Question 3
Find the greatest real number $\alpha$ for which there exists an infinite sequence of natural numbers $(a_n)_{n\ge1}$ satisfying both:
1. $a_n>1997^n$ for every positive integer $n$.
2. For every $n\ge2$, $u_n\ge a_n^\alpha$, where $u_n$ is the greatest common divisor of all numbers $a_i+a_k$ with positive integers $i,k$ satisfying $i+k=n$.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
1. $a_n>1997^n$ for every positive integer $n$.
2. For every $n\ge2$, $u_n\ge a_n^\alpha$, where $u_n$ is the greatest common divisor of all numbers $a_i+a_k$ with positive integers $i,k$ satisfying $i+k=n$.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.