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HMMT 二月 2000 · ORAL 赛 · 第 4 题

HMMT February 2000 — ORAL Round — Problem 4

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [40] On an n × n chessboard, numbers are written on each square so that the number in a square is the average of the numbers on the adjacent squares. Show that all the numbersare the same.
解析

英文解析

  1. Let m be the smallest number, written on some square S . Then clearly all of theadjacent squares to S must have m on them (they can’t have anything smaller, since mis the smallest, and can’t have anything larger since m is the average of those adjacentnumbers). Since this applies to every square on which m is written, all of the squaresmust have m on them. Otherwise there will be some “boundary” square with m on it,
    which will not have m written on one of its neighbours, a contradiction.