AoPS Academy
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 
. Prove that for every positive integer 
at least half of the numbers 
are even.
be a triangle with 
as incenter.The incircle touches 
at 
.Let 
be the antipode of on the incircle.Make a tangent at to incircle.Let it meet 
at 
respectively.Let the other tangent from 
meet the other tangent from 
at 
.Prove that 
meets 
at the midpoint of
be a triangle with all angles acute, and 
its orthocenter. The bisectors of 
and 
intersect at a point . Show that is collinear with the midpoints of ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
and of ![$[AH]$](http://latex.artofproblemsolving.com/0/3/b/03b8986ebe750b377f987f87b41a1dbc4c128e17.png)
.
be a triangle with all angles acute, and its orthocenter. The bisectors of and intersect at a point . Show that is collinear with the midpoints of and of .
and 
be the altitudes from 
and 
in 
.
.
is cyclic, and similarly 
is cyclic, so 
is cyclic. Let's call the circumcircle 
, then the circumcenter is 
(since 
and is the midpoint of ).
bisects 
, so is the midpoint of arc 
of .
, so 
is cyclic with circumcenter 
, where is the midpoint of 
.
. But we also have 
(since is the midpoint of arc of ), so , , and are collinear.
as , respectively and the feet of the altitude from 
to as 
.
.
. This implies quadrilaterals 
are cyclic, giving 
Then using LoS, 
as desired. is the circumcenter of 
, we have 
. Similarly, with the circumcenter of 
, 
. This is enough to solve the problem.
