Let $ABCD$ be a convex quadrilateral and $A',B'C',D'$ be the midpoints of $AB,BC,CD,DA$, respectively. Let $a,b,c,d$ denote the areas of quadrilaterals into which lines $A'C'$ and $B'D'$ divide the quadrilateral $ABCD$ (where a corresponds to vertex $A$ etc.). Prove that $a+c = b+d$.

Given a triangle with sides of lengths $a,b,c$, prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}< 2$.

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 $A$?