[/quote]
,
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?
be the Fermat point of a 
.
,
,
are concurrent and the point of concurrence is 
,
be a quadrilateral such that the excircles 
and 
of triangles 
and 
touching their sides 
and 
respectively touch the extension of 
at the same point 
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 
and 
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 
.
.
is an integer. Prove that 
are integers.
Y by ImSh95
10. Let 
be an odd number and let 
be an integer such that 
. IN a sports tournament, 
players take part in a series of contests. In each contest, 
players participate, and the scores obtained by the players are the numbers
in some order. Each possible subset of players takes part together in exactly one contest. let the final score of player 
be 
, for each 
. Define 
to be the smallest difference between the final scores of two players, i.e.,
Determine, with proof, the maximum possible value of .
be an odd number and let 
be an integer such that 
. IN a sports tournament, 
players take part in a series of contests. In each contest, 
players participate, and the scores obtained by the players are the numbers
in some order. Each possible subset of players takes part together in exactly one contest. let the final score of player 
be 
, for each 
. Define 
to be the smallest difference between the final scores of two players, i.e.,
Determine, with proof, the maximum possible value of .
