SMO 2013 Junior Round 2

Let $a<b<c<d<e$ be real numbers. Among the $10$ sums of the pairs of these numbers, the least three are $32$, $36$ and $37$ while the largest two are $48$ and $51$. Find all possible values of $e$.

In the triangle $ABC$, points $D,E,F$ are on the sides $BC,CA$ and $AB$, respectively, such that $FE$ is parallel to $BC$ and $DF$ is parallel to $CA$. Let $P$ be the intersection of $BE$ and $DF$, and $Q$ the intersection of $FE$ and $AD$. Prove that $PQ$ is parallel to $AB$.

Find all primes that can be written both as a sum of two primes and as a difference of two primes.

Let $a$ and $b$ be positive integers with $a>b>2$. Prove that $\dfrac{2^a+1}{2^b-1}$ is not an integer.

Six musicians gathered at a chamber music festival. At each scheduled concert some of the musicians played while the others listened as members of the audience. What is the least number of such concerts which would need to be scheduled so that for every two musicians each must play for the other in some concert?