This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 3597

2008 National Olympiad First Round, 3
Tags: polynomial , algebra
Let $P(x) = 1-x+x^2-x^3+\dots+x^{18}-x^{19}$ and $Q(x)=P(x-1)$. What is the coefficient of $x^2$ in polynomial $Q$? $ \textbf{(A)}\ 840 \qquad\textbf{(B)}\ 816 \qquad\textbf{(C)}\ 969 \qquad\textbf{(D)}\ 1020 \qquad\textbf{(E)}\ 1140 $

2021 Indonesia MO, 5
Tags: polynomial , algebra , inequalities
Let $P(x) = x^2 + rx + s$ be a polynomial with real coefficients. Suppose $P(x)$ has two distinct real roots, both of which are less than $-1$ and the difference between the two is less than $2$. Prove that $P(P(x)) > 0$ for all real $x$.

2009 USA Team Selection Test, 3
Tags: number theory , least common multiple , algebra , greatest common divisor , geometry , geometric transformation , polynomial
For each positive integer $ n$, let $ c(n)$ be the largest real number such that \[ c(n) \le \left| \frac {f(a) \minus{} f(b)}{a \minus{} b}\right|\] for all triples $ (f, a, b)$ such that --$ f$ is a polynomial of degree $ n$ taking integers to integers, and --$ a, b$ are integers with $ f(a) \neq f(b)$. Find $ c(n)$. [i]Shaunak Kishore.[/i]

2014-2015 SDML (High School), 12
Tags: polynomial , algebra , trigonometry
Which of the following polynomials with integer coefficients has $\sin\left(10^{\circ}\right)$ as a root? $\text{(A) }4x^3-4x+1\qquad\text{(B) }6x^3-8x^2+1\qquad\text{(C) }4x^3+4x-1\qquad\text{(D) }8x^3+6x-1\qquad\text{(E) }8x^3-6x+1$

2020 IMO Shortlist, A2
Tags: polynomial , linear combination , algebra
Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

2009 China Team Selection Test, 2
Tags: polynomial , algebra
Find all complex polynomial $ P(x)$ such that for any three integers $ a,b,c$ satisfying $ a \plus{} b \plus{} c\not \equal{} 0, \frac{P(a) \plus{} P(b) \plus{} P(c)}{a \plus{} b \plus{} c}$ is an integer.

2020 HK IMO Preliminary Selection Contest, 11
Tags: algebra , vieta , polynomial
Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$.

2010 IberoAmerican Olympiad For University Students, 6
Tags: polynomial , search , number theory , algebra
Prove that, for all integer $a>1$, the prime divisors of $5a^4-5a^2+1$ have the form $20k\pm1,k\in\mathbb{Z}$. [i]Proposed by Géza Kós.[/i]

1980 Yugoslav Team Selection Test, Problem 2
Tags: polynomial , algebra , number theory
Let $a,b,c,m$ be integers, where $m>1$. Prove that if $$a^n+bn+c\equiv0\pmod m$$for each natural number $n$, then $b^2\equiv0\pmod m$. Must $b\equiv0\pmod m$ also hold?

1993 AIME Problems, 4
Tags: inequalities , polynomial , matrix , symmetry , linear algebra , algebra
How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?

1998 Denmark MO - Mohr Contest, 2
Tags: polynomial , trinomial , inequalities , algebra
For any real number$m$, the equation $$x^2+(m-2)x- (m+3)=0$$ has two solutions, denoted $x_1 $and $ x_2$. Determine $m$ such that $x_1^2+x_2^2$ is the minimum possible.

1975 AMC 12/AHSME, 5
Tags: binomial theorem , polynomial , algebra
The polynomial $ (x\plus{}y)^9$ is expanded in decreasing powers of $ x$. The second and third terms have equal values when evaluated at $ x\equal{}p$ and $ y\equal{}q$, where $ p$ and $ q$ are positive numbers whose sum is one. What is the value of $ p$? $ \textbf{(A)}\ 1/5 \qquad \textbf{(B)}\ 4/5 \qquad \textbf{(C)}\ 1/4 \qquad \textbf{(D)}\ 3/4 \qquad \textbf{(E)}\ 8/9$

2021 Hong Kong TST, 2
Tags: polynomial , algebra , root
Let $f(x)$ be a polynomial with rational coefficients, and let $\alpha$ be a real number. If \[\alpha^3-2019\alpha=(f(\alpha))^3-2019f(\alpha)=2021,\] prove that $(f^n(\alpha))^3-2019f^n(\alpha)=2021$ for any positive integer $n$. (Here, we define $f^n(x)=\underbrace{f(f(f\cdots f}_{n\text{ times}}(x)\cdots ))$.)

1991 AMC 12/AHSME, 20
Tags: geometry , logarithm , vieta , polynomial , algebra
The sum of all real $x$ such that $(2^{x} - 4)^{3} + (4^{x} - 2)^{3} = (4^{x} + 2^{x} - 6)^{3}$ is $ \textbf{(A)}\ 3/2\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 5/2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 7/2 $

2014 Iran MO (3rd Round), 5
Tags: number theory , polynomial , algebra , ratio
Can an infinite set of natural numbers be found, such that for all triplets $(a,b,c)$ of it we have $abc + 1 $ perfect square? (20 points )

1989 China Team Selection Test, 3
Tags: algebra , polynomial , combinatorial geometry , gauss , derivative , calculus
Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$

1986 Bulgaria National Olympiad, Problem 2
Tags: polynomial , quadratic , algebra
Let $f(x)$ be a quadratic polynomial with two real roots in the interval $[-1,1]$. Prove that if the maximum value of $|f(x)|$ in the interval $[-1,1]$ is equal to $1$, then the maximum value of $|f'(x)|$ in the interval $[-1,1]$ is not less than $1$.

1966 IMO Longlists, 35
Tags: irrational number , polynomial , algebra , root
Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.

2011 Dutch IMO TST, 4
Tags: integer polynomial , polynomial , algebra
Determine all integers $n$ for which the polynomial $P(x) = 3x^3-nx-n-2$ can be written as the product of two non-constant polynomials with integer coeffcients.

2024 Mathematical Talent Reward Programme, 6
Tags: polynomial , algebra
Show that there exists an integer polynomial $P$ such that $P(1) = 2024$ and the set of prime divisors of {$P(2^k)$},$k=0,1,2,.....$ is an infinite set.

1952 Moscow Mathematical Olympiad, 232
Tags: polynomial , divide , algebra
Prove that for any integer $a$ the polynomial $3x^{2n}+ax^n+2$ cannot be divided by $2x^{2m}+ax^m+3$ without a remainder.

2003 Tournament Of Towns, 2
Tags: polynomial , algebra
$P(x)$ is a polynomial with real coefficients such that $P(a_1) = 0, P(a_{i+1}) = a_i$ ($i = 1, 2,\ldots$) where $\{a_i\}_{i=1,2,\ldots}$ is an infinite sequence of distinct natural numbers. Determine the possible values of degree of $P(x)$.

1999 Kazakhstan National Olympiad, 2
Tags: algebra , polynomial
Prove that for any odd $ n $ there exists a unique polynomial $ P (x) $ $ n $ -th degree satisfying the equation $ P \left (x- \frac {1} {x} \right) = x ^ n- \frac {1} {x ^ n}. $ Is this true for any natural number $ n $?

2024 China Team Selection Test, 10
Tags: polynomial , algebra
Let $M$ be a positive integer. $f(x):=x^3+ax^2+bx+c\in\mathbb Z[x]$ satisfy $|a|,|b|,|c|\le M.$ $x_1,x_2$ are different roots of $f.$ Prove that $$|x_1-x_2|>\frac 1{M^2+3M+1}.$$ [i]Created by Jingjun Han[/i]

2018 Purple Comet Problems, 15
Tags: polynomial , algebra
There are integers $a_1, a_2, a_3,...,a_{240}$ such that $x(x + 1)(x + 2)(x + 3) ... (x + 239) =\sum_{n=1}^{240}a_nx^n$. Find the number of integers $k$ with $1\le k \le 240$ such that ak is a multiple of $3$.