Pre Rmo 2013 Question Paper Set A
NATIONAL BOARD FOR HIGHER MATHEMATICS
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 of the three to the product of the
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 of the three to the product of the