[/quote]
,
and 
.
such that
non-negative real numbers,but 
0![\[1\leq \frac{a+b}{a+4b+c}+\frac{b+c}{b+4c+a}+\frac{c+a}{c+4a+b}\leq \frac{4}{3}\]](http://latex.artofproblemsolving.com/8/b/7/8b7d7b8cbcab8f2eed31e17aac717fa3c70c92f1.png)
with 
.
be orthocenter 
and 
midpoint 
.
perpendicular to 
.
be a point on the segment 
such that 
and point 
be the intersection of 
and 
.
concyclic
be an integer. Consider an 
chessboard consisting of 
unit squares. A configuration of 
rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer 
such that, for each peaceful configuration of 
square which does not contain a rook on any of its 
unit squares.
students. Let, for 
,
student have 
many friends. For 
let us define 
to be the number of students who have strictly more than 
friends. Show that 
such that:
with all 
is real number
be a line. Let 
at points 
respectively. Given that these six points on 
,
are coaxal.
specifies the greatest common divisor of 
and 
.
,
be the least prime that does not divide 
,
to be the product of all primes less than 
For 
,
.
defined by 
and ![\[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \]](http://latex.artofproblemsolving.com/8/2/5/825be3bbef6a29de69aead03392ca7215093f320.png)
for 
.
.
and 
Prove that
Y by Adventure10
We have four white equilateral triangles of 
cm on each side and join them by their sides to obtain a triangular base pyramid. At each edge of the pyramid we mark two red dots that divide it into three equal parts. Number the red dots, so that when you scroll them in the order they were numbered, result a path with the smallest possible perimeter. How much does that path measure?
cm on each side and join them by their sides to obtain a triangular base pyramid. At each edge of the pyramid we mark two red dots that divide it into three equal parts. Number the red dots, so that when you scroll them in the order they were numbered, result a path with the smallest possible perimeter. How much does that path measure?
