[/quote]
,
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 
.
and 
then prove 
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 
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 
and incenter 
and let 
,
,
are selected on sides 
,
such that 
,
,
.
intersects 
other than 
and 
meet on 
Y by
Let 
be a triangle and 
be a point that differs from 
. Let 
be the reflection of 
through 
, and 
be the reflection of 
through 
. Similarly, we define 
. Let 
be the line through and perpendicular to 
. Define similarly 
.
a) Assume that is the orthocenter of 
, show that the respective reflection of 
through the bisector of angles 
are the same line.
b) Assume that is the nine-point center of , show that the respective reflection of through 
concur.
be a triangle and 
be a point that differs from 
. Let 
be the reflection of 
through 
, and 
be the reflection of 
through 
. Similarly, we define 
. Let 
be the line through and perpendicular to 
. Define similarly 
.a) Assume that
is the orthocenter of 
, show that the respective reflection of 
through the bisector of angles 
are the same line.b) Assume that
is the nine-point center of , show that the respective reflection of through 
concur.This post has been edited 1 time. Last edited by CheshireOrb, Apr 1, 2021, 6:47 PM
be reflections of 
in 
.
is isogonal conjugate of the line through 
be center of 
.
.
.
at 
we have 
or 
relabel 
.
are isogonal conjugate in 
.
are isogonal conjugate in 
are isogonal conjugate in 
be reflections of 
in 
be center of 
be reflections of 
are isogonal conjugate in 
isogonal conjugate in 
.
or 
.
then 
so 
concur at a point 
as 
and 
as 
.
,
and 
.
Thus it can be shown
.
in 
be the feet of the perpendicular from 
respectively.
be the lines through 
.
,
and the isogonality follows. Now, to prove that these two lines are isogonal, we just need that 
passes through the circumcenter of 
.
.
is a parallelogram! Let 
be the midpoint of 
.
to 
.
and 
and 
.
and the result follows.
,
