Difference between revisions of "1976 USAMO Problems/Problem 1"
(I'd write a solution but I need the diagram for clarification, I will post what it looks like in the discussion part of this article.) |
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== Problem == | == Problem == | ||
| − | {{ | + | <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 <math>4\times 7</math> chessboard, as shown above, is colored either black or white. Prove that with <i>any</i> 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. | * (a) Suppose that each square of a <math>4\times 7</math> chessboard, as shown above, is colored either black or white. Prove that with <i>any</i> 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. | ||
Revision as of 14:43, 3 January 2009
Problem
![[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]](http://latex.artofproblemsolving.com/b/b/0/bb023b115e30010d4f1daa300d792b67658733a4.png)
- (a) Suppose that each square of a
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
board in which the four corner squares of every rectangle, as described above, are not all of the same color.
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.
board in which the four corner squares of every rectangle, as described above, are not all of the same color.Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
See also
| 1976 USAMO (Problems • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
