Olympiad problems

2014 Postal Coaching, 5

Determine all polynomials $f$ with integer coefficients with the property that for any two distinct primes $p$ and $q$, $f(p)$ and $f(q)$ are relatively prime.

2009 China National Olympiad, 1

Tags: algebra , symmetry , polynomial , absolute value , inequalities , function

Given an integer $ n > 3.$ Let $ a_{1},a_{2},\cdots,a_{n}$ be real numbers satisfying $ min |a_{i} \minus{} a_{j}| \equal{} 1, 1\le i\le j\le n.$ Find the minimum value of $ \sum_{k \equal{} 1}^n|a_{k}|^3.$

2025 India STEMS Category C, 5

Tags: polynomial , algebra

Let $P \in \mathbb{R}[x]$. Suppose that the multiset of real roots (where roots are counted with multiplicity) of $P(x)-x$ and $P^3(x)-x$ are distinct. Prove that for all $n\in \mathbb{N}$, $P^n(x)-x$ has at least $\sigma(n)-2$ distinct real roots. (Here $P^n(x):=P(P^{n-1}(x))$ with $P^1(x) = P(x)$, and $\sigma(n)$ is the sum of all positive divisors of $n$). [i]Proposed by Malay Mahajan[/i]

2010 Princeton University Math Competition, 8

Tags: polynomial , algebra

Let $p$ be a polynomial with integer coefficients such that $p(15)=6$, $p(22)=1196$, and $p(35)=26$. Find an integer $n$ such that $p(n)=n+82$.

1996 Moldova Team Selection Test, 5

Tags: algebra , polynomial

Find all polynomials $P(X)$ of fourth degree with real coefficients that verify the properties: [b]a)[/b] $P(-x)=P(x), \forall x\in\mathbb{R};$ [b]b)[/b] $P(x)\geq0, \forall x\in\mathbb{R};$ [b]c)[/b] $P(0)=1;$ [b]d)[/b] $P(X)$ has exactly two local minimums $x_1$ and $x_2$ such that $|x_1-x_2|=2.$

2011 Putnam, B6

Tags: polynomial , algebra

Let $p$ be an odd prime. Show that for at least $(p+1)/2$ values of $n$ in $\{0,1,2,\dots,p-1\},$ \[\sum_{k=0}^{p-1}k!n^k \quad \text{is not divisible by }p.\]

1988 USAMO, 5

Tags: polynomial , algebra

A polynomial product of the form \[(1-z)^{b_1}(1-z^2)^{b_2}(1-z^3)^{b_3}(1-z^4)^{b_4}(1-z^5)^{b_5}\cdots(1-z^{32})^{b_{32}},\] where the $b_k$ are positive integers, has the surprising property that if we multiply it out and discard all terms involving $z$ to a power larger than $32$, what is left is just $1-2z$. Determine, with proof, $b_{32}$.

2011 China Second Round Olympiad, 2

Tags: polynomial , number theory , geometric transformation , algebra , geometry

For any integer $n\ge 4$, prove that there exists a $n$-degree polynomial $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ satisfying the two following properties: [b](1)[/b] $a_i$ is a positive integer for any $i=0,1,\ldots,n-1$, and [b](2)[/b] For any two positive integers $m$ and $k$ ($k\ge 2$) there exist distinct positive integers $r_1,r_2,...,r_k$, such that $f(m)\ne\prod_{i=1}^{k}f(r_i)$.

1979 Polish MO Finals, 6

Tags: polynomial , algebra

A polynomial $w$ of degree $n > 1$ has $n$ distinct zeros $x_1,x_2,...,x_n$. Prove that: $$\frac{1}{w'(x_1)}+\frac{1}{w'(x_2)}+...···+\frac{1}{w'(x_n)}= 0.$$

PEN K Problems, 6

Tags: functional equation , polynomial , function , algebra

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f^{(19)}(n)+97f(n)=98n+232.\]

1988 Czech And Slovak Olympiad IIIA, 5

Tags: algebra , polynomial

Find all numbers $a \in (-2, 2)$ for which the polynomial $x^{154}-ax^{77}+1$ is a multiple of the polynomial $x^{14}-ax^{7}+1$.

1981 Romania Team Selection Tests, 1.

Tags: polynomial , algebra

Consider the polynomial $P(X)=X^{p-1}+X^{p-2}+\ldots+X+1$, where $p>2$ is a prime number. Show that if $n$ is an even number, then the polynomial \[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\] is divisible by $X^2+1$. [i]Mircea Becheanu[/i]

2003 Iran MO (3rd Round), 29

Tags: algebra , polynomial , induction

Let $ c\in\mathbb C$ and $ A_c \equal{} \{p\in \mathbb C[z]|p(z^2 \plus{} c) \equal{} p(z)^2 \plus{} c\}$. a) Prove that for each $ c\in C$, $ A_c$ is infinite. b) Prove that if $ p\in A_1$, and $ p(z_0) \equal{} 0$, then $ |z_0| < 1.7$. c) Prove that each element of $ A_c$ is odd or even. Let $ f_c \equal{} z^2 \plus{} c\in \mathbb C[z]$. We see easily that $ B_c: \equal{} \{z,f_c(z),f_c(f_c(z)),\dots\}$ is a subset of $ A_c$. Prove that in the following cases $ A_c \equal{} B_c$. d) $ |c| > 2$. e) $ c\in \mathbb Q\backslash\mathbb Z$. f) $ c$ is a non-algebraic number g) $ c$ is a real number and $ c\not\in [ \minus{} 2,\frac14]$.

2023 IMC, 3

Tags: polynomial , algebra , functional equation

Find all polynomials $P$ in two variables with real coefficients satisfying the identity $$P(x,y)P(z,t)=P(xz-yt,xt+yz).$$

2013 Hanoi Open Mathematics Competitions, 3

Tags: floor function , polynomial , algebra

What is the largest integer not exceeding $8x^3 +6x - 1$, where $x =\frac12 \left(\sqrt[3]{2+\sqrt5} + \sqrt[3]{2-\sqrt5}\right)$ ? (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

2001 India IMO Training Camp, 2

Tags: polynomial , algebra , function

Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.

2018 Brazil Team Selection Test, 5

Tags: polynomial , number theory , algebra

Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.

1953 AMC 12/AHSME, 4

Tags: polynomial , algebra

The roots of $ x(x^2\plus{}8x\plus{}16)(4\minus{}x)\equal{}0$ are: $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 0,4 \qquad\textbf{(C)}\ 0,4,\minus{}4 \qquad\textbf{(D)}\ 0,4,\minus{}4,\minus{}4 \qquad\textbf{(E)}\ \text{none of these}$

2011 All-Russian Olympiad, 3

Tags: polynomial , number theory , integration , modular arithmetic , algebra , calculus

Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that exist infinitely many positive integers $a, b, c$ such that $P(a)=P(b)=P(c)$. [i]A. Golovanov[/i]

2013 ISI Entrance Examination, 6

Tags: algebra , polynomial , limit

Let $p(x)$ and $q(x)$ be two polynomials, both of which have their sum of coefficients equal to $s.$ Let $p,q$ satisfy $p(x)^3-q(x)^3=p(x^3)-q(x^3).$ Show that (i) There exists an integer $a\geq1$ and a polynomial $r(x)$ with $r(1)\neq0$ such that \[p(x)-q(x)=(x-1)^ar(x).\] (ii) Show that $s^2=3^{a-1},$ where $a$ is described as above.

2003 SNSB Admission, 1

Tags: polynomial , algebra

Does exist polynoms of one variable that are irreducible over the field of integers, have degree $ 60 $ and have multiples of the form $ X^n-1? $ If so, how many of them?

2013 Romania National Olympiad, 4

Tags: group theory , abstract algebra , superior algebra , polynomial , algebra

Given $n\ge 2$ a natural number, $(K,+,\cdot )$ a body with commutative property that $\underbrace{1+...+}_{m}1\ne 0,m=2,...,n,f\in K[X]$ a polynomial of degree $n$ and $G$ a subgroup of the additive group $(K,+,\cdot )$, $G\ne K.$Show that there is $a\in K$ so$f(a)\notin G$.

2013 VJIMC, Problem 3

Tags: polynomial , number theory , algebra

Prove that there is no polynomial $P$ with integer coefficients such that $P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$.

2004 All-Russian Olympiad, 3

Tags: polynomial , algebra , calculus

The polynomials $ P(x)$ and $ Q(x)$ are given. It is known that for a certain polynomial $ R(x, y)$ the identity $ P(x) \minus{} P(y) \equal{} R(x, y) (Q(x) \minus{} Q(y))$ applies. Prove that there is a polynomial $ S(x)$ so that $ P(x) \equal{} S(Q(x)) \quad \forall x.$

1947 Moscow Mathematical Olympiad, 123

Tags: polynomial , remainder , algebra

Find the remainder after division of the polynomial $x+x^3 +x^9 +x^{27} +x^{81} +x^{243}$ by $x-1$.