[/quote]
,
of an equilateral triangle 
,
, points 
and 
so that 
.
and 
is equal to 
.
. 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
over 
.
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)
Y by
In the triangle 
, the segment 
is the angle bisector. The 
-exscribed circle with center at the point 
touches the side of the 
at the point 
and the extension of sides 
and 
at points 
and 
, respectively. It turned out that the length of the segment 
is equal to the radius of this exscribed circle. Prove that the line 
bisects the segment 
.
, the segment 
is the angle bisector. The 
-exscribed circle with center at the point 
touches the side of the 
at the point 
and the extension of sides 
and 
at points 
and 
, respectively. It turned out that the length of the segment 
is equal to the radius of this exscribed circle. Prove that the line 
bisects the segment 
.This post has been edited 1 time. Last edited by parmenides51, Dec 29, 2021, 12:13 PM
if F is foot of altitude from C to AB, because 
,
then 
it suffices to show that 
,
by the angle bisector theorem, so it suffices to show 
.
,
so the equation becomes 
which is clearly true.
