From a well-known lemma, $A$, $I$, $A_0$ and $C$, $I$, $C_0$ are collinear. Also, $A_{0}C_{0}||AC$. Thus, the orthocenter of triangle $A_0BC_0$, the midpoint of $A_{0}C_{0}$, $I$ and the midpoint of $AC$ are collinear.

kymkozan wrote: From a well-known lemma, $A$, $I$, $A_0$ and $C$, $I$, $C_0$ are collinear. Also, $A_{0}C_{0}||AC$. Thus, the orthocenter of triangle $A_0BC_0$, the midpoint of $A_{0}C_{0}$, $I$ and the midpoint of $AC$ are collinear. How can you prove these well known lemmas cause I struggle a bit..??

Good application of Iran Lemma. And overall it's a quality olympiad, probably on a par with Sharygin.