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NATIONAL BOARD FOR HIGHER MATHEMATICSAND

HOMI BHABHA CENTRE FOR SCIENCE EDUCATIONTATA INSTITUTE OF FUNDAMENTAL RESEARCH

Pre-REGIONAL MATHEMATICAL OLYMPIAD, 2014Mumbai Region

October 12, 2014

QUESTION PAPER SET: B

There are 20 questions in this question paper. Each question carries 5 marks.

Answer all questions.

Time allotted: 2.5 hours.

QUESTIONS

1. The first term of a sequence is 2014. Each succeeding term is the sum of the cubes of thedigits of the previous term. What is the 2014th term of the sequence?

2. In a triangle with integer side lengths, one side is three times as long as a second side, andthe length of the third side is 17. What is the greatest possible perimeter of the triangle?

3. If real numbersa, b, c, d, e satisfy

a + 1 =b + 2 =c + 3 =d + 4 =e + 5 =a + b + c + d + e + 3,

what is the value ofa2 + b2 + c2 + d2 + e2?

4. Let ABCD be a convex quadrilateral with perpendicular diagonals. IfAB = 20, BC= 70andCD= 90, then what is the value ofDA?

5. A natural number k is such that k2

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12. One morning, each member of Manjuls family drank an 8-ounce mixture of coffee and milk.The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank

1/7-th of the total amount of milk and 2/17-th of the total amount of coffee. How manypeople are there in Manjuls family?

13. For natural numbersx and y, let (x, y) denote the greatest common divisor ofx and y. Howmany pairs of natural numbers x and y withx y satisfy the equation xy = x + y+ (x, y)?

14. For how many natural numbersnbetween 1 and 2014 (both inclusive) is 8n

9999 nan integer?

15. Let ABCD be a convex quadrilateral with DAB = BDC = 90. Let the incircles oftriangles ABD and BCD touch BD at P and Q, respectively, with P lying in between Band Q. IfAD = 999 and PQ = 200 then what is the sum of the radii of the incircles oftrianglesABD andBDC?

16. Let x1, x2, , x2014 be real numbers different from 1, such that x1 + x2+ +x2014 = 1and x1

1 x1+

x21 x2

+ + x20141 x2014

= 1 .

What is the value of

x211 x1

+ x221 x2

+ x231 x3

+ + x220141 x2014

?

17. In a triangle ABC, let I denote the incenter. Let the lines AI,BI and CI intersect theincircle atP,Q and R, respectively. IfBAC= 40, what is the value ofQPRin degrees?

18. What is the number of ordered pairs (A,B) whereA and B are subsets of{1, 2, . . . ,5}suchthat neitherA B norB A?

19. For a natural number b, let N(b) denote the number of natural numbers a for which theequation x2 + ax + b = 0 has integer roots. What is the smallest value of b for whichN(b) = 6?

20. Let fbe a one-to-one function from the set of natural numbers to itself such that f(mn) =f(m)f(n) for all natural numbers m and n. What is the least possible value off(999)?

Pre-RMO 2014 results will be put up on HBCSE website by 31/10/2014.

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