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January 4, 2016

Regional Math Olympiad 2015 | West Bengal Region

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  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

End of Question Paper


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