PE-MO 2021

a) Determine if it's possible write $6$ positive rational numbers, pairwise distinct, in a circle such that each one is equal to the product of your neighbor numbers.b) Determine if it's possible write $8$ positive rational numbers, pairwise distinct, in a circle such that each one is equal to the product of your neighbor numbers.

The numbers $1$ to $25$ will be written in a table $5 \times 5$ . First, Ana chooses $k$ of these numbers( $1$ to $25$ ), and write in some cells. Then, Enrique writes the remaining numbers with the following goal: The product of the numbers in some column/row is a perfect square.a) Prove that if $k=5$ , Ana can avoid Enrique to reach his goal.b) Prove that if $k=4$ , Enrique can reach his goal.

Let $M,N,P$ be points in the sides $BC,AC,AB$ of $\triangle ABC$ respectively. The quadrilateral $MCNP$ has an incircle of radius $r$ , if the incircles of $\triangle BPM$ and $\triangle ANP$ also have the radius $r$ . Prove that $$AP\cdot MP=BP\cdot NP$$