[/quote]
,
be reals such that 
Show that
Let 
and 
Prove that
and 
Find the value range of 
and 
with real coefficients satisfying
such that![\[
4k^3 + 4k + 1
\]](http://latex.artofproblemsolving.com/f/3/5/f355e456cd47ecfdb13597d52fec3d848b67c594.png)
.
and 
of 
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 let the second intersection point of the circumcircle of 
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 
.
lies between the points 
and 
.
be a circle that passes through 
and that is tangent to 
which lies on different semi-planes with 
with respect to the line 
at points 
.
and 
lies on the circumcircle of 
and 
lies on 
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 
such that the inequality 
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 
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 
Y by
There are 
people at a round table. Some of them are knights who always tell the truth, and the rest are liars who always tell lies. Each of those sitting said the phrase: “among the 
people sitting clockwise from where I sit there are as many knights as among the people seated counterclockwise from where I sit”. For what 
could this happen?
people at a round table. Some of them are knights who always tell the truth, and the rest are liars who always tell lies. Each of those sitting said the phrase: “among the 
people sitting clockwise from where I sit there are as many knights as among the people seated counterclockwise from where I sit”. For what 
could this happen?
