IGO 2014

The inscribed circle of $\triangle ABC$ touches $BC, AC$ and $AB$ at $D,E$ and $F$ respectively. Denote the perpendicular foots from $F, E$ to $BC$ by $K, L$ respectively. Let the second intersection of these perpendiculars with the incircle be $M, N$ respectively. Show that $\frac{{{S}_{\triangle BMD}}}{{{S}_{\triangle CND}}}=\frac{DK}{DL}$

by Mahdi Etesami Fard

Each of Mahdi and Morteza has drawn an inscribed $93$ - gon. Denote the first one by $A_1A_2…A_{93}$ and the second by $B_1B_2…B_{93}$. It is known that $A_iA_{i+1} // B_iB_{i+1}$ for $1 \le i \le 93$ ($A_{93} = A_1, B_{93} = B_1$). Show that $\frac{A_iA_{i+1} }{ B_iB_{i+1}}$ is a constant number independent of $i$.

by Morteza Saghafian

In a triangle ABC we have $\angle C = \angle A + 90^o$. The point $D$ on the continuation of $BC$ is given such that $AC = AD$. A point $E$ in the side of $BC$ in which $A$ doesn't lie is chosen such that $\angle EBC = \angle A, \angle EDC = \frac{1}{2} \angle A$. Prove that $\angle CED = \angle ABC$.

by Morteza Saghafian

Two points $X, Y$ lie on the arc $BC$ of the circumcircle of $\triangle ABC$ (this arc does not contain $A$) such that $\angle BAX = \angle CAY$. Let $M$ denotes the midpoint of the chord $AX$. Show that $BM +CM > AY$.

by Mahan Tajrobekar