GR-HMS 2013

Let the sequence of real numbers $(a_n),n=1,2,3...$ with $a_1=2$ and $a_n=\left(\frac{n+1}{n-1} \right)\left(a_1+a_2+...+a_{n-1} \right),n\geq 2$ . Find the term $a_{2013}$ .

Solve in integers the following equation: $$y=2x^2+5xy+3y^2$$

We define the sets $A_1,A_2,...,A_{160}$ such that $\left|A_{i} \right|=i$ for all $i=1,2,...,160$ . With the elements of these sets we create new sets $M_1,M_2,...M_n$ by the following procedure: in the first step we choose some of the sets $A_1,A_2,...,A_{160}$ and we remove from each of them the same number of elements. These elements that we removed are the elements of $M_1$ . In the second step we repeat the same procedure in the sets that came of the implementation of the first step and so we define $M_2$ . We continue similarly until there are no more elements in $A_1,A_2,...,A_{160}$ , thus defining the sets $M_1,M_2,...,M_n$ . Find the minimum value of $n$ .

Let a triangle $ABC$ inscribed in circle $c(O,R)$ and $D$ an arbitrary point on $BC$ (different from the midpoint).The circumscribed circle of $BOD$ ,which is $(c_1)$ , meets $c(O,R)$ at $K$ and $AB$ at $Z$ .The circumscribed circle of $COD$ $(c_2)$ ,meets $c(O,R)$ at $M$ and $AC$ at $E$ .Finally, the circumscribed circle of $AEZ$ $(c_3)$ ,meets $c(O,R)$ at $N$ .Prove that $\triangle{ABC}=\triangle{KMN}.$