AT-OEMO-FED1 2009

Show that for all positive integer $n$ the following inequality holds $3^{n^2} > (n!)^4$ .

For a positive integers $n,k$ we define k-multifactorial of n as $Fk(n)$ = $(n)$ . $(n-k)$ $(n-2k)$ ... $(r)$ , where $r$ is the reminder when $n$ is divided by $k$ that satisfy $1<=r<=k$ Determine all non-negative integers $n$ such that $F20(n)+2009$ is a perfect square.

There are $n$ bus stops placed around the circular lake. Each bus stop is connected by a road to the two adjacent stops (we call a segment the entire road between two stops). Determine the number of bus routes that start and end in the fixed bus stop A, pass through each bus stop at least once and travel through exactly $n+1$ segments.

Let $D, E$ , and $F$ be respectively the midpoints of the sides $BC, CA$ , and $AB$ of $\vartriangle ABC$ . Let $H_a, H_b, H_c$ be the feet of perpendiculars from $A, B, C$ to the opposite sides, respectively. Let $P, Q, R$ be the midpoints of the $H_bH_c, H_cH_a$ , and $H_aH_b$ respectively. Prove that $PD, QE$ , and $RF$ are concurrent.