Jose and Maria play the following game: Maria writes $2019$ positive integers different on the blackboard. Jose deletes some of them (possibly none, but not all) and write to the left of each of the remaining numbers a sign $+$or a sign $-$. Then the sum written on the board is calculated. If the result is a multiple of $2019$, Jose wins the game, if not, Maria wins. Determine which of the two has a winning strategy.

A group of $10$ friends attend an amusement park. Each has visited three different attractions . Leaving the park and talking to each other, they found that any pair of friends visited at least one attraction in common. Determine what could be the minimum number of friends who could walk in the most visited attraction.

On a circle $\omega$ with center O and radius $r$ three different points $A, B$ and $C$ are chosen. Let $\omega_1$ and $\omega_2$ be the circles that pass through $A$ and are tangent to line $BC$ at points $B$ and $C$, respectively. (a) Show that the product of the areas of $\omega_1$ and $\omega_2$ is independent of the choice of the points $A, B$ and $C$. (b) Determine the minimum value that the sum of the areas of $\omega_1$ and $\omega_2$ can take and for what configurations of points $A, B$ and $C$ on $\omega$ this minimum value is reached.