and 
,
.
.
and 
on 
and 
are on the same side of 
be midpoint of segment 
be intersection of 
with 
and 
that verify the following equation for all 
:
be positive real integer such that 
Prove that
such that 
and 
that satisfy the equation 
.
be a circle, and let a line 
intersect 
Circles 
and 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
and 
respectively.
the segment 
has a length of 
and 
find the length of segment 
and 
Prove that
.
Prove that
and 
Prove that
,
,
is not greater than 
.
Y by MelonGirl
Given a prime number 
. Consider on the coordinate plane 
the set M of such points 
with integer coordinates that 
, 
. Prove that we can note 
different points of the set 
so that any 
of them are not vertices of the parallelogram, and any 
of them do not lie on the same line.
. Consider on the coordinate plane 
the set M of such points 
with integer coordinates that 
, 
. Prove that we can note 
different points of the set 
so that any 
of them are not vertices of the parallelogram, and any 
of them do not lie on the same line.
for all 
Let 
that 
where 
and 
this becomes 
so we'd have 
ie. 
or 
so there are no parallelograms.
and 
that 
which rearranges to 
ie. 
which is again not possible.
