[/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 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 
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
For each integer 
we consider the last digit different from zero in the decimal expansion of 
. The infinite sequence of these digits starts with 
. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.
we consider the last digit different from zero in the decimal expansion of 
. The infinite sequence of these digits starts with 
. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.
analysis, and thus it suffices to look at the last digit mod t. Then we can verify the sequence repeats, and this gives (I think?) All even numbers.
