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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)
+2 w
and 
Prove that
Let 
and Prove that
Let 
and Prove that
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 
, the digits in the base-
representation of 
are all greater than .
and be positive integers with 
. Let 
be a polynomial of degree with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers 
such that the polynomial 
divides , the product 
is zero. Prove that has a nonreal root.
line segment be tangent to 
which is an ellipse at 
with focus as 
and 
such that 
is concyclic. Let 
and 
be other tangents from 
and 
. Prove that 
is concyclic
over the perpendicular bisector of 
to point 
. Clearly also lies on 
, since 
is an isosceles trapezoid. But then we see that 
thus 
and 
are isogonal in 
. But it is a well known fact that and are isogonal in 
, so we see that lies on 
. Thus and 
are reflections of each other about the perpendicular bisector of implying that 
is a cyclic isosceles trapezoid as desired.
