HMMT February, 2023

Algebra Difficulty 5.0 AIME, harder Prove it United States

Problem:
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 $a+b+c$ .

Solution

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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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.