After Part 1, here's the second one. This time the problems are from RMO, conducted in various states of India. Have fun, and please post hints/solutions as comments, like, re-share, and enjoy!
1 ** Two boxes contain between them total 65 balls of several different sizes. Each ball is of any one colour from white, black, red, or yellow. If we take any 5 balls of the same colour, at least two of them always have same size. Prove that there are at least 3 balls which have the same box, same colour, and the same size.
2 A square sheet of paper ABCD is so folded that B falls on the mid-point of CD. Prove that the crease will divide BC in the ratio 5:3.
3 'N' is a 50 digit number. All digits of N are 1, except the digit from the left. Find it, given that 13 divides N.
4 If , and , and , prove that is a square.
5 ABCD is a cyclic quadrilateral with perpendicular diagonals AC and BD, intersecting at E. Prove that , where O is the center of the circle.
6 Let ABCD be a rectangle with and . Suppose is the radius of the circle passing through A and B and touching CD, and is the radius of the circle passing through B and C and touching AD. Show that .