The natural number $a_n$ is obtained by writing together and ordered, in decimal notation , all natural numbers between $1$ and $n$. So we have for example that $a_1 = 1$,$a_2 = 12$, $a_3 = 123$, $. . .$ , $a_{11} = 1234567891011$, $...$ . Find all values of $n$ for which $a_n$ is not divisible by $3$.

For a triangle $\vartriangle ABC$, determine whether or not there exists a point $P$ on the interior of $\vartriangle ABC$ in such a way that every straight line through $P$ divides the triangle $\vartriangle ABC$ in two polygons of equal area.

The giraffe is a chess piece that moves $4$ squares in one direction and then a box in a perpendicular direction. What is the smallest value of $n$ such that the giraffe that starts from a corner on an $n \times n$ board can visit all the squares of said board?

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?

Determine all triples $(x, y, z)$ of nonnegative real numbers that verify the following system of equations: $$x^2 - y = (z -1)^2 $$$$y^2 - z = (x -1)^2$$$$z^2 - x = (y - 1)^2$$

Let $P_1$ and $P_2$ be two non-parallel planes in space, and $A$ a point that does not It is in none of them. For each point $X$, let $X_1$ denote its reflection with respect to $P_1$, and $X_2$ its reflection with respect to $P_2$. Determine the locus of points $X$ for the which $X_1, X_2$ and $A$ are collinear.