[/quote]
,
. Prove that: 
,
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
In a scalene triangle 
, the points 
and 
lie on the lines 
and 
, respectively, such that the circumcircles of triangles 
and 
are tangent to the line 
. Let 
be the intersection of and 
. Prove that 
is perpendicular to the Euler line of triangle .
, the points 
and 
lie on the lines 
and 
, respectively, such that the circumcircles of triangles 
and 
are tangent to the line 
. Let 
be the intersection of and 
. Prove that 
is perpendicular to the Euler line of triangle .This post has been edited 2 times. Last edited by parmenides51, Aug 20, 2021, 1:15 PM
Reason: replaced isosceles with scalene, see #5
Reason: replaced isosceles with scalene, see #5
. Below, I bashed the correct parameters for 
.
,
and 
,
are 
and 
when written in barycentric coordinates wrt 
and so 
.
,
.
is 
and 
and 
,
.
so the conditions on the perpendicularity are (up to permutation) 
.
is isosceles, point 
,
and the perpendicularity holds.
seems to be a nice choice, it is enough to prove that 
.
is on side 
and point 
is on side 
and the line 
cuts 
be the circumcenter and the orthocenter, respectively of 
both lie on the circle 
that is the reflection of 
in 
cut 
at 
is the polar of 
is the polar of 
is the polar of 
)
with altitude 
.
is the 
-
,
lies on 
cut 
then as 
,
collinear. But obviously 
,
and 
cut 
at 
cut 
at 
collinear we will be done since as 
mean 
collinear and by the Lemma we obtain the desired perpendicularity.
.
.
(1).
with 
: 
(2)
:
(3)
or 
,
and 
and it is the foot of the perpendicular from the orthocenter 
,
