58th Ukrainian National Mathematical Olympiad

Clearly, their sum is greater than $2$, thus their sum is odd. Since all three numbers cannot be odd (since their product is $80$), then two of the numbers are even and one is odd. There are only two odd divisors of $80=2^4\cdot 5$: $1$ and $5$. Consider these cases.

$c=1$, the following is possible:

$a=2$, $b=40$, $a+b+c=43$ is prime. $a=4$, $b=20$, $a+b+c=25$ is not prime. $a=8$, $b=10$, $a+b+c=19$ is prime and less than $43$.

$c=5$, the following is possible: $a=2$, $b=8$, $a+b+c=15$ is not prime.