AoPS Academy
?
be a triangle and let 
and 
be variable points on sides 
and 
, respectively, such that 
. Let 
denote the point on 
such that 
is parallel to 
, and let 
denote the point on 
such that 
is parallel to 
. Prove that as 
vary, the circumcircle of 
passes through a fixed point, other than 
.
![\[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\]](http://latex.artofproblemsolving.com/7/9/e/79e388eba20412905964dc2b4ebb491f3c79143b.png)
and the sums ![\[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\]](http://latex.artofproblemsolving.com/3/c/4/3c44766edcc9e5269f0d3396386e41676cbdf42c.png)
For any pair define 
and 
, 
.
, 
such that the sequences 
and 
are unbounded while the sequence 
is bounded?
, 
?
real find continuous 
from real numbers to real numbers satisfying the following equation:
real find continuous from real numbers to real numbers satisfying the following equation:
be the assertion.
is injective.
, 
and 
give injectivity.
gives 
. 
gives 
. 
gives 
. is continuous, so 
. Plugging this in the original equation, 
and 
are the only solutions.
also works.
we have 
We also have 
Now we have 
From is continuous we claim that for all 
and 
is constant.
so that 
or 
or 
are solutions.
is bijective as 
shows.
with 
and turn it into cauchy to see is linear, from where we can check for solutions
satisfy that 
.
