Note that $a^5\equiv a\mod 5$, so since $5\nmid a,b$ we must have $a$ and $b$ are $1$ and $4$ or $2$ and $3$ modulo $5$ in some order. Either way, the minimum possible value of $a+b$ is $\boxed{5}$, achieved when $(a,b)$ is a permutation of $(1,4)$ or $(2,3)$.

parmenides51 wrote: Let a, b be positive integers such that $5 \nmid a, b$ and $5 \mid a^5+b^5$. What is the minimum possible value of $a + b$? The actual contest asks for "Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$. What is the minimum possible value of $a + b$?" This is slightly harder but easily overkill by LTE.

I had a typo so for the original problem posted (with the typo) parmenides51 wrote: Let a, b be positive integers such that $5 \nmid a, b$ and $5 \mid a^5+b^5$. What is the minimum possible value of $a + b$? the reply was brainiacmaniac31 wrote: Note that $a^5\equiv a\mod 5$, so since $5\nmid a,b$ we must have $a$ and $b$ are $1$ and $4$ or $2$ and $3$ modulo $5$ in some order. Either way, the minimum possible value of $a+b$ is $\boxed{5}$, achieved when $(a,b)$ is a permutation of $(1,4)$ or $(2,3)$. I am correcting the wording according to above post, #3, I checked my source and it's my typo, not the source's

We have $5\mid 5^5\mid a^5+b^5\implies 5\mid a+b $ by $FLT $. We must have $\nu_5 (a^5+b^5)\ge 5$ and by $LTE $ $$\nu_5 (a^5+b^5)=\nu_5 (a+b)+1\ge 5$$So $\nu_5 (a+b)\ge 4$ and we get the minimum value is $a+b=5^4$.