Found problems: 3597
2018 CIIM, Problem 2
Let $p(x)$ and $q(x)$ non constant real polynomials of degree at most $n$ ($n > 1$). Show that there exists a non zero polynomial $F(x,y)$ in two variables with real coefficients of degree at most $2n-2,$ such that $F(p(t),q(t)) = 0$ for every $t\in \mathbb{R}$.
2014 Indonesia MO, 3
Suppose that $k,m,n$ are positive integers with $k \le n$. Prove that:
\[\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1\]
2010 India IMO Training Camp, 2
Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.
1952 Poland - Second Round, 1
Find the necessary and sufficient conditions that the real numbers $ a $, $ b $, $ c $ should satisfy so that the equation
$$x^3 + ax^2 + bx + c = 0$$
has three real roots creating an arithmetic progression.
2016 Switzerland Team Selection Test, Problem 2
Find all polynomial functions with real coefficients for which $$(x-2)P(x+2)+(x+2)P(x-2)=2xP(x)$$ for all real $x$
2015 Saint Petersburg Mathematical Olympiad, 1
Is there a quadratic trinomial $f(x)$ with integer coefficients such that $f(f(\sqrt{2}))=0$ ?
[i]A. Khrabrov[/i]
1994 Moldova Team Selection Test, 4
Let $P(x)$ be a polynomial with at most $n{}$ real coefficeints. Prove that if $P(x)$ has integer values for $n+1$ consecutive values of the argument, then $P(m)\in\mathbb{Z},\forall m\in\mathbb{Z}.$
2014 Iran Team Selection Test, 3
prove for all $k> 1$ equation $(x+1)(x+2)...(x+k)=y^{2}$ has finite solutions.
2019 Korea - Final Round, 5
Find all pairs $(p,q)$ such that the equation $$x^4+2px^2+qx+p^2-36=0$$ has exactly $4$ integer roots(counting multiplicity).
2021 Brazil Team Selection Test, 3
A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?
2017 Saudi Arabia Pre-TST + Training Tests, 4
Does there exist an integer $n \ge 3$ and an arithmetic sequence $a_0, a_1, ... , a_n$ such that the polynomial $a_nx^n +... + a_1x + a_0$ has $n$ roots which also form an arithmetic sequence?
2023 Olimphíada, 4
We say that a prime $p$ is $\textit{philé}$ if there is a polynomial $P$ of non-negative integer coefficients smaller than $p$ and with degree $3$, that is, $P(x) = ax^3 + bx^2 + cx + d$ where $a, b, c, d < p$, such that $$\{P(n) | 1 \leq n \leq p\}$$ is a complete residue system modulo $p$. Find all $\textit{philé}$ primes.
Note: A set $A$ is a complete residue system modulo $p$ if for every integer $k$, with $0 \leq k \leq p - 1$, there exists an element $a \in A$ such that $$p | a-k.$$
2006 Kazakhstan National Olympiad, 2
Product of square trinomials $ x ^ 2 + a_1x + b_1 $, $ x ^ 2 + a_2x + b_2 $, $ \dots $, $ x ^ 2 + a_n x + b_n $ equals polynomial $ P (x) = x ^ {2n} + c_1x ^ {2n-1} + c_2x ^ {2n-2} + \dots + c_ {2n-1} x + c_ {2n} $, where the coefficients $ c_1 $, $ c_2 $, $ \dots $, $ c_ {2n} $ are positive. Prove that for some $ k $ ($ 1 \leq k \leq n $) the coefficients $ a_k $ and $ b_k $ are positive.
1974 Bulgaria National Olympiad, Problem 2
Let $f(x)$ and $g(x)$ be non-constant polynomials with integer positive coefficients, $m$ and $n$ are given natural numbers. Prove that there exists infinitely many natural numbers $k$ for which the numbers
$$f(m^n)+g(0),f(m^n)+g(1),\ldots,f(m^n)+g(k)$$
are composite.
[i]I. Tonov[/i]
2013 AMC 12/AHSME, 25
Let $G$ be the set of polynomials of the form
\[P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,\]
where $c_1,c_2,\cdots, c_{n-1}$ are integers and $P(z)$ has $n$ distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$?
${ \textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D}}\ 992\qquad\textbf{(E)}\ 1056 $
2006 Indonesia MO, 2
Let $ a,b,c$ be positive integers. If $ 30|a\plus{}b\plus{}c$, prove that $ 30|a^5\plus{}b^5\plus{}c^5$.
2015 Purple Comet Problems, 20
For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$. Find the minimum possible value for $|p(15)|$.
2014 BAMO, 3
Suppose that for two real numbers $x$ and $y$ the following equality is true:
$$(x+ \sqrt{1+ x^2})(y+\sqrt{1+y^2})=1$$
Find (with proof) the value of $x+y$.
1979 Bulgaria National Olympiad, Problem 4
For each real number $k$, denote by $f(k)$ the larger of the two roots of the quadratic equation
$$(k^2+1)x^2+10kx-6(9k^2+1)=0.$$Show that the function $f(k)$ attains a minimum and maximum and evaluate these two values.
Russian TST 2019, P3
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.
2007 QEDMO 5th, 5
Let $ a$, $ b$, $ c$ be three integers. Prove that there exist six integers $ x$, $ y$, $ z$, $ x^{\prime}$, $ y^{\prime}$, $ z^{\prime}$ such that
$ a\equal{}yz^{\prime}\minus{}zy^{\prime};\ \ \ \ \ \ \ \ \ \ b\equal{}zx^{\prime}\minus{}xz^{\prime};\ \ \ \ \ \ \ \ \ \ c\equal{}xy^{\prime}\minus{}yx^{\prime}$.
2005 Georgia Team Selection Test, 4
Find all polynomials with real coefficients, for which the equality
\[ P(2P(x)) \equal{} 2P(P(x)) \plus{} 2(P(x))^{2}\]
holds for any real number $ x$.
1992 IMO Shortlist, 12
Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose
\[ f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R}\]
Prove that there exists a polynomial $ h$ with $ f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.$
2017 China Team Selection Test, 5
A(x,y), B(x,y), and C(x,y) are three homogeneous real-coefficient polynomials of x and y with degree 2, 3, and 4 respectively. we know that there is a real-coefficient polinimial R(x,y) such that $B(x,y)^2-4A(x,y)C(x,y)=-R(x,y)^2$. Proof that there exist 2 polynomials F(x,y,z) and G(x,y,z) such that $F(x,y,z)^2+G(x,y,z)^2=A(x,y)z^2+B(x,y)z+C(x,y)$ if for any x, y, z real numbers $A(x,y)z^2+B(x,y)z+C(x,y)\ge 0$
2014 Contests, 2
Find all real non-zero polynomials satisfying $P(x)^3+3P(x)^2=P(x^{3})-3P(-x)$ for all $x\in\mathbb{R}$.