A class has $24$ students. Each group consisting of three of the students meet, and choose one of the other $21$ students, A, to make him a gift. In this case, A considers each member of the group that offered him a gift as being his friend. Prove that there is a student that has at least $10$ friends.

Consider a cyclic quadrilateral $ABCD$ in which $BC = CD$ and $AB < AD$. Let $E$ be a point on the side $AD$ and $F$ a point on the line $BC$ such that $AE = AB = AF$. Prove that $EF \parallel BD$.

Let $a, b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that $$\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 $$When equality holds?

Determine all pairs $(k, n)$ of positive integers that satisfy $$1! + 2! + ... + k! = 1 + 2 + ... + n.$$