Olympiad problems

2013 Online Math Open Problems, 42

Find the remainder when \[\prod_{i=0}^{100}(1-i^2+i^4)\] is divided by $101$. [i]Victor Wang[/i]

2018 Romanian Masters in Mathematics, 2

Tags: algebra , polynomial

Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying $$P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$$

2022 South Africa National Olympiad, 6

Tags: algebra , polynomial

Show that there are infinitely many polynomials P with real coefficients such that if x, y, and z are real numbers such that $x^2+y^2+z^2+2xyz=1$, then $$P\left(x\right)^2+P\left(y\right)^2+P\left(z\right)^2+2P\left(x\right)P\left(y\right)P\left(z\right) = 1$$

2007 Italy TST, 3

Tags: function , polynomial , functional equation , algebra

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

2008 Mongolia Team Selection Test, 3

Tags: inequalities , algebra , polynomial , three variable inequality

Find the maximum number $ C$ such that for any nonnegative $ x,y,z$ the inequality $ x^3 \plus{} y^3 \plus{} z^3 \plus{} C(xy^2 \plus{} yz^2 \plus{} zx^2) \ge (C \plus{} 1)(x^2 y \plus{} y^2 z \plus{} z^2 x)$ holds.

2021 Iran RMM TST, 3

Tags: algebra , polynomial

We call a polynomial $P(x)=a_dx^d+...+a_0$ of degree $d$ [i]nice[/i] if $$\frac{2021(|a_d|+...+|a_0|)}{2022}<max_{0 \le i \le d}|a_i|$$ Initially Shayan has a sequence of $d$ distinct real numbers; $r_1,...,r_d \neq \pm 1$. At each step he choose a positive integer $N>1$ and raises the $d$ numbers he has to the exponent of $N$, then delete the previous $d$ numbers and constructs a monic polynomial of degree $d$ with these number as roots, then examine whether it is nice or not. Prove that after some steps, all the polynomials that shayan produces would be nice polynomials Proposed by [i]Navid Safaei[/i]

2014 Canadian Mathematical Olympiad Qualification, 1

Tags: function , algebra , polynomial , number theory , greatest common divisor

Let $f : \mathbb{Z} \rightarrow \mathbb{Z}^+$ be a function, and define $h : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}^+$ by $h(x, y) = \gcd (f(x), f(y))$. If $h(x, y)$ is a two-variable polynomial in $x$ and $y$, prove that it must be constant.

2010 Indonesia TST, 1

Tags: polynomial , relatively prime , number theory , algebra

find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \] for every natural numbers $ a $ and $ b $

PEN B Problems, 5

Tags: modular arithmetic , algebra , polynomial , vieta , symmetry , search , induction

Let $p$ be an odd prime. If $g_{1}, \cdots, g_{\phi(p-1)}$ are the primitive roots $\pmod{p}$ in the range $1<g \le p-1$, prove that \[\sum_{i=1}^{\phi(p-1)}g_{i}\equiv \mu(p-1) \pmod{p}.\]

2013 Online Math Open Problems, 30

Tags: algebra , polynomial , number theory , relatively prime , area of a triangle , geometry

Let $P(t) = t^3+27t^2+199t+432$. Suppose $a$, $b$, $c$, and $x$ are distinct positive reals such that $P(-a)=P(-b)=P(-c)=0$, and \[ \sqrt{\frac{a+b+c}{x}} = \sqrt{\frac{b+c+x}{a}} + \sqrt{\frac{c+a+x}{b}} + \sqrt{\frac{a+b+x}{c}}. \] If $x=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Evan Chen[/i]

2014 Iran MO (3rd Round), 5

Tags: ratio , algebra , polynomial , number theory

Can an infinite set of natural numbers be found, such that for all triplets $(a,b,c)$ of it we have $abc + 1 $ perfect square? (20 points )

1983 IMO Longlists, 38

Tags: polynomial , algebra

Let $\{u_n \}$ be the sequence defined by its first two terms $u_0, u_1$ and the recursion formula \[u_{n+2 }= u_n - u_{n+1}.\] [b](a)[/b] Show that $u_n$ can be written in the form $u_n = \alpha a^n + \beta b^n$, where $a, b, \alpha, \beta$ are constants independent of $n$ that have to be determined. [b](b)[/b] If $S_n = u_0 + u_1 + \cdots + u_n$, prove that $S_n + u_{n-1}$ is a constant independent of $n.$ Determine this constant.

2019 India IMO Training Camp, P1

Tags: number theory , polynomial

Determine all non-constant monic polynomials $f(x)$ with integer coefficients for which there exists a natural number $M$ such that for all $n \geq M$, $f(n)$ divides $f(2^n) - 2^{f(n)}$ [i] Proposed by Anant Mudgal [/i]

2016 India Regional Mathematical Olympiad, 3

Tags: number theory , algebra , polynomial

Find all integers $k$ such that all roots of the following polynomial are also integers: $$f(x)=x^3-(k-3)x^2-11x+(4k-8).$$

2016 Postal Coaching, 1

Tags: polynomial , algebra , combinatorics , blackboard

If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x), f(x)g(x), f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3 - 3x^2 + 5$ and $x^2 - 4x$ are written on the blackboard. Can we write a nonzero polynomial of the form $x^n - 1$ after a finite number of steps? Justify your answer.

2018 CHMMC (Fall), 4

Tags: algebra , polynomial

Find the sum of the real roots of $f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4$.

2019 Dutch IMO TST, 1

Tags: trinomial , quadratic trinomial , algebra , polynomial

Let $P(x)$ be a quadratic polynomial with two distinct real roots. For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$. Show that at least one of the roots of $P$ is negative.

2005 AIME Problems, 13

Tags: polynomial , quadratic , vieta , function , algebra

Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17$. Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2$, find the product $n_1\cdot n_2$.

1988 Balkan MO, 2

Tags: polynomial , algebra

Find all polynomials of two variables $P(x,y)$ which satisfy \[P(a,b) P(c,d) = P (ac+bd, ad+bc), \forall a,b,c,d \in \mathbb{R}.\]

2017 India IMO Training Camp, 1

Tags: algebra , polynomial

Suppose $f,g \in \mathbb{R}[x]$ are non constant polynomials. Suppose neither of $f,g$ is the square of a real polynomial but $f(g(x))$ is. Prove that $g(f(x))$ is not the square of a real polynomial.

2002 Kazakhstan National Olympiad, 6

Tags: polynomial , algebra

Find all polynomials $ P (x) $ with real coefficients that satisfy the identity $ P (x ^ 2) = P (x) P (x + 1) $.

2018 Germany Team Selection Test, 1

Tags: algebra , polynomial , 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.

KoMaL A Problems 2022/2023, A. 835

Tags: algebra , functional equation , polynomial

Let $f^{(n)}(x)$ denote the $n^{\text{th}}$ iterate of function $f$, i.e $f^{(1)}(x)=f(x)$, $f^{(n+1)}(x)=f(f^{(n)}(x))$. Let $p(n)$ be a given polynomial with integer coefficients, which maps the positive integers into the positive integers. Is it possible that the functional equation $f^{(n)}(n)=p(n)$ has exactly one solution $f$ that maps the positive integers into the positive integers? [i]Submitted by Dávid Matolcsi and Kristóf Szabó, Budapest[/i]

1998 Italy TST, 4

Tags: algebra , polynomial

Find all polynomials $P(x) = x^n +a_1x^{n-1} +...+a_n$ whose zeros (with their multiplicities) are exactly $a_1,a_2,...,a_n$.

2015 Mediterranean Mathematical Olympiad, 1

Tags: number theory , polynomial

Let $P(x)=x^4-x^3-3x^2-x+1.$ Prove that there are infinitely many positive integers $n$ such that $P(3^n)$ is not a prime.