Olympiad problems

2018 Centroamerican and Caribbean Math Olympiad, 5

Let $n$ be a positive integer, $1<n<2018$. For each $i=1, 2, \ldots ,n$ we define the polynomial $S_i(x)=x^2-2018x+l_i$, where $l_1, l_2, \ldots, l_n$ are distinct positive integers. If the polynomial $S_1(x)+S_2(x)+\cdots+S_n(x)$ has at least an integer root, prove that at least one of the $l_i$ is greater or equal than $2018$.

1974 Bulgaria National Olympiad, Problem 2

Tags: Polynomials , number theory

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]

2002 USAMO, 3

Tags: AMC , Polynomials , USAMO , lagrange s interpolation

Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots.

PEN Q Problems, 12

Tags: algebra , polynomial , Gauss , Polynomials

Prove that if the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are all distinct, then the polynomial \[(x-a_{1})^{2}(x-a_{2})^{2}\cdots (x-a_{n})^{2}+1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

2014 Israel National Olympiad, 5

Tags: number theory , algebra , Polynomials , Integer Polynomial , integer polynomials , polynomial

Let $p$ be a polynomial with integer coefficients satisfying $p(16)=36,p(14)=16,p(5)=25$. Determine all possible values of $p(10)$.

1991 French Mathematical Olympiad, Problem 5

Tags: polynomial , Polynomials , algebra , geometry , roots of unity , complex numbers

(a) For given complex numbers $a_1,a_2,a_3,a_4$, we define a function $P:\mathbb C\to\mathbb C$ by $P(z)=z^5+a_4z^4+a_3z^3+a_2z^2+a_1z$. Let $w_k=e^{2ki\pi/5}$, where $k=0,\ldots,4$. Prove that $$P(w_0)+P(w_1)+P(w_2)+P(w_3)+P(w_4)=5.$$(b) Let $A_1,A_2,A_3,A_4,A_5$ be five points in the plane. A pentagon is inscribed in the circle with center $A_1$ and radius $R$. Prove that there is a vertex $S$ of the pentagon for which $$SA_1\cdot SA_2\cdot SA_3\cdot SA_4\cdot SA_5\ge R^5.$$

2015 BMT Spring, 6

Tags: Polynomials , algebra

The roots of the equation $x^5-180x^4+Ax^3+Bx^2+Cx+D=0$ are in geometric progression. The sum of their reciprocals is $20$. Compute $|D|$.

1978 Putnam, B3

Tags: Putnam , Polynomials , Recurrence , Sequences

The sequence $(Q_{n}(x))$ of polynomials is defined by $$Q_{1}(x)=1+x ,\; Q_{2}(x)=1+2x,$$ and for $m \geq 1 $ by $$Q_{2m+1}(x)= Q_{2m}(x) +(m+1)x Q_{2m-1}(x),$$ $$Q_{2m+2}(x)= Q_{2m+1}(x) +(m+1)x Q_{2m}(x).$$ Let $x_n$ be the largest real root of $Q_{n}(x).$ Prove that $(x_n )$ is an increasing sequence and that $\lim_{n\to \infty} x_n =0.$

2001 Moldova National Olympiad, Problem 3

Tags: functional equation , fe , Polynomials , algebra

Find all polynomials $P(x)$ with real coefficieints such that $P\left(x^2\right)=P(x)P(x-1)$ for all $x\in\mathbb R$.

2016 Serbia Additional Team Selection Test, 1

Tags: algebra , polynomial , Polynomials

Let $P_0(x)=x^3-4x$. Sequence of polynomials is defined as following:\\ $P_{n+1}=P_n(1+x)P_n(1-x)-1$.\\ Prove that $x^{2016}|P_{2016}(x)$.

1998 Slovenia National Olympiad, Problem 2

Tags: fe , functional equation , Polynomials

Find all polynomials $p$ with real coefficients such that for all real $x$ $$(x-8)p(2x)=8(x-1)p(x).$$

2022 Abelkonkurransen Finale, 4b

Tags: algebra , Polynomials

Do there exist $2022$ polynomials with real coefficients, each of degree equal to $2021$, so that the $2021 \cdot 2022 + 1$ coefficients in their product are equal?

2008 VJIMC, Problem 1

Tags: complex numbers , Polynomials , algebra , polynomial

Find all complex roots (with multiplicities) of the polynomial $$p(x)=\sum_{n=1}^{2008}(1004-|1004-n|)x^n.$$

PEN Q Problems, 13

Tags: algebra , polynomial , algorithm , Polynomials

On Christmas Eve, 1983, Dean Jixon, the famous seer who had made startling predictions of the events of the preceding year that the volcanic and seismic activities of $1980$ and $1981$ were connected with mathematics. The diminishing of this geological activity depended upon the existence of an elementary proof of the irreducibility of the polynomial \[P(x)=x^{1981}+x^{1980}+12x^{2}+24x+1983.\] Is there such a proof?

2015 BMT Spring, 2

Tags: algebra , Polynomials

Let $g(x)=1+2x+3x^2+4x^3+\ldots$. Find the coefficient of $x^{2015}$ of $f(x)=\frac{g(x)}{1-x}$.

2015 IMC, 10

Tags: college contests , Polynomials , Inequality , IMC2015

Let $n$ be a positive integer, and let $p(x)$ be a polynomial of degree $n$ with integer coefficients. Prove that $$ \max_{0\le x\le1} \big|p(x)\big| > \frac1{e^n}. $$ Proposed by Géza Kós, Eötvös University, Budapest

2024 India IMOTC, 10

Tags: Polynomials , algebra , polynomial

Let $r>0$ be a real number. We call a monic polynomial with complex coefficients $r$-[i]good[/i] if all of its roots have absolute value at most $r$. We call a monic polynomial with complex coefficients [i]primordial[/i] if all of its coefficients have absolute value at most $1$. a) Prove that any $1$-good polynomial has a primordial multiple. b) If $r>1$, prove that there exists an $r$-good polynomial that does not have a primordial multiple. [i]Proposed by Pranjal Srivastava[/i]

2011 VJIMC, Problem 1

Tags: Polynomials , number theory

(a) Is there a polynomial $P(x)$ with real coefficients such that $P\left(\frac1k\right)=\frac{k+2}k$ for all positive integers $k$? (b) Is there a polynomial $P(x)$ with real coefficients such that $P\left(\frac1k\right)=\frac1{2k+1}$ for all positive integers $k$?

2020 IMC, 4

Tags: IMC , Polynomials , algebra , polynomial , college contests , IMC 2020

A polynomial $p$ with real coefficients satisfies $p(x+1)-p(x)=x^{100}$ for all $x \in \mathbb{R}.$ Prove that $p(1-t) \ge p(t)$ for $0 \le t \le 1/2.$

2019 Belarusian National Olympiad, 9.3

Tags: inequalities , Polynomials , algebra , function

Positive real numbers $a$ and $b$ satisfy the following conditions: the function $f(x)=x^3+ax^2+2bx-1$ has three different real roots, while the function $g(x)=2x^2+2bx+a$ doesn't have real roots. Prove that $a-b>1$. [i](V. Karamzin)[/i]

1987 Bulgaria National Olympiad, Problem 1

Tags: algebra , Polynomials , Sequences , polynomial

Let $f(x)=x^n+a_1x^{n-1}+\ldots+a_n~(n\ge3)$ be a polynomial with real coefficients and $n$ real roots, such that $\frac{a_{n-1}}{a_n}>n+1$. Prove that if $a_{n-2}=0$, then at least one root of $f(x)$ lies in the open interval $\left(-\frac12,\frac1{n+1}\right)$.

2014 BMT Spring, 2

Tags: algebra , Polynomials

Find the smallest positive value of $x$ such that $x^3-9x^2+22x-16=0$.

2001 China Team Selection Test, 1

Tags: algebra , Polynomials , polynomial

Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer

2016 Tournament Of Towns, 6

Tags: Polynomials , number theory , algebra

$N $ different numbers are written on blackboard and one of these numbers is equal to $0$.One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between $-2016$ and $2016$ were written on blackboard(and some other numbers maybe). Find the smallest possible value of $N $.

Kvant 2025, M2828

Tags: algebra , Polynomials , polynomial

Maxim has guessed a polynomial $f(x)$ of degree $n$. Sasha wants to guess it (knowing $n$). During a turn, Sasha can name a certain segment $[a;b]$ and Maxim will give in response the maximum value of $f(x)$ on the segment $[a;b]$. Will Sasha be able to guess $f(x)$ in a finite number of steps? [i]M. Didin[/i]