IMO 1973

Establish if there exists a finite set $M$ of points in space, not all situated in the same plane, so that for any straight line $d$ which contains at least two points from M there exists another straight line $d'$ , parallel with $d,$ but distinct from $d$ , which also contains at least two points from $M$ .

Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation $$x^{4} + ax^{3} + bx^{2} + ax + 1 = 0$$ has at least one real root.

$G$ is a set of non-constant functions $f$ . Each $f$ is defined on the real line and has the form $f(x)=ax+b$ for some real $a,b$ . If $f$ and $g$ are in $G$ , then so is $fg$ , where $fg$ is defined by $fg(x)=f(g(x))$ . If $f$ is in $G$ , then so is the inverse $f^{-1}$ . If $f(x)=ax+b$ , then $f^{-1}(x)= \frac{x-b}{a}$ . Every $f$ in $G$ has a fixed point (in other words we can find $x_f$ such that $f(x_f)=x_f$ . Prove that all the functions in $G$ have a common fixed point.

Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that:

a.) $a_{k} < b_{k}$ for all $k = 1, \ldots, n,$ b.) $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$ c.) $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$