VWO 1994

Let $a,b,c>0$ the sides of a right triangle. Find all real $x$ for which $a^x>b^x+c^x$ , with $a$ is the longest side.

Determine all integer solutions (a,b,c) with $c\leq 94$ for which: $(a+\sqrt c)^2+(b+\sqrt c)^2 = 60 + 20\sqrt c$

Two regular tetrahedrons $A$ and $B$ are made with the 8 vertices of a unit cube. (this way is unique)

What's the volume of $A\cup B$ ?

Let $(f_i)$ be a sequence of functions defined by: $f_1(x)=x, f_n(x) = \sqrt{f_{n-1}(x)}-\dfrac14$ . ( $n\in \mathbb{N}, n\ge2$ ) (a) Prove that $f_n(x) \le f_{n-1}(x)$ for all x where both functions are defined. (b) Find for each $n$ the points of $x$ inside the domain for which $f_n(x)=x$ .