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December 6, 2015

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

5 comments on “RMO 2015 West Bengal”

  1. In MP Region RMO, in question No. 4 above (RMO -West Bengal), 28 is given in place of 36. Is it printing error or there is a solution to it.

  2. No. The question paper is correct.
    If you see other region's papers, you will see that in some paper, 36 is replaced by 32, 28.
    Similarly, If you see other region's papers then you will understand that the polynomial p(x)=x^2+ax+b in question 2 has been repeated in other papers
    They just twisted the problems a bit.

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