AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be an integer, 
a real number, and 
be positive real numbers such that 
. Prove 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)
× is coloured black and white like a chessboard. The following steps are permitted: Choose a rectangle inside the board (consisting of entire cells)whose side lengths are both odd or both even, but not both equal to 
, and invert the colours of all cells inside the rectangle. Determine the values of for which it is possible to make all the cells have the same colour in a finite number of such steps.
a grid of 9*9, we choose a little square in of this grid three times, we can choose three times the same.
and a positive real number 
, non-negative real numbers 
satisfy 
. Define 
as the number of elements in the following collection of subsets:
.
, from 
outside the circle, draw tangents 
(
are tangential points) and secant 
(
lie on circle 
, 
is between and 
). 
intersects 
at 
; 
is midpoint of 
. The line crossing , paralleling with 
, intersects at 
. 
intersects 
at 
. Let 
lie inside the triangle 
such that: 
=
. From draw chords 
, 
, 
on circle . Prove that: 
is a isosceles triangle
with 
colors. We call a vertex rainbow if it's connected to all of the colors. At most how many rainbows can exist?
be an integer. You have two 
boards. Each board contains the numbers to 
inclusive, one number per square, arbitrarily arranged on each board. A move consists of exchanging two rows or two columns on the first board (no moves can be made on the second board). Show that it is possible to make a sequence of moves such that for all 
and 
, the number that is in the 
row and 
column of the first board is different from the number that is in the row and column of the second board.
be real numbers such that 
and 
Prove that
Equality holds when 
Equality holds when 
Equality holds when 
are pairwise distinct 
numbers and 
, find the minimum value of![\[
\left( \frac{a}{b - c} + \frac{b}{c - a} + \frac{c}{a - b} \right)^2 + a^2 + b^2 + c^2.
\]](http://latex.artofproblemsolving.com/3/6/9/3694d92e77e5049af2ec32f1dc1879dcc74f5936.png)
, there exists some positive integer 
that makes the following equation have no integer root 
.
are nonnegative because 
. We aim to show that there must exist a less than 
that cannot be equal to the left-hand side of the given equation. We count the number of possible values of each so that 
.
, assume for the sake of contradiction that 
. When we multiply this by itself 
times we see that 
, which we don't want. to be less than 
. There are possible values for in the range 
. When we take the product of for all such that , we get ![\[ \prod\limits_{i=1}^n 2^{2^{n-i}} = 2^{\sum\limits_{i=1}^n 2^{n-i}} = 2^{2^n - 1}\]](http://latex.artofproblemsolving.com/e/4/d/e4d0e5a2f1977c4e12bec6fa560fbe9166f30418.png)
This value is an upper bound for the amount of tuples 
such that the LHS of the given equation is less than . However, this value 
is less than the 
possible values of less than . Thus, by pigeonhole there must exist a less than that cannot be equal to the LHS of the given equation.
.
Where 
.
be real numbers such that 
and 
Prove that
is always false. If it turns out this equation can be true for some large