Difference between revisions of "2002 USAMO Problems/Problem 6"
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I have an <math>n \times n</math> sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let <math>b(n)</math> be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants <math>c</math> and <math>d</math> such that <center><math>\dfrac{1}{7} n^2 - cn \leq b(n) \leq \dfrac{1}{5} n^2 + dn</math></center> for all <math>n > 0</math>. | I have an <math>n \times n</math> sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let <math>b(n)</math> be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants <math>c</math> and <math>d</math> such that <center><math>\dfrac{1}{7} n^2 - cn \leq b(n) \leq \dfrac{1}{5} n^2 + dn</math></center> for all <math>n > 0</math>. | ||
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== See also == | == See also == | ||
Revision as of 17:19, 16 July 2014
Problem
I have an 
sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let 
be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants 
and 
such that
for all 
.
Solution
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Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See also
| 2002 USAMO (Problems • Resources) | ||
| Preceded by Problem 5 |
Followed by Last question | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
