In a square of side $10$ cm , the vertices are joined to the midpoints on the opposite sides, as shown in the figure. How much does the area of the colored region measure?

A five-digit integer is said to be balanced i f the sum of any three of its digits is divisible by any of the other two. How many balanced numbers are there?

The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?

On a board with $3$ columns and $4$ rows, each of the $12$ squares will be painted green or white. In the first and last row, the number of squares painted green must be the same. Furthermore, in the first and last column, the number of squares painted green must also be unequal. How many different ways can you paint the board?

Let $[ABC]$ be a acute-angled triangle and its circumscribed circle $\Gamma$. Let $D$ be the point on the line $AB$ such that $A$ is the midpoint of the segment $[DB]$ and $P$ is the point of intersection of $CD$ with $\Gamma$. Points $W$ and $L$ lie on the smaller arcs $\overarc{BC}$ and $\overarc{AB}$, respectively, and are such that $\overarc{BW} = \overarc{LA }= \overarc{AP}$. The $LC$ and $AW$ lines intersect at $Q$. Shows that $LQ = BQ$.

A metro network with $n \ge 2$ stations, where each station is connected to each of the others by a one-way line, is said to be dispersed i f there are two stations $A$ and $B$ such that it is not possible to go from $A$ to $B$ through is from the network. If a network is dispersed, but it is possible to choose a station $A$ and reverse the direction of all lines to and from $A$ so that the new network is no longer dispersed, the network is said to be correctable. Indicates all integers $n$ for which there is a network with $n$ stations, dispersed and not correctable.