[/quote]
,
be positive real numbers. Prove the inequality![\[
\frac{x_1 + 3x_2}{x_2 + x_3}
\;+\;
\frac{x_2 + 3x_3}{x_3 + x_4}
\;+\;
\frac{x_3 + 3x_4}{x_4 + x_1}
\;+\;
\frac{x_4 + 3x_1}{x_1 + x_2}
\;\ge\;8.
\]](http://latex.artofproblemsolving.com/5/c/0/5c062a37f6ff120232ae87eb0093a4ce73ee0eb1.png)
,
is a positive integer.
with exactly 
elements such that the sum of its elements is a prime number.
with at most four natural divisors, which have the property that for any two distinct proper divisors 
and 
of 
,
divides 
satisfying the equation 
with complex coefficients such that 
for any complex number 
and 
be two fixed points in the plane. For each point 
of the plane, outside of the line 
,
be the barycenter of the triangle 
.
.
be a tree on 
leaves. Suppose that there exists a subset of at least 
vertices of 
be an acute scalene triangle. Denote by 
the circle with diameter 
be the contact points of the tangents from 
lie on opposite sides of 
and 
lie on opposite sides of 
.
be the circle with diameter 
,
,
the circle with diameter 
.
are concurrent.
intersects the line 
at 
and 
is the midpoint of 
.
and 
intersect on ![\[x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0\]](http://latex.artofproblemsolving.com/b/9/d/b9d355433e896e8b97f28c2e4235140fd2348e77.png)
correct to four decimal places.
be an integer, 
a real number, and 
be positive real numbers such that 
.![\[
\frac{1 + x_1^k}{1 + x_2} + \frac{1 + x_2^k}{1 + x_3} + \cdots + \frac{1 + x_n^k}{1 + x_1} \geq n.
\]](http://latex.artofproblemsolving.com/0/8/d/08de0c74a4e36b50a64d17875d3fd93eeb5b52de.png)
be positive real numbers. Prove that there exists a cyclic permutation 
of 
such that for all positive real numbers 
the following holds:![\[
\frac{a}{x\,a + y\,b + z\,c}
\;+\;
\frac{b}{x\,b + y\,c + z\,a}
\;+\;
\frac{c}{x\,c + y\,a + z\,b}
\;\ge\;
\frac{3}{x + y + z}.
\]](http://latex.artofproblemsolving.com/1/5/2/15261b3f712c2cc8ffbeb92a9711e06ad0992b5d.png)
Y by
A chessboard with some unit squares marked is called a 
if for any pair of rows 
, a rook placed on a marked square in row 
can reach a marked square in row 
in several moves by only moving to marked squares above, below, or to the right of its current position.
Consider a chessboard with 220 rows and 12 columns, where exactly 9 unit squares in each row are marked. Regardless of how the marked squares are chosen, if it is possible to delete
columns and rearrange the remaining columns to form a determine the maximum possible value of .
if for any pair of rows 
, a rook placed on a marked square in row 
can reach a marked square in row 
in several moves by only moving to marked squares above, below, or to the right of its current position.Consider a chessboard with 220 rows and 12 columns, where exactly 9 unit squares in each row are marked. Regardless of how the marked squares are chosen, if it is possible to delete
columns and rearrange the remaining columns to form a determine the maximum possible value of .
=
