Olympiad problems

2024 Baltic Way, 18

An infinite sequence $a_1, a_2,\ldots$ of positive integers is such that $a_n \geq 2$ and $a_{n+2}$ divides $a_{n+1} + a_n$ for all $n \geq 1$. Prove that there exists a prime which divides infinitely many terms of the sequence.

divisor bound sequence number theory prime number prime numbers sequences

2022 Azerbaijan Junior National Olympiad, C4

There is a $8*8$ board and the numbers $1,2,3,4,...,63,64$. In all the unit squares of the board, these numbers are places such that only $1$ numbers goes to only one unit square. Prove that there is atleast $4$ $2*2$ squares such that the sum of the numbers in $2*2$ is greater than $100$.

bound combinatorics

2024 Baltic Way, 17

Do there exist infinitely many quadruples $(a,b,c,d)$ of positive integers such that the number $a^{a!} + b^{b!} - c^{c!} - d^{d!}$ is prime and $2 \leq d \leq c \leq b \leq a \leq d^{2024}$?

bound fermat's little theorem number theory divisibility prime number Fermat's little theorem prime numbers

2023 Bulgaria EGMO TST, 5

The positive integers $x_1$, $x_2$, $\ldots$, $x_5$, $x_6 = 144$ and $x_7$ are such that $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n=1,2,3,4$. Determine the value of $x_7$.

bound equation calculate algebra number theory