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HMMT 二月 2002 · 冲刺赛 · 第 3 题

HMMT February 2002 — Guts Round — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [5] How many elements are in the set obtained by transforming { (0 , 0) , (2 , 0) } 14 times?
解析
  1. How many elements are in the set obtained by transforming { (0 , 0) , (2 , 0) } 14 times? Solution: Transforming it k ≥ 1 times yields the diamond { ( n, m ) : | n − 1 | + | m | ≤ k +1 } with the points (1 , k ) , (1 , k + 1) , (1 , − k ) , (1 , − k − 1) removed (this can be seen inductively). 2 2 So we get ( k + 1) + k − 4 lattice points, making the answer 477 .