1976 USAMO 第 1 题

$$[asy] void fillsq(int x, int y){ fill((x,y)--(x+1,y)--(x+1,y+1)--(x,y+1)--cycle, mediumgray); } int i; fillsq(1,0);fillsq(4,0);fillsq(6,0); fillsq(0,1);fillsq(1,1);fillsq(2,1);fillsq(4,1);fillsq(5,1); fillsq(0,2);fillsq(2,2);fillsq(4,2); fillsq(0,3);fillsq(1,3);fillsq(4,3);fillsq(5,3); for(i=0; i<=7; ++i){draw((i,0)--(i,4),black+0.5);} for(i=0; i<=4; ++i){draw((0,i)--(7,i),black+0.5);} draw((3,1)--(3,3)--(7,3)--(7,1)--cycle,black+1); [/asy]$$

- (a) Suppose that each square of a $4\times 7$ chessboard, as shown above, is colored either black or white. Prove that with any such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board such as the one outlined in the figure) whose four distinct unit corner squares are all of the same color.

- (b) Exhibit a black-white coloring of a $4\times 6$ board in which the four corner squares of every rectangle, as described above, are not all of the same color.

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[asy] void fillsq(int x, int y){ fill((x,y)--(x+1,y)--(x+1,y+1)--(x,y+1)--cycle,mediumgray); } int 我; fillsq(1,0);fillsq(4,0);fillsq(6,0); fillsq(0,1);fillsq(1,1);fillsq(2,1);fillsq(4,1);fillsq(5,1); fillsq(0,2);fillsq(2,2);fillsq(4,2); fillsq(0,3);fillsq(1,3);fillsq(4,3);fillsq(5,3); for(i=0; i<=7; ++i){绘制((i,0)--(i,4),黑色+0.5);} for(i=0; i<=4; ++i){绘制((0,i)--(7,i),黑色+0.5);} 绘制((3,1)--(3,3)--(7,3)--(7,1)--循环,黑色+1); [/asy]

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- (a) 假设 $4\times 7$ 棋盘的每个方格(如上所示)的颜色为黑色或白色。证明对于任何此类着色，棋盘必须包含一个矩形(由棋盘的水平线和垂直线形成，如图所示)，其四个不同的单元角方块都是相同的颜色。
- (b) 展示 $4\times 6$ 棋盘的黑白着色，其中每个矩形的四个角正方形，如上所述，并不都是相同的颜色。