HMMT 二月 2020 · 冲刺赛 · 第 8 题

HMMT February 2020 — Guts Round — Problem 8

[5] Tessa picks three real numbers x, y, z and computes the values of the eight expressions of the form ± x ± y ± z . She notices that the eight values are all distinct, so she writes the expressions down in increasing order. For example, if x = 2, y = 3, z = 4, then the order she writes them down is − x − y − z, + x − y − z, − x + y − z, − x − y + z, + x + y − z, + x − y + z, − x + y + z, + x + y + z. How many possible orders are there? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2020, February 15, 2020 — GUTS ROUND Organization Team Team ID# 1 √

解析

[5] Tessa picks three real numbers x, y, z and computes the values of the eight expressions of the form ± x ± y ± z . She notices that the eight values are all distinct, so she writes the expressions down in increasing order. For example, if x = 2, y = 3, z = 4, then the order she writes them down is − x − y − z, + x − y − z, − x + y − z, − x − y + z, + x + y − z, + x − y + z, − x + y + z, + x + y + z. How many possible orders are there? Proposed by: Yuan Yao Answer: 96 3 Solution: There are 2 = 8 ways to choose the sign for each of x , y , and z . Furthermore, we can order | x | , | y | , and | z | in 3! = 6 different ways. Now assume without loss of generality that 0 < x < y < z . Then there are only two possible orders depending on the sign of x + y − z : − x − y − z, + x − y − z, − x + y − z, − x − y + z, x + y − z, x − y + z, − x + y + z, x + y + z − x − y − z, + x − y − z, − x + y − z, x + y − z, − x − y + z, x − y + z, − x + y + z, x + y + z Thus, the answer is 8 · 6 · 2 = 96. 1 √