Found problems: 3597
2013 ELMO Shortlist, 2
For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$?
[i]Proposed by Andre Arslan[/i]
2017 Federal Competition For Advanced Students, 1
Determine all polynomials $P(x) \in \mathbb R[x]$ satisfying the following two conditions :
(a) $P(2017) = 2016$ and
(b) $(P(x) + 1)^2 = P(x^2 + 1)$ for all real numbers $x$.
[i]proposed by Walther Janous[/i]
2012 ELMO Shortlist, 8
Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$.
[i]Victor Wang.[/i]
2007 Stanford Mathematics Tournament, 11
The polynomial $R(x)$ is the remainder upon dividing $x^{2007}$ by $x^2-5x+6$. $R(0)$ can be expressed as $ab(a^c-b^c)$. Find $a+c-b$.
2025 All-Russian Olympiad, 9.5
Let \( P_1(x) \) and \( P_2(x) \) be monic quadratic trinomials, and let \( A_1 \) and \( A_2 \) be the vertices of the parabolas \( y = P_1(x) \) and \( y = P_2(x) \), respectively. Let \( m(g(x)) \) denote the minimum value of the function \( g(x) \). It is known that the differences \( m(P_1(P_2(x))) - m(P_1(x)) \) and \( m(P_2(P_1(x))) - m(P_2(x)) \) are equal positive numbers. Find the angle between the line \( A_1A_2 \) and the $x$-axis. \\
2017 IFYM, Sozopol, 8
Find all polynomials $P\in \mathbb{R}[x]$, for which $P(P(x))=\lfloor P^2 (x)\rfloor$ is true for $\forall x\in \mathbb{Z}$.
2018 AMC 12/AHSME, 22
Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$?
$\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286 $
2024 Switzerland Team Selection Test, 3
Determine all monic polynomial with integer coefficient $P$ such that for every integer $a,b$ there exists integer $c$ so that
\[P(a)P(b)=P(c)\]
2011 China Second Round Olympiad, 2
For any integer $n\ge 4$, prove that there exists a $n$-degree polynomial $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$
satisfying the two following properties:
[b](1)[/b] $a_i$ is a positive integer for any $i=0,1,\ldots,n-1$, and
[b](2)[/b] For any two positive integers $m$ and $k$ ($k\ge 2$) there exist distinct positive integers $r_1,r_2,...,r_k$, such that $f(m)\ne\prod_{i=1}^{k}f(r_i)$.
PEN D Problems, 13
Let $\Gamma$ consist of all polynomials in $x$ with integer coefficients. For $f$ and $g$ in $\Gamma$ and $m$ a positive integer, let $f \equiv g \pmod{m}$ mean that every coefficient of $f-g$ is an integral multiple of $m$. Let $n$ and $p$ be positive integers with $p$ prime. Given that $f,g,h,r$ and $s$ are in $\Gamma$ with $rf+sg\equiv 1 \pmod{p}$ and $fg \equiv h \pmod{p}$, prove that there exist $F$ and $G$ in $\Gamma$ with $F \equiv f \pmod{p}$, $G \equiv g \pmod{p}$, and $FG \equiv h \pmod{p^n}$.
2017 China Team Selection Test, 6
For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.
2010 Argentina Team Selection Test, 5
Let $p$ and $q$ be prime numbers. The sequence $(x_n)$ is defined by $x_1 = 1$, $x_2 = p$ and $x_{n+1} = px_n - qx_{n-1}$ for all $n \geq 2$.
Given that there is some $k$ such that $x_{3k} = -3$, find $p$ and $q$.
2007 Today's Calculation Of Integral, 200
Evaluate the following definite integral.
\[\int_{0}^{\pi}\frac{\cos nx}{2-\cos x}dx\ (n=0,\ 1,\ 2,\ \cdots)\]
2013 Brazil Team Selection Test, 4
Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying
(i) $A$ and $B$ are disjoint;
(ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$.
Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)
2004 Austria Beginners' Competition, 3
Determine the value of the parameter $m$ such that the equation $(m-2)x^2+(m^2-4m+3)x-(6m^2-2)=0$ has real solutions, and the sum of the third powers of these solutions is equal to zero.
2018 Greece National Olympiad, 3
Let $n,m$ be positive integers such that $n<m$ and $a_1, a_2, ..., a_m$ be different real numbers.
(a) Find all polynomials $P$ with real coefficients and degree at most $n$ such that:
$|P(a_i)-P(a_j)|=|a_i-a_j|$ for all $i,j=\{1, 2, ..., m\}$ such that $i<j$.
(b) If $n,m\ge 2$ does there exist a polynomial $Q$ with real coefficients and degree $n$ such that:
$|Q(a_i)-Q(a_j)|<|a_i-a_j|$ for all $i,j=\{1, 2, ..., m\}$ such that $i<j$
Edit: See #3
1977 Germany Team Selection Test, 2
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
2018 Nordic, 4
Let $f = f(x,y,z)$ be a polynomial in three variables $x$, $y$, $z$ such that $f(w,w,w) = 0$ for all $w \in \mathbb{R}$. Show that there exist three polynomials $A$, $B$, $C$ in these same three variables such that $A + B + C = 0$ and \[ f(x,y,z) = A(x,y,z) \cdot (x-y) + B(x,y,z) \cdot (y-z) + C(x,y,z) \cdot (z-x). \] Is there any polynomial $f$ for which these $A$, $B$, $C$ are uniquely determined?
2016 Lusophon Mathematical Olympiad, 3
Suppose a real number $a$ is a root of a polynomial with integer coefficients $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$. Let $G=|a_n|+|a_{n-1}|+...+|a_1|+|a_0|$. We say that $G$ is a [i]gingado [/i] of $a$.
For example, as $2$ is root of $P(x)=x^2-x-2$, $G=|1|+|-1|+|-2|=4$, we say that $4$ is a [i]gingado[/i] of $2$. What is the fourth largest real number $a$ such that $3$ is a [i]gingado [/i] of $a$?
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 $P(m)\in\mathbb{Z},\forall m\in\mathbb{Z}.$
2004 IMC, 1
Let $A$ be a real $4\times 2$ matrix and $B$ be a real $2\times 4$ matrix such that
\[ AB = \left(%
\begin{array}{cccc}
1 & 0 & -1 & 0 \\
0 & 1 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & -1 & 0 & 1 \\
\end{array}%
\right). \]
Find $BA$.
1960 Putnam, A5
Find all polynomials $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients.
2005 Croatia National Olympiad, 2
Let $P(x)$ be a monic polynomial of degree $n$ with nonnegative coefficients and the free term equal to $1$. Prove that if all the roots of $P(x)$ are real, then $P(x) \geq (x+1)^{n}$ holds for every $x \geq 0$.
2014 USAMO, 1
Let $a$, $b$, $c$, $d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.
2012 IMC, 4
Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1,x_2,\dots,x_n$ satisfying
\[x_1(1-x_2)=x_2(1-x_3)=\dots=x_n(1-x_1)=a.\]
[i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]