OPM 1996

Prove that given three positive numbers, we can choose two of them, say $x$ and $y,$ with $x >y$ such that $$\frac{x-y}{1 +xy }<1.$$ Prove also that if the number $1$ that appears in the second member of the previous inequality is replaced by a lower number, even if very close to $1$ , the previous proposition is false.

A box contains $900$ cards numbered from $100$ to $999$ . Paulo randomly takes a certain number of cards from the box and calculates, for each card, the sum of the digits written on it. How many cards does Paulo need to take out of the box to be sure of finding at least three cards whose digit sums are the same?

Consider a right-angled triangle whose legs are $1$ cm long. Suppose that each point of the triangle was assigned a color from the set of Brown, Blue, Green and Orange colors. It proves that, whatever way this was done, there is at least one pair of points of the same color at a distance equal to or greater than $2-\sqrt 2$ cm from each other.

In a regular polygon with $134$ sides, $67$ diagonals are drawn so that exactly one diagonal emerges from each vertex. We call the length of a diagonal the number of sides of the polygon included between the vertices of the diagonal and which is less than or equal to $67$ . If we order the lengths of the diagonals in ascending order, we obtain a succession of $67$ numbers $(d_1,d_2,...,d_{67})$ . It will be possible to draw diagonals such that

a) $(d_1,d_2,...,d_{67})=\underbrace{2 ... 2}_{6},\underbrace{3 ... 3}_{61}$ ?

b) $(d_1,d_2,...,d_{67}) =\underbrace{3 ... 3}_{8},\underbrace{6 ... 6}_{55}.\underbrace{8 ... 8}_{4}$ ?