Found problems: 3597
2009 Bulgaria National Olympiad, 4
Let $ n\ge 3$ be a natural number. Find all nonconstant polynomials with real coeficcietns $ f_{1}\left(x\right),f_{2}\left(x\right),\ldots,f_{n}\left(x\right)$, for which
\[ f_{k}\left(x\right)f_{k+ 1}\left(x\right) = f_{k +1}\left(f_{k + 2}\left(x\right)\right), \quad 1\le k\le n,\]
for every real $ x$ (with $ f_{n +1}\left(x\right)\equiv f_{1}\left(x\right)$ and $ f_{n + 2}\left(x\right)\equiv f_{2}\left(x\right)$).
2022 Federal Competition For Advanced Students, P2, 4
Decide whether for every polynomial $P$ of degree at least $1$, there exist infinitely many primes that divide $P(n)$ for at least one positive integer $n$.
[i](Walther Janous)[/i]
2017 Taiwan TST Round 3, 2
Prove that there exists a polynomial with integer coefficients satisfying the following conditions:
(a)$f(x)=0$ has no rational root.
(b) For any positive integer $n$, there always exists an integer $m$ such that $n\mid f(m)$.
1957 Miklós Schweitzer, 9
[b]9.[/b] Find all pairs of linear polynomials $f(x)$, $g(x)$ with integer coefficients for which there exist two polynomials $u(x)$, $v(x)$ with integer coefficients such that $f(x)u(x)+g(x)v(x)=1$. [b](A. 8)[/b]
2013 Bosnia Herzegovina Team Selection Test, 2
The sequence $a_n$ is defined by $a_0=a_1=1$ and $a_{n+1}=14a_n-a_{n-1}-4$,for all positive integers $n$.
Prove that all terms of this sequence are perfect squares.
2020 LIMIT Category 1, 1
Find all polynomial $P(x)$ with degree $\leq n$and non negative coefficients such that $$P(x)P(\frac{1}{x})\leq P(1)^2$$ for all positive $x$. Here $n$ is a natuaral number
2014 Contests, 901
Given the polynomials $P(x)=px^4+qx^3+rx^2+sx+t,\ Q(x)=\frac{d}{dx}P(x)$, find the real numbers $p,\ q,\ r,\ s,\ t$ such that $P(\sqrt{-5})=0,\ Q(\sqrt{-2})=0$ and $\int_0^1 P(x)dx=-\frac{52}{5}.$
2012 France Team Selection Test, 2
Determine all non-constant polynomials $X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0$ with integer coefficients for which the roots are exactly the numbers $a_0,a_1,\ldots ,a_{n-1}$ (with multiplicity).
1980 Poland - Second Round, 4
Prove that if $ a $ and $ b $ are real numbers and the polynomial $ ax^3 - ax^2 + 9bx - b $ has three positive roots, then they are equal.
1994 AIME Problems, 3
The function $f$ has the property that, for each real number $x,$ \[ f(x)+f(x-1) = x^2. \] If $f(19)=94,$ what is the remainder when $f(94)$ is divided by 1000?
2020 Tuymaada Olympiad, 2
All non-zero coefficients of the polynomial $f(x)$ equal $1$, while the sum of the coefficients is $20$. Is it possible that thirteen coefficients of $f^2(x)$ equal $9$?
[i](S. Ivanov, K. Kokhas)[/i]
2010 Tuymaada Olympiad, 1
Baron Münchausen boasts that he knows a remarkable quadratic triniomial with positive coefficients. The trinomial has an integral root; if all of its coefficients are increased by $1$, the resulting trinomial also has an integral root; and if all of its coefficients are also increased by $1$, the new trinomial, too, has an integral root. Can this be true?
2009 Romania Team Selection Test, 3
Show that there are infinitely many pairs of prime numbers $(p,q)$ such that $p\mid 2^{q-1}-1$ and $q\mid 2^{p-1}-1$.
1985 IMO Shortlist, 12
A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by
\[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\]
for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$
1985 Putnam, B1
Let $k$ be the smallest positive integer for which there exist distinct integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ such that the polynomial $$p(x)=\left(x-m_{1}\right)\left(x-m_{2}\right)\left(x-m_{3}\right)\left(x-m_{4}\right)\left(x-m_{5}\right)$$ has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ for which this minimum $k$ is achieved.
2005 International Zhautykov Olympiad, 1
Prove that the equation $ x^{5} \plus{} 31 \equal{} y^{2}$ has no integer solution.
OMMC POTM, 2021 12
Let $r,s,t$ be the roots of $x^3+6x^2+7x+8$. Find
$$(r^2+s+t)(s^2+t+r)(t^2+r+s).$$
[i]Proposed by Evan Chang (squareman), USA[/i]
2012 Belarus Team Selection Test, 3
Given a polynomial $P(x)$ with positive real coefficients.
Prove that $P(1)P(xy) \ge P(x)P(y)$ for all $x\ge1, y \ge 1$.
(K. Gorodnin)
1962 Poland - Second Round, 2
What conditions should real numbers $ a $, $ b $, $ c $, $ d $, $ e $, $ f $ meet in order for a polynomial of second degree $$ax^2 + 2bxy + cy^2 + 2dx + 2ey + f$$ was the product of two first degree polynomials with real coefficients ?
2007 IMC, 6
Let $ f \ne 0$ be a polynomial with real coefficients. Define the sequence $ f_{0}, f_{1}, f_{2}, \ldots$ of polynomials by $ f_{0}= f$ and $ f_{n+1}= f_{n}+f_{n}'$ for every $ n \ge 0$. Prove that there exists a number $ N$ such that for every $ n \ge N$, all roots of $ f_{n}$ are real.
2004 Kurschak Competition, 2
Find the smallest positive integer $n\neq 2004$ for which there exists a polynomial $f\in\mathbb{Z}[x]$ such that the equation $f(x)=2004$ has at least one, and the equation $f(x)=n$ has at least $2004$ different integer solutions.
2018 ELMO Shortlist, 1
Determine all nonempty finite sets of positive integers $\{a_1, \dots, a_n\}$ such that $a_1 \cdots a_n$ divides $(x + a_1) \cdots (x + a_n)$ for every positive integer $x$.
[i]Proposed by Ankan Bhattacharya[/i]
2002 AMC 10, 18
For how many positive integers $n$ is $n^3-8n^2+20n-13$ a prime number?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{more than 4}$
1971 All Soviet Union Mathematical Olympiad, 157
a) Consider the function $$f(x,y) = x^2 + xy + y^2$$ Prove that for the every point $(x,y)$ there exist such integers $(m,n)$, that $$f((x-m),(y-n)) \le 1/2$$
b) Let us denote with $g(x,y)$ the least possible value of the $f((x-m),(y-n))$ for all the integers $m,n$. The statement a) was equal to the fact $g(x,y) \le 1/2$.
Prove that in fact, $$g(x,y) \le 1/3$$
Find all the points $(x,y)$, where $g(x,y)=1/3$.
c) Consider function $$f_a(x,y) = x^2 + axy + y^2 \,\,\, (0 \le a \le 2)$$
Find any $c$ such that $g_a(x,y) \le c$.
Try to obtain the closest estimation.
2015 India Regional MathematicaI Olympiad, 2
Let $P_1(x) = x^2 + a_1x + b_1$ and $P_2(x) = x^2 + a_2x + b_2$ be two quadratic polynomials with integer coeffcients. Suppose $a_1 \ne a_2$ and there exist integers $m \ne n$ such that $P_1(m) = P_2(n), P_2(m) = P_1(n)$. Prove that $a_1 - a_2$ is even.