Find all pairs of positive integers $(m,n)$ such that $$lcm(1,2,\dots,n)=m!$$where $lcm(1,2,\dots,n)$ is the smallest positive integer multiple of all $1,2,\dots n-1$ and $n$.

Philipe has two congruent regular $4046$-gons. He invites Philomena to a trick he's planning: he'll give her one of the $4046$-gons and she will paint in red $2023$ vertices of her polygon. Without knowing which vertices she chose, he'll paint $k$ vertices of the remaining polygon. After this, he wants to rotate the polygons so that each painted vertices from Philomena's polygon is corresponding to some painted vertice from Philipe's polygon. What is the minimal value of $k$ for which he can choose his vertices so that, no matter how she paints her polygon, the trick is always possible.

A sequence of positive real numbers $a_1, a_2, \dots$ is called $\textit{phine}$ if it satisfies $$a_{n+2}=\frac{a_{n+1}+a_{n-1}}{a_n},$$for all $n\geq2$. Is there a $\textit{phine}$ sequence such that, for every real number $r$, there is some $n$ for which $a_n>r$?

Let $ABC$ be a triangle, $I$ its incenter and $I_a$ its $A$-excenter. Let $\omega$ be its circuncircle and $D$ be the intersection of $AI$ and $\omega$. Let some line $r$ through $D$ cut $BC$ in $E$ and $\omega$ in $F$. The lines $IE$ and $I_aE$ intersect $I_aF$ and $IF$ in $P$ and $Q$, respectively. Furthermore, the circles $PII_a$ and $QII_a$ intersect $I_aE$ and $IE$ in $R$ and $S$, respectively. Prove that there is a circle passing through $F,E,R$ and $S$.