[/quote]
,
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
and 
then prove 
be the set of positive real numbers.
such that there exists a function 
with the property:![\[
f(y^2f(x) + y + c) = xf(x+y^2)
\]](http://latex.artofproblemsolving.com/e/3/0/e30ee6d9f69cc43422a5c2b9813192a632d359cb.png)
for all 
.
be non-negative real numbers such that 
Prove that
and 
.
,![\[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]](http://latex.artofproblemsolving.com/0/c/e/0ce097b0b507403c959c2992cae1217ecdd7e581.png)
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
such that 
holds for all integers 
be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each 
or 
.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
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 
.
and circumcenter 
.
,
,
.
with diameter 
,
at a point 
.
meets line 
at 
.
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
Y by jhu08, HamstPan38825, Adventure10, Mango247
Let 
be a scalene triangle with circumcircle 
, and let 
,
,
be the points where its incircle meets 
, 
, 
respectively. Let the circumcircles of 
, 
, and 
meet a second time at 
respectively. Prove that the perpendiculars from 
to 
respectively are concurrent.
Proposed by Michael Kural
be a scalene triangle with circumcircle 
, and let 
,
,
be the points where its incircle meets 
, 
, 
respectively. Let the circumcircles of 
, 
, and 
meet a second time at 
respectively. Prove that the perpendiculars from 
to 
respectively are concurrent.Proposed by Michael Kural
,
,
.
is the radical axis of 
and 
and that 
is the radical axis of 
and 
and 
,
is the incenter, 
is a cyclic quadrilateral, so 
and 
.
and 
.
so by Carnot's Theorem the perpendiculars to 
from 
respectively are concurrent as desired. 
to 
at 
,
.
,
,
to 
,
be incenter and circumcenter of 
.
.
be antipode of 
.
and thus, 
and we are done by sharky-devil lemma.
)
, we need to show 
.(Easy to see 
)
. This is just straight forward complex numbers with 
as the unit circle and 
,etc
and we're done by symmetry . 
be isogonal conjugate of 
wrt 
form a cyclic hexagon.
are 
respectively and feet from 
.
and thus 
is cyclic. Similarly we obtain that 
and 
are cyclic. Suppose those circles are pairwise different, then by radical axis, we should have 
concurrent, which is clearly false, thus in fact 
is cyclic hexagon. 
,
and 
.
be the isogonal conjugate of 
.
be the feet from 
.
is cyclic hexagon, however 
is also cyclic. Hence, 
,
and 
,
meet 
respectively, the antipodes of 
respectively, and 
,
,
are rectangles.
we know 
passes though 
are concurrent at 
.
respectively. As a result, we know 
concur at the image of 
and 
by Radical Axis. Then, 
and we are done by Carnot's Theorem. 
)
denote the antipodes of 
is the diameter of 
are collinear.
are collinear, so 
.
