Olympiad problems

2004 Miklós Schweitzer, 7

Suppose that the closed subset $K$ of the sphere $$S^2=\{ (x,y,z)\in \mathbb{R}^3\colon x^2+y^2+z^2=1 \}$$ is symmetric with respect to the origin and separates any two antipodal points in $S^2 \backslash K$. Prove that for any positive $\varepsilon$ there exists a homogeneous polynomial $P$ of odd degree such that the Hausdorff distance between $$Z(P)=\{ (x,y,z)\in S^2 \colon P(x,y,z)=0\}$$ and $K$ is less than $\varepsilon$.

2020 Jozsef Wildt International Math Competition, W27

Tags: integer polynomial , algebra , polynomial , number theory

Let $$P(x)=a_0x^n+a_1x^{n-1}+\ldots+a_n$$ where $a_0,\ldots,a_n$ are integers. Show that if $P$ takes the value $2020$ for four distinct integral values of $x$, then $P$ cannot take the value $2001$ for any integral value of $x$. [i]Proposed by Ángel Plaza[/i]

2000 Austrian-Polish Competition, 1

Tags: floor function , sum , polynomial , algebra

Find all polynomials $P(x)$ with real coefficients having the following property: There exists a positive integer n such that the equality $$\sum_{k=1}^{2n+1}(-1)^k \left[\frac{k}{2}\right] P(x + k)=0$$ holds for infinitely many real numbers $x$.

2010 Purple Comet Problems, 25

Tags: algebra , polynomial , number theory , relatively prime

Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$. There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$. Find $m+n$.

2010 Math Prize For Girls Problems, 16

Tags: quadratic , function , analytic geometry , algebra , polynomial

Let $P$ be the quadratic function such that $P(0) = 7$, $P(1) = 10$, and $P(2) = 25$. If $a$, $b$, and $c$ are integers such that every positive number $x$ less than 1 satisfies \[ \sum_{n = 0}^\infty P(n) x^n = \frac{ax^2 + bx + c}{{(1 - x)}^3}, \] compute the ordered triple $(a, b, c)$.

Russian TST 2020, P1

Tags: number theory , polynomial

Let $P(x)$ be a polynomial taking integer values at integer inputs. Are there infinitely many natural numbers that are not representable in the form $P(k)-2^n$ where $n{}$ and $k{}$ are non-negative integers? [i]Proposed by F. Petrov[/i]

2009 Canadian Mathematical Olympiad Qualification Repechage, 3

Tags: polynomial , algebra

Prove that there does not exist a polynomial $f(x)$ with integer coefficients for which $f(2008) = 0$ and $f(2010) = 1867$.

2021 Latvia TST, 2.3

Tags: algebra , polynomial , linear combination

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

2002 All-Russian Olympiad, 1

Tags: algebra , polynomial , quadratic , ring theory , equation , functional equation

The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.

1984 Bulgaria National Olympiad, Problem 4

Tags: polynomial , algebra

Let $a,b,a_2,\ldots,a_{n-2}$ be real numbers with $ab\ne0$ such that all the roots of the equation $$ax^n-ax^{n-1}+a_2x^{n-2}+\ldots+a_{n-2}x^2-n^2bx+b=0$$are positive and real. Prove that these roots are all equal.

2023 Thailand Online MO, 2

Tags: algebra , polynomial , polynomial roots

Let $P(x)$ be a polynomial with real coefficients. Prove that not all roots of $x^3P(x)+1$ are real.

1992 IMO Longlists, 44

Tags: algebra , polynomial

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

1995 Czech And Slovak Olympiad IIIA, 6

Tags: sidelenghts , polynomial , algebra , parameter

Find all real parameters $p$ for which the equation $x^3 -2p(p+1)x^2+(p^4 +4p^3 -1)x-3p^3 = 0$ has three distinct real roots which are sides of a right triangle.

2013 India National Olympiad, 3

Tags: calculus , integration , inequalities , function , algebra , polynomial

Let $a,b,c,d \in \mathbb{N}$ such that $a \ge b \ge c \ge d $. Show that the equation $x^4 - ax^3 - bx^2 - cx -d = 0$ has no integer solution.

2021 JHMT HS, 7

Tags: graphing lines , slope , polynomial

A line passing through $(20,21)$ intersects the curve $y = x^3-2x^2-3x+5$ at three distinct points $A, B,$ and $C,$ such that $B$ is the midpoint of $\overline{AC}$. The slope of this line is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2008 Mathcenter Contest, 2

Tags: algebra , polynomial

Find all polynomials $P(x)$ which have the properties: 1) $P(x)$ is not a constant polynomial and is a mononic polynomial. 2) $P(x)$ has all real roots and no duplicate roots. 3) If $P(a)=0$ then $P(a|a|)=0$ [i](nooonui)[/i]

2003 IMC, 6

Tags: algebra , polynomial , real analysis

Let $ p=\sum\limits_{k=0}^n a_kX^k\in R[X] $ a polynomial such that all his roots lie in the half plane $ \{z\in C| Re(z)<0 \}. $ Prove that $ a_ka_{k+3}<a_{k+1}a_{k+2}, $ for every k=0,1,2...,n-3.

2022 239 Open Mathematical Olympiad, 1

Tags: algebra , game , polynomial

Egor and Igor take turns (Igor starts) replacing the coefficients of the polynomial \[a_{99}x^{99} + \cdots + a_1x + a_0\]with non-zero integers. Egor wants the polynomial to have as many different integer roots as possible. What is the largest number of roots he can always achieve?

1966 Poland - Second Round, 2

Tags: algebra , polynomial , cubic

Prove that if two cubic polynomials with integer coefficients have an irrational root in common, then they have another common irrational root.

2020 Estonia Team Selection Test, 3

Tags: number theory , polynomial

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

1995 Romania Team Selection Test, 3

Tags: algebra , polynomial , integer polynomial , irreducible

Let $f$ be an irreducible (in $Z[x]$) monic polynomial with integer coefficients and of odd degree greater than $1$. Suppose that the modules of the roots of $f$ are greater than $1$ and that $f(0)$ is a square-free number. Prove that the polynomial $g(x) = f(x^3)$ is also irreducible

2016 Saudi Arabia GMO TST, 3

Tags: divide , divisor , polynomial , algebra

Find all polynomials $P,Q \in Z[x]$ such that every positive integer is a divisor of a certain nonzero term of the sequence $(x_n)_{n=0}^{\infty}$ given by the conditions: $x_0 = 2016$, $x_{2n+1} = P(x_{2n})$, $x_{2n+2} = Q(x_{2n+1})$ for all $n \ge 0$