AoPS Academy
. Let the perpendicular line from 
to 
meets 
at points 
, respectively, and the foot from 
to 
is 
. 
intersects line 
at 
, 
intersects line at 
, and lines 
intersect at 
.
is tangent to 
.
denote the number of prime factors of 
, counted with multiplicity. Evaluate ![\[ \sum_{n=1}^{1989} (-1)^{\Omega(n)}\left\lfloor \frac{1989}{n} \right \rfloor. \]](http://latex.artofproblemsolving.com/2/0/4/2043dca9f2ff2adfd3c8d49fa4dd278871968db0.png)
of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of 
as elements of 
. Prove that 
. or .
+1 w
that satisfy the equation
intersects 
again at 
and circumcircle of 
intersects 
again at 
. It is clear that 
and 
. Hence, triangles 
and 
are similar and 
. Using law of sines for triangles 
and 
we got the following relations. 
. But 
. Hence, 
. Thus 
(we have used the fact of similarity 
and 
). Hence, 
and it means that quadrilateral 
is cyclic. So, 
. But 
and 
. So, 
as desired.
(i dont remember exactly if it was 8032038 or some other number, so correct me on this)
(i posted this at 12:29) lol
?
and 
,
and 
and 
are such that 
and 
.
.