and 
positive real numbers. Prove that![\[\frac{1+x_{1}^2}{1+x_{1}x_{2}}+\frac{1+x_{2}^2}{1+x_{2}x_{3}}+\cdots+\frac{1+x_{n}^2}{1+x_{n}x_{1}}\geq n.\]](http://latex.artofproblemsolving.com/e/9/e/e9e73a14f4dfa6315e7562998a9394d371847139.png)
and 
Prove that
Let 
and 
Let 
and 
Let 
and 
Show that![\[\sum_{k=1}^n\frac{1}{1+\sum_{j\neq k}x_j}\le\frac n{1+(n-1)\sqrt[n]{x_1x_2\cdots x_n}}.\]](http://latex.artofproblemsolving.com/e/4/c/e4ca07e5003fad8937af3c44dcbc9906fa48e5b4.png)
with 
the following inequality holds:
and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
,
are all greater than 
and 
.
be a polynomial of degree 
such that the polynomial 
divides 
is zero. Prove that 
Y by Adventure10, Rijul saini
Let 
a point inside 
. Let 
where 
and 
. (All measures are in degrees.)
It can easily be shown that elements of
and 
are interchangeable in their own sets and . e.g. if is a model, then 
is also a model.
I have found five models.
1)
2)
3)
4)
5)
The url http://jsbin.com/uyolet contains a calculator for this kind of problems.
If
or two times of them are integers in degrees, the complete solution set out of above models will be
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a point inside 
. Let 
where 
and 
. (All measures are in degrees.)It can easily be shown that elements of
and 
are interchangeable in their own sets and . e.g. if is a model, then 
is also a model.I have found five models.
1)
2)
3)
4)
5)
The url http://jsbin.com/uyolet contains a calculator for this kind of problems.
If
or two times of them are integers in degrees, the complete solution set out of above models will be6)
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