[/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 
.
be a scalene triangle with orthocenter 
and circumcenter 
,
,
.
with diameter 
,
at a point 
.
meets line 
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 Adventure10, Mango247
Diagonals 
and 
of equilateral trapezium 
(
) are orthogonal and intersect each other at point 
. Let 
and 
be altitudes of trapezium. Denote by 
and 
be the midpoints of 
and 
respectively. Prove that 
.
(I. Nagel, Evpatoria)
and 
of equilateral trapezium 
(
) are orthogonal and intersect each other at point 
. Let 
and 
be altitudes of trapezium. Denote by 
and 
be the midpoints of 
and 
respectively. Prove that 
.(I. Nagel, Evpatoria)
.
.
and![$$[ABP]+[DCQ]=\frac{[ABM]+[ABO]}{2}+\frac{[DCN]+[DCO]}{2}=\frac{2-(1-x)^2}{4}<\frac{1}{2}=[AOD].$$](http://latex.artofproblemsolving.com/d/e/8/de85395408341900132d25cc75a1e00109ef0c61.png)
