CZ-SK-MO 2025

Real numbers $a,b,c,d$ satisfy $$a+b+c+d=0, \frac1a+\frac1b+\frac1c+\frac1d=0.$$ How many equalities $$ab=cd, ac=bd, ad=bc$$ can simultaneously hold? Determine all possibilities.

Let $ABC$ be an acute triangle, $H$ its orthocenter, $O$ circumcenter and $M$ midpoint of $BC.$ Denote $D\neq A$ the intersection of line $AH$ and the circumcircle $\omega,$ $E\neq D$ intersection of line $DM$ and $\omega.$ Finally, let $F\neq E$ be the intersection of line $AE$ and the circumcircle of $OME.$ Show that $FH=FA.$