1. Let ABC be a right-angled triangle with . Let BD is the altitude from B on AC. Let P,Q and Ibe the incenters of triangles ABD,CBD and ABC respectively.Show that circumcenter of triangle PIQ lie on the hypotenuse AC.

2. For any natural number n > 1 write the finite decimal expansion of (for example we write as its infinite decimal expansion not 0.5). Determine the length of non-periodic part of the (infinite) decimal expansion of .

3. Find all real functions such that

4. There are four basketball players A,B,C,D. Initially the ball is with A. The ball is always passed from one person to a different person. In how many ways can the ball come back to A after (for example , or ).

5. Let ABCD be a convex quadrilateral.Let diagonals AC and BD intersect at P. Let PE,PF,PG and PH are altitudes from P on the side AB,BC,CD and DA respectively. Show that ABCD has a incircle if and only if .

6. Show that from a set of 11 square integers one can select six numbers such that (mod 12)