and 
be on segment 
of an acute triangle 
such that 
and 
.
and 
be the points on 
and 
,
and 
.
and 
is on the circumference of triangle 
be a function satisfying![\[
f(f(x)) = x^2 - 6x + 6
\quad \text{for all} \quad x \in \mathbb{Z}.
\]](http://latex.artofproblemsolving.com/8/a/e/8aebed1e260316d675ca7d104d83fabcd8b5717e.png)
Given that![\[
f(i) < f(i+1) \quad \text{for} \quad i = 0, 1, 2, 3, 4, 5,
\]](http://latex.artofproblemsolving.com/9/f/9/9f92574f9fe14e2e30c3e51e194c661fbd4ece07.png)
find the value of![\[
f(0) + f(1) + f(2) + \cdots + f(6).
\]](http://latex.artofproblemsolving.com/4/0/7/4075d7efa78dda4c46a5cf764fb8852e8c0fb09b.png)
such that for all 
the followings hold:
.
lie in the interiors of the sides 
respectively, such that 
is perpendicular to 
and 
is perpendicular to 
.
on that line.)
be the circumcentre of triangle 
.
and 
be four distinct points in the plane.
and 
are distinct and parallel, then
,
plus the oriented angle 
e
.
equals 0°, then
a
and 
Let 
Prove that ![$$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$](http://latex.artofproblemsolving.com/7/8/e/78e428cbc99ce4e9448f5c4cec86f50567123812.png)
be reals such that 
. Prove that
specifies the greatest common divisor of 
and 
.
and 
Prove that
Let 
Prove that
![\[
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0
\]](http://latex.artofproblemsolving.com/7/8/f/78f8e6c2570684631dc288144e13568d71e68ec8.png)
cannot completely cover a square of side length 
,
,
.
to meet the circumcircle of 
at 
,
.
Y by
We will say that four circles form an Olympian flower if their centers are vertices of a cyclic quadrilateral and every two consecutive circles are tangent. We will call these touchpoints as counting touchpoints . Let 
be a quadrilateral with right angles at 
and 
. Show that there exists exactly one set of four circles that form an Olympian flower whose counting touchpoints are precisely 
and and calculate the radii of the circles in terms of the side lengths of the quadrilateral.
be a quadrilateral with right angles at 
and 
. Show that there exists exactly one set of four circles that form an Olympian flower whose counting touchpoints are precisely 
and and calculate the radii of the circles in terms of the side lengths of the quadrilateral.This post has been edited 2 times. Last edited by parmenides51, Dec 16, 2022, 12:01 PM
