[/quote]
,
prove that![\[ \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{d^2} + \frac{d^2}{a^2}\geq\frac{a}{d} + \frac{b}{a} + \frac{c}{b} + \frac{d}{c}\]](http://latex.artofproblemsolving.com/0/0/4/004e7f148bbb0c797688f5195aa4002dbc316470.png)
If you know same tasks, link them please
be the Fermat point of a 
.
,
,
are concurrent and the point of concurrence is 
,
(
well). i want to study so if you guys have some easy or normal problems please send me
?
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 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 
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 
.
.
Y by
In the cells of a 
board, Julián writes all the integers from 
to 
(inclusive) in any order of his choice, without repeating numbers. For every three consecutive cells on the board, the cell containing the middle value of the three numbers in those cells is marked. For example, if the three numbers are 
, 
and 
, then the cell with is marked. Let 
be the sum of all the numbers in the marked cells. Determine the minimum value that can take.
Note: Each marked number contributes to the sum exactly once, but it can be marked multiple times.
board, Julián writes all the integers from 
to 
(inclusive) in any order of his choice, without repeating numbers. For every three consecutive cells on the board, the cell containing the middle value of the three numbers in those cells is marked. For example, if the three numbers are 
, 
and 
, then the cell with is marked. Let 
be the sum of all the numbers in the marked cells. Determine the minimum value that can take.Note: Each marked number contributes to the sum
exactly once, but it can be marked multiple times.
