p1. The sequence of numbers $123456789101112131415...$ is obtained by writing the positive integers in order, one after the other. What position is $2$ in the first time does $2019$ appear in succession? p2. Consider a square in the plane with vertices $$(a_1, b_1), (a_2, b_2), (a_3, b_3), (a_4, b_4)$$where $a_i$, $b_i$ are integer numbers for each $i = 1, ..., 4$. Suppose the area of the square is a power of $3$. Prove that its sides are parallel to the axes. p3. Prove that for every integer $n> 2$, it is true that $$\frac{1}{n + 1}+ \frac{1}{n + 2}+ ...+ \frac{1}{2n}< \frac56$$ p4. $ABC$ is a triangle of area $4$ with circumcenter $O$ and $M$ is the midpoint of $AO$. We choose the points $P, Q$ on the sides $AB$ and $AC$ respectively such that $M$ is at $PQ$ and segments $BC$ and $PQ$ are parallel. Suppose the area of the triangle $APQ$ is $ 1$. Calculate the angle $BAC$.