Recently, I came across this exercise:
Suppose that $a$ and $b$ are odd numbers. Prove that only for finitely many positive integers $j$ does $2^j$ divide $a^j+b^j$.
I tried to solve it using basic mathematics (i.e. congruences modulo powers of $2$), but I could not prove the statement. Considering that it comes from some high school math competition, I think it should be solvable with very little mathematical background. What am I missing?