and 
Prove that
Let 
and 
Prove that
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
Prove that
Let 
Prove that
Let 
and 
Prove that
Let 
Prove that
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
,
,
,
,
.
.
be reals such that 
Prove that
Where 
and 
Prove that 
.
Prove that 
,
,
Point 
is chosen on the circle with radius 
is chosen on the circle with radius 
is chosen on the circle with radius 
,
in terms of 
.
of moving 
unit in the positive direction and probability 
of moving 
.
Y by
Given two natural numbers 
, Xavier and Ze play the following game. First, Xavier writes 
consecutive numbers of his choice; then, repeat some of them, also of his choice, until he has 
numbers, with the condition that the sum of the numbers written is an even number. Ze wins the game if he manages to separate the numbers into two groups with the same amount. Otherwise, Xavier wins. For example, for 
and 
, if Xavier wrote the numbers 
, Ze could win, separating these numbers into groups 
and 
. For what values of and can Xavier guarantee victory?
, Xavier and Ze play the following game. First, Xavier writes 
consecutive numbers of his choice; then, repeat some of them, also of his choice, until he has 
numbers, with the condition that the sum of the numbers written is an even number. Ze wins the game if he manages to separate the numbers into two groups with the same amount. Otherwise, Xavier wins. For example, for 
and 
, if Xavier wrote the numbers 
, Ze could win, separating these numbers into groups 
and 
. For what values of and can Xavier guarantee victory?
