Friday, Jan 20th

Problem
Julian writes down 5 positive integers such that their sum equals their product. Which numbers could have Julian writen down?

Solution
Let $a,b,c,d,e$ be the numbers Julian wrote down and without loss of generality suppose that $a\leq b\leq c\leq d\leq e$. If $a>1$, then $a+b+c+d+e\leq 5e$ and $abcde\geq 2^4 e$. Thus Julian must had written at least one 1. If $b>1$ something similar happens, as $a+b+c+d+e<5e$ and $abcde\geq 8e$. Hence $b=1$. If $c=1$ we have that $d=3$ and $e=3$ or $d=2$ and $e=5$ are solutions. If $c=2$, $d=2$ and $e=2$. If $c>2$, then $a+b+c+d+e<2e+5<4e$ and $abcde\geq 9e$, hence there are no solutions.

Therefore, the only solutions are $(1,1,1,2,5), (1,1,1,3,3), (1,1,2,2,2)$ and their permutations.