Olympiad problems

2019 Moldova Team Selection Test, 9

Tags: polynomial , algebra

Find all polynomials $P(X)$ with real coefficients such that if real numbers $x,y$ and $z$ satisfy $x+y+z=0,$ then the points $\left(x,P(x)\right), \left(y,P(y)\right), \left(z,P(z)\right)$ are all colinear.

2016 Estonia Team Selection Test, 6

Tags: arc , coloring , algebra , combinatorics , polynomial

A circle is divided into arcs of equal size by $n$ points ($n \ge 1$). For any positive integer $x$, let $P_n(x)$ denote the number of possibilities for colouring all those points, using colours from $x$ given colours, so that any rotation of the colouring by $ i \cdot \frac{360^o}{n}$ , where i is a positive integer less than $n$, gives a colouring that differs from the original in at least one point. Prove that the function $P_n(x)$ is a polynomial with respect to $x$.

2021/2022 Tournament of Towns, P5

Tags: polynomial , algebra

What is the maximal possible number of roots on the interval (0,1) for a polynomial of degree 2022 with integer coefficients and with the leading coefficient equal to 1?

2010 Tuymaada Olympiad, 4

Tags: floor function , polynomial , limit , number theory , algebra

Prove that for any positive real number $\alpha$, the number $\lfloor\alpha n^2\rfloor$ is even for infinitely many positive integers $n$.

2011 AIME Problems, 15

Tags: quadratic , polynomial , algebra , probability , floor function

Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by the square of a prime. Find $a+b+c+d+e$.

2010 AMC 10, 21

Tags: vieta , polynomial , algebra

The polynomial $ x^3\minus{}ax^2\plus{}bx\minus{}2010$ has three positive integer zeros. What is the smallest possible value of $ a$? $ \textbf{(A)}\ 78 \qquad \textbf{(B)}\ 88 \qquad \textbf{(C)}\ 98 \qquad \textbf{(D)}\ 108 \qquad \textbf{(E)}\ 118$

2018 India National Olympiad, 4

Tags: divisibility , roots of a polynomial , polynomial , algebra

Find all polynomials with real coefficients $P(x)$ such that $P(x^2+x+1)$ divides $P(x^3-1)$.

1972 Polish MO Finals, 3

Tags: inequalities , polynomial , algebra

Prove that there is a polynomial $P(x)$ with integer coefficients such that for all $x$ in the interval $\left[ \frac{1}{10} , \frac{9}{10}\right]$ we have $$\left|P(x) -\frac12 \right| < \frac{ 1}{1000 }.$$

2013 Iran MO (3rd Round), 4

Tags: polynomial , vieta , diophantine equation , pell equation , number theory , algebra

Prime $p=n^2 +1$ is given. Find the sets of solutions to the below equation: \[x^2 - (n^2 +1)y^2 = n^2.\] (25 points)

2022 MMATHS, 9

Tags: polynomial , sum of roots , algebra

Let $f$ be a monic cubic polynomial such that the sum of the coefficients of $f$ is $5$ and such that the sum of the roots of $f$ is $1$. Find the absolute value of the sum of the cubes of the roots of $f$.

2011 Vietnam National Olympiad, 1

Tags: algebra , quadratic , modular arithmetic , polynomial , number theory

Define the sequence of integers $\langle a_n\rangle$ as; \[a_0=1, \quad a_1=-1, \quad \text{ and } \quad a_n=6a_{n-1}+5a_{n-2} \quad \forall n\geq 2.\] Prove that $a_{2012}-2010$ is divisible by $2011.$

2011 USA Team Selection Test, 3

Tags: algebra , polynomial , function , number theory

Let $p$ be a prime. We say that a sequence of integers $\{z_n\}_{n=0}^\infty$ is a [i]$p$-pod[/i] if for each $e \geq 0$, there is an $N \geq 0$ such that whenever $m \geq N$, $p^e$ divides the sum \[\sum_{k=0}^m (-1)^k {m \choose k} z_k.\] Prove that if both sequences $\{x_n\}_{n=0}^\infty$ and $\{y_n\}_{n=0}^\infty$ are $p$-pods, then the sequence $\{x_ny_n\}_{n=0}^\infty$ is a $p$-pod.

2021 Thailand Mathematical Olympiad, 10

Tags: polynomial , algebra

Let $d\geq 13$ be an integer, and let $P(x) = a_dx^d + a_{d-1}x^{d-1} + \dots + a_1x+a_0$ be a polynomial of degree $d$ with complex coefficients such that $a_n = a_{d-n}$ for all $n\in\{0,1,\dots,d\}$. Prove that if $P$ has no double roots, then $P$ has two distinct roots $z_1$ and $z_2$ such that $|z_1-z_2|<1$.

2012 USA Team Selection Test, 1

Tags: polynomial , algebra

Consider (3-variable) polynomials \[P_n(x,y,z)=(x-y)^{2n}(y-z)^{2n}+(y-z)^{2n}(z-x)^{2n}+(z-x)^{2n}(x-y)^{2n}\] and \[Q_n(x,y,z)=[(x-y)^{2n}+(y-z)^{2n}+(z-x)^{2n}]^{2n}.\] Determine all positive integers $n$ such that the quotient $Q_n(x,y,z)/P_n(x,y,z)$ is a (3-variable) polynomial with rational coefficients.

2013 NIMO Summer Contest, 4

Tags: vieta , algebra , polynomial

Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \][i]Proposed by Evan Chen[/i]

2024 Saint Petersburg Mathematical Olympiad, 4

Tags: number theory , algebra , polynomial

Let's consider all possible quadratic trinomials of the form $x^2 + ax + b$, where $a$ and $b$ are positive integers not exceeding some positive integer $N$. Prove that the number of pairs of such trinomials having a common root does not exceed $N^2$.

2007 Balkan MO Shortlist, N3

Tags: trigonometry , algebra , polynomial

i thought that this problem was in mathlinks but when i searched i didn't find it.so here it is: Find all positive integers m for which for all $\alpha,\beta \in \mathbb{Z}-\{0\}$ \[ \frac{2^m \alpha^m-(\alpha+\beta)^m-(\alpha-\beta)^m}{3 \alpha^2+\beta^2} \in \mathbb{Z} \]

2015 Iran MO (3rd round), 5

Tags: polynomial , function , functional equation , algebra

Find all polynomials $p(x)\in\mathbb{R}[x]$ such that for all $x\in \mathbb{R}$: $p(5x)^2-3=p(5x^2+1)$ such that: $a) p(0)\neq 0$ $b) p(0)=0$

2007 China Team Selection Test, 1

Tags: algebra , polynomial , geometry

When all vertex angles of a convex polygon are equal, call it equiangular. Prove that $ p > 2$ is a prime number, if and only if the lengths of all sides of equiangular $ p$ polygon are rational numbers, it is a regular $ p$ polygon.

2010 Iran MO (3rd Round), 3

Tags: polynomial , search , inequalities , algebra

prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)

STEMS 2021 Math Cat B, Q2

Tags: polynomial , power of 2 , prime number , divisibility , kobayashi

Determine all non-constant monic polynomials $P(x)$ with integer coefficients such that no prime $p>10^{100}$ divides any number of the form $P(2^n)$

1976 IMO Longlists, 11

Tags: algebra , recurrence relation , root , chebyshev polynomial , polynomial

Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.

2004 Austrian-Polish Competition, 10

Tags: polynomial , algebra

For each polynomial $Q(x)$ let $M(Q)$ be the set of non-negative integers $x$ with $0 < Q(x) < 2004.$ We consider polynomials $P_n(x)$ of the form \[P_n(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_{n-1} \cdot x + 1\] with coefficients $a_i \in \{ \pm1\}$ for $i = 1, 2, \ldots, n-1.$ For each $n = 3^k, k > 0$ determine: a.) $m_n$ which represents the maximum of elements in $M(P_n)$ for all such polynomials $P_n(x)$ b.) all polynomials $P_n(x)$ for which $|M(P_n)| = m_n.$

2013 Greece Team Selection Test, 1

Tags: polynomial , algebra

Determine whether the polynomial $P(x)=(x^2-2x+5)(x^2-4x+20)+1$ is irreducible over $\mathbb{Z}[X]$.

2002 Italy TST, 3

Tags: algebra , number theory , vieta , polynomial , relatively prime

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that $(\text{i})$ $x$ and $y$ are relatively prime; $(\text{ii})$ $x$ divides $y^2+m;$ $(\text{iii})$ $y$ divides $x^2+m.$