is ![$\left[\sqrt{\alpha n} + \frac{1}{\beta}\right]$](http://latex.artofproblemsolving.com/1/2/b/12b45d9d0f5268002da2adae8dac73390a106d96.png)
(Here ![$\left[.\right]$](http://latex.artofproblemsolving.com/3/c/a/3ca27308567a179890adebe5721a274c19ace40f.png)
denotes GIF) then
Let 
and 
then the number of numbers out of the first 
natural numbers which are divisible by 
or 
is
N
be a triangle and let 
and 
be the circle through 
and 
which is tangent to the incircle of the triangle 
and 
are defined similarly. The circles 
distinct from 
and 
are defined similarly. Prove that the lines 
and 
are concurrent at a point on the line 
.
and 
Prove that
satisfying![\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]](http://latex.artofproblemsolving.com/a/e/6/ae65e3d53931fe9d60d0de76dc79540ccf44c26e.png)
and let 
denote the largest real number satisfying 
for all positive integers 
.
?
Y by Adventure10, Mango247
Let 
be a regular hexagon of side lengths 
and 
its centre, Join cto each of the six vertices , thus getting 
unit line segments in all. Find the number of closed paths from (i) to (ii) 
to each of length 
be a regular hexagon of side lengths 
and 
its centre, Join cto each of the six vertices , thus getting 
unit line segments in all. Find the number of closed paths from (i) to (ii) 
to each of length 
