p1. Within an equilateral triangle of side $4$, $10$ points are marked. Show that there are two of these points at a distance less than or equal to $\sqrt3$. p2. A certain country has $n$ cities and $\frac{n^2-3n + 4}{2}$ direct flights between some city pairs. Suppose there are no two cities with more than one direct flight between them and that direct flights can be made in either direction. Show that can be reached from any city to any other in the country through some combination of flights. p3. Consider a grid board of $m \times n$ cells. How many squares are crossed by a diagonal line of the board? Clarification: We say that a line crosses a square when it passes through it's interior. p4. The numbers are written on a blackboard $$1\,\,\,2\,\,\,3\,\,\,4\,\,\,5\,\,\,6\,\,\,... \,\,\,21 \,\,\,22$$We want to place symbols $+$ or $-$ in front of each number and consider the result of the corresponding sum. a) Is it possible to place the symbols so that the resulting sum is $0$? b) Show that there is some result that can be obtained by placing the symbols $+$ or $-$ in at least $16000$ different ways. p5. On the squares of a board of $5\times 5$ are written in a disorderly way the numbers from $ 1$ to $25$. Show that there exists a row such that the product of its squares is divisible by $32$. p6. To promote the $25$ years of the National Mathematical Olympiad, the olympic commission made smaller posters and postcards, using exactly the same image. On Professor Cort's desk is there is a poster stretched, and on this, without sticking out, is a postal letter. Professor Cortes notices a very particular phenomenon: there is a point in both images that is located exactly in the same position on the desk. Show that such a phenomenon always exists. Determine if it is possible for there to be more than one point with such a property. PS. Easier versions of P1, P3 were posted as Juniors P1, P3.