[/quote]
,
and 
Prove that
Let 
Prove that
Let 
and 
Prove that
,
chess board, except for the 
central square, there is a checker. In one move, a checker can jump over adjacent checker (diagonally), and this adjacent checker (that was jumped over) is removed. Is it possible to come up with a sequence of moves, so that only one checker is left on the board?
and 
of triangles 
and 
touching their sides 
respectively touch the extension of 
.
meets 
,
and 
.
and 
is right
with 
and 
.
?
that it is possible to form a triangle from segments of length 
.
times and obtain the numbers 
so that segments with lengths 
Prove that
which satisfy 
![$ \frac{a+b+c}{3} \ge \sqrt[10]{\frac{a^3+b^3+c^3}{3}}$](http://latex.artofproblemsolving.com/c/d/a/cdae05ad34239366b0dd3921243522c711b6ae58.png)
method, I can assum 
![$ \frac{2x+\frac{1}{x^2}}{3} \ge \sqrt[10]{ \frac{2x^3+ \frac{1}{x^6}}{3}} $](http://latex.artofproblemsolving.com/c/d/4/cd481672bee1e87d8c97ac623436e0ec75cfa73a.png)
For a positive integer 
,
be the set of bijective functions from 
to itself. For a pair of positive integers 
such that 
,
,
if 
,
?
and a permutation 
?
be the set of non-negative integers and 
be the set of positive real numbers. Let 
be a function such that 
and 
for all integers 
,
for all integers 
.
.
Y by
In a convex quadrilateral 
we draw the bisectors 
, 
, 
, 
of external angles 
, 
, 
, 
respectively. The intersection points of the lines and , and , and , and are designated by 
, 
, 
, 
. It is known that 
perpendiculars drawn from on 
, from om 
, from on 
intersect at one point. Prove that the quadrilateral is cyclic.
we draw the bisectors 
, 
, 
, 
of external angles 
, 
, 
, 
respectively. The intersection points of the lines and , and , and , and are designated by 
, 
, 
, 
. It is known that 
perpendiculars drawn from on 
, from om 
, from on 
intersect at one point. Prove that the quadrilateral is cyclic.
and 
,
.
is the midpoint of 
are among 
.
,
at 
respectively. There is a circle centered at 
touching 
,
is that of 
,
