- \(2\) circles \(\Gamma\) and \(\sum,\) with centers \(O\) and \(O’,\) respectively, are such that \(O’\) lies on \(\Gamma.\) Let \(A\) be a point on \(\sum,\) and let \(M\) be the midpoint of \(AO’.\) Let \(B\) be another point on \(\sum,\) such that \(AB~||~OM.\) Then prove that the midpoint of \(AB\) lies on \(\Gamma.\)
**SOLUTION: Here** - Let \(P(x)=x^2+ax+b\) be a quadratic polynomial where \(a,b\) are real numbers. Suppose \(\langle P(-1)^2,P(0)^2,P(1)^2\rangle\) be an \(AP\) of positive integers. Prove that \(a,b\) are integers.
**SOLUTION: Here** - Show that there are infinitely many triples \((x,y,z)\) of positive integers, such that \(x^3+y^4=z^{31}.\)
**SOLUTION: Here** - Suppose \(36\) objects are placed along a circle at equal distances. In how many ways can \(3\) objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite.
**SOLUTION:****Here** - Let \(ABC\) be a triangle with circumcircle \(\Gamma\) and incenter \(I.\) Let the internal angle bisectors of \(\angle A,\angle B,\angle C\) meet \(\Gamma\) in \(A’,B’,C’\) respectively. Let \(B’C’\) intersect \(AA’\) at \(P,\) and \(AC\) in \(Q.\) Let \(BB’\) intersect \(AC\) in \(R.\) Suppose the quadrilateral \(PIRQ\) is a kite; that is, \(IP=IR\) and \(QP=QR.\) Prove that \(ABC\) is an equilateral triangle.
**SOLUTION:****Here** - Show that there are infinitely many positive real numbers, which are not integers, such that \(a\left(3-\{a\}\right)\) is an integer. (Here, \(\{a\}\) is the fractional part of \(a.\) For example\(,~\{1.5\}=0.5;~\{-3.4\}=1-0.4=0.6.\))
**SOLUTION:****Here**

**End of Question Paper**