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 
and 
,
intersects lines 
and 
at points 
.
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 Adventure10, Mango247
Consider a circle 
. The point 
lies outside and a line is drawn through , cutting at distinct points 
and 
. Circles 
and 
are drawn through which are tangent to at and respectively.Prove that the difference of the radii of and is independent of the positions of ,
and .
. The point 
lies outside and a line is drawn through , cutting at distinct points 
and 
. Circles 
and 
are drawn through which are tangent to at and respectively.Prove that the difference of the radii of and is independent of the positions of ,
and .
and 
are joined.
are tangent to each other, 
are collinear.
is a parallelogram.
which is independent of the positions of 
(proved)
and 
which are both radii.
and similarly 
(radii) and so 
(
