Difference between revisions of "2021 USAJMO Problems/Problem 3"
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Revision as of 12:22, 16 April 2021
Problem
An equilateral triangle 
of side length 
is given. Suppose that 
equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside , such that each unit equilateral triangle has sides parallel to , but with opposite orientation. (An example with 
is drawn below.)
![[asy] draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle,linewidth(0.5)); filldraw((0.45,0.55)--(0.65,0.55)--(0.55,0.55-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); filldraw((0.54,0.3)--(0.34,0.3)--(0.44,0.3-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); [/asy]](http://latex.artofproblemsolving.com/b/a/b/bab00fe73ccc6121244ce7f6bbd5f89bced7f2d8.png)
Prove that![\[n \leq \frac{2}{3} L^{2}.\]](http://latex.artofproblemsolving.com/9/5/f/95fe9c1b574a4e693e5ad5190a56fcb62aa26701.png)
Solution
See Also
| 2021 USAJMO (Problems • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAJMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
