![$$\iiint_{[0,1]^3}\frac{x^2y^2z^2}{x^2y^2+y^2z^2+x^2z^2} = 3\iiint_{[0,1]^3}\frac{x^4y^2z^2}{y^2+y^2z^2+z^2}.$$](http://latex.artofproblemsolving.com/1/e/5/1e5f9a03da5b9a74f61d679a8d133c65720477cc.png)
Tried experimenting with a cyclic sum of the RH integrand but couldn't get anywhere due to reduction of degree with the single squared terms in denominator. Explanations much appreciated!
,
and for any 
define![\[ u_{i + 1} = \frac{u_i}{1 + n u_i^2}, \]](http://latex.artofproblemsolving.com/1/8/4/18486d744b8e249208b7f9ce114bd13405d0121e.png)
then prove that![\[ \lim_{n \to +\infty} \sum_{i = 1}^n u_i = \sqrt{3}. \]](http://latex.artofproblemsolving.com/3/f/0/3f0fd9af5fc46654f5f4dd85b6e8c9a8b57848a3.png)
![\[ u_{i + 1} = \frac{u_i}{1 + (n - i) u_i^2}, \]](http://latex.artofproblemsolving.com/b/3/6/b36f69e9c75970adacdd33e6f0228894e20ae466.png)
then prove that![\[ \lim_{n \to +\infty} \sum_{i = 1}^n u_i = 1 + \frac{\pi}{4}. \]](http://latex.artofproblemsolving.com/9/6/6/9662b0a4427f7d127485e704943d5f6b2eb19f63.png)
[/list]
![\[\cos(x) \le \frac{\sin^3(x)}{x^3} \]](http://latex.artofproblemsolving.com/c/d/3/cd376d934c081c70eaabc031e3ff7ef4e4d4cf4f.png)
for 
,
Y by Adventure10, Mango247
Let us consider f as the following function : ![$ f(x)= \frac {x - \sqrt{2}}{ [x \sqrt{2} ] } $](//latex.artofproblemsolving.com/a/9/f/a9f640a2ba0f2c58a1a7085e843e82ae41c72951.png)
1- Find the definition domain
2-Prove that
is continous on 
3-Study the continuity of on 
4-Then draw the geometric representation of on ![$ ] \frac {1}{ \sqrt{2} } ; 2 \sqrt{2} [ $](//latex.artofproblemsolving.com/d/0/4/d04f852d2494e9cd545a1ff20462e7283595c9ce.png)
![$ f(x)= \frac {x - \sqrt{2}}{ [x \sqrt{2} ] } $](http://latex.artofproblemsolving.com/a/9/f/a9f640a2ba0f2c58a1a7085e843e82ae41c72951.png)
1- Find the definition domain
2-Prove that
is continous on 
3-Study the continuity of
on 
4-Then draw the geometric representation of
on ![$ ] \frac {1}{ \sqrt{2} } ; 2 \sqrt{2} [ $](http://latex.artofproblemsolving.com/d/0/4/d04f852d2494e9cd545a1ff20462e7283595c9ce.png)
.
is continuous at 
.
is not continuous at 
.
?
notation) ......corrections made above. 
