[/quote]
,
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 
and let the second intersection point of the circumcircle of 
be 
.
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 
.
be a polygon formed by the edges of an infinite chessboard, which does not intersect itself. Let the numbers 
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 
has to be constant.
,
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 
)
be two concentric circles. Let 
be any two equilateral triangles inscribed in 
are any two points on ![\[ P'A^2 + P'B^2 + P'C^2 = A'P^2 + B'P^2 + C'P^2. \]](http://latex.artofproblemsolving.com/2/a/7/2a73de753f0332329b743d494272d3fb85f60ef9.png)
(meaning 
takes pairs of positive real numbers to positive real numbers) such that for any real numbers 
,
holds; and
then 
.
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.
