[/quote]
,
be an acute triangle and let 
be an interior point for it such that 
.
When does equality hold?
,
,
and circumcentre 
.
be the midpoint of 
and let 
be a point such that 
is a parallellogram. Suppose that there exists a point 
on 
to 
such that 
.
be the midpoint of 
.
,
.
are the midpoints of 
and 
respectively. Let 
are points on the minor arcs 
and 
of the circumcircle of triangle 
.
and 
are tangent if and only if 
.
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
and 
on 
and 
.
with 
and 
are 
and 
with 
and 
are 
,
and 
are concurrent.
be fixed points on this order on a line. Let 
be a variable circle through 
and 
at the points 
and 
and 
with 
passes through a fixed point independent of 
,
,
satisfying 
?
the sum of all positive divisors of 
be the set of all lines in the plane and 
points in the plane. For a line 
let 
be the number of distinct points among the orthogonal projections of the points from 
.
be an integer. Alice and Bob play the following game on the vertices of a regular 
rounds: In Alice’s turn on the 
-
position clockwise or anticlockwise. If at the end of any person’s turn the two tokens are on the same vertex, then Alice wins the game, otherwise Bob wins. Decide for each value of 
is not a perfect cube of an integer.
such that if 
denotes the number of positive integers less than or equal to 
be an integer and denote 
.
we denote Im 
,
and 
.
I
+ 
.
with 
.![\[
(x^{2}-yz)(y^{2}-zx)(z^{2}-xy).
\]](http://latex.artofproblemsolving.com/a/8/7/a87fae27584012a867bc0a90190c0317f9f7ab07.png)
so that:
with all real numbers 
satisfies 
Y by
There is a family of circles 
forming a figure 
, and a family of equilateral triangles 
with parallel sides and of the same orientation forming a figure 
. It is known that for every triangle 
we have that area 
area . Prove that: area 
area .
forming a figure 
, and a family of equilateral triangles 
with parallel sides and of the same orientation forming a figure 
. It is known that for every triangle 
we have that area 
area . Prove that: area 
area .This post has been edited 4 times. Last edited by parmenides51, Oct 31, 2021, 10:17 PM
