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HMMT 二月 2008 · TEAM2 赛 · 第 1 题

HMMT February 2008 — TEAM2 Round — Problem 1

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 10 ] (Distributive law) Prove that ( x ⊕ y ) z = x z ⊕ y z for all x, y, z ∈ R ∪ {∞} . n
解析

英文解析

  1. [ 10 ] (Distributive law) Prove that ( x ⊕ y ) z = x z ⊕ y z for all x, y, z ∈ R ∪ {∞} .
    Solution: This is equivalent to proving thatmin( x, y ) + z = min( x + z, y + z ) .
    Consider two cases. If x ≤ y , then LHS = x + z and RHS = x + z . If x > y , then LHS = y + zand RHS = y + z . It follows that LHS = RHS . n