my solution to no.5: Let the E,F be the feet of perp. from O to AC,AB respectively, E’,F’ the feet of perp from I to AC,AB respectively. Then EE’=AM/2, FF’=AN/2. Translate EE’,FF’ to OM’,ON’. Then AMN and OM’N’ are similar with ratio of similarity equals 1/2. Note that O,M’I,N’ are concyclic, thus M’N’=OIsinA. Therefore MN=2OIsinA. The result follows by applying Euler Theorem. However, during the competition, I stupidly apply Ptolemy’s Theorem to OM’IN’, which results in quite complicated computation requiring some trigo identities….

Anyway, I still got it correct