# KVS JMO Question Paper

**Junior Mathematics Olympiad**

**KENDRIYA VIDYALAYA SANGATHAN**

**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: 3x

^{2}-3xy + y

^{2}=7 and 2x

^{2}- 3xy + 2y

^{2}= 14 5. Let x be the LCM of 3

^{2002}– 1 and 3

^{2002}+ 1. Find the last digit of x. 6. Let f

_{0}(x)=1/1-x and f

_{n}(x)= f

_{0}(f

_{n-1}-(x)) Where n= 1,2,3…. Calculate f

_{2009}(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 E

_{1}and E

_{2}have a common tangent which touches E

_{1}at P and E

_{2}at Q. These two circles meet at M and N where N is nearer to PQ than M. The line PN meets the circle E

_{2}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 a

^{4}+ b

^{4}+ c

^{4}>= abc( a + b + c)

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