You want to color a flag like the one shown in the following image, for which four different colors are available. Two regions of the flag that share a side (or a segment of a side) must have different colors. The flag cannot be flipped, rotated, or reflected. How many different flags can be colored with these conditions?

Let $a,b,c,d, e$ be real numbers such that they simultaneously satisfy the following equations $$a+b+c+d+e=8$$$$a^2+b^2+c^2+d^2+e^2=16$$Determine the smallest and largest value that $a$ can take.

A scalene acute triangle $ABC$ is drawn on the plane, in which $BC$ is the longest side. Points $P$ and $D$ are constructed, the first inside $ABC$ and the second outside, so that $\angle ABC = \angle CBD$, $\angle ACP = \angle BCD$ and that the area of triangle $ABC$ is equal to the area of quadrilateral $BPCD$. Prove that triangles $BCD$ and $ACP$ are similar.

The letters $A,B,C$ and $D$ each represent a different digit, so each of the four-digit numbers $ABCD$, $BCDA$, $CDAB$ and $DABC$ satisfy that its least prime divisor is equal to $11$. Determine all possible values of the sum $$ABCD +BCDA+CDAB+DABC$$and for each possible value of said sum, give an example of a choice of digits $A,B,C$ and $D$ with which to obtain that value and which satisfies the conditions established above.

Let $ABC$ be a scalene triangle, $\Gamma$ its circumscribed circle and $H$ the point where the altitudes of triangle $ABC$ meet. The circumference with center at $H$ passing through $A$ cuts $\Gamma$ at a second point $D$. In the same way, the circles with center at $H$ and passing through $B$ and $C$ cut $\Gamma$ again at points $E$ and $F$, respectively. Prove that $H$ is also the point in which the altitudes of the triangle $DEF$ meet.

Let $n > 1$ be a natural number. There are $n$ bulbs in a line, each of which can be on or off. Every minute, simultaneously, all the lit bulbs turn off and the unlit bulbs that were adjacent to exactly one lit bulb turn on. Determine for what values of $n$ there is an initial arrangement such that if this process is followed indefinitely, all the lights will never be off.