Regional Math Olympiad 2015 | West Bengal Region

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

End of Question Paper