After seeing that my friends on Brilliant are eagerly preparing for RMO, I wish to help them and therefore I am posting the RMO question paper, I also want to know about the correct way of solving these problems (my second purpose for posting these questions). Please post solution also.
Two boxes contain between them balls of several different sizes. Each ball is white, black, red or yellow. If you take any five balls of the same colour, at least of them will always be of the same size (radius). Prove that there are at least three balls which lie in the same box have the same colour and have the same size (radius).
For all positive real numbers prove that .
A square sheet of paper is so folded that falls on the mid-point of of . Prove that the crease will divide in the ratio .
Find the remainder when is divided by .
is any point inside a . The perimeter of the triangle . Prove that
is a -digit number (in a decimal scale). All digits except the digit (from the left) are 1.If is divisible by , find the digit.
A census-man on duty visited a house in 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”. There upon the census man said, “In that case, please give me the sum of the ages of every possible pair of you”. They gave the sums as follows: . The census-man took these figures and happily went away. How did he calculate the individual ages of the ladies from these figures?
If the circumcenter and centroid of a triangle coincide, prove that the triangle must be equilateral.