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Problem 1053

AMC 12 late, AIME early

Geometry Difficulty 4.9 Find the answer HMMT February

Let $W, S$ be as in problem 32. Let $A$ be the least positive integer such that an acute triangle with side lengths $S, A$ , and $W$ exists. Find $A$ .

Your answer

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

Official solution

There are two solutions to the alphametic in problem 32: $36 \times 686=24696$ and $86 \times 636=54696$ . So $(W, S)$ may be $(3,2)$ or $(8,5)$ . If $(W, S)=(3,2)$ , then by problem (3) $A=3$ , but then by problem $31 W=4$ , a contradiction. So, $(W, S)$ must be $(8,5)$ . By problem $33, A=7$ , and this indeed checks in problem 31.

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