at 
meet at 
.
meets sides 
and 
and 
respectively. Let 
is perpendicular to 
.
for which 
is the square of an integer
of natural numbers with 
of 
if the product of the elements of 
![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
.
and 
.
and 
meet at 
.
,
,
,
to all have integer lengths.
Y by
Let 
be an inner point of an acute-angled triangle 
. Let 
and 
be the projections of on the sides 
and 
respectively . Let 
be the intersection of the perpendiculars on 
and 
from points
and 
respectilvey. Let 
be the projection of on . Show that points 
and lie on a circle.
be an inner point of an acute-angled triangle 
. Let 
and 
be the projections of on the sides 
and 
respectively . Let 
be the intersection of the perpendiculars on 
and 
from points
and 
respectilvey. Let 
be the projection of on . Show that points 
and lie on a circle.
