How many multiples of $11$ of four digits, of the form $\overline{abcd}$, satisfy that $a\neq b, b\neq c$ and $c\neq a$?

Let $\omega$ be a circle. For each $n$, let $A_n$ be the area of a regular $n$-sided polygon circumscribed to $\omega$ and $B_n$ the area of a regular $n$-sided polygon inscribed in $\omega$ . Try that $3A_{2015} + B_{2015}> 4A_{4030}$

Ten students are seated around a circular table. The teacher has a list of fifteen problems and each student is given six problems, in such a way that each problem is given exactly four times and any two students they have at most three problems in common. Prove that no matter how the teacher distributes the problems, there will always be two students sitting next to each other who have at least one problem in common.

Let $n$ be a positive integer. Andrés has $n+1$ cards and each of them has a positive integer written, in such a way that the sum of the $n+1$ numbers is $3n$. Show that Andrés can place one or more cards in a red box and one or more cards in a blue box in such a way that the sum of the numbers of the cards in the red box is equal to twice the sum of the numbers of the cards in the blue box. Clarification: Some of Andrés's letters can be left out of the boxes.

Find all positive integers $n$ for which $2^n + 2021n$ is a perfect square.

Two circles $\omega_1$ and $\omega_2$, which have centers $O_1$ and $O_2$, respectively, intersect at $A$ and $B$. A line $\ell$ that passes through $B$ cuts to $\omega_1$ again at $C$ and cuts to $\omega_2$ again in $D$, so that points $C, B, D$ appear in that order. The tangents of $\omega_1$ and $\omega_2$ in $C$ and $D$, respectively, intersect in $E$. Line $AE$ intersects again to the circumscribed circumference of the triangle $AO_1O_2$ in $F$. Try that the length of the $EF$ segment is constant, that is, it does not depend on the choice of $\ell$.