RIOPLATENSE 1992

Let $f:Z \to N -\{0\}$ such that: $f(x + y)f(x-y) = (f(x)f(y))^2$ and $f(1)\ne 1$ .

Provethat $\log_{f(1)}f(z)$ is a perfect square for every integer $z$ .

Determine the integers $0 \le a \le b \le c \le d$ such that: $$2^n= a^2 + b^2 + c^2 + d^2.$$

Let $D$ be the center of the circumcircle of the acute triangle $ABC$ . If the circumcircle of triangle $ADB$ intersects $AC$ (or its extension) at $M$ and also $BC$ (or its extension) at $N$ , show that the radii of the circumcircles of $\triangle ADB$ and $\triangle MNC$ are equal.

On the planet Mars there are $100$ states that are in dispute. To achieve a peace situation, blocs must be formed that meet the following two conditions: (1) Each block must have at most $50$ states. (2) Every pair of states must be together in at least one block. Find the minimum number of blocks that must be formed.

Definition: A natural number is abundant if the sum of its positive divisors is greater than its double.

Find an odd abundant number and prove that there are infinitely many odd abundant numbers.