[/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 
respectively, and the intersections of 
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 Amir Hossein, ValidName, scimaths, megarnie, Adventure10, Mango247
There are 
mutually external circles drawn on a blackboard, such that no two are tangent and no three share a common tangent. A tangent segment is a line segment that is a common tangent to two circles, starting at one tangent point and ending at the other one. Luciano is drawing tangent segments on the blackboard, one at a time, so that no tangent segment intersects any other circles or previously drawn tangent segments. Luciano keeps drawing tangent segments until no more can be drawn.
Find all possible numbers of tangent segments when Luciano stops drawing.
mutually external circles drawn on a blackboard, such that no two are tangent and no three share a common tangent. A tangent segment is a line segment that is a common tangent to two circles, starting at one tangent point and ending at the other one. Luciano is drawing tangent segments on the blackboard, one at a time, so that no tangent segment intersects any other circles or previously drawn tangent segments. Luciano keeps drawing tangent segments until no more can be drawn.Find all possible numbers of tangent segments when Luciano stops drawing.
(let 
)
segments such that no more can be drawn. Choose a circle and continuously dilate and translate it, together with its segments, while keeping the rest of the picture the same. Do this until either a new segment becomes available or an old segment becomes intersected (wlog assume the latter). If the old segment becomes intersected by a circle, we easily see that at this moment 3 circles must have a common tangent. If the old segment becomes intersected by another segment, assume these 2 segments are tangent to the same circle. (We can assume this wlog if we assume all circles are always 
cm apart). Then again at this moment, 3 circles must have a common tangent.
-
segments.
cm apart (where 
is very big), and continuously deform it circle-by-circle to the configuration previously explained, we see that 
.
.
,
is the number of non-circular faces then 
so 
,
.
,
.
is 
pocket knives.
pocket knives, and WLOG 
in that order and the 4th one 
.
through both 
and 
.
.
pocket knives, and each face of 
.![\[ V+F-E=2 \rightarrow E = 3n-3, \]](http://latex.artofproblemsolving.com/a/b/c/abc67477a0045e86a5ded6b39bf3665da226450c.png)
as desired.
