AoPS Academy
. 
lie on a moving circle 
. intersects 
at 
and 
at 
. 
intersect at 
. 
intersect at 
. 
is the perpendicular image of with respect to 
. Prove that 
and the orthocenter 
are collinear.
be a positive integer. Let 
be the number of pairs of integers 
satisfying 
for odd values of 
. Let 
be the number of pairs of integers satisfying . Prove that 
.
be an acute triangle with 
, let be the midpoint of the side 
, and let 
be the circumcircle of the triangle 
. The tangent of at crosses the line 
at . Let 
be the circumcenter of the triangle 
. Prove that midpoint of the segment 
lies on .
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
. Also, set 
(can scale). Then, by weighted AM-HM, we need to show![\[a(xa+yb+zc)+b(xb+yc+za)+c(xc+ya+zb)\le \frac{x+y+z}3\]](http://latex.artofproblemsolving.com/9/3/d/93d8f3ea4c42c2cc726d97a303a9590e9d08f619.png)
![\[x(a^2+b^2+c^2)+ y(ab+bc+ca)+z(ab+bc+ca)\le \frac{x+y+z}3\]](http://latex.artofproblemsolving.com/d/c/5/dc53184ebb58ace2d5c5fb6bae07279c54fe5ac9.png)
Since 
, both sides are weighted means of 
. But since 
, the weight for is at least 
and the other two weights are at most . And since is less than or equal to 
, we are done.
