Olympiad problems

2015 Miklos Schweitzer, 5

Tags: algebra , polynomial

Let $f(x) = x^n+x^{n-1}+\dots+x+1$ for an integer $n\ge 1.$ For which $n$ are there polynomials $g, h$ with real coefficients and degrees smaller than $n$ such that $f(x) = g(h(x)).$

2006 China Team Selection Test, 3

Tags: polynomial , linear algebra , vector , number theory , greatest common divisor , algebra , modular arithmetic

Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is: \[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\] Find the least possible value of $n$.

2001 Taiwan National Olympiad, 5

Tags: polynomial , induction , number theory , function , algebra

Let $f(n)=\sum_{k=0}^{n-1}x^ky^{n-1-k}$ with, $x$, $y$ real numbers. If $f(n)$, $f(n+1)$, $f(n+2)$, $f(n+3)$, are integers for some $n$, prove $f(n)$ is integer for all $n$.

2006 Iran MO (3rd Round), 3

Tags: polynomial , algebra

Find all real $x,y,z$ that \[\left\{\begin{array}{c}x+y+zx=\frac12\\ \\ y+z+xy=\frac12\\ \\ z+x+yz=\frac12\end{array}\right.\]

IMSC 2023, 6

Tags: algebra , polynomial

Find all polynomials $P(x)$ with integer coefficients, such that for all positive integers $m, n$, $$m+n \mid P^{(m)}(n)-P^{(n)}(m).$$ [i]Proposed by Navid Safaei, Iran[/i]

2017 Bulgaria National Olympiad, 5

Tags: polynomial , algebra , polynomial equation

Let $n$ be a natural number and $f(x)$ be a polynomial with real coefficients having $n$ different positive real roots. Is it possible the polynomial: $$x(x+1)(x+2)(x+4)f(x)+a$$ to be presented as the $k$-th power of a polynomial with real coefficients, for some natural $k\geq 2$ and real $a$?

2015 Mediterranean Mathematical Olympiad, 1

Tags: polynomial , number theory

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.

1985 IMO Longlists, 88

Tags: algebra , binomial theorem , polynomial , calculus

Determine the range of $w(w + x)(w + y)(w + z)$, where $x, y, z$, and $w$ are real numbers such that \[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\]

2002 USAMO, 3

Tags: polynomial , lagrange s interpolation

Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots.

2016 Serbia Additional Team Selection Test, 1

Tags: polynomial , algebra

Let $P_0(x)=x^3-4x$. Sequence of polynomials is defined as following:\\ $P_{n+1}=P_n(1+x)P_n(1-x)-1$.\\ Prove that $x^{2016}|P_{2016}(x)$.

2009 USA Team Selection Test, 8

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

Fix a prime number $ p > 5$. Let $ a,b,c$ be integers no two of which have their difference divisible by $ p$. Let $ i,j,k$ be nonnegative integers such that $ i \plus{} j \plus{} k$ is divisible by $ p \minus{} 1$. Suppose that for all integers $ x$, the quantity \[ (x \minus{} a)(x \minus{} b)(x \minus{} c)[(x \minus{} a)^i(x \minus{} b)^j(x \minus{} c)^k \minus{} 1]\] is divisible by $ p$. Prove that each of $ i,j,k$ must be divisible by $ p \minus{} 1$. [i]Kiran Kedlaya and Peter Shor.[/i]

2017 Moldova Team Selection Test, 2

Tags: polynomial , algebra

Let $$f(X)=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}$$be a polynomial with real coefficients which satisfies $$a_{n}\geq a_{n-1}\geq \cdots \geq a_{1}\geq a_{0}>0.$$Prove that for every complex root $z$ of this polynomial, we have $|z|\leq 1$.

2001 Mongolian Mathematical Olympiad, Problem 1

Tags: polynomial , functional equation

Prove that for every positive integer $n$ there exists a polynomial $p(x)$ of degree $n$ with real coefficients, having $n$ distinct real roots and satisfying $$p(x)p(4-x)=p(x(4-x))$$

2006 Cuba MO, 1

Tags: algebra , polynomial

Determine all monic polynomials $P(x)$ of degree $3$ with coefficients integers, which are divisible by $x-1$, when divided by $ x-5$ leave the same remainder as when divided by$ x+5$ and have a root between $2$ and $3$.

1986 IMO Shortlist, 7

Tags: calculus , polynomial , algebra , system of equations

Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations: \[\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.\] Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$

2014 Romania Team Selection Test, 2

Tags: algebra , polynomial

Let $p$ be an[color=#FF0000] odd [/color]prime number. Determine all pairs of polynomials $f$ and $g$ from $\mathbb{Z}[X]$ such that \[f(g(X))=\sum_{k=0}^{p-1} X^k = \Phi_p(X).\]

1997 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: polynomial , algebra

Find all positive integers $n$ with the following property: there exists a polynomial $P_n(x)$ of degree $n$, with integer coefficients, such that $P_n(0)=0$ and $P_n(x)=n$ for $n$ distinct integer solutions.

2010 Indonesia TST, 1

Tags: polynomial , number theory , algebra , relatively prime

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 $

2014 Dutch IMO TST, 5

Tags: polynomial , arithmetic sequence , integration , ratio , calculus , algebra

Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$

2023 LMT Fall, 11

Tags: polynomial , algebra

Find the number of degree $8$ polynomials $f (x)$ with nonnegative integer coefficients satisfying both $f (1) = 16$ and $f (-1) = 8$.

2013 China National Olympiad, 2

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

For any positive integer $n$ and $0 \leqslant i \leqslant n$, denote $C_n^i \equiv c(n,i)\pmod{2}$, where $c(n,i) \in \left\{ {0,1} \right\}$. Define \[f(n,q) = \sum\limits_{i = 0}^n {c(n,i){q^i}}\] where $m,n,q$ are positive integers and $q + 1 \ne {2^\alpha }$ for any $\alpha \in \mathbb N$. Prove that if $f(m,q)\left| {f(n,q)} \right.$, then $f(m,r)\left| {f(n,r)} \right.$ for any positive integer $r$.

2016 Taiwan TST Round 3, 1

Tags: number theory , polynomial , algebra

Let $n$ be a positive integer. Find the number of odd coefficients of the polynomial $(x^2-x+1)^n$.

The Golden Digits 2024, P3

Tags: number theory , polynomial

Let $p$ be a prime number and $\mathcal{A}$ be a finite set of integers, with at least $p^k$ elements. Denote by $N_{\text{even}}$ the number of subsets of $\mathcal{A}$ with even cardinality and sum of elements divisible by $p^k$. Define $N_{\text{odd}}$ similarly. Prove that $N_{\text{even}}\equiv N_{\text{odd}}\bmod{p}.$

2023 Germany Team Selection Test, 1

Tags: algebra , polynomial

Let $P$ be a polynomial with integer coefficients. Assume that there exists a positive integer $n$ with $P(n^2)=2022$. Prove that there cannot be a positive rational number $r$ with $P(r^2)=2024$.

2014 Indonesia MO, 4

Tags: polynomial , calculus , algebra , integration

Determine all polynomials with integral coefficients $P(x)$ such that if $a,b,c$ are the sides of a right-angled triangle, then $P(a), P(b), P(c)$ are also the sides of a right-angled triangle. (Sides of a triangle are necessarily positive. Note that it's not necessary for the order of sides to be preserved; if $c$ is the hypotenuse of the first triangle, it's not necessary that $P(c)$ is the hypotenuse of the second triangle, and similar with the others.)