NL-BXMO-TST 2019

Prove that for each positive integer $n$ there are at most two pairs $(a, b)$ of positive integers with following two properties: (i) $a^2 + b = n$ , (ii) $a+b$ is a power of two, i.e. there is an integer $k \ge 0$ such that $a+b = 2^k$ .

Let $\Delta ABC$ be a triangle with an inscribed circle centered at $I$ . The line perpendicular to $AI$ at $I$ intersects $\odot (ABC)$ at $P,Q$ such that, $P$ lies closer to $B$ than $C$ . Let $\odot (BIP) \cap \odot (CIQ) =S$ . Prove that, $SI$ is the angle bisector of $\angle PSQ$

Let $x$ and $y$ be positive real numbers.

1. Prove: if $x^3 - y^3 \ge 4x$ , then $x^2 > 2y$ .
2. Prove: if $x^5 - y^3 \ge 2x$ , then $x^3 \ge 2y$ .

Do there exist a positive integer $k$ and a non-constant sequence $a_1, a_2, a_3, ...$ of positive integers such that $a_n = gcd(a_{n+k}, a_{n+k+1})$ for all positive integers $n$ ?