In a triangle $ABC$, $AB > AC$, the external bisector of angle $A$ meets the circumcircle of triangle $ABC$ at $E$, and $F$ is the foot of the perpendicular from $E$ onto $AB$. Prove that $2AF = AB - AC$

Find all prime numbers $p$ such that $5^p + 12^p$ is a perfect square

There are $n$ blue points and $n$ red points on a straight line. Prove that the sum of all distances between pairs of points of the same colour is less than or equal to the sum of all distances between pairs of points of different colours

Find all functions $f : R \to R$ such for any $x, y \in R,$ $$(x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2)$$

In a triangle $ABC$, $\angle C = 60^o$, $D, E, F$ are points on the sides $BC, AB, AC$ respectively, and $M$ is the intersection point of $AD$ and $BF$. Suppose that $CDEF$ is a rhombus. Prove that $DF^2 = DM \cdot DA$

Let $n$ be any integer $\ge 2$. Prove that $\sum 1/pq = 1/2$, where the summation is over all integers$ p, q$ which satisfy $0 < p < q \le n$,$ p + q > n$, $(p, q) = 1$.