[/quote]
,
defined on all real numbers and taking real values such that ![\[f(f(y)) + f(x - y) = f(xf(y) - x),\]](http://latex.artofproblemsolving.com/f/5/2/f528fc4047e21247663d4993913eadf346315be7.png)
for all real numbers 
and 
Find minimum 
is inscribed into a circle 
.
be the centroid of 
,
be an altitude of this triangle. A ray 
meets 
.
is tangent to 
denote the set of positive integers. For any positive integer 
,
is called 
for all 
.
be real numbers such that 
Prove that
Let 
Prove that
Prove that
is written on the blackboard. Every time, Carmela can erase the number 
on the black board and replace it with a new number 
,
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
be a triangle inscribed in a circle, where the tangents to the circle at points 
.
and 
.
and 
intersect at point 
.
be the reflection of 
and 
intersect at point 
,
.
.
such that for all positive integer 
,
Where 
denotes the number of divisors of 
.
,
such that for all positive integer 
.
then![\[\sum |\pm z_1\pm z_2\pm z_3|^2=2^3\sum_{i=1}^3|z_k|^2.\]](http://latex.artofproblemsolving.com/6/d/1/6d1cd28a47704aa0cac2a6c6b572816fc1db790a.png)
Generalize the problem.
and 
be their respective pedal triangles. Let![\[ X_1=P_2Q_3\cap P_3Q_2,\quad X_2=P_1Q_3\cap P_3Q_1,\quad X_3=P_1Q_2\cap P_2Q_1 \]](http://latex.artofproblemsolving.com/7/f/e/7fe3ec4f69ec074635b2202daad9371e3bb8c72d.png)
Prove that the points 
,
and 
lie on the line 
.
be the intersections of 
with the sides of 
are the midpoints of 
,
,
and 
.
![$ a,b\in[\frac{1}{2},1] $](http://latex.artofproblemsolving.com/0/d/2/0d24f742e1f7db553b3a8795c061c796f07a897f.png)
.
Let ![$ a,b\in[\frac{1}{2},2] $](http://latex.artofproblemsolving.com/a/8/d/a8d4c8205bac0e34129f582a8f255355eda12e0b.png)
.
Y by Adventure10
Let 
, 
, 
, 
, 
be points such that 
is a cyclic quadrilateral and 
is a parallelogram. The diagonals 
and 
intersect at 
and the rays 
and 
intersect at 
. Prove that 
.
(4th Middle European Mathematical Olympiad, Team Competition, Problem 6)
, 
, 
, 
, 
be points such that 
is a cyclic quadrilateral and 
is a parallelogram. The diagonals 
and 
intersect at 
and the rays 
and 
intersect at 
. Prove that 
.(4th Middle European Mathematical Olympiad, Team Competition, Problem 6)
on 
and 
(case 
is similar). It's easy to prove that 
is inscribed quadrilateral, and 
.
hence 
is similar to 
which gives 
,
the rest is obvious by chasing angle
,
is cyclic. Therefore, 
.
.
.
and 
are similar. Therefore, their attitudes are in the same ratio as the respective bases, namely 
.
,
.
(angles with perpendicular rays). Now we have the desired 
,
,
and 
.
is tangent to the circumcircle of triangle 
,
.
and 
are similar, 
.
.
is also midpoint of the other diagonal 
.
and lines 
are parallel. In particular, we have 
.
.
,
and 
are similar and hence 
,
be the points of intersection of the lines 
and 
respectively.
are concyclic and 
and take 
on 
.
,
.
(1).
,
maps to 
.
