Let $A$ and $B$ be two fixed points of a given circle and $XY$ a diameter of this circle. Find the locus of the intersection points of lines $AX$ and $BY$ . ($BY$ is not a diameter of the circle). Albanian National Mathematical Olympiad 2010---12 GRADE Question 1.

We denote $N_{2010}=\{1,2,\cdots,2010\}$ (a)How many non empty subsets does this set have? (b)For every non empty subset of the set $N_{2010}$ we take the product of the elements of the subset. What is the sum of these products? (c)Same question as the (b) part for the set $-N_{2010}=\{-1,-2,\cdots,-2010\}$. Albanian National Mathematical Olympiad 2010---12 GRADE Question 2.

(a)Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity. (b)What is the smallest area possible of pentagons with integral coordinates. Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.

The sequence of Fibonnaci's numbers if defined from the two first digits $f_1=f_2=1$ and the formula $f_{n+2}=f_{n+1}+f_n$, $\forall n \in N$. (a) Prove that $f_{2010} $ is divisible by $10$. (b) Is $f_{1005}$ divisible by $4$? Albanian National Mathematical Olympiad 2010---12 GRADE Question 4.

All members of the senate were firstly divided into $S$ senate commissions . According to the rules, no commission has less that $5$ senators and every two commissions have different number of senators. After the first session the commissions were closed and new commissions were opened. Some of the senators now are not a part of any commission. It resulted also that every two senators that were in the same commission in the first session , are not any more in the same commission. (a)Prove that at least $4S+10$ senators were left outside the commissions. (b)Prove that this number is achievable. Albanian National Mathematical Olympiad 2010---12 GRADE Question 5.