be non-negative real numbers.Prove that :![\[ \sqrt{x^4+y^4 } +(2+\sqrt{2})xy \geq x^2+y^2 \]](http://latex.artofproblemsolving.com/6/2/3/623d50afffd81b2783ff98568791164b53747567.png)
from the interval ![$[-2,10]$](http://latex.artofproblemsolving.com/d/6/8/d68464eed4c2fe38f134793ffca4a671614f8634.png)
such that their sum is 
.
.
.
such that![\[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\geqslant (a_1x_1+a_2x_2+\ldots +a_nx_n)^2.\]](http://latex.artofproblemsolving.com/4/d/2/4d269e5b682be6778e89237aa91da5e0399042a7.png)
Y by
The circles 
and 
intersect at the distinct points 
and 
. Points 
and 
belong respectively to the circles and so that the chords ![$[MA]$](//latex.artofproblemsolving.com/d/d/f/ddf91ec100609c83be5c154de4e74c7649687ba6.png)
and ![$[MB]$](//latex.artofproblemsolving.com/6/5/c/65cd8f377128d62f97243a68e9ad7956a303d43d.png)
are tangent at point to the circles and , respectively. To prove it that the angles 
and 
are equal.
and 
intersect at the distinct points 
and 
. Points 
and 
belong respectively to the circles and so that the chords ![$[MA]$](http://latex.artofproblemsolving.com/d/d/f/ddf91ec100609c83be5c154de4e74c7649687ba6.png)
and ![$[MB]$](http://latex.artofproblemsolving.com/6/5/c/65cd8f377128d62f97243a68e9ad7956a303d43d.png)
are tangent at point to the circles and , respectively. To prove it that the angles 
and 
are equal.
