Find the integers $a, b, c \geq 1$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$.
We can assume, by symmetry, that we have $a \leq b \leq c$. Thus, if $a>3$, we have $1 / c \leq 1 / b \leq 1 / a4$, we get $1 / b + 1 / c < 1 / 2$. Therefore, we have $b=4$ or $b=3$. If $b=4$, then $c=4$. If $b=3$, then $c=6$.

Finally, the solutions are $(a, b, c)=(3,3,3)$ or $(2,4,4)$ or $(2,3,6)$ and their permutations.