Let $ABC$ be an equilateral triangle with side length $8$, and let $D$, $E$, and $F$ be points on the sides $BC$, $CA$, and $AB$ respectively. Given that $BD = 2$ and $\angle ADE = \angle DEF = 60^\circ$, calculate the length of segment $AF$.

Ana and Beto play the following game with a stick of length $15$. Ana starts, and on her first turn, she cuts the stick into two pieces with integer lengths. Then, on each player's turn, they must cut one of the pieces, of their choice, into two new pieces with integer lengths. The player who, on their turn, leaves at least one piece with length equal to $1$ loses. Determine which of the two players has a winning strategy.

a) Find an example of an infinite list of numbers of the form $a + n \cdot d$, with $n \geqslant 0$, where $a$ and $d$ are positive integers, such that no number in the list is equal to the $k$-th power of an integer, for all $k = 2, 3, 4, \dots$. b) Find an example of an infinite list of numbers of the form $a + n \cdot d$, with $n \geqslant 0$, where $a$ and $d$ are positive integers, such that no number in the list is equal to the square of an integer, but the list contains infinitely many numbers that are equal to the cubes of positive integers.

Find all pairs $(a, b)$ of positive rational numbers such that $$\sqrt{a}+\sqrt{b} = \sqrt{2+\sqrt{3}}.$$

Let $A_1A_2\cdots A_n$ be a regular polygon with $n$ sides, $n \geqslant 3$. Initially, there are three ants standing at vertex $A_1$. Every minute, two ants simultaneously move to an adjacent vertex, but in different directions (one clockwise and the other counterclockwise), and the third stays at its current vertex. Determine all the values of $n$ for which, after some time, the three ants can meet at the same vertex of the polygon, different from $A_1$.

A list of $7$ numbers is constructed using the following procedure: each number in the list is equal to the sum of the previous number and the previous number written in reverse order. For example, if a number in the list is $23544$, the next number is $68076 = 23544 + 44532$. (It is forbidden for any number in the list to start with $0$, although the reversed numbers may start with $0$.) Decide whether it is possible to choose the first number of the list so that the seventh number is a prime number.