Determine all integer values of n for which the number $\frac{14n+25}{2n+1}$ 'is a perfect square.

In triangle $[ABC]$, the bisector in $C$ and the altitude passing through $B$ intersect at point $D$. Point $E$ is the symmetric of point $D$ wrt $BC$ and lies on the circle circumscribed to the triangle $[ABC]$. Prove that the triangle is $[ABC]$ isosceles.

In an athletics tournament, five teams participate. Each athlete has a shirt numbered with a positive integer, and all athletes on the same team have different numbers. Each athlete participates in a single event and only one athlete from each team is present in each event. Emídio noticed that the sum of the athletes' jersey numbers in each event is always $20$. What is the maximum number of athletes in the tournament?

Numbers from $1$ to $8$ are placed on the vertices of a cube, one on each of the eight vertices, so that the sum of the numbers on any three vertices of the same face is greater than $9$. Determines the minimum value that the sum of the numbers on one side can have.

Let $[ABCD]$ be a convex quadrilateral with $AB = 2, BC = 3, CD = 7$ and $\angle B = 90^o$, for which there is a inscribed circle. Determine the radius of this circle.

In a building whose floors are numbered $1$ to $8$, the builder wants to place elevators so that, for every choice of two floors, there are always at least three elevators that stop on those floors. Furthermore, each elevator can only stop at a maximum of $5$ floors. What is the minimum number of elevators that need to be placed?