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10th KVS Junior Mathematics Olympiad (JMO) – 2007 M.M. 100 Time : 3 hours Note : Attempt all questions. 1. Solve | x-1 | + | x | + | x + 1 | = x + 2 2. Find the greatest number of four digits which when divided by 3, 5, 7, 9 leaves remainders 1, 3, 5, 7 respectively. 3. A printer numbers the pages of a book starting with 1. He uses 3189 digits in all. How many pages does the book have ? 4. ABCD is a parallelogram. P, Q, R and S are points on sides AB, BC, CD and DA respectively such that AP=DR. If the area of the parallelogram is 16 cm2, find the area of the quadrilateral PQRS. 5. ABC is a right angle triangle with B = 90o. M is the mid point of AC and BM = 117 cm. Sum of the lengths of sides AB and BC is 30 cm. Find the area of the triangle ABC. 6. Solve : ( a x ) (a x ) ( a x ) (a x ) 7. a x Without actually calculating, find which is greater : 31 11 or 17 14 8. Show that there do not exist any distinct natural numbers a, b, c, d such that a3 + b3 = c3 + d3 and a + b = c + d 9. Find the largest prime factor of : 312 + 212 – 2.66 10. If only downward motion along lines is allowed, what is the total number of paths from point P to point Q in the figure below ? P Q