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KVS JMO Question Paper

Junior Mathematics Olympiad


12th KVS Junior Mathematics Olympiad – 2009

Time: 3 Hours                                                          M.M. 100

NOTE: Attempt all questions. All questions carry equal marks. The use of electronic devices are strictly prohibited. 1. Consider the following multiplication in decimal notations (999).(abc)= def132, determine the digits a,b,c,d,e,f. 2. Find the greatest number of 4 digits, which when divided by 3,5,7, and 9 leaves remainder 1,3,5 and 7 respectively. 3. If n is a positive integer such that n/810   = 0d25d25 where d is a single digit in decimal base. Find ‘n’. 4. Solve in integers: 3x2-3xy + y2=7 and 2x2– 3xy + 2y2 = 14 5. Let x be the LCM of 32002 – 1 and 32002 + 1. Find the last digit of x. 6.  Let f0(x)=1/1-x and fn(x)= f0(fn-1-(x)) Where n= 1,2,3…. Calculate f2009(2009) 7.  Triangles ABC and DAG are two isosceles triangles with BAC = 20° and ADC= 100°. Show that AB = BC + CD. 8.  Two intersecting circles E1 and E2 have a common tangent which touches E1 at P and E2 at Q. These two circles meet at M and N where N is nearer to PQ than M. The line PN meets the circle E2 again at R. Prove that MQ bisects angle PMR. 9.  AB is a line segment of length 24 cm. and C is its middle point. On AB, AC and CB semi circles are described. Determine the radius of the circle which touches all the three semi circles. 10.  Prove that a4 + b4+ c4 >= abc( a + b + c)