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• There are 20 questions in this question paper. Each question carries 5 marks. • Answer all questions • Time allotted: 2 hours QUESTIONS 1. What is the smallest positive integer k such that k(33 + 43 + 53 ) = an for some positive integers a and n, with n > 1?

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Pre Rmo 2013 Question Paper Set ANATIONAL BOARD FOR HIGHER MATHEMATICSANDHOMI BHABHA CENTRE FOR SCIENCE EDUCATIONTATA INSTITUTE OF FUNDAMENTAL RESEARCHPre-REGIONAL MATHEMATICAL OLYMPIAD, 2013Mumbai RegionOctober 20, 2013QUESTION PAPER SET: A There are 20 questions in this question paper. Each question carries 5 marks. Answer all questions. Time allotted: 2 hours.QUESTIONS1. What is the smallest positive integer k such that k(33 + 43 + 53 ) = an for some positive integers a and n, with n > 1?n2. Let Sn =k=01 . What is the value ofk+1+ k991?n=1 Sn + Sn13. It is given that the equation x2 + ax + 20 = 0 has integer roots. What is the sum of all possible values of a?4. Three points X, Y, Z are on a striaght line such that XY = 10 and XZ = 3. What is the product of all possible values of Y Z?5. There are n 1 red balls, n green balls and n + 1 blue balls in a bag. The number of ways of choosing two balls from the bag that have different colours is 299. What is the value of n? 6. Let S(M ) denote the sum of the digits of a positive integer M written in base 10. Let N be the smallest positive integer such that S(N ) = 2013. What

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