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Problem 1022

AMC 12 late, AIME early

Algebra Difficulty 4.9 Find the answer HMMT February

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.

Official 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 $2+4+1=7$ .

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.