Olympiad problems

2014 Contests, 3

The sequence $(a_n)$ is defined with the recursion $a_{n + 1} = 5a^6_n + 3a^3_{n-1} + a^2_{n-2}$ for $n\ge 2$ and the set of initial values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$. (That is, the initial values are these three numbers in any order.) Show that the sequence contains no sixth power of a natural number.

2017 Federal Competition For Advanced Students, P2, 3

Tags: number theory , rational , recurrence relation , number theory with sequences

Let $(a_n)_{n\ge 0}$ be the sequence of rational numbers with $a_0 = 2016$ and $a_{n+1} = a_n + \frac{2}{a_n}$ for all $n \ge 0$. Show that the sequence does not contain a square of a rational number. Proposed by Theresia Eisenkölbl

2015 Balkan MO Shortlist, N2

Tags: recurrence relation , number theory with sequences , sequence , divide , number theory , algebra

Sequence $(a_n)_{n\geq 0}$ is defined as $a_{0}=0, a_1=1, a_2=2, a_3=6$, and $ a_{n+4}=2a_{n+3}+a_{n+2}-2a_{n+1}-a_n, n\geq 0$. Prove that $n^2$ divides $a_n$ for infinite $n$. (Romania)

2017 Iran MO (3rd round), 2

Tags: number theory , number theory with sequences , sequence

Consider a sequence $\{a_i\}^\infty_{i\ge1}$ of positive integers. For all positvie integers $n$ prove that there exists infinitely many positive integers $k$ such that there is no pair $(m,t)$ of positive integers where $m>n$ and $$kn+a_n=tm(m+1)+a_m$$

2007 Korea Junior Math Olympiad, 1

Tags: number theory , number theory with sequences , integer sequence , multiple , sequence

A sequence $a_1,a_2,...,a_{2007}$ where $a_i \in\{2,3\}$ for $i = 1,2,...,2007$ and an integer sequence $x_1,x_2,...,x_{2007}$ satisfies the following: $a_ix_i + x_{i+2 }\equiv 0$ ($mod 5$) , where the indices are taken modulo $2007$. Prove that $x_1,x_2,...,x_{2007}$ are all multiples of $5$.