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 $$AP\cdot MP=BP\cdot NP$$

Let $n\geq 3$ be a positive integer and a circle $\omega$ is given. A regular polygon(with $n$ sides) $P$ is drawn and your vertices are in the circle $\omega$ and these vertices are red. One operation is choose three red points $A,B,C$, such that $AB=BC$ and delete the point $B$. Prove that one can do some operations, such that only two red points remain in the circle.