[/quote]
,
and 
be positive integers and let 
be a set of 
of 
![\[ k < \frac {1}{2} + \sqrt {2 \cdot n}
\]](http://latex.artofproblemsolving.com/d/a/5/da56146bf1ad38f257e203e4a0231bd604060a7a.png)
.
and 
. Prove that
be a positive integer. Consider a figure of a equilateral triangle of side 
small equilateral triangles of side 
.
vertices of the small triangles, such that for every integer 
,
,
is the center of the inscribed circle, point 
lies on the side of 
,
.
is equal to the diameter of the circle inscribed in triangle 
from the set of non-negative integers to itself such that 
,
,
.
be the set of real numbers. Let 
be a function such that ![\[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\]](http://latex.artofproblemsolving.com/5/d/3/5d3476fd9195d085d0030d6036a9fde1cf6756a7.png)
for every 
.
.
for every 
or 
for every 
be an integer. Ion draws a regular 
such that 
coprime, 
and 
.
where 
is a parallelogram.
.
is connected to the midpoint of ![$[CT]$](http://latex.artofproblemsolving.com/f/3/f/f3f8e9913dc696e543e6a34d169ec3deb276c84d.png)
,
to the midpoint of ![$[DT]$](http://latex.artofproblemsolving.com/6/d/d/6dd49550ab42ec2f047eaf4b357ac4e7d1387b1e.png)
,
to the midpoint of ![$[AT]$](http://latex.artofproblemsolving.com/6/e/3/6e397da3c2a38b457f7b3504c5cb2a488a64f3a6.png)
and point ![$[BT]$](http://latex.artofproblemsolving.com/6/a/4/6a4b1bcea7921e762a359321ab861b6937f04782.png)
.
.
.
.
be the intersection of diagonals 
and 
.
and 
.
is perpendicular to 
.
have real roots and real coefficients with 
.
?
,
.
.
(I won't be proving this here). This new information allows us to utilize the mean inequality chain as follows:
Similarly, applying AM-HM yields 
.
.
Y by
Let 
have angles 
and 
such that 
and 
. Moreover, suppose that the product of the side lengths of the triangle is equal to its area. Let 
denote the circumcircle of 
. Let 
intersect 
at 
, 
intersect 
at 
, and 
intersect 
at 
. If the area of 
can be written as 
for relatively prime integers 
and 
and squarefree 
, find the sum of all prime factors of , counting multiplicities (so the sum of prime factors of 
is 
), given that has 
divisors.
have angles 
and 
such that 
and 
. Moreover, suppose that the product of the side lengths of the triangle is equal to its area. Let 
denote the circumcircle of 
. Let 
intersect 
at 
, 
intersect 
at 
, and 
intersect 
at 
. If the area of 
can be written as 
for relatively prime integers 
and 
and squarefree 
, find the sum of all prime factors of , counting multiplicities (so the sum of prime factors of 
is 
), given that has 
divisors.
,
of the circumcircle.
.
.
.
,
.
