MathLabs

第1题

In plane, 点s 满足整数 coordinates 是顶点 的unit 平方s, colored alternately black 且white as 上a chess棋盘. F或正整数 m,n m,n , consider a right 三角形 满足整数-coordinate 顶点 且legs 的长度s m,n m,n along 平方 边. 设 S1 S_1 be black 面积 且 S2 S_2 white 面积. Define f(m,n)=∣S1−S2∣ f(m,n)=|S_1-S_2| 译文：. (a) Calculate f(m,n) f(m,n) 译文： when m,n m,n 是both even 或both odd. (b) 证明 f(m,n)≤max⁡(m,n)/2 f(m,n)\le\max(m,n)/2. (c) 证明 no 常数 C C 译文： satisfies f(m,n)<C f(m,n)<C 对所有 m,n m,n .
第 3/3 步：构造 unbounded excess
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通俗地说

构造 unbounded excess

f(n+1,n)=n−16f(n+1,n)=\dfrac{n-1}{6}
详细分析

译文：Take even n n 且comp是三角形s 满足legs (n,n)(n,n) 且 (n+1,n)(n+1,n). The first has zero excess. Along added strip, 和ming similar white 三角形s gives white 面积 (2n+1)/12(2n+1)/12; black 面积 是 n/2−(2n+1)/12 n/2-(2n+1)/12, so new excess 是 (n−1)/6(n-1)/6, which tends 到infinity.