[/quote]
,
be a set consisting of non-negative integers such that:
,
,
and 
.
such that![\[
f\left(\frac{x^2 - f(x)}{f(x) - 1}\right) = x
\]](http://latex.artofproblemsolving.com/8/0/0/8009a4709ffd6708023d67db2b544e605d1fbe68.png)
for all real numbers 
satisfying 
.
and 
such that![\[ {a^2+b\over b^2-a}\quad\mbox{and}\quad{b^2+a\over a^2-b} \]](http://latex.artofproblemsolving.com/e/3/b/e3bc96af9f635e1c10e21dc91894c8afbf113a6d.png)
.
and 
lie on segments 
and 
,
and 
intersect at point 
.
and 
intersect at a second point 
intersect at a second point 
.
lies on segment 
such that 
.
and 
are similar.
segments 
are parallel. Angle bisectors of 
and 
meet at 
and 
meet at 
is cyclic
and excentres 
and 
,
and 
respectively. The incircle touches 
and 
and 
respectively. Prove that the circles 
and 
have a common point other than 
and 
,![$\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$](http://latex.artofproblemsolving.com/b/f/a/bfa17e8bb4cc05a84ed6a757f506d6d0ed2b0081.png)
and circumcenter 
.
is a parallelogram. Prove that 
.
be the midpoint of side 
.
at 
.
is tangent to 
,
.
,
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 
intersect the circumcircle of 
and 
,
and 
and 
are parallelograms (with 
,
)
be the point of intersection of lines 
and 
.
,
be a scalene triangle with circumcenter 
,
,
at points 
.
intersects the circumcircle of 
.
and 
intersect 
and 
respectively. Prove that the lines 
,
,
are concurrent.
Y by
A non-degenerate triangle 
is given in the plane, let 
be the set of points which lie strictly inside it. Also let 
be the set of circles in the plane. For a point 
, let 
be the reflection of 
in sides 
respectively. Define a function 
such that 
is the circumcircle of 
. Is 
injective?
Note: The function is called injective if for any 
, 
is given in the plane, let 
be the set of points which lie strictly inside it. Also let 
be the set of circles in the plane. For a point 
, let 
be the reflection of 
in sides 
respectively. Define a function 
such that 
is the circumcircle of 
. Is 
injective?Note: The function
is called injective if for any 
, 
This post has been edited 1 time. Last edited by Project_Donkey_into_M4, Apr 20, 2025, 6:24 PM
