AT-OEMO-FED1 2019

We consider the two sequences $(a_n)_{n\ge 0}$ and $(b_n) _{n\ge 0}$ of integers, which are given by $a_0 = b_0 = 2$ and $a_1= b_1 = 14$ and for $n\ge 2$ they are defined as $a_n = 14a_{n-1} + a_{n-2}$ , $b_n = 6b_{n-1}-b_{n-2}$ . Determine whether there are infinite numbers that occur in both sequences

Let $ABC$ be a triangle and $I$ its incenter. The circle passing through $A, C$ and $I$ intersect the line $BC$ for second time at point $X$ . The circle passing through $B, C$ and $I$ intersects the line $AC$ for second time at point $Y$ . Show that the segments $AY$ and $BX$ have equal length.

Let $n\ge 2$ be an integer. Ariane and Bérénice play a game on the number of the residue classes modulo $n$ . At the beginning there is the residue class $1$ on each piece of paper. It is the turn of the player whose turn it is to replace the current residue class $x$ with either $x + 1$ or by $2x$ . The two players take turns, with Ariane starting. Ariane wins if the residue class $0$ is reached during the game. Bérénice wins if she can prevent that permanently. Depending on $n$ , determine which of the two has a winning strategy.

Find all pairs $(a, b)$ of real numbers such that $a \cdot \lfloor b \cdot n\rfloor = b \cdot \lfloor a \cdot n \rfloor$ applies to all positive integers $n$ . (For a real number $x, \lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$ .)