RMO 1990 Original Question Paper

1. Two boxes contain between them 65 balls of several different sizes. Each ball is white, black, red or yellow. If you take any 5 balls of the same colour at least two of them will always be of the same size (radius). Prove that there are at least

2. For all positive real numbers a, b, c prove that ≥

3. A square sheet of paper ABCD is so folded that B falls on the mid-point M of CD. Prove that the crease will divide BC in the ratio 5 : 3

4. Find the remainder when 2¹⁹⁹⁰ is divided by 1990.

5. P is any point inside a triangle ABC. The perimeter of the triangle AB + BC + CA = 2s. Prove that s < AP + BP + CP < 2s.

6. N is a 50 digit number (in the decimal scale). All digits except the 26th digit (from the left) are 1. If N is divisible by 13, find the 26th digit.

7. A censusman on duty visited a house which the lady inmates declined to reveal their individual ages, but said — “we do not mind giving you the sum of the ages of any two ladies you may choose”. Thereupon the censusman said — “In that case please give me the sum of the ages of every possible pair of you”. The gave the sums as follows : 30, 33, 41, 58, 66, 69. The censusman took these figures and happily went away. How did he calculate the individual ages of the ladies from these figures.

8. If the circumcenter and centroid of a triangle coincide, prove that the triangle must be equilateral.
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