# Regional Math Olympiad 2015 (West Bengal Region)

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

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