[/quote]
,
such that![\[ f(x) + f(y) + 2xy = f(x + y) \]](http://latex.artofproblemsolving.com/e/3/d/e3dfbb279ff146cc0923115fe9d3f8d9e3488e17.png)
for all nonnegative real numbers 
and 
.
(from the set of real numbers to itself) where
for all reals 
and 
be two triangles inscribed in the same circle, centered at 
,
.
with respect to triangle 
.
.
lie in this order on the circle and form a convex, non-degenerate hexagon.
and 
be positive integers posessing the following property: the equation
holds for all positive integers 
.
for some integer 
.
be a triangle with circumcircle 
and 
a line without common points with 
the foot of the perpendicular from the center of 
intersect 
different from 
,
and 
have a common point different from 
have the property that both![\[
\left\lfloor \sqrt{2n - 1} \right\rfloor \quad \text{and} \quad \left\lfloor \sqrt{3n + 2} \right\rfloor
\]](http://latex.artofproblemsolving.com/d/e/c/dec527c4673470e7023d6aed00fac90268bd2fcc.png)
are consecutive perfect squares.
is an integer.
,
be the foot of the altitude from 
.
be a point in the interior of the segment 
.
be the point on the segment 
such that 
.
be the point on the segment 
such that 
.
be the point of intersection of 
and 
.
.
be a point on segment 
.
be a fixed circle passing through 
be a variable point on 
be the intersection of the tangent to the circumcircle of 
at 
and the tangent to the circumcircle of 
at 
.
Y by
See the image times
Summary of the problem:the time is wrong
Page URL: https://artofproblemsolving.com/community/c10h3546056
Steps to reproduce:impossible
Expected behavior:the time should update
Frequency:once
Operating system(s):Idk
Browser(s), including version:safari
Additional information
Summary of the problem:the time is wrong
Page URL: https://artofproblemsolving.com/community/c10h3546056
Steps to reproduce:impossible
Expected behavior:the time should update
Frequency:once
Operating system(s):Idk
Browser(s), including version:safari
Additional information
This post has been edited 1 time. Last edited by Speedysolver1, Apr 12, 2025, 10:39 AM
.
