Home > Geometry, IMO, Olympiad > Romanian TST II 2011 Problem 1

## Romanian TST II 2011 Problem 1

A square of sidelength $\ell$ is contained in the unit square whose centre is not interior to the former. Show that $\ell \leq 1/2$.

Romanian TST 2011

Proof: There is a line through the center of the big square which does not cut the smaller square. This line intersects two of the sides of the square in their interior and makes with them an angle $\alpha$. Using the formula of the maximal square inscribed in a triangle we get that

$\displaystyle \ell \leq \frac{1}{2}\frac{\sin\alpha + \cos \alpha}{\sin \alpha \cos \alpha +1} \leq \frac{1}{2}$