Maths Olympiad (very hard)

1.
Let triangle ABC be isosceles with AB=BC and let D be a point on its circumcircle.
Denote by T the circle with centre A and radius AB. Let E be the intersection point of AC and BD. Suppose that the circumcircle of triangle CDE intersects T at points C and G. Prove that as the point D varies along the circumcircle of triangle ABC, the line EG passes through a fixed point.
2.
A magician and Siu Ming play a game. There are N sheets of paper on a table. The magician writes some positive integers on each sheet of paper. The magician then
tells Siu Ming to think of a positive integer n not greater than 2013. The game is
played as follows: at hte start of the game, the magician does not know the positive integer n. On each sheet of paper, Siu Ming only needs to tell the magician whether or not n is there. After being told which sheets of paper n appears, the magician
has to say one number. If this number is what Siu Ming was thinking of, then the magician wins, otherwise he loses. For which values of N will the magician
guarantee a win ?
3.
For any positive real numbers a,b,c,k satisfying a+b+c=3, prove that
(2a^2+1)/(ka+1)+(2b^2+1)/(kb+1)+(2c^2+1)/(kc+1) ≧ (3k+3)/(k^2-2k+3)
Find all cases of equality.
4.
Solve the equation
2^m+m^2=3^n
where m,n are nonnegative integers.