[/quote]
,
.
are said to be friends if the following conditions are true:
elements.[/quote]
be a circle on the plane. Let 
and 
be circles which are internally tangent to 
and 
respectively and let the intersection points of 
and 
.
and 
respectively. Let the second intersection point of the line 
and 
and 
be 
and let the intersection points of 
and 
be 
.
be a scalene triangle with incenter 
and incircle 
be 
respectively. Let the line 
intersect the circumcircle of 
.
and 
.
be a circle that passes through 
and that is tangent to 
at points 
.
and 
lies on the circumcircle of 
and 
lies on 
people at a party. Prove that there are two people such that, of the remaining 
people, there are at least 
of them, each of whom either knows both or else knows neither of the two. Assume that knowing is a symmetric relation, and that 
denotes the greatest integer less than or equal to 
.
represent the number of unit squares that have exactly 
edges on the boundary of 
holds for each polygon constructed with these conditions.
such that, for all 
,
is defined as, 
and for all 
,
then 
.
.
,
is also an arithmetic sequence, show that the sequence 
,
are coprime positive integers. Find all 
such that:
such that![\[f(f(f(\frac{x+y}{2}))+x+y)=f(x)+f(y)+f(\frac{x+y}{2})\]](http://latex.artofproblemsolving.com/6/2/4/624afa78b68f810aaaecf0e1e2251047beeddc36.png)
for all rational numbers 
.
in term of a function 
such that 
vertices, each edge has one of the colors 
,
,
.
,
groups such that any two vertices connected by an edge of color 
are in different groups, find the minimum possible value of 
.
be given pairwise coprime positive integers where 
.
be positive integers. We call 
to be a grandson of 
for all functions 
,
for all non-negative real numbers 
.
and 
be sequences of real numbers for which 
and
for all positive integers 
for all 
)
Y by
Let 
be a semicircle with diameter 
and midpoint 
. Let 
be a point on different from 
and 
.
The circle
touches in a point 
, the segment 
in a point 
, and additionally the segment 
. The circle 
touches in a point 
and additionally the segments 
and .
Show that the lines
and 
are perpendicular.
be a semicircle with diameter 
and midpoint 
. Let 
be a point on different from 
and 
.The circle
touches in a point 
, the segment 
in a point 
, and additionally the segment 
. The circle 
touches in a point 
and additionally the segments 
and .Show that the lines
and 
are perpendicular.
