# Pre-RMO-Solved-Paper-2012.pdf

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NATIONAL BOARD FOR HIGHER MATHEMATICS AND HOMI BHABHA CENTRE FOR SCIENCE EDUCATION TATA INSTITUTE OF FUNDAMENTAL RESEARCH Pre-REGIONAL MATHEMATICAL OLYMPIAD, 2012 Mumbai Region October 7, 2012 QUESTION PAPER SET: A ã There are 20 questions in this question paper. Each question
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NATIONAL BOARD FOR HIGHER MATHEMATICS AND HOMI BHABHA CENTRE FOR SCIENCE EDUCATIONTATA INSTITUTE OF FUNDAMENTAL RESEARCHPre-REGIONAL MATHEMATICAL OLYMPIAD, 2012 Mumbai RegionOctober 7, 2012QUESTION PAPER SET: A ã  There are 20 questions in this question paper. Each question carries 5 marks. ã  Answer all questions. ã  Time allotted: 2 hours. QUESTIONS 1. Rama was asked by her teacher to subtract 3 from a certain number and then divide theresult by 9. Instead, she subtracted 9 and then divided the result by 3. She got 43 as theanswer. What would have been her answer if she had solved the problem correctly?2. A triangle with perimeter 7 has integer side lengths. What is the maximum possible area of such a triangle?3. For how many pairs of positive integers ( x,y ) is  x + 3 y  = 100?4. The letters  R ,  M  , and  O  represent whole numbers. If   R × M   × O  = 240,  R × O  + M   = 46and  R + M   × O  = 64, what is the value of   R + M   + O ?5. Let  S  n  =  n 2 + 20 n + 12,  n  a positive integer. What is the sum of all possible values of   n  forwhich  S  n  is a perfect square?6. A postman has to deliver ﬁve letters to ﬁve diﬀerent houses. Mischievously, he posts oneletter through each door without looking to see if it is the correct address. In how manydiﬀerent ways could he do this so that exactly two of the ﬁve houses receive the correctletters?7. In ∆ ABC  , we have  AC   =  BC   = 7 and  AB  = 2. Suppose that  D  is a point on line  AB  suchthat  B  lies between  A  and  D  and  CD  = 8. What is the length of the segment  BD ?8. In rectangle  ABCD ,  AB  = 5 and  BC   = 3. Points  F   and  G  are on line segment  CD  so that DF   = 1 and  GC   = 2. Lines  AF   and  BG  intersect at  E  . What is the area of ∆ AEB ?9. Suppose that 4 X 1 = 5 , 5 X 2 = 6 , 6 X 3 = 7 ,..., 126 X 123 = 127 , 127 X 124 = 128. What is thevalue of the product  X  1 X  2 ...X  124 ?10.  ABCD  is a square and  AB  = 1. Equilateral triangles  AYB  and  CXD  are drawn such that X   and  Y   are inside the square. What is the length of   XY  ?11. Let  P  ( n ) = ( n + 1)( n + 3)( n + 5)( n + 7)( n + 9). What is the largest integer that is a divisorof   P  ( n ) for all positive even integers  n ?12. If  1   2011 + √  2011 2 − 1= √  m −√  n , where  m  and  n  are positive integers, what is the valueof   m + n ?1  www.examrace.com  13. If   a  =  b − c ,  b  =  c − d ,  c  =  d − a  and  abcd  = 0 then what is the value of   ab  +  bc  +  cd  +  da ?14.  O  and  I   are the circumcentre and incentre of ∆ ABC   respectively. Suppose  O  lies in theinterior of ∆ ABC   and  I   lies on the circle passing through  B ,  O , and  C  . What is themagnitude of    BAC   in degrees?15. How many non-negative integral values of   x  satisfy the equation  x 5  =  x 7  ?(Here [ x ] denotes the greatest integer less than or equal to  x . For example [3 . 4] = 3 and[ − 2 . 3] = − 3.)16. Let  N   be the set of natural numbers. Suppose  f   :  N   →  N   is a function satisfying thefollowing conditions.(a)  f  ( mn ) =  f  ( m ) f  ( n );(b)  f  ( m )  < f  ( n ) if   m < n ;(c)  f  (2) = 2.What is the value of  20  k =1 f  ( k )?17. Let  x 1 ,x 2 ,x 3  be the roots of the equation  x 3 +3 x +5 = 0. What is the value of the expression  x 1  + 1 x 1  x 2  + 1 x 2  x 3  + 1 x 3  ?18. What is the sum of the squares of the roots of the equation  x 2 − 7[ x ] + 5 = 0?(Here [ x ] denotes the greatest integer less than or equal to  x . For example [3 . 4] = 3 and[ − 2 . 3] = − 3.)19. How many integer pairs ( x,y ) satisfy  x 2 + 4 y 2 − 2 xy − 2 x − 4 y − 8 = 0?20.  PS   is a line segment of length 4 and  O  is the midpoint of   PS  . A semicircular arc is drawnwith  PS   as diameter. Let  X   be the midpoint of this arc.  Q  and  R  are points on the arc PXS   such that  QR  is parallel to  PS   and the semicircular arc drawn with  QR  as diameteris tangent to  PS  . What is the area of the region  QXROQ  bounded by the two semicirculararcs? END OF QUESTION PAPER 2  www.examrace.com  Pre-RMO 2012 Answer Key for Set A 1. 152. 3 √  743. 334. 205. 166. 207. 38. 12.59. 3.510. √  3 − 111. 1512. 201113. 0.514. 6015. 916. 21017.  − 29518. 9219. 620. 2 π − 21  www.examrace.com
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