1 Let and be two circles touching each other externally at R. Let and be the centres of $latex (\Gamma_1)$ and , respectively. Let be a line which is tangent to at P and passing through , and let be the line tangent to at Q and passing through . Let . If KP=KQ then prove that the triangle PQR is equilateral.

4 Let N be an integer greater than 1 and let be the number of non empty subsets S of ({1,2,…..,n}) with the property that the average of the elements of S is an integer.Prove that is always even.

5 In an acute triangle ABC, let O,G,H be its circumcentre, centroid and orthocenter. Let $latex (D\in BC, E\in CA)$ and . Let F be the midpoint of AB. If the triangles ODC, HEA, GFB have the same area, find all the possible values of (angle C).

6 Let a,b,c,x,y,z be six positive real numbers satisfying x+y+z=a+b+c and xyz=abc. Further, suppose that and a<b<c. Prove that a=x,b=y and c=z.

CONGRATULATIONS FROM BRAZIL EXCELLENT BLOG INVITE EVERYONE TO VISIT MY BLOG OF MATHEMATICS http://postulanteuniversitario.blogspot.com