IR-MO-R2 2020

let $x,y,z$ be positive reals , such that $x+y+z=1399$ find the $$\max( [x]y + [y]z + [z]x )$$ ( $[a]$ is the biggest integer not exceeding $a$ )

let $\omega_1$ be a circle with $O_1$ as its center , let $\omega_2$ be a circle passing through $O_1$ with center $O_2$ let $A$ be one of the intersection of $\omega_1$ and $\omega_2$ let $x$ be a line tangent line to $\omega_1$ passing from $A$ let $\omega_3$ be a circle passing through $O_1,O_2$ with its center on the line $x$ and intersect $\omega_2$ at $P$ (not $O_1$ ) prove that the reflection of $P$ through $x$ is on $\omega_1$