add up to zero. Find the maximum of 
in term of 
'
'
.
be an odd prime and 
be integers so that the integers 
are divisible by 
denote the number of positive divisors of the natural number 
.
.
Y by Adventure10, Mango247
Let 
be a triangle with 
. Let 
be the midpoint of 
, 
the midpoint of 
and 
the foot of the perpendicular from to 
. Prove that 
.
be a triangle with 
. Let 
be the midpoint of 
, 
the midpoint of 
and 
the foot of the perpendicular from to 
. Prove that 
.
,
,
and 
.
,
.![\[ \angle{ACB}=\angle{ABC}=\angle{ADN}+\angle{DAN}=180^o-\angle{AND} \]](http://latex.artofproblemsolving.com/6/7/6/6766995d49937a3f9e3ce7041c6d31bb04cd35db.png)
are cyclic, thus 
.
for which the quadrilateral 
is a rectangle. Observe that 
and the points 
belong to the circle with the diameter 
,
are cyclically.
on the side ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, a point 
and the projection ![$NA\perp NC\Longleftrightarrow \frac{MA}{MD}=\frac{DC}{DB}\Longleftrightarrow [ABM]=[CDM]$](http://latex.artofproblemsolving.com/f/7/1/f715b78e9f21f498ce1437fba9389c3429dc9019.png)
.
(
and 
)
and this is true since 
