SAUDI ARABIAN MATHEMATICAL COMPETITIONS

Let $S$ be the final sum, then the necessary conditions are $3|S$, $10|S$. One can check that for every positive integer $n$ then $S(n)\equiv n(\mathrm{mod}9)$ which implies that for all ways to insert the plus signs, we have $$S\equiv 1+1+1+\cdots +1\equiv 0(\mathrm{mod}3).$$ Then the condition that divisible by $3$ is always satisfied. Suppose that we inserted $m$ signs in total, then we have $m+1$ parts. Each part is a positive integer that ends with $1$ so the rightmost digit of $S$ is equal to or congruent to $m+1$ modulo $10$. Then we must have $m\in \{9,19,29\}$. It is easy to see that the given condition is also the sufficiency condition. Denote $a_1,a_2,\dots ,a_{m+1}$ as the number of digits of each part that partitioned from $m$ signs then $$a_1+a_2+a_3+\cdots +a_{m+1}=30$$ This equation has $\left(\genfrac{}{}{0ex}{}{29}{m}\right)$ solutions. Therefore the number of ways to insert the plus signs is $\left(\genfrac{}{}{0ex}{}{29}{9}\right)+\left(\genfrac{}{}{0ex}{}{29}{19}\right)+\left(\genfrac{}{}{0ex}{}{29}{29}\right)$.