KURSCHAK 1992

Define for $n$ given positive reals the strange mean as the sum of the squares of these numbers divided by their sum. Decide which of the following statements hold for $n=2$ :

a) The strange mean is never smaller than the third power mean.

b) The strange mean is never larger than the third power mean.

c) The strange mean, depending on the given numbers, can be larger or smaller than the third power mean.

Which statement is valid for $n=3$ ?

For any positive integer $k$ define $f_1(k)$ as the square of the digital sum of $k$ in the decimal system, and $f_{n}(k)=f_1(f_{n-1}(k))$ $\forall n>1$ . Compute $f_{1992}(2^{1991})$ .

Consider finitely many points in the plane such that no three are collinear. Prove that we can paint the points with two colors such that there is no half-plane that contains exactly three points such that those three points have the same color.