HMMT 二月 2006 · 冲刺赛 · 第 33 题

HMMT February 2006 — Guts Round — Problem 33

[10] 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th IX HARVARD-MIT MATHEMATICS TOURNAMENT, 25 FEBRUARY 2006 — GUTS ROUND

解析

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 . Answer: 7 Solution: There are two solutions to the alphametic in problem 32: 36 × 686 = 24696 and 86 × 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.