HMMT November, 2022

Number theory Difficulty 5.0 AIME, harder Prove it United States

Problem:
A triple of positive integers $(a, b, c)$ is tasty if $\operatorname{lcm}(a, b, c) \mid a+b+c-1$ and $a<b<c$ . Find the sum of $a+b+c$ across all tasty triples.

Solution

Solution:
The condition implies $c \mid b+a-1$ . WLOG assume $c>b>a$ ; since $b+a-1<2c$ we must have $b+a-1=c$ . Substituting into $b \mid a+c-1$ and $a \mid c+b-1$ gives
$\begin{aligned} & b \mid 2a-2 \\ & a \mid 2b-2 \end{aligned}$
Since $2a-2<2b$ we must either have $a=1$ (implying $a=b$ , bad) or $2a-2=b \Longrightarrow a \mid 4a-6 \Longrightarrow a=2,3,6$ . If $a=2$ then $b=2$ . Otherwise, if $a=3$ we get $(3,4,6)$ and if $a=6$ we get $(6,10,15)$ , so answer is $13+31=44$ .

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