Olympiad problems

2019 Lusophon Mathematical Olympiad, 5

a) Show that there are five integers $A, B, C, D$, and $E$ such that $2018 = A^5 + B^5 + C^5 + D^5 + E^5$ b) Show that there are no four integers $A, B, C$ and $D$ such that $2018 = A^5 + B^5 + C^5 + D^5$

2007 Thailand Mathematical Olympiad, 7

Tags: algebra , sum of powers

Let $a, b, c$ be complex numbers such that $a+b+c = 1$, $a^2+b^2+c^2 = 2$ and $a^3+b^3+c^3 = 3$. Find the value of $a^4 + b^4 + c^4$.

1985 Spain Mathematical Olympiad, 4

Tags: number theory , positive integer , prime , sum of powers

Prove that for each positive integer $k $ there exists a triple $(a,b,c)$ of positive integers such that $abc = k(a+b+c)$. In all such cases prove that $a^3+b^3+c^3$ is not a prime.

2000 Tournament Of Towns, 3

Tags: inequalities , algebra , sum of powers

Prove the inequality $$ 1^k+2^k+...+n^k \le \frac{n^{2k}-(n-1)^k}{n^k-(n-1)^k}$$ (L Emelianov)

2018 Greece JBMO TST, 3

Tags: algebra , combinatorics , maximum , sum , sum of powers

$12$ friends play a tennis tournament, where each plays only one game with any of the other eleven. Winner gets one points. Loser getos zero points, and there is no draw. Final points of the participants are $B_1, B_2, ..., B_{12}$. Find the largest possible value of the sum $\Sigma_3=B_1^3+B_2^3+ ... + B_{12}^3$ .

1997 Czech and Slovak Match, 5

Tags: number theory , integer , sum of powers

The sum of several integers (not necessarily distinct) equals $1492$. Decide whether the sum of their seventh powers can equal (a) $1996$; (b) $1998$.

2018 China Team Selection Test, 4

Tags: modular arithmetic , arithmetic sequence , number theory , primitive root , sum of powers

Let $p$ be a prime and $k$ be a positive integer. Set $S$ contains all positive integers $a$ satisfying $1\le a \le p-1$, and there exists positive integer $x$ such that $x^k\equiv a \pmod p$. Suppose that $3\le |S| \le p-2$. Prove that the elements of $S$, when arranged in increasing order, does not form an arithmetic progression.

2010 Bundeswettbewerb Mathematik, 4

Tags: power of 2 , sum of powers , combinatorics

Find all numbers that can be expressed in exactly $2010$ different ways as the sum of powers of two with non-negative exponents, each power appearing as a summand at most three times. A sum can also be made from just one summand.

1990 Nordic, 1

Tags: number theory , sum of powers

Let $m, n,$ and $p$ be odd positive integers. Prove that the number $\sum\limits_{k=1}^{{{(n-1)}^{p}}}{{{k}^{m}}}$ is divisible by $n$

1989 All Soviet Union Mathematical Olympiad, 499

Tags: irrational , sum , sum of powers , rational , algebra

Do there exist two reals whose sum is rational, but the sum of their $n$ th powers is irrational for all $n > 1$? Do there exist two reals whose sum is irrational, but the sum of whose $n$ th powers is rational for all $n > 1$?

2012 QEDMO 11th, 6

Tags: number theory , remainder , sum of powers

Let $p$ be an odd prime number. Prove that $$1^{p-1} + 2^{p-1} +...+ (p-1)^{p-1} \equiv p + (p-1)! \mod p^2$$

1996 Nordic, 2

Tags: number theory , algebra , integer , sum of powers

Determine all real numbers $x$, such that $x^n+x^{-n}$ is an integer for all integers $n$.

2004 Thailand Mathematical Olympiad, 17

Tags: number theory , remainder , sum of powers

Compute the remainder when $1^{2547} + 2^{2547} +...+ 2547^{2547}$ is divided by $25$.

2014 Saudi Arabia GMO TST, 2

Tags: sum of powers , number theory , divide , divisible

Let $p$ be a prime number. Prove that there exist infinitely many positive integers $n$ such that $p$ divides $1^n + 2^n +... + (p + 1)^n.$

2014 IMAC Arhimede, 4

Tags: polynomial , algebra , sum of powers , sum , rational

Let $n$ be a natural number and let $P (t) = 1 + t + t^2 + ... + t^{2n}$. If $x \in R$ such that $P (x)$ and $P (x^2)$ are rational numbers, prove that $x$ is rational number.

2018 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory , coprime , coprime numbers , sum of powers , divide

Determine all the triples $\{a, b, c \}$ of positive integers coprime (not necessarily pairwise prime) such that $a + b + c$ simultaneously divides the three numbers $a^{12} + b^{12}+ c^{12}$, $ a^{23} + b^{23} + c^{23} $ and $ a^{11004} + b^{11004} + c^{11004}$