such that p is a prime, 
and 
is a divisor of 
.
such that![\[ \frac {2^n + 1}{n^2}
\]](http://latex.artofproblemsolving.com/9/2/2/922444e7669c326efcad5fde3ea00f8c2d9fa5d8.png)
is an integer.
on a circle and points 
and 
on segment 
Find the ratio 
,
is a point on arc 
is maximized.
and 
be the altitudes. Let the points 
and 
fulfil the following requirements:
is the circumcentre of triangle 
and 
are equal to the circumradius of triangle 
has the same direction as the oriented segment 
.
has the same direction as 
,
has the same direction as 
Y by Adventure10
Let 
be a regular trihedral pyramid. The points 
are chosen at lateral edges 
respectively such that the planes 
and 
are parallel. Let 
be the circumcenter of 
. Prove that 
is perpendicular to the plane 
.
(M. Kurylo, Lypova Dolyna)
be a regular trihedral pyramid. The points 
are chosen at lateral edges 
respectively such that the planes 
and 
are parallel. Let 
be the circumcenter of 
. Prove that 
is perpendicular to the plane 
.(M. Kurylo, Lypova Dolyna)
