Olympiad problems

1987 IMO Longlists, 30

Consider the regular $1987$-gon $A_1A_2 . . . A_{1987}$ with center $O$. Show that the sum of vectors belonging to any proper subset of $M = \{OA_j | j = 1, 2, . . . , 1987\}$ is nonzero.

VI Soros Olympiad 1999 - 2000 (Russia), 11.5

Tags: polynomial , algebra

Find all polynomials $P(x)$ with real coefficients such that for all real $x$ holds the equality $$(1 + 2x)P(2x) = (1 + 2^{1999}x)P(x) .$$

2007 Czech-Polish-Slovak Match, 1

Tags: polynomial , algebra , complex number

Find all polynomials $P$ with real coefficients satisfying $P(x^2)=P(x)\cdot P(x+2)$ for all real numbers $x.$

1959 Poland - Second Round, 1

Tags: polynomial , algebra

What necessary and sufficient condition should the coefficients $ a $, $ b $, $ c $, $ d $ satisfy so that the equation $$ax^3 + bx^2 + cx + d = 0$$ has two opposite roots?

2011 Uzbekistan National Olympiad, 4

Tags: polynomial , algebra , function

Does existes a function $f:N->N$ and for all positeve integer n $f(f(n)+2011)=f(n)+f(f(n))$

2000 German National Olympiad, 2

Tags: algebra , polynomial , min , inequalities

For an integer $n \ge 2$, find all real numbers $x$ for which the polynomial $f(x) = (x-1)^4 +(x-2)^4 +...+(x-n)^4$ takes its minimum value.

2014 IMC, 3

Tags: polynomial , real analysis , algebra

Let $n$ be a positive integer. Show that there are positive real numbers $a_0, a_1, \dots, a_n$ such that for each choice of signs the polynomial $$\pm a_nx^n\pm a_{n-1}x^{n-1} \pm \dots \pm a_1x \pm a_0$$ has $n$ distinct real roots. (Proposed by Stephan Neupert, TUM, München)

2017 Morocco TST-, 6

Tags: number theory , polynomial , functional equation , sum of digits

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

2003 Federal Competition For Advanced Students, Part 2, 1

Tags: polynomial , algebra , number theory

Consider the polynomial $P(n) = n^3 -n^2 -5n+ 2$. Determine all integers $n$ for which $P(n)^2$ is a square of a prime. [hide="Remark."]I'm not sure if the statement of this problem is correct, because if $P(n)^2$ be a square of a prime, then $P(n)$ should be that prime, and I don't think the problem means that.[/hide]

2009 Moldova Team Selection Test, 2

Tags: pigeonhole principle , geometric transformation , greatest common divisor , algebra , geometry , number theory , polynomial

$ f(x)$ and $ g(x)$ are two polynomials with nonzero degrees and integer coefficients, such that $ g(x)$ is a divisor of $ f(x)$ and the polynomial $ f(x)\plus{}2009$ has $ 50$ integer roots. Prove that the degree of $ g(x)$ is at least $ 5$.

2000 ITAMO, 6

Tags: polynomial , algebra , integer polynomial

Let $p(x)$ be a polynomial with integer coefficients such that $p(0) = 0$ and $0 \le p(1) \le 10^7$. Suppose that there exist positive integers $a,b$ such that $p(a) = 1999$ and $p(b) = 2001$. Determine all possible values of $p(1)$. (Note: $1999$ is a prime number.)

2013 India Regional Mathematical Olympiad, 6

Tags: factorial , polynomial , root , greatest common divisor , number theory

Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$, where $a$ and $b$ are nonzero real numbers. Suppose that the roots of the equation $P(x)=0$ are the reciprocals of the roots of the equation $Q(x)=0$. Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013!+1)$ and $Q(2013!+1)$.

1989 Vietnam National Olympiad, 2

Tags: algebra , polynomial , inequalities

The sequence of polynomials $ \left\{P_n(x)\right\}_{n\equal{}0}^{\plus{}\infty}$ is defined inductively by $ P_0(x) \equal{} 0$ and $ P_{n\plus{}1}(x) \equal{} P_n(x)\plus{}\frac{x \minus{} P_n^2(x)}{2}$. Prove that for any $ x \in [0, 1]$ and any natural number $ n$ it holds that $ 0\le\sqrt x\minus{} P_n(x)\le\frac{2}{n \plus{} 1}$.

2023 Belarusian National Olympiad, 11.6

Tags: polynomial , algebra

Let $a$ be some integer. Prove that the polynomial $x^4(x-a)^4+1$ can not be a product of two non-constant polynomials with integer coefficients

2013 Mediterranean Mathematics Olympiad, 1

Tags: algebra , polynomial

Do there exist two real monic polynomials $P(x)$ and $Q(x)$ of degree 3,such that the roots of $P(Q(X))$ are nine pairwise distinct nonnegative integers that add up to $72$? (In a monic polynomial of degree 3, the coefficient of $x^{3}$ is $1$.)

2012 Iran Team Selection Test, 2

Tags: polynomial , algebra , pigeonhole principle

Do there exist $2000$ real numbers (not necessarily distinct) such that all of them are not zero and if we put any group containing $1000$ of them as the roots of a monic polynomial of degree $1000$, the coefficients of the resulting polynomial (except the coefficient of $x^{1000}$) be a permutation of the $1000$ remaining numbers? [i]Proposed by Morteza Saghafian[/i]

1997 All-Russian Olympiad, 1

Tags: modular arithmetic , algebra , quadratic , polynomial

Do there exist two quadratic trinomials $ax^2 +bx+c$ and $(a+1)x^2 +(b + 1)x + (c + 1)$ with integer coeficients, both of which have two integer roots? [i]N. Agakhanov[/i]

1988 IMO Longlists, 3

Tags: polynomial , function , algebra , generating functions , binomial coefficient

Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial \[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n. \]

2017 Latvia Baltic Way TST, 4

Tags: polynomial , algebra , consecutive

The values of the polynomial $P(x) = 2x^3-30x^2+cx$ for any three consecutive integers are also three consecutive integers. Find these values.

2018 Romania National Olympiad, 4

Tags: superior algebra , polynomial , irreducible

For any $k \in \mathbb{Z},$ define $$F_k=X^4+2(1-k)X^2+(1+k)^2.$$ Find all values $k \in \mathbb{Z}$ such that $F_k$ is irreducible over $\mathbb{Z}$ and reducible over $\mathbb{Z}_p,$ for any prime $p.$ [i]Marius Vladoiu[/i]

2001 Iran MO (2nd round), 1

Tags: polynomial , algebra

Find all polynomials $P$ with real coefficients such that $\forall{x\in\mathbb{R}}$ we have: \[ P(2P(x))=2P(P(x))+2(P(x))^2. \]

2007 Italy TST, 3

Tags: functional equation , algebra , function , polynomial

Find all $f: R \longrightarrow R$ such that \[f(xy+f(x))=xf(y)+f(x)\] for every pair of real numbers $x,y$.

2022 Indonesia MO, 2

Tags: number theory , algebra , polynomial

Let $P(x)$ be a polynomial with integer coefficient such that $P(1) = 10$ and $P(-1) = 22$. (a) Give an example of $P(x)$ such that $P(x) = 0$ has an integer root. (b) Suppose that $P(0) = 4$, prove that $P(x) = 0$ does not have an integer root.

2005 Junior Balkan Team Selection Tests - Romania, 6

Tags: polynomial , algebra , geometry

Let $ABC$ be an equilateral triangle and $M$ be a point inside the triangle. We denote by $A'$, $B'$, $C'$ the projections of the point $M$ on the sides $BC$, $CA$ and $AB$ respectively. Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent if and only if $M$ belongs to an altitude of the triangle.

1978 Putnam, B5

Tags: algebra , polynomial

Find the largest $a$ for which there exists a polynomial $$P(x) =a x^4 +bx^3 +cx^2 +dx +e$$ with real coefficients which satisfies $0\leq P(x) \leq 1$ for $-1 \leq x \leq 1.$