SMO 2006

In the triangle $ABC,\angle A=\frac{\pi}{3},D,M$ are points on the line $AC$ and $E,N$ are points on the line $AB$ such that $DN$ and $EM$ are the perpendicular bisectors of $AC$ and $AB$ respectively. Let $L$ be the midpoint of $MN$ . Prove that $\angle EDL=\angle ELD$

Show that any representation of 1 as the sum of distinct reciprocals of numbers drawn from the arithmetic progression $\{2,5,8,11,...\}$ such as given in the following example must have at least eight terms: $$1=\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{20}+\frac{1}{41}+\frac{1}{110}+\frac{1}{1640}$$

Consider the sequence $p_{1},p_{2},...$ of primes such that for each $i\geq2$ , either $p_{i}=2p_{i-1}-1$ or $p_{i}=2p_{i-1}+1$ . Show that any such sequence has a finite number of terms.

Let $a,b,n$ be positive integers. Prove that $n!$ divides $$b^{n-1}a(a+b)(a+2b)...(a+(n-1)b)$$