[/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 governor elections, there were three candidates: 
, 
, and 
. In the first round, received 
of the votes that were cast between and . No candidate obtained the majority needed to win in the first round, and was the one who received the least votes of the three, so there was a runoff between and . The voters for the runoff were the same as in the first round, except for 
of those who voted for , who chose not to participate in the runoff; 
is an integer, 
. It is also known that all those who voted for in the first round also voted for him again in the runoff, but it is unknown what those who voted for in the first round did.
A journalist claims that, knowing all this, one can infer with certainty who will win the runoff. Determine for which values of the journalist is telling the truth.
Note: The winner of the runoff is the one who receives more than half of the total votes cast in the runoff.
, 
, and 
. In the first round, received 
of the votes that were cast between and . No candidate obtained the majority needed to win in the first round, and was the one who received the least votes of the three, so there was a runoff between and . The voters for the runoff were the same as in the first round, except for 
of those who voted for , who chose not to participate in the runoff; 
is an integer, 
. It is also known that all those who voted for in the first round also voted for him again in the runoff, but it is unknown what those who voted for in the first round did.A journalist claims that, knowing all this, one can infer with certainty who will win the runoff. Determine for which values of
the journalist is telling the truth.Note: The winner of the runoff is the one who receives more than half of the total votes cast in the runoff.
