RIOPLATENSE 2012

An integer $n$ is called apocalyptic if the addition of $6$ different positive divisors of $n$ gives $3528$ . For example, $2012$ is apocalyptic, because it has six divisors, $1$ , $2$ , $4$ , $503$ , $1006$ and $2012$ , that add up to $3528$ .

Find the smallest positive apocalyptic number.

A rectangle is divided into $n^2$ smaller rectangle by $n - 1$ horizontal lines and $n - 1$ vertical lines. Between those rectangles there are exactly $5660$ which are not congruent. For what minimum value of $n$ is this possible?

Let $T$ be a non-isosceles triangle and $n \ge 4$ an integer . Prove that you can divide $T$ in $n$ triangles and draw in each of them an inner bisector so that those $n$ bisectors are parallel.

Find all real numbers $x$ , such that: a) $\lfloor x \rfloor + \lfloor 2x \rfloor +...+ \lfloor 2012x \rfloor = 2013$ b) $\lfloor x \rfloor + \lfloor 2x \rfloor +...+ \lfloor 2013x \rfloor = 2014$

Let $a \ge 2$ and $n \ge 3$ be integers . Prove that one of the numbers $a^n+ 1 , a^{n + 1}+ 1 , ... , a^{2 n-2}+ 1$ does not share any odd divisor greater than $1$ with any of the other numbers.

In each square of a $100 \times 100$ board there is written an integer. The allowed operation is to choose four squares that form the figure or any of its reflections or rotations, and add $1$ to each of the four numbers. The aim is, through operations allowed, achieving a board with the smallest possible number of different residues modulo $33$ . What is the minimum number that can be achieved with certainty?