AoPS Academy
, prove that there exist infinitely many pairs of positive integers 
(
) such that![\[
\tau(m^k) \tau(m^k + 1) \cdots \tau(n^k - 1) \tau(n^k)
\]](http://latex.artofproblemsolving.com/b/8/6/b863243eaace4276de6b12b7c5a449f1306b0145.png)
is a perfect square, where 
denotes the number of positive divisors of 
.
be a triangle and let 
be a point in its interior. Lines 
, 
, 
intersect sides 
, 
, 
at 
, 
, 
, respectively. Prove that![\[ [PAF]+[PBD]+[PCE]=\frac{1}{2}[ABC] \]](http://latex.artofproblemsolving.com/b/6/f/b6f492b82d7b43d9a5a0a7f9e81205cb8e732bc9.png)
lies on at least one of the medians of triangle . (Here ![$[XYZ]$](http://latex.artofproblemsolving.com/c/7/6/c7608a3e97e44fc5bcf0cd62a1f7dfc0a0a2e7d4.png)
denotes the area of triangle 
.)
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 
.
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 