Find the range of values that can be taken by :
are positive reals.
weakly periodic if it is continuous and 
for all 
.
for all 
for all 
![\[\alpha =1-\cfrac{1}{2a_1-\cfrac{1}{2a_2-\cfrac{1}{2a_3-\cdots}}}\]](http://latex.artofproblemsolving.com/3/0/9/309e0411f481b6d77ccdfe8f18af9852074375aa.png)
where coefficients 
are positive integers, infinitely many of which are greater than 
.
at least half of the numbers 
are even.
be a triangle with 
as incenter.The incircle touches 
at 
.
be the antipode of 
at 
respectively.Let the other tangent from 
meet the other tangent from 
at 
.
meets 
at the midpoint of 
Y by Mango247, Mango247, Mango247
The vertices of a regular 
-gon are painted red, blue and yellow. In one step, you can select two adjacent vertices of different colors and repaint them with a third color. Is it always possible to get a polygon in a few such steps, all vertices of which have a common color?
-gon are painted red, blue and yellow. In one step, you can select two adjacent vertices of different colors and repaint them with a third color. Is it always possible to get a polygon in a few such steps, all vertices of which have a common color?
