AoPS Academy
be a positive integer and consider the set 
.
of numbers not previously chosen such that for every distinct 
, neither 
divides 
nor divides .
, all multiples of every element in , including those in itself, are removed from 
. until no moves are possible., which player has a winning strategy if any
be an acute triangle. Let 
be the reflection of point 
over the line 
. Let 
and 
be the circumcenter and the orthocenter of triangle , respectively, and let 
be the midpoint of segment 
. Let 
and 
be the points where the reflection of line 
with respect to line 
intersects the circumcircle of triangle , where point lies on the arc not containing . If 
is a point on the line 
such that 
, prove that 
.![$p(x)\in\mathbb{R}[x]$](http://latex.artofproblemsolving.com/9/a/e/9aecb18edc17f7391a44db6e4f9d1502ab1d92f7.png)
such that for all 
:
such that:
be a triangle. Circle 
passes through , meets segments 
and 
again at points and respectively, and intersects segment at 
and 
such that lies between 
and . The tangent to circle 
at and the tangent to circle 
at meet at point 
. Suppose that points and are distinct. Prove that line 
is parallel to .
be an integer greater than zero. Then, the value of the sum of the reciprocals of the cubes of the roots of the equation![\[
mx^4 + 8x^3 - 139x^2 - 18x + 9 = 0
\]](http://latex.artofproblemsolving.com/a/3/a/a3a9bca412e927632e4921c51f2a6593409031d9.png)
is equal to:
be the reciprocals of the roots. Then they solve 
.
and 
are elementary symmetric polynomials give: