Difference between revisions of "2004 IMO Problems/Problem 3"
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| − | Define a | + | == Problem == |
| + | Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure. | ||
| − | + | <asy> | |
| + | unitsize(0.5 cm); | ||
| + | |||
| + | draw((0,0)--(1,0)); | ||
| + | draw((0,1)--(1,1)); | ||
| + | draw((2,1)--(3,1)); | ||
| + | draw((0,2)--(3,2)); | ||
| + | draw((0,3)--(3,3)); | ||
| + | draw((0,0)--(0,3)); | ||
| + | draw((1,0)--(1,3)); | ||
| + | draw((2,1)--(2,3)); | ||
| + | draw((3,1)--(3,3)); | ||
| + | </asy> | ||
| + | |||
| + | Determine all <math>m \times n</math> rectangles that can be covered without gaps and without overlaps with hooks such that; | ||
| + | (a) the rectangle is covered without gaps and without overlaps, | ||
| + | (b) no part of a hook covers area outside the rectangle. | ||
| + | |||
| + | ==Solution== | ||
| + | {{solution}} | ||
| + | |||
| + | ==See Also== | ||
| + | |||
| + | {{IMO box|year=2004|num-b=2|num-a=4}} | ||
Latest revision as of 23:53, 18 November 2023
Problem
Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure.
![[asy] unitsize(0.5 cm); draw((0,0)--(1,0)); draw((0,1)--(1,1)); draw((2,1)--(3,1)); draw((0,2)--(3,2)); draw((0,3)--(3,3)); draw((0,0)--(0,3)); draw((1,0)--(1,3)); draw((2,1)--(2,3)); draw((3,1)--(3,3)); [/asy]](http://latex.artofproblemsolving.com/e/f/4/ef4cdb21baa20a696547d85c9c5773637ad93506.png)
Determine all 
rectangles that can be covered without gaps and without overlaps with hooks such that;
(a) the rectangle is covered without gaps and without overlaps,
(b) no part of a hook covers area outside the rectangle.
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
See Also
| 2004 IMO (Problems) • Resources | ||
| Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
| All IMO Problems and Solutions | ||
