# Pre Rmo 2013 Question Paper Set A

AND

HOMI BHABHA CENTRE FOR SCIENCE EDUCATION

TATA INSTITUTE OF FUNDAMENTAL RESEARCH

Pre-REGIONAL MATHEMATICAL OLYMPIAD, 2013

Mumbai Region

October 20, 2013

QUESTION PAPER SET: A

• 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?

n

√

2. Let Sn =

k=0

1

√ . What is the value of

k+1+ k

99

1

?

n=1 Sn + Sn−1

3. 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 is the value of S(5N + 2013)? 7. Let Akbar and Birbal together have n marbles, where n > 0. Akbar says to Birbal, “ If I give you some marbles then you will have twice as many marbles as I will have.” Birbal says to Akbar, “ If I give you some marbles then you will have thrice as many marbles as I will have.”

What is the minimum possible value of n for which the above statements are true? 8. Let AD and BC be the parallel sides of a trapezium ABCD. Let P and Q be the midpoints of the diagonals AC and BD. If AD = 16 and BC = 20, what is the length of P Q? 1

9. In a triangle ABC, let H, I and O be the orthocentre, incentre and circumcentre, respectively. If the points B, H, I, C lie on a circle, what is the magnitude of ∠BOC in degrees? 10. Carol was given three numbers and was asked to add the largest...

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