Ukrainian National Mathematical Olympiad

It follows from the problem statement that $\frac{ab}{c}$ is a positive integer. Then there should exist positive integers $m,n,x,y$ such that $c=mn$, $a=mx$, $b=ny$. This implies that $d=\frac{ab}{c}=xy$, and so $a+b+c+d=mx+ny+mn+xy=(n+x)(m+y)$, which is, obviously, not prime.