Olympiad problems

1997 Czech and Slovak Match, 5

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

2023 Cono Sur Olympiad, 1

Tags: sum of powers , inequalities

A list of $n$ positive integers $a_1, a_2,a_3,\ldots,a_n$ is said to be [i]good[/i] if it checks simultaneously: $\bullet a_1<a_2<a_3<\cdots<a_n,$ $\bullet a_1+a_2^2+a_3^3+\cdots+a_n^n\le 2023.$ For each $n\ge 1$, determine how many [i]good[/i] lists of $n$ numbers exist.

2018 Greece JBMO TST, 3

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

$12$ 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 $\Sigma_3=B_1^3+B_2^3+ ... + B_{12}^3$ .

2005 Estonia National Olympiad, 2

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

Let $a, b$ and $c$ be arbitrary integers. Prove that $a^2 + b^2 + c^2$ is divisible by $7$ when $a^4 + b^4 + c^4$ divisible by $7$.

2009 JBMO Shortlist, 4

Tags: sum of powers , number theory

Determine all prime numbers $p_1, p_2,..., p_{12}, p_{13}, p_1 \le p_2 \le ... \le p_{12} \le p_{13}$, such that $p_1^2+ p_2^2+ ... + p_{12}^2 = p_{13}^2$ and one of them is equal to $2p_1 + p_9$.

2001 Regional Competition For Advanced Students, 1

Tags: digit , power , sum of powers , sum , number theory

Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?

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 $n > 1$?

2008 Korea Junior Math Olympiad, 6

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

If $d_1,d_2,...,d_k$ are all distinct positive divisors of $n$, we define $f_s(n) = d_1^s+d_2^s+..+d_k^s$. For example, we have $f_1(3) = 1 + 3 = 4, f_2(4) = 1 + 2^2 + 4^2 = 21$. Prove that for all positive integers $n$, $n^3f_1(n) - 2nf_9(n) + n^2f_3(n)$ is divisible by $8$.

2018 Rioplatense Mathematical Olympiad, Level 3, 3

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

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}$

2009 Thailand Mathematical Olympiad, 7

Tags: sum of powers , power , number theory

Let $a, b, c$ be real numbers, and define $S_n = a^n + b^n + c^n$ for positive integers $n$. Suppose that $S_1, S_2, S_3$ are integers satisfying $6 | 5S_1 - 3S_2 - 2S_3$. Show that $S_n$ is an integer for all positive integers $n$.

2000 Tournament Of Towns, 3

Tags: sum of powers , inequalities , algebra

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

2014 IMAC Arhimede, 4

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

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.

1990 Nordic, 1

Tags: sum of powers , number theory

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 $n$

2003 Junior Balkan Team Selection Tests - Romania, 2

Tags: prime , consecutive , divide , sum of powers , number theory

Consider the prime numbers $n_1< n_2 <...< n_{31}$. Prove that if $30$ divides $n_1^4 + n_2^4+...+n_{31}^4$, then among these numbers one can find three consecutive primes.

2005 iTest, 39

Tags: sum of powers , number theory

What is the smallest positive integer that when raised to the $6^{th}$ power, it can be represented by a sum of the $6^{th}$ powers of distinct smaller positive integers?

1996 Nordic, 2

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

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

2019 Irish Math Olympiad, 6

Tags: consecutive , sum of powers , number theory

The number $2019$ has the following nice properties: (a) It is the sum of the fourth powers of fuve distinct positive integers. (b) It is the sum of six consecutive positive integers. In fact, $2019 = 1^4 + 2^4 + 3^4 + 5^4 + 6^4$ (1) $2019 = 334 + 335 + 336 + 337 + 338 + 339$ (2) Prove that $2019$ is the smallest number that satises [b]both [/b] (a) and (b). (You may assume that (1) and (2) are correct!)

1998 Abels Math Contest (Norwegian MO), 3

Tags: sum , sum of powers , number theory

Let $n$ be a positive integer. (a) Prove that $1^5 +3^5 +5^5 +...+(2n-1)^5$ is divisible by $n$. (b) Prove that $1^3 +3^3 +5^3 +...+(2n-1)^3$ is divisible by $n^2$.

1996 Estonia National Olympiad, 4

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

Prove that, for each odd integer $n \ge 5$, the number $1^n+2^n+...+15^n$ is divisible by $480$.

2019 Tournament Of Towns, 2

Tags: number theory , sum of powers , divisibility

Consider two positive integers $a$ and $b$ such that $a^{n+1} + b^{n+1}$ is divisible by $a^n + b^n$ for infinitely many positive integers $n$. Is it necessarily true that $a = b$? (Boris Frenkin)

1999 Tournament Of Towns, 2

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

Let $d = a^{1999} + b^{1999} + c^{1999}$ , where $a, b$ and $c$ are integers such that $a + b + c = 0$. (a) May it happen that $d = 2$? (b) May it happen that $d$ is prime? (V Senderov)

2002 Junior Balkan Team Selection Tests - Romania, 1

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

Let $n$ be an even positive integer and let $a, b$ be two relatively prime positive integers. Find $a$ and $b$ such that $a + b$ is a divisor of $a^n + b^n$.

2018 China Team Selection Test, 4

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

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.

1985 Spain Mathematical Olympiad, 4

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

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.

2001 Estonia National Olympiad, 4

Tags: composite , sum of powers , prime , number theory

Prove that for any integer $a > 1$ there is a prime $p$ for which $1+a+a^2+...+ a^{p-1}$ is composite.