Olympiad problems

1993 Czech And Slovak Olympiad IIIA, 4

The sequence ($a_n$) of natural numbers is defined by $a_1 = 2$ and $a_{n+1}$ equals the sum of tenth powers of the decimal digits of $a_n$ for all $n \ge 1$. Are there numbers which appear twice in the sequence ($a_n$)?

2003 Junior Balkan Team Selection Tests - Romania, 2

Tags: primes , number theory , consecutive , divides , Sum of powers

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.

2012 QEDMO 11th, 6

Tags: number theory , remainder , Sum of powers

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

2019 Irish Math Olympiad, 6

Tags: number theory , Sum of powers , consecutive

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!)

1995 Singapore MO Open, 5

Tags: Sum of powers , inequalities , algebra

Let $a, b, c, d$ be four positive real numbers. Prove that $$a^{10} + b^{10}+c^{10} + d^{10} \ge (0.1a + 0.2b + 0.3c + 0.4d)^{10} + (0.4a + 0.3b + 0.2c + 0.ld)^{10} + (0.2a + 0.4b + 0.1c + 0.3d)^{10} + (0.3a + 0.1b + 0.4c + 0.2d)^{10}$$

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)

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.

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)

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.

2009 JBMO Shortlist, 4

Tags: JBMO , number theory , Sum of powers

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 $2p_1 + p_9$.

2012 IFYM, Sozopol, 5

Tags: number theory , Sequences , Sum of powers

We are given the following sequence: $a_1=8,a_2=20,a_{n+2}=a_{n+1}^2+12a_n a_{n+1}+11a_n$. Prove that none of the members of the sequence can be presented as a sum of three seventh powers of natural numbers.

2023 Cono Sur Olympiad, 1

Tags: inequalities , Sum of powers , cono sur

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.

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 $n$.

2005 iTest, 39

Tags: number theory , Sum of powers

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?

1998 Abels Math Contest (Norwegian MO), 3

Tags: Sum , number theory , Sum of powers

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 $n^2$.

1995 Romania Team Selection Test, 1

Tags: Sum , Sum of powers , inequalities , algebra

Let $a_1, a_2,...., a_n$ be distinct positive integers. Prove that $(a_1^5 + ...+ a_n^5) + (a_1^7 + ...+ a_n^7) \ge 2(a_1^3 + ...+ a_n^3)^2$ and find the cases of equality.

1999 Tournament Of Towns, 2

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

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)

1970 Swedish Mathematical Competition, 1

Tags: number theory , Sum of powers

Show that infinitely many positive integers cannot be written as a sum of three fourth powers of integers.

2019 Lusophon Mathematical Olympiad, 5

Tags: number theory , Sum of powers

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 $2018 = A^5 + B^5 + C^5 + D^5$

1996 Swedish Mathematical Competition, 5

Tags: number theory , power of 2 , Sum of powers

Let $n \ge 1$. Prove that it is possible to select some of the integers $1,2,...,2^n$ so that for each $p = 0,1,...,n - 1$ the sum of the $p$-th powers of the selected numbers is equal to the sum of the $p$-th powers of the remaining numbers.

2001 Regional Competition For Advanced Students, 1

Tags: number theory , Digit , Sum of powers , Sum , powers

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?

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) $1998$.

2013 Balkan MO Shortlist, N4

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

Let $p$ be a prime number greater than $3$. Prove that the sum $1^{p+2} + 2^{p+2} + ...+ (p-1)^{p+2}$ is divisible by $p^2$.

2008 Korea Junior Math Olympiad, 6

Tags: number theory , Divisors , Sum of powers , Sum , divisible

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 $8$.

2005 Estonia National Olympiad, 2

Tags: divisible , divides , number theory , Sum of powers

Let $a, b$, and $n$ be integers such that $a + b$ is divisible by $n$ and $a^2 + b^2$ is divisible by $n^2$. Prove that $a^m + b^m$ is divisible by $n^m$ for all positive integers $m$.