HMMT 二月 2006 · 冲刺赛 · 第 33 题
HMMT February 2006 — Guts Round — Problem 33
题目详情
- [10] Let W , S be as in problem 32. Let A be the least positive integer such that an acutetriangle with side lengths S , A , and W exists. Find A .
IX HARVARD-MIT MATHEMATICS TOURNAMENT, 25 FEBRUARY 2006 — GUTS ROUNDth
英文原题
[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 .
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IX th HARVARD-MIT MATHEMATICS TOURNAMENT, 25 FEBRUARY 2006 — GUTS ROUND
解析
英文解析
- Let W , S be as in problem 32. Let A be the least positive integer such that an acutetriangle 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 byproblem (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.