NO-ABEL 1993

If $a,b,c,d$ are real numbers with $b < c < d$ , prove that $(a+b+c+d)^2 > 8(ac+bd)$ .

The Fermat-numbers are defined by $F_n = 2^{2^n}+1$ for $n\in N$ . (a) Prove that $F_n = F_{n-1}F_{n-2}....F_1F_0 +2$ for $n > 0$ . (b) Prove that any two different Fermat numbers are coprime

Each of the $8$ vertices of a given cube is given a value $1$ or $-1$ . Each of the $6$ faces is given the value of product of its four vertices. Let $A$ be the sum of all the $14$ values. Which are the possible values of $A$ ?