Let $ n$ be a natural number. A cube of edge $ n$ may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for $ n$.

Let $\triangle{ABC}$ be a triangle, $D$ the midpoint of $BC$, and $M$ be the midpoint of $AD$. The line $BM$ intersects the side $AC$ on the point $N$. Show that $AB$ is tangent to the circuncircle to the triangle $\triangle{NBC}$ if and only if the following equality is true: \[\frac{{BM}}{{MN}} =\frac{({BC})^2}{({BN})^2}.\]

We have a grid of $k^2-k+1$ rows and $k^2-k+1$ columns, where $k=p+1$ and $p$ is prime. For each prime $p$, give a method to put the numbers 0 and 1, one number for each square in the grid, such that on each row there are exactly $k$ 0's, on each column there are exactly $k$ 0's, and there is no rectangle with sides parallel to the sides of the grid with 0s on each four vertices.

Given a natural number $n \geq 2$, consider all the fractions of the form $\frac{1}{ab}$, where $a$ and $b$ are natural numbers, relative primes and such that: $a < b \leq n$, $a+b>n$. Show that for each $n$, the sum of all this fractions are $\frac12$.

Three tokens $A$, $B$, $C$ are, each one in a vertex of an equilateral triangle of side $n$. Its divided on equilateral triangles of side 1, such as it is shown in the figure for the case $n=3$ Initially, all the lines of the figure are painted blue. The tokens are moving along the lines painting them of red, following the next two rules: (1) First $A$ moves, after that $B$ moves, and then $C$, by turns. On each turn, the token moves over exactly one line of one of the little triangles, form one side to the other. (2) Non token moves over a line that is already painted red, but it can rest on one endpoint of a side that is already red, even if there is another token there waiting its turn. Show that for every positive integer $n$ it is possible to paint red all the sides of the little triangles.

There are $n$ different points $A_1,... , A_n$ in the plane and at each point $A_i$ is assigned a real number $\lambda_i$, distinct from zero, in such way that $(\overline{A_i A_j})^2 = \lambda_i + \lambda_j$ for all $i,j$ with $i\ne j$ . Show that: (1) $n \le 4$ (2) If $n=4$, then $\frac{1}{\lambda_1} + \frac{1}{\lambda_2} + \frac{1}{\lambda_3}+ \frac{1}{\lambda_4} = 0$