Let be a sequence $ \left( s_n\right)_{n\geqslant 0} $ of positive real numbers, with $ s_0 $ being the golden ratio, and defined as $$ s_{n+2}=\frac{1+s_{n+1}}{s_n} . $$ Establish the necessary and sufficient condition under which $ \left( s_n\right)_{n\geqslant 0} $ is convergent.

Let $a, b, c$ be positive real numbers. Show that $$a^5+b^5+c^5 \geq 5abc(b^2-ac)$$ and determine when the equality occurs.

Let $ABC$ be a regular triangle. The line passing through the midpoint of $AB$ and parallel to $AC$ meets the minor arc $AB$ of the circumcircle at point $K$. Prove that the ratio $AK:BK$ is equal to the ratio of the side and the diagonal of a regular pentagon.

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that \[ {AM\over AC}={CN\over CE}=r. \] Determine $r$ if $B,M$ and $N$ are collinear.

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that \[ {AM\over AC}={CN\over CE}=r. \] Determine $r$ if $B,M$ and $N$ are collinear.

The points $D$ and $E$ lie on the side $(BC)$ of the triangle $ABC$ such that $D$ is between $B$ and $E.$ A point $R$ on the segment $(AE)$ is called [i]remarkable[/i] if the lines $PQ$ and $BC$ are parallel, where $\{P\}=DR \cap AC$ and $\{Q\}=CR \cap AB.$ A point $R'$ on the segment $(AD)$ is called [i]remarkable[/i] if the lines $P'Q'$ and $BC$ are parallel, where $\{P'\}=BR' \cap AC$ and $\{Q'\}=ER' \cap AB.$ a) If there exists a remarkable point on the segment $(AE),$ prove that any point of the segment $(AE)$ is remarkable. b) If each of the segments $(AD)$ and $(AE)$ contains a remarkable point, prove that $BD=CE=\varphi \cdot DE,$ where $\varphi= \frac{1+\sqrt{5}}{2}$ is the golden ratio.

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that \[ {AM\over AC}={CN\over CE}=r. \] Determine $r$ if $B,M$ and $N$ are collinear.

We say that a prime $p$ is $n$-$\textit{rephinado}$ if $n | p - 1$ and all $1, 2, \ldots , \lfloor \sqrt[\delta]{p}\rfloor$ are $n$-th residuals modulo $p$, where $\delta = \varphi+1$. Are there infinitely many $n$ for which there are infinitely many $n$-$\textit{rephinado}$ primes? Notes: $\varphi =\frac{1+\sqrt{5}}{2}$. We say that an integer $a$ is a $n$-th residue modulo $p$ if there is an integer $x$ such that $$x^n \equiv a \text{ (mod } p\text{)}.$$

Let $\phi = \frac{1+\sqrt5}{2}$. Prove that a positive integer appears in the list $$\lfloor \phi \rfloor , \lfloor 2 \phi \rfloor, \lfloor 3\phi \rfloor ,... , \lfloor n\phi \rfloor , ... $$ if and only if it appears exactly twice in the list $$\lfloor 1/ \phi \rfloor , \lfloor 2/ \phi \rfloor, \lfloor 3/\phi \rfloor , ... ,\lfloor n/\phi \rfloor , ... $$