AoPS Academy
be a positive integer. Prove that there exists a positive integer 
such that exactly 
of the integers 
are squarefree.
be an acute triangle with 
, midpoint 
of side 
, altitude 
(
), and orthocenter 
. A circle passes through points 
and 
, is tangent to line 
, and intersects the circumcircle of triangle at a second point 
. The circumcircle of triangle 
intersects line 
at a second point 
. Prove that the lines 
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 
.
, with two of the points, say 
in its interior.
meet two side of the triangle (recall that no three points are collinear), say 
and 
. Thus 
are in some order the vertices of a convex quadrilateral, and we are done.
points be 
, 
, 
, 
, and 
. We first consider , , , and . If each of these is in the exterior of the triangle determined by the other 
points, we are done by 
. If one of these is in the interior, however, WLOG let it be . We now proceed with casework based on the position of . is in the exterior of triangle 
, the lines , , and 
split the plane into regions. If is in the region determined by and , then 
is convex, and analogous reasoning applies to the other 
regions. is in the interior of triangle , extend 
bast to intersect 
at , and define 
and 
similarly. Then, if is in the interior of triangle 
or 
, 
or 
is convex respectively. Analogous reasoning applies to the other pairs of triangles. 
?
points forming a convex quadrilateral.
