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

2009 Postal Coaching, 3
Tags: trigonometry , algebra , polynomial
Find all real polynomial functions $f : R \to R$ such that $f(\sin x) = f(\cos x)$.

PEN Q Problems, 3
Tags: polynomial
Let $n \ge 2$ be an integer. Prove that if $k^2 + k + n$ is prime for all integers $k$ such that $0 \leq k \leq \sqrt{\frac{n}{3}}$, then $k^2 +k + n$ is prime for all integers $k$ such that $0 \leq k \leq n - 2$.

2009 Thailand Mathematical Olympiad, 6
Tags: polynomial , algebra
Find all polynomials of the form $P(x) = (-1)^nx^n + a_1x^{n-1} + a_2x^{n-2} + ...+ a_{n-1}x + a_n$ with the following two properties: (i) $\{a_1, a_2, . . . , a_n-1, a_n\} =\{0, 1\}$, and (ii) all roots of $P(x)$ are distinct real numbers

1994 Baltic Way, 5
Tags: algebra , polynomial
Let $p(x)$ be a polynomial with integer coefficients such that both equations $p(x)=1$ and $p(x)=3$ have integer solutions. Can the equation $p(x)=2$ have two different integer solutions?

2014 Irish Math Olympiad, 9
Tags: polynomial , algebra
Let $n$ be a positive integer and $a_1,...,a_n$ be positive real numbers. Let $g(x)$ denote the product $(x + a_1)\cdot ... \cdot (x + a_n)$ . Let $a_0$ be a real number and let $f(x) = (x - a_0)g(x)= x^{n+1} + b_1x^n + b_2x^{n-1}+...+ b_nx + b_{n+1}$ . Prove that all the coeffcients $b_1,b_2,..., b_{n+1}$ of the polynomial $f(x)$ are negative if and only if $a_0 > a_1 + a_2 +...+ a_n$.

1976 Chisinau City MO, 121
Tags: odd , integer , algebra , polynomial
Prove that the polynomial $P (x)$ with integer coefficients, taking odd values for $x = 0$ and $x= 1$, has no integer roots.

2019 SG Originals, Q7
Tags: polynomial , number theory
Let $n$ be a natural number. A sequence is $k-$complete if it contains all residues modulo $n^k$. Let $Q(x)$ be a polynomial with integer coefficients. For $k\ge 2$, define $Q^k(x)=Q(Q^{k-1}(x))$, where $Q^1(x)=Q(x)$. Show that if $$0,Q(0),Q^2(0),Q^3(0),\ldots $$is $2018-$complete, then it is $k-$complete for all positive integers $k$. [i]Proposed by Ma Zhao Yu[/i]

1961 AMC 12/AHSME, 22
Tags: polynomial , quadratic , algebra
If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by: ${{ \textbf{(A)}\ 3x^2-x+4 \qquad\textbf{(B)}\ 3x^2-4 \qquad\textbf{(C)}\ 3x^2+4 \qquad\textbf{(D)}\ 3x-4 }\qquad\textbf{(E)}\ 3x+4 } $

2016 NIMO Problems, 5
Tags: algebra , polynomial
Find the constant $k$ such that the sum of all $x \ge 0$ satisfying $\sqrt{x}(x+12)=17x-k$ is $256.$ [i]Proposed by Michael Tang[/i]

2011 NIMO Problems, 7
Tags: greatest common divisor , algebra , polynomial , number theory
Let $P(x) = x^2 - 20x - 11$. If $a$ and $b$ are natural numbers such that $a$ is composite, $\gcd(a, b) = 1$, and $P(a) = P(b)$, compute $ab$. Note: $\gcd(m, n)$ denotes the greatest common divisor of $m$ and $n$. [i]Proposed by Aaron Lin [/i]

1979 IMO Shortlist, 3
Tags: vieta , algebra , functional equation , polynomial
Find all polynomials $f(x)$ with real coefficients for which \[f(x)f(2x^2) = f(2x^3 + x).\]

2016 Iran MO (3rd Round), 1
Tags: algebra , polynomial
Let $P(x) \in \mathbb {Z}[X]$ be a polynomial of degree $2016$ with no rational roots. Prove that there exists a polynomial $T(x) \in \mathbb {Z}[X]$ of degree $1395$ such that for all distinct (not necessarily real) roots of $P(x)$ like $(\alpha ,\beta):$ $$T(\alpha)-T(\beta) \not \in \mathbb {Q}$$ Note: $\mathbb {Q}$ is the set of rational numbers.

2000 Harvard-MIT Mathematics Tournament, 6
Tags: vieta , polynomial , algebra
If integers $m,n,k$ satisfy $m^2+n^2+1=kmn$, what values can $k$ have?

2006 USAMO, 3
Tags: polynomial , modular arithmetic , algebra
For integral $m$, let $p(m)$ be the greatest prime divisor of $m.$ By convention, we set $p(\pm 1) = 1$ and $p(0) = \infty.$ Find all polynomials $f$ with integer coefficients such that the sequence \[ \{p \left( f \left( n^2 \right) \right) - 2n \}_{n \geq 0} \] is bounded above. (In particular, this requires $f \left (n^2 \right ) \neq 0$ for $n \geq 0.$)

1959 AMC 12/AHSME, 44
Tags: vieta , algebra , polynomial
The roots of $x^2+bx+c=0$ are both real and greater than $1$. Let $s=b+c+1$. Then $s:$ $ \textbf{(A)}\ \text{may be less than zero}\qquad\textbf{(B)}\ \text{may be equal to zero}\qquad$ $\textbf{(C)}\ \text{must be greater than zero}\qquad\textbf{(D)}\ \text{must be less than zero}\qquad $ $\textbf{(E)}\text{ must be between -1 and +1} $

2001 China Team Selection Test, 3
Tags: minimization , calculus , polynomial , function , algebra
Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.

2013 Stanford Mathematics Tournament, 6
Tags: algebra , polynomial
Compute the largest root of $x^4-x^3-5x^2+2x+6$.

1997 Romania Team Selection Test, 1
Tags: polynomial , algebra , inequalities
Let $P(X),Q(X)$ be monic irreducible polynomials with rational coefficients. suppose that $P(X)$ and $Q(X)$ have roots $\alpha$ and $\beta$ respectively, such that $\alpha + \beta $ is rational. Prove that $P(X)^2-Q(X)^2$ has a rational root. [i]Bogdan Enescu[/i]

2015 Switzerland Team Selection Test, 6
Tags: algebra , function , functional equation , polynomial
Find all polynomial function $P$ of real coefficients such that for all $x \in \mathbb{R}$ $$P(x)P(x+1)=P(x^2+2)$$

1989 IMO Shortlist, 5
Tags: root , diophantine equation , equation , algebra , polynomial
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

2025 Canada National Olympiad, 3
Tags: polynomial , algebra
A polynomial $c_dx^d+c_{d-1}x^{d-1}+\dots+c_1x+c_0$ with degree $d$ is [i]reflexive[/i] if there is an integer $n\ge d$ such that $c_i=c_{n-i}$ for every $0\le i\le n$, where $c_i=0$ for $i>d$. Let $\ell\ge 2$ be an integer and $p(x)$ be a polynomial with integer coefficients. Prove that there exist reflexive polynomials $q(x)$, $r(x)$ with integer coefficients such that \[(1+x+x^2+\dots+x^{\ell-1})p(x)=q(x)+x^\ell r(x)\]

VI Soros Olympiad 1999 - 2000 (Russia), 9.2
Tags: polynomial , algebra
Can the equation $x^3 + ax^2 + bx + c = 0$ have only negative roots , if we know that $a+2b+4c=- \frac12 $?

2007 Germany Team Selection Test, 1
Tags: number theory , algebra , polynomial
Let $ k \in \mathbb{N}$. A polynomial is called [i]$ k$-valid[/i] if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.) [b]a.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often. [b]b.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .

2007 Canada National Olympiad, 3
Tags: functional inequalities , polynomial , algebra , function
Suppose that $ f$ is a real-valued function for which \[ f(xy)+f(y-x)\geq f(y+x)\] for all real numbers $ x$ and $ y$. a) Give a non-constant polynomial that satisfies the condition. b) Prove that $ f(x)\geq 0$ for all real $ x$.

2018 Middle European Mathematical Olympiad, 2
Tags: root , algebra , polynomial
Let $P(x)$ be a polynomial of degree $n\geq 2$ with rational coefficients such that $P(x) $ has $ n$ pairwise different reel roots forming an arithmetic progression .Prove that among the roots of $P(x) $ there are two that are also the roots of some polynomial of degree $2$ with rational coefficients .