Olympiad problems

1984 IMO Longlists, 13

Tags: polynomial , diophantine equation , equation , quadratic , number theory

Prove: (a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$ (b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$

2023 China Team Selection Test, P20

Tags: polynomial , number theory

Let $a,b,d$ be integers such that $\left|a\right| \geqslant 2$, $d \geqslant 0$ and $b \geqslant \left( \left|a\right| + 1\right)^{d + 1}$. For a real coefficient polynomial $f$ of degree $d$ and integer $n$, let $r_n$ denote the residue of $\left[ f(n) \cdot a^n \right]$ mod $b$. If $\left \{ r_n \right \}$ is eventually periodic, prove that all the coefficients of $f$ are rational.

2011 Postal Coaching, 1

Tags: algebra , number theory , polynomial

Prove that, for any positive integer $n$, there exists a polynomial $p(x)$ of degree at most $n$ whose coeﬃcients are all integers such that, $p(k)$ is divisible by $2^n$ for every even integer $k$, and $p(k) -1$ is divisible by $2^n$ for every odd integer $k$.

2025 China Team Selection Test, 12

Tags: polynomial , algebra

Let $ P(x), Q(x) $ be non-constant real polynomials, such that for all positive integer $ m $, there exists a positive integer $ n $ satisfy $ P(m) = Q(n) $. Prove that (1) If $\deg Q \mid \deg P$, then there exists real polynomial $ h(x) $ $ x $, satisfy $ P(x) = Q(h(x)) $ holds for all real number $x.$ (2) $\deg Q \mid \deg P$.

2022 Indonesia TST, A

Tags: inequalities , polynomial , algebra

Given a monic quadratic polynomial $Q(x)$, define \[ Q_n (x) = \underbrace{Q(Q(\cdots(Q(x))\cdots))}_{\text{compose $n$ times}} \] for every natural number $n$. Let $a_n$ be the minimum value of the polynomial $Q_n(x)$ for every natural number $n$. It is known that $a_n > 0$ for every natural number $n$ and there exists some natural number $k$ such that $a_k \neq a_{k+1}$. (a) Prove that $a_n < a_{n+1}$ for every natural number $n$. (b) Is it possible to satisfy $a_n < 2021$ for every natural number $n$? [i]Proposed by Fajar Yuliawan[/i]

1991 Irish Math Olympiad, 5

Tags: algebra , polynomial

Find all polynomials $f(x) = x^{n} + a_{1}x^{n-1} + \cdots + a_{n}$ with the following properties (a) all the coefficients $a_{1}, a_{2}, ..., a_{n}$ belong to the set $\{ -1, 1 \}$; and (b) all the roots of the equation $f(x)=0$ are real.

2012 Bulgaria National Olympiad, 3

Tags: polynomial , algebra

We are given a real number $a$, not equal to $0$ or $1$. Sacho and Deni play the following game. First is Sasho and then Deni and so on (they take turns). On each turn, a player changes one of the “*” symbols in the equation: \[*x^4+*x^3+*x^2+*x^1+*=0\] with a number of the type $a^n$, where $n$ is a whole number. Sasho wins if at the end the equation has no real roots, Deni wins otherwise. Determine (in term of $a$) who has a winning strategy

1980 Vietnam National Olympiad, 2

Tags: algebra , polynomial , integration , calculus

Can the equation $x^3-2x^2-2x+m = 0$ have three different rational roots?

1996 Romania National Olympiad, 2

Tags: polynomial , algebra

Find all polynomials $p_n(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ ($n\geq 2$) with real and non-zero coeficients s.t. $p_n(x)-p_1(x)p_2(x)...p_{n-1}(x)$ be a constant polynomial. ;)

2017 IMO Shortlist, A1

Tags: polynomial , algebra , ineqstd

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

2001 239 Open Mathematical Olympiad, 5

Tags: polynomial , number theory

Let $P(x)$ be a monic polynomial with integer coefficients of degree $10$. Prove that there exist distinct positive integers $a,b$ not exceeding $101$ such that $P(a)-P(b)$ is divisible by $101$.

2010 AMC 12/AHSME, 23

Tags: number theory , modular arithmetic , polynomial , algebra

The number obtained from the last two nonzero digits of $ 90!$ is equal to $ n$. What is $ n$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 68$

2010 Turkey MO (2nd round), 2

Tags: number theory , modular arithmetic , algebra , polynomial

For integers $a$ and $b$ with $0 \leq a,b < {2010}^{18}$ let $S$ be the set of all polynomials in the form of $P(x)=ax^2+bx.$ For a polynomial $P$ in $S,$ if for all integers n with $0 \leq n <{2010}^{18}$ there exists a polynomial $Q$ in $S$ satisfying $Q(P(n)) \equiv n \pmod {2010^{18}},$ then we call $P$ as a [i]good polynomial.[/i] Find the number of [i]good polynomials.[/i]

1999 Bosnia and Herzegovina Team Selection Test, 6

Tags: polynomial , root , imaginary , algebra

It is given polynomial $$P(x)=x^4+3x^3+3x+p, (p \in \mathbb{R})$$ $a)$ Find $p$ such that there exists polynomial with imaginary root $x_1$ such that $\mid x_1 \mid =1$ and $2Re(x_1)=\frac{1}{2}\left(\sqrt{17}-3\right)$ $b)$ Find all other roots of polynomial $P$ $c)$ Prove that does not exist positive integer $n$ such that $x_1^n=1$

2018 India Regional Mathematical Olympiad, 2

Tags: polynomial , algebra

Find the set of all real values of $a$ for which the real polynomial equation $P(x)=x^2-2ax+b=0$ has real roots, given that $P(0)\cdot P(1)\cdot P(2)\neq 0$ and $P(0),P(1),P(2)$ form a geometric progression.

PEN Q Problems, 5

Tags: algebra , induction , polynomial

(Eisentein's Criterion) Let $f(x)=a_{n}x^{n} +\cdots +a_{1}x+a_{0}$ be a nonconstant polynomial with integer coefficients. If there is a prime $p$ such that $p$ divides each of $a_{0}$, $a_{1}$, $\cdots$,$a_{n-1}$ but $p$ does not divide $a_{n}$ and $p^2$ does not divide $a_{0}$, then $f(x)$ is irreducible in $\mathbb{Q}[x]$.

2007 Finnish National High School Mathematics Competition, 2

Tags: algebra , polynomial

Determine the number of real roots of the equation \[x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x +\frac{5}{2}= 0.\]

2017 Hanoi Open Mathematics Competitions, 4

Tags: algebra , polynomial

Let a,b,c be three distinct positive numbers. Consider the quadratic polynomial $P (x) =\frac{c(x - a)(x - b)}{(c -a)(c -b)}+\frac{a(x - b)(x - c)}{(a - b)(a - c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}+ 1$. The value of $P (2017)$ is (A): $2015$ (B): $2016$ (C): $2017$ (D): $2018$ (E): None of the above.

2001 District Olympiad, 1

Tags: polynomial , matrix , algebra , linear algebra

Let $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A)=d\neq 0$ and $\det(A+dA^*)=0$. Prove that $\det(A-dA^*)=4$. [i]Daniel Jinga[/i]

2012 India IMO Training Camp, 2

Tags: rotation , algebra , polynomial , inequalities

Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that \[\text{max}_{|z|=1}\{|P(z)|\}\ge\sqrt{2|a_ma_n|+\sum_{k=m}^{n} |a_k|^2}\]

1990 Austrian-Polish Competition, 6

Tags: algebra , integer polynomial , polynomial

$p(x)$ is a polynomial with integer coefficients. The sequence of integers $a_1, a_2, ... , a_n$ (where $n > 2$) satisfies $a_2 = p(a_1), a_3 = p(a_2), ... , a_n = p(a_{n-1}), a_1 = p(a_n)$. Show that $a_1 = a_3$.

2020 Azerbaijan IMO TST, 2

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 $b$.

2013 Stanford Mathematics Tournament, 9

Tags: algebra , factoring polynomials , polynomial

Let $a=-\sqrt{3}+\sqrt{5}+\sqrt{7}, b=\sqrt{3}-\sqrt{5}+\sqrt{7}, c=\sqrt{3}+\sqrt{5}-\sqrt{7}$. Evaluate \[\frac{a^4}{(a-b)(a-c)}+\frac{b^4}{(b-c)(b-a)}+\frac{c^4}{(c-a)(c-b)}.\]

1976 AMC 12/AHSME, 19

Tags: analytic geometry , algorithm , graphing lines , algebra , polynomial , slope , modular arithmetic

A polynomial $p(x)$ has remainder three when divided by $x-1$ and remainder five when divided by $x-3$. The remainder when $p(x)$ is divided by $(x-1)(x-3)$ is $\textbf{(A) }x-2\qquad\textbf{(B) }x+2\qquad\textbf{(C) }2\qquad\textbf{(D) }8\qquad \textbf{(E) }15$

1994 AIME Problems, 13

Tags: polynomial , vieta , trigonometry , algebra

The equation \[ x^{10}+(13x-1)^{10}=0 \] has 10 complex roots $r_1, \overline{r_1}, r_2, \overline{r_2}, r_3, \overline{r_3}, r_4, \overline{r_4}, r_5, \overline{r_5},$ where the bar denotes complex conjugation. Find the value of \[ \frac 1{r_1\overline{r_1}}+\frac 1{r_2\overline{r_2}}+\frac 1{r_3\overline{r_3}}+\frac 1{r_4\overline{r_4}}+\frac 1{r_5\overline{r_5}}. \]