RMO 2015 West Bengal

Here's the RMO 2015 question paper of west bengal region, which was held on 6th december, in I.S.I. Kolkata.

1. $ 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
2. 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
3. Show that there are infinitely many triples $ (x,y,z)$ of positive integers, such that $ x^3+y^4=z^{31}.$
   SOLUTION: Here
4. 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
5. 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
6. 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