Let $a<b<c<d<e$ be real numbers. Among the $10$ sums of the pairs of these numbers, the least $3$ are $32,36,37$, while the largest two are $48$ and $51$. Find all possible values of $e$
2013 Singapore Junior Math Olympiad
2nd Round
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 prime numbers which can be presented as a sum of two primes and difference of two primes at the same time.
Let $a,b,$ be positive integers and $a>b>2$. Prove that $\frac{2^a+1}{2^b-1}$ is never an integer
$6$ 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 every $2$ musicians each must play for the other in some concert?