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-c3d96c0fd94b05f3ece3c8c121b79b3e3b5614ba58c84955eb766d23a0fd44fa | A polynomial $f(x) = (x-a)(x-b)(x-c) \cdots (x-z)(x-1)(x-2)(x-3) \cdots (x-1000)$ for $x > 1000$ is expanded. What is the sum of the roots of $f$? | [
"-500500"
] | [
"A polynomial $f(x) = (x-a)(x-b)(x-c) \\cdots (x-z)(x-1)(x-2)(x-3) \\cdots (x-1000)$ for $x > 1000$ is expanded. What is the sum of the roots of $f$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cb080e9a3d06731cba9910ba243989aba82f3d348ee27bb2cad68a473ae310c5 | A polynomial $f(x) = x^4 + ax^3 + bx^2 + cx + 1$ has coefficients $a$, $b$, and $c$ determined by two players A and B moving alternately (A goes first). Player B aims to ensure that $f(x)$ has at least one real zero. If both players play optimally, find the sum of all possible values of $b$ such that B can guarantee a ... | [
"no_valid_answer_found"
] | [
"A polynomial $f(x) = x^4 + ax^3 + bx^2 + cx + 1$ has coefficients $a$, $b$, and $c$ determined by two players A and B moving alternately (A goes first). Player B aims to ensure that $f(x)$ has at least one real zero. If both players play optimally, find the sum of all possible values of $b$ such that B can guarant... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-847f89daa79ad6dbaaf6ca6f94ed589138dcba3bbe46ac0d4588841cc496c15c | A polynomial $f(x)$ satisfies the equation $f(x^2) - x^3f(x) = 2(x^3 - 1)$ for every real number $x$. What is the degree of $f(x)$? | [
"3"
] | [
"A polynomial $f(x)$ satisfies the equation $f(x^2) - x^3f(x) = 2(x^3 - 1)$ for every real number $x$. What is the degree of $f(x)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-afdd82dbf48dfafb4f538c6124e9647a691636f8612dd7f029d9abd8a5959e33 | A polynomial $p(x)$ is called self-centered if it has integer coefficients and $p(100) = 100.$ If $p(x)$ is a self-centered polynomial, what is the maximum number of integer solutions $k$ to the equation $p(k) = k^3$? | [
"10"
] | [
"A polynomial $p(x)$ is called self-centered if it has integer coefficients and $p(100) = 100.$ If $p(x)$ is a self-centered polynomial, what is the maximum number of integer solutions $k$ to the equation $p(k) = k^3$?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-25fe0a358bd04c4bbadfb508eb453db1c21e2df349223b6b21c90ec09c02d1db | A polynomial $p(x)$ leaves a remainder of $-1$ when divided by $x - 1,$ a remainder of 3 when divided by $x - 2,$ and a remainder of 4 when divided by $x + 3.$ Let $r(x)$ be the remainder when $p(x)$ is divided by $(x - 1)(x - 2)(x + 3).$ Find $r(6).$ | [
"40"
] | [
"A polynomial $p(x)$ leaves a remainder of $-1$ when divided by $x - 1,$ a remainder of 3 when divided by $x - 2,$ and a remainder of 4 when divided by $x + 3.$ Let $r(x)$ be the remainder when $p(x)$ is divided by $(x - 1)(x - 2)(x + 3).$ Find $r(6).$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5ee23170edf69a1dd2246932ed2366c323e32584eec9670a12c2b41c37120d4c | A polynomial $p(x)$ of degree $n \geq 2$ has exactly $n$ real roots, counted with multiplicity. The coefficient of $x^n$ is $1$, all the roots are less than or equal to $1$, and $p(2) = 3^n$. What values can $p(1)$ take? | [
"[0, 2^n]",
"\\([0, 2^n]\\)"
] | [
"A polynomial $p(x)$ of degree $n \\geq 2$ has exactly $n$ real roots, counted with multiplicity. The coefficient of $x^n$ is $1$, all the roots are less than or equal to $1$, and $p(2) = 3^n$. What values can $p(1)$ take?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 64,
"tir": 19
} | 83 | 0 | 0 |
omr-ws-38b3df5504088d24275fb24741a911a65133591f0e299c4cdf7eabc8a088c5c2 | A polynomial \( F(x) \) of degree \( n \) satisfies \( F(k) = F_k \) for \( k = 0, 1, 2, \ldots, n \), where \( F_k \) is the Fibonacci sequence with \( F_0 = 0 \) and \( F_1 = 1 \). Compute \( F(n+1) \). | [
"\\( F(n+1) = F_{n+1} (1 + (-1)^{n+1}) \\)"
] | [
"A polynomial \\( F(x) \\) of degree \\( n \\) satisfies \\( F(k) = F_k \\) for \\( k = 0, 1, 2, \\ldots, n \\), where \\( F_k \\) is the Fibonacci sequence with \\( F_0 = 0 \\) and \\( F_1 = 1 \\). Compute \\( F(n+1) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 8,
"tir": 17
} | 41 | 0 | 0 |
omr-ws-1dbfca04ae7adc4ea63abc0d162b868e421d444861e28016a61f34c9397aedbc | A polynomial \( P \) has integer coefficients and \( P(3) = 4 \) and \( P(4) = 3 \). For how many \( x \) might we have \( P(x) = x \)? | [
"0"
] | [
"A polynomial \\( P \\) has integer coefficients and \\( P(3) = 4 \\) and \\( P(4) = 3 \\). For how many \\( x \\) might we have \\( P(x) = x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-907cc5e0aaa8c9b7aefd5d8db50edab1d818c39ef1b2520e56f9cf75fd00bbf9 | A polynomial \( P \) has integral coefficients and at least 9 different integral roots. Let \( n \) be an integer such that \( |P(n)| < 2017 \). Find the sum of all possible values of \( P(n) \). | [
"0"
] | [
"A polynomial \\( P \\) has integral coefficients and at least 9 different integral roots. Let \\( n \\) be an integer such that \\( |P(n)| < 2017 \\). Find the sum of all possible values of \\( P(n) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4fd3db2c18fdb0bea385df45da70ad459c60ca3ad9a46f662ccff3b2f7085a30 | A polynomial \( P \) has roots \( \frac{1}{4}, \frac{1}{2}, 2, 4 \). The product of the roots is 1, and \( P(1) = 1 \). Find \( P(0) \). | [
"\\(\\frac{8}{9}\\)"
] | [
"A polynomial \\( P \\) has roots \\( \\frac{1}{4}, \\frac{1}{2}, 2, 4 \\). The product of the roots is 1, and \\( P(1) = 1 \\). Find \\( P(0) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 58
} | 89 | 0 | 0 |
omr-ws-82b7d168f51ec29c297bfebebc6b63e0993198ed4621eee6aa68dbadc450e5e9 | A polynomial \( P \) of degree 2015 satisfies the equation \( P(n) = \frac{1}{n^2} \) for \( n = 1, 2, \dots, 2016 \). Find \( \lfloor 2017P(2017) \rfloor \). | [
"8"
] | [
"A polynomial \\( P \\) of degree 2015 satisfies the equation \\( P(n) = \\frac{1}{n^2} \\) for \\( n = 1, 2, \\dots, 2016 \\). Find \\( \\lfloor 2017P(2017) \\rfloor \\)."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 2
} | 2 | 0 | 0 |
omr-ws-7d9af65b5cd7b2e5012186e5fda670b968640ccaedff779f0aa75240fbe8adfd | A polynomial \( P \) of degree 5 has the values \( P(1) = 0 \), \( P(3) = 1 \), \( P(9) = 2 \), \( P(27) = 3 \), \( P(81) = 4 \), and \( P(243) = 5 \). Find the coefficient of the term \( x \) in this polynomial. | [
"\\(\\frac{1}{5 \\cdot \\ln(3)}\\)"
] | [
"A polynomial \\( P \\) of degree 5 has the values \\( P(1) = 0 \\), \\( P(3) = 1 \\), \\( P(9) = 2 \\), \\( P(27) = 3 \\), \\( P(81) = 4 \\), and \\( P(243) = 5 \\). Find the coefficient of the term \\( x \\) in this polynomial."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-80d5d2e69e9460deb388eea0b08161200bbe554e8bea1414f3e0e5d1198783b8 | A polynomial \( P \) with real coefficients is called *great* if for some integer \( a > 1 \) and for all integers \( x \), there exists an integer \( z \) such that \( aP(x) = P(z) \). Find all *great* polynomials. | [
"\\( P(x) = \\alpha (x + q)^k \\) with \\( \\alpha \\in \\mathbb{R}, q \\in \\mathbb{Q} \\)"
] | [
"A polynomial \\( P \\) with real coefficients is called *great* if for some integer \\( a > 1 \\) and for all integers \\( x \\), there exists an integer \\( z \\) such that \\( aP(x) = P(z) \\). Find all *great* polynomials."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-f85c02586d441d281365f0ea5242d14c47f4b28c0df75713644829c98159cea9 | A polynomial \( P(x) \) has 2 roots and positive integer coefficients. Given that \( P(0) = 5 \) and \( P(1) = 9 \), find the sum of all possible values of \( P(2) \). | [
"51"
] | [
"A polynomial \\( P(x) \\) has 2 roots and positive integer coefficients. Given that \\( P(0) = 5 \\) and \\( P(1) = 9 \\), find the sum of all possible values of \\( P(2) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-d174d121e19f4d65bd0e8244abf56ee1d9fceb142c6430f58623bef87840c2af | A polynomial \( P(x) \) has integer coefficients. What is the least number of distinct integer solutions to \( P(x) = 2016 \) that must exist so that for any such polynomial, there are no integer solutions to \( P(x) = 4031 \)? | [
"6"
] | [
"A polynomial \\( P(x) \\) has integer coefficients. What is the least number of distinct integer solutions to \\( P(x) = 2016 \\) that must exist so that for any such polynomial, there are no integer solutions to \\( P(x) = 4031 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-c481f88e8463298f1c51be090e7270b3e2485a6d16c80fbb93f2821794df5af8 | A polynomial \( P(x) \) is called spicy if all of its coefficients are nonnegative integers less than 9. How many spicy polynomials satisfy \( P(3) = 2019 \)? | [
"162"
] | [
"A polynomial \\( P(x) \\) is called spicy if all of its coefficients are nonnegative integers less than 9. How many spicy polynomials satisfy \\( P(3) = 2019 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-4505b7ac580a4c50291c4e3b094acff55627ceaa3bed65a8aacdeb6d5bcb5901 | A polynomial \( P(x) \) is defined such that for \( n = \{-2, 1, 2, 3\} \), \( P(n) = \frac{1}{n^2} - \frac{1}{n^3} \). Additionally, \( P(x) \) is of the lowest degree and is monic. Determine the polynomial \( P(x) \). | [
"\\( P(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \\)"
] | [
"A polynomial \\( P(x) \\) is defined such that for \\( n = \\{-2, 1, 2, 3\\} \\), \\( P(n) = \\frac{1}{n^2} - \\frac{1}{n^3} \\). Additionally, \\( P(x) \\) is of the lowest degree and is monic. Determine the polynomial \\( P(x) \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d9a49238d2f9e5af02e2b450bbdfa1a4dff01831e7530b32fced9c660c58f6d2 | A polynomial \( P(x) \) leaves a remainder of \( 2x \) when divided by \( (x-1)^2 \), and a remainder of \( 3x \) when divided by \( (x-2)^3 \). Find the sum of the \( x^3 \) coefficient and the \( x^2 \) coefficient of \( P(x) \). | [
"39"
] | [
"A polynomial \\( P(x) \\) leaves a remainder of \\( 2x \\) when divided by \\( (x-1)^2 \\), and a remainder of \\( 3x \\) when divided by \\( (x-2)^3 \\). Find the sum of the \\( x^3 \\) coefficient and the \\( x^2 \\) coefficient of \\( P(x) \\)."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 24
} | 70 | 0 | 0 |
omr-ws-ae6b4cde899578297deb621e99c0a5b15acd3ede7fb8135e3210c88585d7c9a9 | A polynomial \( P(x) \) leaves remainders 10 and \( 2x - 3 \) when divided by \( (x-2) \) and \( (x+1)^2 \) respectively. Find the remainder when \( P(x) \) is divided by \( (x-2)(x+1)^2 \). | [
"\\( x^2 + 4x - 2 \\)"
] | [
"A polynomial \\( P(x) \\) leaves remainders 10 and \\( 2x - 3 \\) when divided by \\( (x-2) \\) and \\( (x+1)^2 \\) respectively. Find the remainder when \\( P(x) \\) is divided by \\( (x-2)(x+1)^2 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-f15a23c80ce5b9c29a85e3a0e7f7da621bb7feb3d7bf1e155bafa6829ee182f5 | A polynomial \( P(x) \) of degree 2005 satisfies \( P(k) = 2^k \) for \( k = 0, 1, \ldots, 2003 \). Find \( P(2004) \). | [
"2^{2004} - 1"
] | [
"A polynomial \\( P(x) \\) of degree 2005 satisfies \\( P(k) = 2^k \\) for \\( k = 0, 1, \\ldots, 2003 \\). Find \\( P(2004) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-e97f8bc796c2a6d0c16d18eb1ae5fe5b5005d8e5d41cbd724d6676f1b9644494 | A polynomial \( P(x) \) of degree 2005 satisfies \( P(k) = 2^k \) for \( k = 0, 2, \ldots, 2003 \). Find \( P(2004) \). | [
"2^{2004} - 1"
] | [
"A polynomial \\( P(x) \\) of degree 2005 satisfies \\( P(k) = 2^k \\) for \\( k = 0, 2, \\ldots, 2003 \\). Find \\( P(2004) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-cc3bc06f1c33b7cd3d8d6521c13b7cb0877641e41ba9378db2a0e62cb943c3b5 | A polynomial \( P(x) \) of degree 2013 with real coefficients fulfills the relation \( P(n) = 3^n \) for \( n = 0, 1, 2, 3, \ldots, 2013 \). Evaluate \( P(2014) \). | [
"\\( 3^{2014} - 2^{2014} \\)"
] | [
"A polynomial \\( P(x) \\) of degree 2013 with real coefficients fulfills the relation \\( P(n) = 3^n \\) for \\( n = 0, 1, 2, 3, \\ldots, 2013 \\). Evaluate \\( P(2014) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-aeb13430f7ef244e2a8760e993b2bf5ee6b250778727a5f3f7e09bfdf533119d | A polynomial \( P(x) \) of degree 6 is divisible by its derivative \( P'(x) \), and \( P'(x) \) is divisible by \( x-1 \). Let \( a_0 \) be the coefficient of the term of highest degree of \( P(x) \). Determine all roots of the equation \( P(x) = a_0 \). | [
"\\frac{3}{2} - \\frac{\\sqrt{3}}{2}i"
] | [
"A polynomial \\( P(x) \\) of degree 6 is divisible by its derivative \\( P'(x) \\), and \\( P'(x) \\) is divisible by \\( x-1 \\). Let \\( a_0 \\) be the coefficient of the term of highest degree of \\( P(x) \\). Determine all roots of the equation \\( P(x) = a_0 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-9dae9fa9e64662e4e2a4a569fe5c8b08f635dca6c4b53c84e78e3752df3cadab | A polynomial \( P(x) \) of degree 6 is divisible by its derivative \( P'(x) \), and \( P'(x) \) is divisible by \( x-1 \). Let \( a_0 \) be the coefficient of the term of highest degree of \( P(x) \). Determine the area of the polygon whose vertices are the roots of \( P(x) \). | [
"0"
] | [
"A polynomial \\( P(x) \\) of degree 6 is divisible by its derivative \\( P'(x) \\), and \\( P'(x) \\) is divisible by \\( x-1 \\). Let \\( a_0 \\) be the coefficient of the term of highest degree of \\( P(x) \\). Determine the area of the polygon whose vertices are the roots of \\( P(x) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6aa52d80f745e78e3207d18965d4413e587afd0e65e43d83541c1ea588d63e5e | A polynomial \( P(x) \) of degree 7 satisfies \( P(n) = 2^n \) for \( n = 0 \) to 7. Find \( P(10) \). | [
"968"
] | [
"A polynomial \\( P(x) \\) of degree 7 satisfies \\( P(n) = 2^n \\) for \\( n = 0 \\) to 7. Find \\( P(10) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 21
} | 43 | 0 | 0 |
omr-ws-f88b2772df095da92ace81f953a6151a0199a9eb0557cc85182ff2ae533e9f65 | A polynomial \( P(x) \) of degree 8 satisfies \( P(n) = \frac{1}{n} \) for \( n = 1, 2, 3, \ldots, 9 \). Find \( P(10) \). | [
"\\( \\frac{1}{5} \\)"
] | [
"A polynomial \\( P(x) \\) of degree 8 satisfies \\( P(n) = \\frac{1}{n} \\) for \\( n = 1, 2, 3, \\ldots, 9 \\). Find \\( P(10) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-30bdcf456249a47536b6970cf59c6dce904d4ec785dc1e96e72f2dcc80b3c82e | A polynomial \( P(x) \) of degree \( n \) satisfies \( P(0) = 0 \), \( P(1) = \frac{1}{2} \), \( P(2) = \frac{2}{3} \), ..., \( P(n) = \frac{n}{n+1} \). Find \( P(n+1) \). | [
"\\(\\frac{(-1)^{n+1} + n + 1}{n + 2}\\)"
] | [
"A polynomial \\( P(x) \\) of degree \\( n \\) satisfies \\( P(0) = 0 \\), \\( P(1) = \\frac{1}{2} \\), \\( P(2) = \\frac{2}{3} \\), ..., \\( P(n) = \\frac{n}{n+1} \\). Find \\( P(n+1) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 18
} | 32 | 0 | 0 |
omr-ws-ed36f000c599e74705f462124a49ce88a82939ab2526ba854eb1832a80f12fd3 | A polynomial \( P(x) \) of degree \( n \) satisfies \( P(k) = 2^k \) for \( k = 0, 1, 2, \ldots, n \). Find the value of \( P(n+1) \). | [
"\\( 2^{n+1} - 1 \\)"
] | [
"A polynomial \\( P(x) \\) of degree \\( n \\) satisfies \\( P(k) = 2^k \\) for \\( k = 0, 1, 2, \\ldots, n \\). Find the value of \\( P(n+1) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 31
} | 63 | 0 | 0 |
omr-ws-755d90c0040227ef532cec52f75796996088fcbedb320a7750a164484ba86577 | A polynomial \( P(x) \) of degree \( n \) satisfies \( P(k) = 2^k \) for all integers \( k \) in the range \([0, n]\). Find \( P(n+1) \). | [
"\\(2^{n+1} - 1\\)"
] | [
"A polynomial \\( P(x) \\) of degree \\( n \\) satisfies \\( P(k) = 2^k \\) for all integers \\( k \\) in the range \\([0, n]\\). Find \\( P(n+1) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-91aa9e656222a41999a56c02811b0f50758e1fa5f2ce2e74d5bcfa5f303b8b4b | A polynomial \( P(x) \) of degree \( n \) satisfies \( P(k) = \frac{k}{k+1} \) for \( k = 0, 1, 2, \ldots, n \). Find \( P(n+1) \). | [
"\\(\\frac{(n+1) + (-1)^{n+1}}{n+2}\\)"
] | [
"A polynomial \\( P(x) \\) of degree \\( n \\) satisfies \\( P(k) = \\frac{k}{k+1} \\) for \\( k = 0, 1, 2, \\ldots, n \\). Find \\( P(n+1) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-1822e73cc788440ce1bc31875dc2d6be15f9f473c6bfadac52107cf76eaab41a | A polynomial \( P(x) \) of degree \( n \ge 5 \) with integer coefficients has \( n \) distinct integer roots, one of which is \( 0 \). Find all integer roots of the polynomial \( P(P(x)) \). | [
"The integer roots of \\( P(P(x)) \\) are the roots of \\( P(x) \\)."
] | [
"A polynomial \\( P(x) \\) of degree \\( n \\ge 5 \\) with integer coefficients has \\( n \\) distinct integer roots, one of which is \\( 0 \\). Find all integer roots of the polynomial \\( P(P(x)) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 14
} | 31 | 0 | 0 |
omr-ws-f06e5540e4b0f8c17b08f723fa113e5216074e4e81d11b3cd217859e6011b537 | A polynomial \( P(x) \) of degree three is known to have \( P(1)=1 \), \( P(2)=1 \), \( P(3)=2 \), and \( P(4)=2 \). Compute \( P(5) \). | [
"-1"
] | [
"A polynomial \\( P(x) \\) of degree three is known to have \\( P(1)=1 \\), \\( P(2)=1 \\), \\( P(3)=2 \\), and \\( P(4)=2 \\). Compute \\( P(5) \\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-c824042689d45962aea229098eb691175c1a091ac982456a4af67323b97936a1 | A polynomial \( P(x) \) satisfies \( P(8) = 144 \), and the following expression is true for all \( x \neq 0, \frac{1}{2} \): \( \frac{8P(x^2 + 2)}{P(4x)} = 1 + \frac{x^3 + 1}{2x - 1} \). What is \( P(10) \)? | [
"240"
] | [
"A polynomial \\( P(x) \\) satisfies \\( P(8) = 144 \\), and the following expression is true for all \\( x \\neq 0, \\frac{1}{2} \\): \\( \\frac{8P(x^2 + 2)}{P(4x)} = 1 + \\frac{x^3 + 1}{2x - 1} \\). What is \\( P(10) \\)?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-3800d1468f71effae11460d20ea6a69381841fce025a7028c2915ecb097b0178 | A polynomial \( P(x) \) satisfies \( \deg(P(x)) = 2n \) and \( P(j) = \frac{2^j}{j+1} \) for all \( j \in \{0, 1, 2, \ldots, 2n\} \). Find \( P(2n+2) \). | [
"\\(\\frac{2^{2n+2} - 1}{2n+3}\\)"
] | [
"A polynomial \\( P(x) \\) satisfies \\( \\deg(P(x)) = 2n \\) and \\( P(j) = \\frac{2^j}{j+1} \\) for all \\( j \\in \\{0, 1, 2, \\ldots, 2n\\} \\). Find \\( P(2n+2) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 4
} | 22 | 0 | 0 |
omr-ws-4881caa0a5ab189aaea7b6c9c5c382070fed16c6782714d5dd93a89cac7afe0c | A polynomial \( P(x) \) satisfies \[ P \left( x + \frac{2}{x} \right) = \frac{x^3 + 1}{x} + \frac{x^3 + 8}{2x^2} + 3. \] The value of \( P(1) \) is ... | [
"\\(\\frac{5}{2}\\)"
] | [
"A polynomial \\( P(x) \\) satisfies\n\\[\nP \\left( x + \\frac{2}{x} \\right) = \\frac{x^3 + 1}{x} + \\frac{x^3 + 8}{2x^2} + 3.\n\\]\nThe value of \\( P(1) \\) is ..."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-988a2f65e8533b3a8c73da93ffd4e25af3f782a025b1d1b2215f7e61187b9bbd | A polynomial \( P(x) \) with integer coefficients returns the values \( 4, 3, 2, 1, 0 \) when \( x \) is \( 1, 2, 3, 4, 5 \) respectively. What is the smallest possible value of \( P(6) \), given that it is positive? | [
"119"
] | [
"A polynomial \\( P(x) \\) with integer coefficients returns the values \\( 4, 3, 2, 1, 0 \\) when \\( x \\) is \\( 1, 2, 3, 4, 5 \\) respectively. What is the smallest possible value of \\( P(6) \\), given that it is positive?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-6ac8c9be58e410a8454b0e787d023f31f1e7591ec30dfc9bd37e44f1f5e16eae | A polynomial \( P(x) \) with integer coefficients satisfies \( P(1) = 17 \), \( P(m) = m^2 + n^2 - mn \), and \( P(n) = mn - 1 \) for some fixed integers \( m \) and \( n \). Find all possible pairs of integers \( (m, n) \). | [
"\\((0, -1), (3, 2), (2, 3), (-1, 0)\\)"
] | [
"A polynomial \\( P(x) \\) with integer coefficients satisfies \\( P(1) = 17 \\), \\( P(m) = m^2 + n^2 - mn \\), and \\( P(n) = mn - 1 \\) for some fixed integers \\( m \\) and \\( n \\). Find all possible pairs of integers \\( (m, n) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-7760381b6c67a96181df18c5bb79da380c3ea21361f0a5776f43780362a7fdbd | A polynomial \( P(x) \) with real coefficients satisfies the condition that for any positive integer \( n \), \( P(n) = 1^8 + 2^8 + \cdots + n^8 \). Let the coefficient of \( x^7 \) in \( P(x) \) be \( a \). Find the value of \( 12a \). | [
"-8"
] | [
"A polynomial \\( P(x) \\) with real coefficients satisfies the condition that for any positive integer \\( n \\), \\( P(n) = 1^8 + 2^8 + \\cdots + n^8 \\). Let the coefficient of \\( x^7 \\) in \\( P(x) \\) be \\( a \\). Find the value of \\( 12a \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f2700d2d854f1327cb4fb592bed0b23e3a56a9a218434868f6ad5d4edf96a3a0 | A polynomial \( Q(x) \) has at least 9 distinct integer roots. Given that there is an integer \( n \) such that \( |Q(n)| \leq 2019 \), calculate the sum of all possible values of \( Q(n) \). | [
"0"
] | [
"A polynomial \\( Q(x) \\) has at least 9 distinct integer roots. Given that there is an integer \\( n \\) such that \\( |Q(n)| \\leq 2019 \\), calculate the sum of all possible values of \\( Q(n) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-e5d7198e5793443121ef4a81f79b9a85c24270218502fda231eac280e460c778 | A polynomial \( f \) with real coefficients satisfies \( f(1) = -1 \), \( f(2) = 3 \), \( f(3) = 9 \), \( f(4) = 16 \), and \( f(5) = 24 \). If \( f \) has the least possible degree, find the value of \( f(7) \). | [
"45"
] | [
"A polynomial \\( f \\) with real coefficients satisfies \\( f(1) = -1 \\), \\( f(2) = 3 \\), \\( f(3) = 9 \\), \\( f(4) = 16 \\), and \\( f(5) = 24 \\). If \\( f \\) has the least possible degree, find the value of \\( f(7) \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-75744f20343a8925e3c8447b144f235de3fd5c764c3c03b757243b7ca71bc132 | A polynomial \( f(x) \) has the remainder \( x + 1 \) when divided by \( x^2 - 4x + 3 \) and the remainder \( 3x - 1 \) when divided by \( x^2 - 3x + 2 \). Find the remainder when \( f(x) \) is divided by \( x^3 - 6x^2 + 11x - 6 \). | [
"\\(-2x^2 + 9x - 5\\)"
] | [
"A polynomial \\( f(x) \\) has the remainder \\( x + 1 \\) when divided by \\( x^2 - 4x + 3 \\) and the remainder \\( 3x - 1 \\) when divided by \\( x^2 - 3x + 2 \\). Find the remainder when \\( f(x) \\) is divided by \\( x^3 - 6x^2 + 11x - 6 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-22d4c3b1886bb4a9fa015c3016f32ea6c3172ed28f87ab52142a2bfb5bff393c | A polynomial \( f(x) \) leaves a remainder of 10 when divided by \( (x-2) \) and a remainder of \( (2x-3) \) when divided by \( (x+1)^2 \). Find the remainder when \( f(x) \) is divided by \( (x-2)(x+1)^2 \). | [
"\\( x^2 + 4x - 2 \\)"
] | [
"A polynomial \\( f(x) \\) leaves a remainder of 10 when divided by \\( (x-2) \\) and a remainder of \\( (2x-3) \\) when divided by \\( (x+1)^2 \\). Find the remainder when \\( f(x) \\) is divided by \\( (x-2)(x+1)^2 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-d974d2809335cae28260c61ed16082ac60efef00393a0695a8e55e4cc974c3b3 | A polynomial \( f(x) \) of degree three has three different real zeros \( a, b, c \) and the coefficient of \( x^3 \) is positive. Calculate the value of \( f'(a) + f'(b) + f'(c) \) in terms of the zeros \( a, b, \) and \( c \). | [
"a^2 + b^2 + c^2 - ab - ac - bc"
] | [
"A polynomial \\( f(x) \\) of degree three has three different real zeros \\( a, b, c \\) and the coefficient of \\( x^3 \\) is positive. Calculate the value of \\( f'(a) + f'(b) + f'(c) \\) in terms of the zeros \\( a, b, \\) and \\( c \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-fec4d1683c698bb7add818e0c065feaa853110b17fe6b331293d501f3f4067b8 | A polynomial \( f(x) \) satisfies \( f(1) = f(2) = f(4) = f(5) = f(6) = f(7) = f(8) = f(9) = 26 \). Find \( f(3) \). | [
"26"
] | [
"A polynomial \\( f(x) \\) satisfies \\( f(1) = f(2) = f(4) = f(5) = f(6) = f(7) = f(8) = f(9) = 26 \\). Find \\( f(3) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-11389a4e7875bbf0300f43123b927d85c220e94543df9438810f52636482fee4 | A polynomial \( f(x) \) with rational coefficients leaves a remainder of 15 when divided by \( x-3 \) and a remainder of \( 2x+1 \) when divided by \( (x-1)^2 \). Find the remainder when \( f(x) \) is divided by \( (x-3)(x-1)^2 \). | [
"\\( 2x^2 - 2x + 3 \\)"
] | [
"A polynomial \\( f(x) \\) with rational coefficients leaves a remainder of 15 when divided by \\( x-3 \\) and a remainder of \\( 2x+1 \\) when divided by \\( (x-1)^2 \\). Find the remainder when \\( f(x) \\) is divided by \\( (x-3)(x-1)^2 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 24
} | 42 | 0 | 0 |
omr-ws-de2959365c451cf579226397acdc1452c52ccaddef14142f994bb86a04d70cb8 | A polynomial \( f(x) \) with real coefficients is called generating if for each polynomial \( \varphi(x) \) with real coefficients, there exists a positive integer \( k \) and polynomials \( g_1(x), \dotsc, g_k(x) \) with real coefficients such that \[ \varphi(x) = f(g_1(x)) + \dotsb + f(g_k(x)). \] Find all generating... | [
"f(x) = ax + b"
] | [
"A polynomial \\( f(x) \\) with real coefficients is called generating if for each polynomial \\( \\varphi(x) \\) with real coefficients, there exists a positive integer \\( k \\) and polynomials \\( g_1(x), \\dotsc, g_k(x) \\) with real coefficients such that\n\\[\n\\varphi(x) = f(g_1(x)) + \\dotsb + f(g_k(x)).\n\... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 1
} | 1 | 0 | 0 |
omr-ws-dcbfb152a383a554afb74153f27e9b450e9bad2614867f0eec0ec2d3985d82ff | A polynomial \( p \) of degree \( n \) satisfies \( p(k) = 2^k \) for all \( 0 \le k \le n \). Find \( p(n+1) \). | [
"\\( 2^{n+1} - 1 \\)"
] | [
"A polynomial \\( p \\) of degree \\( n \\) satisfies \\( p(k) = 2^k \\) for all \\( 0 \\le k \\le n \\). Find \\( p(n+1) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-294eca4f44e962f6e26d9d7ce5dbf0d98e1e692a3865b72a72b3a429e296b488 | A polynomial \( p(x) = ax^3 + bx^2 + cx + d \) has roots at \( 2 \) and \( -3 \). It is also given that \( p'(-3) < 0 \) (the derivative). Find the largest possible value of \( c \). | [
"9"
] | [
"A polynomial \\( p(x) = ax^3 + bx^2 + cx + d \\) has roots at \\( 2 \\) and \\( -3 \\). It is also given that \\( p'(-3) < 0 \\) (the derivative). Find the largest possible value of \\( c \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 1
} | 13 | 0 | 0 |
omr-ws-beb5e5c5398fd3f737a6c72b66d612a7b9794e75558efa314dedc4b04276f54e | A polynomial \( p(x) \) has a remainder of 4 when divided by \( x+6 \), a remainder of 10 when divided by \( x-9 \), and a remainder of 56 when divided by \( 2x+9 \). What is the remainder when \( p(x) \) is divided by \( (x+6)(x-9)(2x+9) \)? | [
"\\( -\\frac{1028}{405} x^2 + \\frac{1082}{135} x + \\frac{2152}{15} \\)"
] | [
"A polynomial \\( p(x) \\) has a remainder of 4 when divided by \\( x+6 \\), a remainder of 10 when divided by \\( x-9 \\), and a remainder of 56 when divided by \\( 2x+9 \\). What is the remainder when \\( p(x) \\) is divided by \\( (x+6)(x-9)(2x+9) \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-30252dbc2c48ebad595c05579061112cc616f7463a1b2bd6448031bacdc38738 | A polynomial \( p(x) \) is called self-centered if it has integer coefficients and \( p(100) = 100 \). If \( p(x) \) is a self-centered polynomial, what is the maximum number of integer solutions \( k \) to the equation \( p(k) = k^3 \)? | [
"10"
] | [
"A polynomial \\( p(x) \\) is called self-centered if it has integer coefficients and \\( p(100) = 100 \\). If \\( p(x) \\) is a self-centered polynomial, what is the maximum number of integer solutions \\( k \\) to the equation \\( p(k) = k^3 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-5b89abe3ffe7991e880e916e2794f517f1be437753eb9ead7957b09a62508269 | A polynomial \( p(x) \) leaves a remainder when divided by \( (x-1) \) and a remainder when divided by \( (x-3) \). What is the remainder when \( p(x) \) is divided by \( (x-1)(x-3) \)? | [
"\\( \\frac{r_2 - r_1}{2} x + \\frac{3r_1 - r_2}{2} \\)"
] | [
"A polynomial \\( p(x) \\) leaves a remainder when divided by \\( (x-1) \\) and a remainder when divided by \\( (x-3) \\). What is the remainder when \\( p(x) \\) is divided by \\( (x-1)(x-3) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 4
} | 22 | 0 | 0 |
omr-ws-8258956c1554471599dd88cab1678c4fea98cdb6b00ba4252840225f41e0c485 | A polynomial \( p(x) \) of degree 7 satisfies \( p(j) = 2^j \) for \( j = 0, 1, \ldots, 7 \). Find the value of \( p(10) \). | [
"968"
] | [
"A polynomial \\( p(x) \\) of degree 7 satisfies \\( p(j) = 2^j \\) for \\( j = 0, 1, \\ldots, 7 \\). Find the value of \\( p(10) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-db891db501242e1fa40e2cc355835bea67b867e98b64b2875245b5f93c603930 | A polynomial \( p(x) \) of degree \( n \) satisfies \( p(0) = 0 \), \( p(1) = \frac{1}{2} \), \( p(2) = \frac{2}{3} \), ..., \( p(n) = \frac{n}{n+1} \). Find \( p(n+1) \). | [
"\\( \\frac{n+1 + (-1)^{n+1}}{n+2} \\)"
] | [
"A polynomial \\( p(x) \\) of degree \\( n \\) satisfies \\( p(0) = 0 \\), \\( p(1) = \\frac{1}{2} \\), \\( p(2) = \\frac{2}{3} \\), ..., \\( p(n) = \\frac{n}{n+1} \\). Find \\( p(n+1) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 24
} | 24 | 0 | 0 |
omr-ws-a2f8a301eede3e581cb04603d1b1fc2652710efc04c9848268896df0947b521b | A polynomial \( p(x) \) of degree \( n \) satisfies \( p(k) = 2^k \) for all \( k = 1, 2, \ldots, n+1 \). Find the value of \( p(n+2) \). | [
"2^{n+2} - 2"
] | [
"A polynomial \\( p(x) \\) of degree \\( n \\) satisfies \\( p(k) = 2^k \\) for all \\( k = 1, 2, \\ldots, n+1 \\). Find the value of \\( p(n+2) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7998c43298681fc40078e018f969b21dff796334c4686288e20b8ee6f52bd696 | A polynomial \( p(x) \) of degree \( n \) satisfies \( p(k) = k^2 \) for \( k = 1, 2, 3, \ldots, n \). Find the value of \( p(n+1) \). | [
"(n+1)^2 + (-1)^n n!"
] | [
"A polynomial \\( p(x) \\) of degree \\( n \\) satisfies \\( p(k) = k^2 \\) for \\( k = 1, 2, 3, \\ldots, n \\). Find the value of \\( p(n+1) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-cd155b85f441134d49d84dc0579620d9a0a4e71305caff8f14e44648f543b5e0 | A polynomial \(P\) has integer coefficients and \(P(3) = 4\) and \(P(4) = 3\). For how many \(x\) may we have \(P(x) = x\)? | [
"0"
] | [
"A polynomial \\(P\\) has integer coefficients and \\(P(3) = 4\\) and \\(P(4) = 3\\). For how many \\(x\\) may we have \\(P(x) = x\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-0d82785bd2dad93fbb9036995124d570dd525ad75997f30bc190b5c1cf17d551 | A polynomial \(p(x)\) is called "good" if \(p(x)^2 + c = p(x^2 + c)\) for a constant complex number \(c\). a) If \(c\) is a rational number but not an integer, find all such polynomials. b) If \(c = a + bi\) where \(i^2 = -1\) and \(a^2 + b^2 > 4\), find all such polynomials. | [
"x^2 + c"
] | [
"A polynomial \\(p(x)\\) is called \"good\" if \\(p(x)^2 + c = p(x^2 + c)\\) for a constant complex number \\(c\\).\na) If \\(c\\) is a rational number but not an integer, find all such polynomials.\nb) If \\(c = a + bi\\) where \\(i^2 = -1\\) and \\(a^2 + b^2 > 4\\), find all such polynomials."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-bbdeaaf11c2117a0ca1c9f29a198d45f7abe0d09a325a6edd53f097bf717a118 | A polynomial equation for the variable $x$ and of the smallest degree with rational coefficients has one root of $\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$ and a second root of $\frac{\sqrt{11}+\sqrt{10}}{\sqrt{11}-\sqrt{10}}$. If the coefficient of the $x$ term in the polynomial equation is $-52$, find the coeffici... | [
"422"
] | [
"A polynomial equation for the variable $x$ and of the smallest degree with rational coefficients has one root of $\\frac{\\sqrt{3}+\\sqrt{2}}{\\sqrt{3}-\\sqrt{2}}$ and a second root of $\\frac{\\sqrt{11}+\\sqrt{10}}{\\sqrt{11}-\\sqrt{10}}$. If the coefficient of the $x$ term in the polynomial equation is $-52$, fi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-b4eaa26d082bc686a055b417ce54dd8476f1844fe635718d98ea00d28853ecd1 | A polynomial function \( P(x) \) with integer coefficients satisfies that \( P(20) = P(13) = 2013 \). What is the maximum number of integer roots that \( P(x) \) can have? | [
"0"
] | [
"A polynomial function \\( P(x) \\) with integer coefficients satisfies that \\( P(20) = P(13) = 2013 \\). What is the maximum number of integer roots that \\( P(x) \\) can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4d6d74f14d6f34c9a2b875eccc39b2cdea175b73f9fedd26a2d512158ce939df | A polynomial has remainder $7$ when divided by $x + 2$ and remainder $x + 3$ when divided by $x^2 + 2$. Determine the remainder when it is divided by $(x + 2)(x^2 + 2)$. | [
"\\( x^2 + x + 5 \\)"
] | [
"A polynomial has remainder $7$ when divided by $x + 2$ and remainder $x + 3$ when divided by $x^2 + 2$. Determine the remainder when it is divided by $(x + 2)(x^2 + 2)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-2a7df0d33002242831f3df4e586921847dd28333d4875160e0507c4fb842c13b | A polynomial is constructed such that its degree is a randomly chosen positive integer from $1$ to $10$, and each of its coefficients are chosen uniformly at random in the interval $[0,1]$. Find the probability that the polynomial has a real root. | [
"\\dfrac{1}{2}"
] | [
"A polynomial is constructed such that its degree is a randomly chosen positive integer from $1$ to $10$, and each of its coefficients are chosen uniformly at random in the interval $[0,1]$. Find the probability that the polynomial has a real root."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-bbb32a2edd13d2febc6830a5ce091bf54e1cc9e6d0e76706ae2bf7ac096d83ce | A polynomial is monic if the leading coefficient is $1$. Suppose $p(x)$ is a cubic monic polynomial with roots $a, b, c$. If $a + b + c = 1$, $a^2 + b^2 + c^2 = 5$, and $a^3 + b^3 + c^3 = 16$, determine $p(x)$. | [
"\\( x^3 - x^2 - 3x - 2 \\)"
] | [
"A polynomial is monic if the leading coefficient is $1$. Suppose $p(x)$ is a cubic monic polynomial with roots $a, b, c$. If $a + b + c = 1$, $a^2 + b^2 + c^2 = 5$, and $a^3 + b^3 + c^3 = 16$, determine $p(x)$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9384ab875a6fab3bc66e99232e4d589038de9cb48f4b0c21884c267590435e17 | A polynomial of degree $13$ is divided by $d(x)$ to give a quotient of degree $7$ and a remainder of $3x^3+4x^2-x+12$. What is $\deg d$? | [
"6"
] | [
"A polynomial of degree $13$ is divided by $d(x)$ to give a quotient of degree $7$ and a remainder of $3x^3+4x^2-x+12$. What is $\\deg d$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9031b398ec3be90ea0ac433985ace96da697043e0ca7fae4a2d79af376a35f16 | A polynomial of degree $3n$ takes the value $0$ at $2, 5, 8, \ldots, 3n-1$, the value $1$ at $1, 4, 7, \ldots, 3n-2$, and the value $2$ at $0, 3, 6, \ldots, 3n$. Its value at $3n+1$ is $730$. Find $n$. | [
"4"
] | [
"A polynomial of degree $3n$ takes the value $0$ at $2, 5, 8, \\ldots, 3n-1$, the value $1$ at $1, 4, 7, \\ldots, 3n-2$, and the value $2$ at $0, 3, 6, \\ldots, 3n$. Its value at $3n+1$ is $730$. Find $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-d363242c856ed1f300d63493dd32882a1d33c9358b1e7947c4b324ece19573e8 | A polynomial of degree $6$ with rational coefficients has $5$ distinct roots: $1, 2-i, 2+i, 1-i, 1+i$. If the polynomial has a double root, what is the sum of all possible values of this double root? | [
"1"
] | [
"A polynomial of degree $6$ with rational coefficients has $5$ distinct roots: $1, 2-i, 2+i, 1-i, 1+i$. If the polynomial has a double root, what is the sum of all possible values of this double root?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b309fc8239dc87eab4caf31679bc7ac0a2ed6baeb2d780a74459398c4e329667 | A polynomial of degree \(3n\) has the value 2 at \(0, 3, 6, \ldots, 3n\), the value 1 at \(1, 4, 7, \ldots, 3n-2\), and the value 0 at \(2, 5, 8, \ldots, 3n-1\). Its value at \(3n+1\) is 730. What is \(n\)? | [
"4"
] | [
"A polynomial of degree \\(3n\\) has the value 2 at \\(0, 3, 6, \\ldots, 3n\\), the value 1 at \\(1, 4, 7, \\ldots, 3n-2\\), and the value 0 at \\(2, 5, 8, \\ldots, 3n-1\\). Its value at \\(3n+1\\) is 730. What is \\(n\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-bbb53e6db00fe11b26166ffe482292fa0149030587cc797fb19ac0d62634d753 | A polynomial only has odd degree terms and contains the point (3,6). What is the remainder when it is divided by $x^2 - 9$? | [
"\\(2x\\)"
] | [
"A polynomial only has odd degree terms and contains the point (3,6). What is the remainder when it is divided by $x^2 - 9$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-68553de78018b1ae83c59b13d8db6733fab4c964f904b7822c1b576ae98a4a99 | A polynomial product of the form \[(1 - z)^{b_1} (1 - z^2)^{b_2} (1 - z^3)^{b_3} (1 - z^4)^{b_4} (1 - z^5)^{b_5} \dotsm (1 - z^{32})^{b_{32}},\]where the $b_k$ are positive integers, has the surprising property that if we multiply it out and discard all terms involving $z$ to a power larger than 32, what is left is jus... | [
"2^{27} - 2^{11}"
] | [
"A polynomial product of the form\n\\[(1 - z)^{b_1} (1 - z^2)^{b_2} (1 - z^3)^{b_3} (1 - z^4)^{b_4} (1 - z^5)^{b_5} \\dotsm (1 - z^{32})^{b_{32}},\\]where the $b_k$ are positive integers, has the surprising property that if we multiply it out and discard all terms involving $z$ to a power larger than 32, what is le... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fd0b01b32eaa63d7fa71f50b159937f5167e0bc08bb55be773ec0b6b55faba11 | A polynomial satisfies \( f(x^2 - 2) = f(x)f(-x) \). Assuming that \( f(x) \neq 0 \) for \( -2 \leq x \leq 2 \), what is the value of \( f(-2) + f(1) \)? | [
"2"
] | [
"A polynomial satisfies \\( f(x^2 - 2) = f(x)f(-x) \\). Assuming that \\( f(x) \\neq 0 \\) for \\( -2 \\leq x \\leq 2 \\), what is the value of \\( f(-2) + f(1) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-668b342e2d785e4b54c15232af8b8c963a3956526c655c9cce9ce8a0d4a12244 | A polynomial with integer coefficients is of the form \[9x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 15 = 0.\]Find the number of different possible rational roots of this polynomial. | [
"16"
] | [
"A polynomial with integer coefficients is of the form\n\\[9x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 15 = 0.\\]Find the number of different possible rational roots of this polynomial."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1089a314a297775f49aeb6e011ae1d1a4c0b92280f493c24782b31ed912660a | A polynomial with integer coefficients is of the form \[x^3 + a_2 x^2 + a_1 x - 11 = 0.\]Enter all the possible integer roots of this polynomial, separated by commas. | [
"-11, -1, 1, 11"
] | [
"A polynomial with integer coefficients is of the form\n\\[x^3 + a_2 x^2 + a_1 x - 11 = 0.\\]Enter all the possible integer roots of this polynomial, separated by commas."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-e6e9bb489598fa946212f9a172bac2f89a39e52a9c760fa35af38349a4c68efe | A polynomial with integer coefficients is of the form \[x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 18.\]You are told that the integer $r$ is a double root of this polynomial. (In other words, the polynomial is divisible by $(x - r)^2.$) Enter all the possible values of $r,$ separated by commas. | [
"-3,-1,1,3"
] | [
"A polynomial with integer coefficients is of the form\n\\[x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 18.\\]You are told that the integer $r$ is a double root of this polynomial. (In other words, the polynomial is divisible by $(x - r)^2.$) Enter all the possible values of $r,$ separated by commas."
] | [
"MATH_training_set"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-446a1bc69c0dfcfbd4df27b6e99697a73dbad5a71f3dbdf05c71db48fdc667f5 | A polynomial with real coefficients of degree \( n \) has at most one coefficient equal to 0 and at least \( n \) coefficients that are equal to each other. If all the roots of the polynomial are real numbers, what is the largest value \( n \) can take? | [
"3"
] | [
"A polynomial with real coefficients of degree \\( n \\) has at most one coefficient equal to 0 and at least \\( n \\) coefficients that are equal to each other. If all the roots of the polynomial are real numbers, what is the largest value \\( n \\) can take?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-185500959c30632daf3bd7191eaee762c3436bff4c3973d0b8222f5752d3aef5 | A polyomino $P$ occupies $n$ cells of an infinite grid of unit squares. In each move, $P$ is lifted and placed back into a new position, possibly rotated and reflected, so that the preceding and the new position have $n-1$ cells in common. We say that $P$ is a caterpillar of area $n$ if, by means of a series of moves, ... | [
"1"
] | [
"A polyomino $P$ occupies $n$ cells of an infinite grid of unit squares. In each move, $P$ is lifted and placed back into a new position, possibly rotated and reflected, so that the preceding and the new position have $n-1$ cells in common. We say that $P$ is a caterpillar of area $n$ if, by means of a series of mo... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-51342a2df6f490d17201cad24b480a9ddcea7e0b56419c6fdde603a0ca2d51ee | A polyomino is a figure consisting of unit squares joined together by their sides. For a given positive integer \( n \geq 3 \), consider an infinite grid of unit square cells. Determine the greatest positive integer \( C \) such that for any coloring of the cells in \( n \) colors, there exists a polyomino within the g... | [
"n-1"
] | [
"A polyomino is a figure consisting of unit squares joined together by their sides. For a given positive integer \\( n \\geq 3 \\), consider an infinite grid of unit square cells. Determine the greatest positive integer \\( C \\) such that for any coloring of the cells in \\( n \\) colors, there exists a polyomino ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-3ae7cefd09526884f26d92743cbf4d3087e62b236843fe4823b72662a7750739 | A polyomino is a geometric figure formed by joining one or more unit squares edge to edge. A frame is a polyomino with one hole such that both its outside boundary and inside boundary are rectangles. Find the largest integer $n$ for which there does not exist a frame with area $n$. | [
"9"
] | [
"A polyomino is a geometric figure formed by joining one or more unit squares edge to edge. A frame is a polyomino with one hole such that both its outside boundary and inside boundary are rectangles. Find the largest integer $n$ for which there does not exist a frame with area $n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-14a496cb3bef4a70b24f5a3ab083c04a34ccb63c6804eba52ad0f4e10f57eb96 | A pomegranite has 20 seeds, with exactly half red and half blue. If Kee Young picks and eats 4 seeds one by one, what is the probability that he ends up with 2 seeds of each color in his stomach? | [
"\\(\\frac{135}{323}\\)"
] | [
"A pomegranite has 20 seeds, with exactly half red and half blue. If Kee Young picks and eats 4 seeds one by one, what is the probability that he ends up with 2 seeds of each color in his stomach?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-87ea9a0bae329fedfed6906da5520c9baaaa961f583f050aa1c2f048cd558eeb | A pond has a shape of a square. After the first frosty day, ice covered all parts of the pond which were $10$ meters away from the edge or less. After the second frosty day – $20$ meters away from the edge or less, after the third day – $30$ meters or less, and so on. It is known that after the first day the area of th... | [
"10"
] | [
"A pond has a shape of a square. After the first frosty day, ice covered all parts of the pond which were $10$ meters away from the edge or less. After the second frosty day – $20$ meters away from the edge or less, after the third day – $30$ meters or less, and so on. It is known that after the first day the area ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-92b7654cc9d03bc867966f5a3dad6d7a6189ed4ef7350c4b3b80ba654b2add31 | A pond has a shape of a square. After the first frosty day, ice covered all parts of the pond which were $10$ meters away from the edge or less. After the second frosty day – $20$ meters away from the edge or less, after the third day – $30$ meters or less, and so on. It is known that after the first day the area of th... | [
"6"
] | [
"A pond has a shape of a square. After the first frosty day, ice covered all parts of the pond which were $10$ meters away from the edge or less. After the second frosty day – $20$ meters away from the edge or less, after the third day – $30$ meters or less, and so on. It is known that after the first day the area ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-b28a13d6562cc4634fd279280afae4c0f83b89ba4d65333bd9311ea114a3b51e | A pond is divided into \(2n\) parts (\(n \geq 5\)), where each part has three neighboring parts. There are \(4n+1\) frogs in the pond. If there are three or more frogs in one part, three of the frogs will jump to the three neighboring parts, respectively. After a series of jumps, let \(S\) be the sum of the number of f... | [
"2n + 1"
] | [
"A pond is divided into \\(2n\\) parts (\\(n \\geq 5\\)), where each part has three neighboring parts. There are \\(4n+1\\) frogs in the pond. If there are three or more frogs in one part, three of the frogs will jump to the three neighboring parts, respectively. After a series of jumps, let \\(S\\) be the sum of t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-9691930ae8c906419339165339dbc4c2a286a8b9316e9fb271ddcae48d4ff017 | A pond is divided into \(2n\) parts (\(n \geq 5\)). Two parts are called neighbors if they share a common side or arc. Each part has three neighbors. There are \(4n + 1\) frogs in the pond. If there are three or more frogs in one part, three of the frogs will jump to the three neighboring parts. After a series of jumps... | [
"5n + 1"
] | [
"A pond is divided into \\(2n\\) parts (\\(n \\geq 5\\)). Two parts are called neighbors if they share a common side or arc. Each part has three neighbors. There are \\(4n + 1\\) frogs in the pond. If there are three or more frogs in one part, three of the frogs will jump to the three neighboring parts. After a ser... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-41c769fa622d920e3e53fe8909d56dce146b6fa46057b4c29dd3dd5fb0fd7703 | A pond is enclosed by a wooden deck that is 3 feet wide. The fence surrounding the deck is 100 feet long. If the pond is circular, what is the diameter of the pond? | [
"\\(\\frac{100}{\\pi} - 6\\)"
] | [
"A pond is enclosed by a wooden deck that is 3 feet wide. The fence surrounding the deck is 100 feet long. If the pond is circular, what is the diameter of the pond?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-7f4d8a3626518df16e732e544152b9aaa0a15896218f4186af122c19cd254c1e | A pond is shaped as an inverted truncated pyramid with a square base. The depth of the pond is 6 meters. The bottom square has a side length of 2 meters, and the top square has a side length of 8 meters. Water is being filled at a rate of $\frac{19}{3}$ cubic meters per hour. At what rate is the water level rising exac... | [
"\\(\\frac{19}{27}\\)"
] | [
"A pond is shaped as an inverted truncated pyramid with a square base. The depth of the pond is 6 meters. The bottom square has a side length of 2 meters, and the top square has a side length of 8 meters. Water is being filled at a rate of $\\frac{19}{3}$ cubic meters per hour. At what rate is the water level risin... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 25
} | 39 | 0 | 0 |
omr-ws-3d4cb03117b4cf2daf363172bbe1381a6e96a1c97898bba89ef08b852a1fbf97 | A pond is shaped as an inverted truncated pyramid with a square base. The depth of the pond is 6 meters. The bottom square has a side length of 2 meters, and the top square has a side length of 8 meters. Water is filled at a rate of \(\frac{19}{3}\) cubic meters per hour. At what rate is the water level rising exactly ... | [
"\\dfrac{19}{27}"
] | [
"A pond is shaped as an inverted truncated pyramid with a square base. The depth of the pond is 6 meters. The bottom square has a side length of 2 meters, and the top square has a side length of 8 meters. Water is filled at a rate of \\(\\frac{19}{3}\\) cubic meters per hour. At what rate is the water level rising ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-07d49846996a1f22e3486d4e8802ecc36137a2763ce2dc0e052f3f4a53dcba58 | A pooh bear has \(2n+1\) honey pots. No matter which one of them he sets aside, he can split the remaining \(2n\) pots into two sets of the same total weight, each consisting of \(n\) pots. If the total weight of all \(2n+1\) pots is \(W\), find the value of \(W\) when \(n=5\) and each pot weighs a natural number. | [
"11"
] | [
"A pooh bear has \\(2n+1\\) honey pots. No matter which one of them he sets aside, he can split the remaining \\(2n\\) pots into two sets of the same total weight, each consisting of \\(n\\) pots. If the total weight of all \\(2n+1\\) pots is \\(W\\), find the value of \\(W\\) when \\(n=5\\) and each pot weighs a n... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f8c36822063cc960c5aeae0232af20d7788afc800f61aa1ac5fd082596da79c6 | A pool containing $2020$ liters of water is being drained by $2020$ pumps which drain at speeds of $1$ liter per hour, $2$ liters per hour, ..., $2020$ liters per hour. However, the pump that drains at $n$ liters per hour stops draining once there remains $n-1$ liters in the pool. Find the total amount of time, in hour... | [
"\\(\\frac{4040}{2021}\\)"
] | [
"A pool containing $2020$ liters of water is being drained by $2020$ pumps which drain at speeds of $1$ liter per hour, $2$ liters per hour, ..., $2020$ liters per hour. However, the pump that drains at $n$ liters per hour stops draining once there remains $n-1$ liters in the pool. Find the total amount of time, in... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-893681ee1665e79183fcf1eb8aef66e6b8a7333833cb62f7767eeca0d2541a4b | A pool is filled through two pipes of different diameters. When both pipes are used, the pool is filled in 6 hours. If only the larger pipe is used, the pool is filled 5 hours faster than if only the smaller pipe is used. Find the time needed to fill the pool using only the smaller pipe. | [
"15"
] | [
"A pool is filled through two pipes of different diameters. When both pipes are used, the pool is filled in 6 hours. If only the larger pipe is used, the pool is filled 5 hours faster than if only the smaller pipe is used. Find the time needed to fill the pool using only the smaller pipe."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-6276fb4a58efb5a0201b483f38368458658f2b21031fd1486048fecc93bf78e1 | A pool is filled with water from two taps. The pool can be filled if the first tap is opened for 10 minutes and the second for 20 minutes. If the first tap is opened for 5 minutes, and the second for 15 minutes, the pool will be $\frac{3}{5}$ filled. How much time would it have taken to fill the whole pool by using eac... | [
"$\\frac{5}{6}$ hour and $\\frac{5}{18}$ hour"
] | [
"A pool is filled with water from two taps. The pool can be filled if the first tap is opened for 10 minutes and the second for 20 minutes. If the first tap is opened for 5 minutes, and the second for 15 minutes, the pool will be $\\frac{3}{5}$ filled. How much time would it have taken to fill the whole pool by usi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d69d96849fd0d9d4491cf6ebd3b445558a3861107ccdaec00150b1f7718947b0 | A pool is filled with water through one pipe 5 hours faster than it is when the water is passed through a second pipe, and 30 hours faster than when through a third pipe. The capacity of the third pipe is $\frac{1}{2.5}$ of the capacity of the first pipe and 24 m$^3$/h less than the capacity of the second pipe. Find th... | [
"60 m$^3$/h and 24 m$^3$/h"
] | [
"A pool is filled with water through one pipe 5 hours faster than it is when the water is passed through a second pipe, and 30 hours faster than when through a third pipe. The capacity of the third pipe is $\\frac{1}{2.5}$ of the capacity of the first pipe and 24 m$^3$/h less than the capacity of the second pipe. F... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d2a9c4e0eec3e556738345660e2735014b89cbe07effdc01b62a5d7400fb4a66 | A pool table is shaped like a square $ABCD$. SuperPablo has a special mission: he must hit a billiard ball, initially placed at vertex $A$, so that, after hitting the sides of the square exactly $2010$ times, the ball arrives at a vertex of the square for the first time. How many possible values are there for the angle... | [
"1004"
] | [
"A pool table is shaped like a square $ABCD$. SuperPablo has a special mission: he must hit a billiard ball, initially placed at vertex $A$, so that, after hitting the sides of the square exactly $2010$ times, the ball arrives at a vertex of the square for the first time. How many possible values are there for the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-f71f173fa8be68799b1e5c869650677a3f88f0db02caf77ff9570c7f41a6541a | A pool was filled by a pipe and a hose together in 5 hours. At another time, it was filled by using the pipe for 7 hours and the hose for 2 hours. How long would it take to fill the pool using either the pipe alone or the hose alone? Use only one variable to find the answer. | [
"Hose = 12.5 hours, Pipe = 8.33 hours"
] | [
"A pool was filled by a pipe and a hose together in 5 hours. At another time, it was filled by using the pipe for 7 hours and the hose for 2 hours. How long would it take to fill the pool using either the pipe alone or the hose alone? Use only one variable to find the answer."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-7e5d336f3ce394b104da8c52c0922adf6f5517ab436fa8f46edc98f5685af29f | A pool with a volume of 20,000 gallons contains water with 0.02% chlorine. Water containing 0.004% chlorine is pumped into the pool at a rate of 10 gallons per minute, and the pool water flows out at the same rate. At what time will the pool water have 0.008% chlorine? | [
"\\( t \\approx 1386.2 \\) minutes."
] | [
"A pool with a volume of 20,000 gallons contains water with 0.02% chlorine. Water containing 0.004% chlorine is pumped into the pool at a rate of 10 gallons per minute, and the pool water flows out at the same rate. At what time will the pool water have 0.008% chlorine?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-d97829553b16590a118b8f1afdc08cf0ec4819f24fac761e067531e4715f681f | A poorly manufactured clock has identical minute and hour hands. The clock stops at some time between 12 am and 12 pm. How many times could the clock have stopped such that it is impossible to determine the exact time after 12 pm? | [
"132"
] | [
"A poorly manufactured clock has identical minute and hour hands. The clock stops at some time between 12 am and 12 pm. How many times could the clock have stopped such that it is impossible to determine the exact time after 12 pm?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-0e06c9cc8978fba70d131aa265fe59f7f290798992e3e6dedd30cd0217b927eb | A population is described by a normal distribution \( N(25, 16) \). If you take a sample of size 36, what is the probability that the sample mean is between 22 and 27? | [
"0.9987"
] | [
"A population is described by a normal distribution \\( N(25, 16) \\). If you take a sample of size 36, what is the probability that the sample mean is between 22 and 27?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 26
} | 49 | 0 | 0 |
omr-ws-532c23c5f3eaa396f0c22ac7bc5f0371785731473cc0b6374fb6d98b8396e670 | A population is modeled by the logistic differential equation \(\frac{dP}{dt} = \frac{P}{5} \left(1 - \frac{P}{12}\right)\). a) If \(P(0) = 3\), what is \(\lim_{t \to \infty} P(t)\)? b) If \(P(0) = 20\), what is \(\lim_{t \to \infty} P(t)\)? | [
"12"
] | [
"A population is modeled by the logistic differential equation \\(\\frac{dP}{dt} = \\frac{P}{5} \\left(1 - \\frac{P}{12}\\right)\\).\na) If \\(P(0) = 3\\), what is \\(\\lim_{t \\to \\infty} P(t)\\)?\nb) If \\(P(0) = 20\\), what is \\(\\lim_{t \\to \\infty} P(t)\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-6235b03496948b3e91e2436583bf4a79fb6b6f450f42c2c0a1473292398707b6 | A population of Monarch butterflies exhibits logistic growth with a carrying capacity of 500 butterflies and a growth rate parameter \( r = 0.1 \) individuals/(individuals x month). What is the maximum growth rate of the population in butterflies per month when the population equals half of the carrying capacity? | [
"6250 butterflies per month."
] | [
"A population of Monarch butterflies exhibits logistic growth with a carrying capacity of 500 butterflies and a growth rate parameter \\( r = 0.1 \\) individuals/(individuals x month). What is the maximum growth rate of the population in butterflies per month when the population equals half of the carrying capacity... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-11cef829af44a5d31fc692e204a8bf2a13dacbebb70744da51fdb7c2dc59da9c | A population starts with a single amoeba. Each amoeba has a 3/4 probability of splitting into two amoebas and a 1/4 probability of dying out without producing offspring. What is the probability that the family tree of the original amoeba will continue indefinitely? | [
"\\dfrac{2}{3}"
] | [
"A population starts with a single amoeba. Each amoeba has a 3/4 probability of splitting into two amoebas and a 1/4 probability of dying out without producing offspring. What is the probability that the family tree of the original amoeba will continue indefinitely?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5114792b1bf279e76b60d7b03bc22448f8a4e6f961411ed246df167ff3c25eed | A portion of a wooden cube is sawed off at each vertex so that a small equilateral triangle is formed at each corner with vertices on the edges of the cube. The $24$ vertices of the new object are all connected to each other by straight lines. How many of these lines (excluding their endpoints) lie entirely in the inte... | [
"42"
] | [
"A portion of a wooden cube is sawed off at each vertex so that a small equilateral triangle is formed at each corner with vertices on the edges of the cube. The $24$ vertices of the new object are all connected to each other by straight lines. How many of these lines (excluding their endpoints) lie entirely in the... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-355f97482982c79217ac474ac6a4b0aa0ca506c7249d7ef06ebd3ec5b6e8d2c2 | A positive factor of \( 60^4 \) is randomly chosen. The probability it has exactly \( 6 \) positive factors can be expressed as \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m + n \). | [
"232"
] | [
"A positive factor of \\( 60^4 \\) is randomly chosen. The probability it has exactly \\( 6 \\) positive factors can be expressed as \\( \\frac{m}{n} \\), where \\( m \\) and \\( n \\) are relatively prime positive integers. Find \\( m + n \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8d334089163e4c1d17d25b0c71c46809b748f71c9b9a8d1d804d227863609460 | A positive five-digit integer is in the form $AB,CBA$; where $A$, $B$ and $C$ are each distinct digits. What is the greatest possible value of $AB,CBA$ that is divisible by eleven? | [
"96,\\!569"
] | [
"A positive five-digit integer is in the form $AB,CBA$; where $A$, $B$ and $C$ are each distinct digits. What is the greatest possible value of $AB,CBA$ that is divisible by eleven?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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