Found problems: 27
2012 Indonesia TST, 1
The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and
$a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$.
Prove that no term in $a_i$ is in the range $[1612, 2012]$.
2003 Poland - Second Round, 6
Each pair $(x, y)$ of nonnegative integers is assigned number $f(x, y)$ according the conditions:
$f(0, 0) = 0$;
$f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$,
$f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$ for $x, y \ge 0$.
Let $n$ be a fixed nonnegative integer and let $a$, $b$ be nonnegative integers such that $f(a, b) = n$. Decide how many numbers satisfy the equation $f(a, x) + f(b, x) = n$.