A three-digit $\overline{abc}$ number is called Ecuadorian if it meets the following conditions: $\bullet$ $\overline{abc}$ does not end in $0$. $\bullet$ $\overline{abc}$ is a multiple of $36$. $\bullet$ $\overline{abc} - \overline{cba}$ is positive and a multiple of $36$. Determine all the Ecuadorian numbers.

Find how many integer values $3\le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.

Let $ABC$ be a triangle and $D$ be a point on segment $AC$. The circumscribed circle of the triangle $BDC$ cuts $AB$ again at $E$ and the circumference circle of the triangle $ABD$ cuts $BC$ again at $F$. Prove that $AE = CF$ if and only if $BD$ is the interior bisector of $\angle ABC$.

Let $ABCD$ be a square. On the segments $AB$, $BC$, $CD$ and $DA$, choose points $E, F, G$ and $H$, respectively, such that $AE = BF = CG = DH$. Let $P$ be the intersection point of $AF$ and $DE$, $Q$ be the intersection point of $BG$ and $AF$, $R$ the intersection point of $CH$ and $BG$, and $S$ the point of intersection of $DE$ and $CH$. Prove that $PQRS$ is a square.

Bored of waiting for his plane to travel to the International Mathematics Olympiad, Daniel began to write powers of $2$ in a list in his notebook as follows: $\bullet$ Starting with the number $1$, Daniel writes the next power of $2$ at the end of his list and reverses the order of the numbers in the list. Let us call such a modification of the list, including the first step, a move. The list in each of the first $4$ moves it looks like this: $$1 \,\,\,\, \to 2, 1 \,\,\,\, \,\,\,\, \to 4, 1, 2 \,\,\,\, \,\,\,\, \to 8, 2, 1, 4$$Daniel plans to carry out operations until his plane arrives, but he is worried let the list grow too. After $2020$ moves, what is the sum of the first $1010$ numbers?

Let $x_0, a, b$ be reals given such that $b > 0$ and $x_0 \ne 0$. For every nonnegative integer $n$ a real value $x_{n+1}$ is chosen that satisfies $$x^2_{n+1}= ax_nx_{n+1} + bx^2_n .$$a) Find how many different values $x_n$ can take. b) Find the sum of all possible values of $x_n$ with repetitions as a function of $n, x_0, a, b$.