58th Ukrainian National Mathematical Olympiad

Clearly, their sum is greater than $2$, so the prime sum has to be odd. Since all three numbers can't be odd simultaneously, since their product is $320$, then two numbers are even and one is odd. There are exactly two odd divisors of $320=2^6\cdot 5$: $1$ and $5$. Consider these cases.

$c=1$, the following is possible:

$a=2$, $b=160$, $a+b+c=163$ is prime. $a=4$, $b=80$, $a+b+c=85$ is not prime. $a=8$, $b=40$, $a+b+c=49$ is not prime. $a=16$, $b=20$, $a+b+c=37$ is prime and less than $163$. $a=32$, $b=10$, $a+b+c=43>37\Rightarrow a+b+c>37$.

$c=5$, the following is possible: $a=2$, $b=32$, $a+b+c=39$ is not prime. $a=4$, $b=16$, $a+b+c=25$ is not prime.