- circles and with centers and respectively, are such that lies on Let be a point on and let be the midpoint of Let be another point on such that Then prove that the midpoint of lies on

**SOLUTION: Here** - Let be a quadratic polynomial where are real numbers. Suppose be an of positive integers. Prove that are integers.

**SOLUTION: Here** - Show that there are infinitely many triples of positive integers, such that

**SOLUTION: Here** - Suppose objects are placed along a circle at equal distances. In how many ways can objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite.

**SOLUTION:****Here** - Let be a triangle with circumcircle and incenter Let the internal angle bisectors of meet in respectively. Let intersect at and in Let intersect in Suppose the quadrilateral is a kite; that is, and Prove that is an equilateral triangle.

**SOLUTION:****Here** - Show that there are infinitely many positive real numbers, which are not integers, such that is an integer. (Here, is the fractional part of For example)

**SOLUTION:****Here**

**End of Question Paper**