Olympiad problems

2021 China Girls Math Olympiad, 4

Call a sequence of positive integers $(a_n)_{n \ge 1}$ a "CGMO sequence" if $(a_n)_{n \ge 1}$ strictly increases, and for all integers $n \ge 2022$, $a_n$ is the smallest integer such that there exists a non-empty subset of $\{a_{1}, a_{2}, \cdots, a_{n-1} \}$ $A_n$ where $a_n \cdot \prod\limits_{a \in A_n} a$ is a perfect square. Proof: there exists $c_1, c_2 \in \mathbb{R}^{+}$ s.t. for any "CGMO sequence" $(a_n)_{n \ge 1}$ , there is a positive integer $N$ that satisfies any $n \ge N$, $c_1 \cdot n^2 \le a_n \le c_2 \cdot n^2$.

2005 Alexandru Myller, 3

Tags: infinite sequence , sequence , algebra

[b]a)[/b] Find the number of infinite sequences of integers $ \left( a_n \right)_{n\ge 1} $ that have the property that $ a_na_{n+2}a_{n+3}=-1, $ for any natural number $ n. $ [b]b)[/b] Prove that there is no infinite sequence of integers $ \left( b_n \right)_{n\ge 1} $ that have the property that $ b_nb_{n+2}b_{n+3}=2005, $ for any natural number $ n. $

2023 Indonesia TST, N

Tags: gcd , integer sequence , infinite sequence , coprime , number theory

Given an integer $a>1$. Prove that there exists a sequence of positive integers \[ n_1, n_2, n_3, \ldots \] Such that \[ \gcd(a^{n_i+1} + a^{n_i} - 1, \ a^{n_j + 1} + a^{n_j} - 1) =1 \] For every $i \neq j$.

2019 Philippine TST, 3

Tags: divisor , bounding , infinite sequence , number theory

Let $a_1, a_2, a_3,\ldots$ be an infinite sequence of positive integers such that $a_2 \ne 2a_1$, and for all positive integers $m$ and $n$, the sum $m + n$ is a divisor of $a_m + a_n$. Prove that there exists an integer $M$ such that for all $n > M$, we have $a_n \ge n^3$.

2025 Junior Macedonian Mathematical Olympiad, 3

Tags: algorithm , prime number , infinite sequence , number theory

Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.

2023 Indonesia TST, N

Tags: number theory , gcd , integer sequence , infinite sequence , coprime

Given an integer $a>1$. Prove that there exists a sequence of positive integers \[ n_1, n_2, n_3, \ldots \] Such that \[ \gcd(a^{n_i+1} + a^{n_i} - 1, \ a^{n_j + 1} + a^{n_j} - 1) =1 \] For every $i \neq j$.