RMO 2011

Let $ABC$ be a triangle. Let $D, E, F$ be points respectively on the segments $BC, CA, AB$ such that $AD, BE, CF$ concur at the point $K$ . Suppose $\frac{BD}{DC} = \frac {BF}{FA}$ and $\angle ADB = \angle AFC$ . Prove that $\angle ABE = \angle CAD$ .

A natural number $n$ is chosen strictly between two consecutive perfect squares. The smaller of these two squares is obtained by subtracting $k$ from $n$ and the larger by adding $l$ to $n.$ Prove that $n-kl$ is a perfect square.

Consider a $20$ -sided convex polygon $K$ , with vertices $A_1, A_2,...,A_{20}$ in that order. Find the number of ways in which three sides of $K$ can be chosen so that every pair among them has at least two sides of $K$ between them. (For example $(A_1A_2, A_4A_5, A_{11}A_{12})$ is an admissible triple while $(A_1A_2, A_4A_5, A_{19}A_{20})$ is not.

Let $ABC$ be a triangle and let $BB_1,CC_1$ be respectively the bisectors of $\angle{B},\angle{C}$ with $B_1$ on $AC$ and $C_1$ on $AB$ , Let $E,F$ be the feet of perpendiculars drawn from $A$ onto $BB_1,CC_1$ respectively. Suppose $D$ is the point at which the incircle of $ABC$ touches $AB$ . Prove that $AD=EF$