are non-negative real numbers such that 
Prove that![\[\sqrt{a^2+6ab}+\sqrt{b^2+6bc}+\sqrt{c^2+6ca} \ge 5\sqrt{1 + \frac{153abc}{50(a+b+c)}}.\]](http://latex.artofproblemsolving.com/4/2/9/4295e20975f0a7c9902a73695bebb665385f01b3.png)
or 
of real numbers such that![\[ a_{m^2 + m + n} = a_{m}^2 + a_m + a_n\]](http://latex.artofproblemsolving.com/b/b/9/bb97053e86e7733fe5c26f7859c87566eb444b52.png)
for all positive integers 
and 
.
be a convex quadrilateral with 
and 
.
and 
be points on segments 
,
intersects lines 
at points 
and 
.
be 
and the midpoint of 
.
and 
lie on a circle.
points are marked on it, dividing it into 
and announces this number to Vasya. To win the game, Vasya needs to color all marked points using 
marked points. Find all 
Y by
In the triangle 
, the segments 
are the heights, 
is their intersection point, 
is the intersection point of 
and 
is the midpoint of the segment 
is the intersection point of the line 
and the side 
. Prove that the line 
is perpendicular to the side 
.
, the segments 
are the heights, 
is their intersection point, 
is the intersection point of 
and 
is the midpoint of the segment 
is the intersection point of the line 
and the side 
. Prove that the line 
is perpendicular to the side 
.
.
,
.
,
,
)
.
and 
.
,![\[
\angle{MTL}=90^\circ+\angle{BAC}-\angle{ACB}
\]](http://latex.artofproblemsolving.com/e/6/5/e65bdb0402c9199716c9d68a0c7ade17746eac45.png)
and![\[
\angle{MSL}=\angle{MKA}+\angle{ACB}.
\]](http://latex.artofproblemsolving.com/9/d/b/9db029da1bf8a72c7ea5a0915319f3ee62e9c46f.png)
.![\[
\angle{MSL}+\angle{MTL}=180^\circ.
\]](http://latex.artofproblemsolving.com/5/9/9/59931f02222f86865b72674fe779d833ebee624d.png)
It follows that 
