VWO 1993

The 20 pupils in a class each send 10 cards to 10 (different) class members. [note: you cannot send a card to yourself.] (a) Show at least 2 pupils sent each other a card. (b) Now suppose we had $n$ pupils sending $m$ cards each. For which $(m,n)$ is the above true? (That is, find minimal $m(n)$ or maximal $n(m)$ )

A jeweler covers the diagonal of a unit square with small golden squares in the following way:

• the sides of all squares are parallel to the sides of the unit square
• for each neighbour is their sidelength either half or double of that square (squares are neighbour if they share a vertex)
• each midpoint of a square has distance to the vertex of the unit square equal to $\dfrac12, \dfrac14, \dfrac18, ...$ of the diagonal. (so real length: $\times \sqrt2$ )
• all midpoints are on the diagonal

(a) What is the side length of the middle square? (b) What is the total gold-plated area?

For $a,b,c>0$ we have: $$-1 < \left(\dfrac{a-b}{a+b}\right)^{1993} + \left(\dfrac{b-c}{b+c}\right)^{1993} + \left(\dfrac{c-a}{c+a}\right)^{1993} < 1$$

Define the sequence $oa_n$ as follows: $oa_0=1, oa_n= oa_{n-1} \cdot cos\left( \dfrac{\pi}{2^{n+1}} \right)$ .

Find $\lim\limits_{n\rightarrow+\infty} oa_n$ .