p1. Consider the sequence of digits obtained from writing the consecutive natural numbers from $1$ to $100,000$: $$1234567891011121314...9999899999100000$$Determine how many times the $2016$ block appears in this sequence. p2. For an equilateral triangle $\triangle ABC$, determine whether or not there is a point $P$ inside $\triangle ABC$ so that any straight line that passes through $P$ divides the triangle $\triangle ABC$ in two polygonal lines of equal length. p3. On a $1000 \times 1000$ squared board, place domino pieces ($2\times 1$ or $1\times 2$), so that each piece of domino covers exactly two squares of the board. Two domino pieces are not allowed to be adjacent, and they are allowed to be touch in a vertex. Determine the maximum number of domino pieces that can be put following these rules. p4. The product $$\frac12 \cdot \frac24 \cdot \frac38 \cdot \frac{4}{16} \cdot ... \cdot \frac{99}{2^{99}} \cdot \frac{100}{2^{100}}$$is written in its most simplified form. What is the last digit of the denominator? p5. Let $\vartriangle ABC$ be an is osceles triangle with $AC = BC$. Let $O$ be the center of the circle circumscribed to the triangle and $I$ the center of the inscribed circle. If $D$ is the point on side $BC$ such that $OD$ is perpendicular to $BI$. Show that $ID$ is parallel to $AC$. p6. Beto plays the following solitaire: initially a machine chooses at random $26$ positive integers between $1$ and $2016$, and writes them on a blackboard (there may be numbers repeated). At each step, Beto chooses some of the numbers written on the blackboard, and they subtract from each of them the same non-negative integer number k with the condition that the resulting numbers remain non-negative. The objective of the game is to achieve that in sometime the $26$ numbers are equal to $0$, in which case the game ends and Beto win. Determine the fewest steps that guarantee Beto victory, without import the $26$ numbers initially chosen.