | 0 |
omr-ws-1526705c80122ef64d9da3877f02ec98f3b47a2f6adabd433e0b335cbf19740d | 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 \( B \) such that \( -B \leq b_n \leq B \) for all \( n \),... | [
"\\dfrac{2}{\\sqrt{4 - k^2}}"
] | [
"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 \\) such that \\( -B \\leq b_n \\l... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
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