Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.9 AIME Find the answer

Suppose $a, b$ , and $c$ are distinct positive integers such that $\sqrt{a \sqrt{b \sqrt{c}}}$ is an integer. Compute the least possible value of $a+b+c$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

First, check that no permutation of $(1,2,3)$ works, so the sum must be more than 6 . Then since $(a, b, c)=(2,4,1)$ has $\sqrt{2 \sqrt{4 \sqrt{1}}}=2$ , the answer must be $2+4+1=7$ .

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