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 Adventure10, Mango247
Points 
, 
, 
, 
are such that no three of them are collinear. Let 
and 
. Let 
, 
and 
denote the circles with diameter 
, 
and 
, respectively. Prove that either 
pass through one point, or no two of them intersect.
, 
, 
, 
are such that no three of them are collinear. Let 
and 
. Let 
, 
and 
denote the circles with diameter 
, 
and 
, respectively. Prove that either 
pass through one point, or no two of them intersect.
