Školjka

%V0 A positive integer $N$ is called balanced, if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P\!\left(x\right) = \left(x+a\right)\left(x+b\right)$. a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P\!\left(1\right)$, $P\!\left(2\right)$, ..., $P\!\left(50\right)$ are balanced. b) Prove that if $P\!\left(n\right)$ is balanced for all positive integers $n$, then $a=b$. Proposed by Jorge Tipe, Peru