that satisfies:![\[
\sum_{i=1}^{n} a_{\left\lfloor \frac{n}{i} \right\rfloor} = n^k
\]](http://latex.artofproblemsolving.com/b/b/e/bbe7fe81e366fe390be5d17193d1581cc40dbece.png)
be a positive integer. Prove that for all integers 
,![\[
\frac{c^{a_n} - c^{a_{n-1}}}{n} \in \mathbb{Z}
\]](http://latex.artofproblemsolving.com/0/4/4/04464c1a350e3f9a929ab51534018498680068fc.png)
Prove that![$$\frac{a}{b}+ \frac{b^2}{c^2} + \frac{kc^2}{a^2}\geq 2\sqrt[4]{4k}$$](http://latex.artofproblemsolving.com/0/9/f/09f1fdf57d62924cd5bae6920b345d84b7ffd9d0.png)
Where 
![$$\frac{a}{b}+ \frac{b^2}{c^2} + \frac{ 2c^2}{a^2}\geq 2\sqrt[4]{8}$$](http://latex.artofproblemsolving.com/0/4/2/042b083074851c68c2810f115e1b6c97a2c53f6f.png)
![$$\frac{a}{b}+ \frac{b^2}{c^2} + \frac{ c^2}{8a^2}\geq \sqrt[4]{8}$$](http://latex.artofproblemsolving.com/b/7/1/b7118d873076582934629efc1170650febbfdda5.png)
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
,
and orthocenter 
.
is constant.
Y by
From the point of intersection of the bisectors of the acute triangle ABC (where AB > AC), the base of the perpendicular descending from the point of intersection of the bisectors of the acute triangle ABC to the side BC is the point D. Find the ratio BD : BA if AD is the bisector of the angle BAC.
