HMMT 二月 2002 · 冲刺赛 · 第 35 题

HMMT February 2002 — Guts Round — Problem 35

[ ± 7] Suppose a, b, c, d are real numbers such that | a − b | + | c − d | = 99; | a − c | + | b − d | = 1 . Determine all possible values of | a − d | + | b − c | . 2 3 x x x

解析

Suppose a, b, c, d are real numbers such that | a − b | + | c − d | = 99; | a − c | + | b − d | = 1 . Determine all possible values of | a − d | + | b − c | . Solution: 99 If w ≥ x ≥ y ≥ z are four arbitrary real numbers, then | w − z | + | x − y | = | w − y | + | x − z | = w + x − y − z ≥ w − x + y − z = | w − x | + | y − z | . Thus, in our case, two of the three numbers | a − b | + | c − d | , | a − c | + | b − d | , | a − d | + | b − c | are equal, and the third one is less than or equal to these two. Since we have a 99 and a 1, the third number must be 99. 2 3 x x x