Difference between revisions of "2004 Indonesia MO Problems/Problem 8"
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A floor with an area of <math>3 \text{ m}^2</math> will be covered by <math>5</math> rugs with various shapes, each having an area of <math>1 \text{ m}^2</math>. Show that there exist <math>2</math> overlapping rugs with the overlapped area at least <math>1/5 \text{ m}^2</math>. | A floor with an area of <math>3 \text{ m}^2</math> will be covered by <math>5</math> rugs with various shapes, each having an area of <math>1 \text{ m}^2</math>. Show that there exist <math>2</math> overlapping rugs with the overlapped area at least <math>1/5 \text{ m}^2</math>. | ||
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| + | ==Solution== | ||
| + | Let the first 3 rugs occupy the entire floor, then the next rug that you add in, by the pigeon hole principle, it must overlap with another rug. Let a, b and c be the overlapping region with rug 1, rug 2 and rug 3 respectively, a+b+c = 1, thus at least one of a, b and c must be greater than 0.2. | ||
Latest revision as of 03:45, 16 June 2024
Problem 8
A floor with an area of 
will be covered by 
rugs with various shapes, each having an area of 
. Show that there exist 
overlapping rugs with the overlapped area at least 
.
Solution
Let the first 3 rugs occupy the entire floor, then the next rug that you add in, by the pigeon hole principle, it must overlap with another rug. Let a, b and c be the overlapping region with rug 1, rug 2 and rug 3 respectively, a+b+c = 1, thus at least one of a, b and c must be greater than 0.2.
