be a positive integer. Prove that there exists a positive integer 
such that exactly 
of the integers 
are squarefree.
be an acute triangle with 
,
of side 
)
.
and 
,
,
.
intersects line 
at a second point 
.
and 
are perpendicular.
be a non-constant monic polynomial with integer coefficients and let 
be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
.
?
Y by Adventure10, Mango247
Given a circle, let 
be a chord that is not a diameter, and let 
be a point on the longer arc . Let 
and 
denote the reflections of 
and 
, respectively, about lines 
and 
, respectively. Prove that the distance between the midpoint of and the midpoint of 
is independent of the choice of .
be a chord that is not a diameter, and let 
be a point on the longer arc . Let 
and 
denote the reflections of 
and 
, respectively, about lines 
and 
, respectively. Prove that the distance between the midpoint of and the midpoint of 
is independent of the choice of .
be midpoints of 
.
be parallelograms. Then 
is also a parallelogram, so 
is also the midpoint of 
.
is fixed, and 
is fixed. So 
is also fixed.
