Let $x,y$ be positive reals such that $x+y=2$. Prove that : $x^3y^3(x^3+y^3) \leq 2$ Source : INMO 2002  My attempt : I started with the left side of the inequality to be proved. $x^3y^3(x^3+y^3) = x^3y^3(x+y)(x^2+y^2-xy) = 2 x^3y^3(x^2+y^2+2xy-3xy)$ $=2x^3y^3(4-3xy)$ How to proceed ? Do I have to some AM-GM or Cauchy-Schwarz on particular set of values ?
If $x,y$ are positive reals such that $x+y = 2$ show that $x^3y^3(x^3+y^3) \leq 2$.   I thought about working backwards on this one. We can try to prove that $x^3y^3(x^3+y^3) \leq 2$ is true by first realizing that $x^3+y^3 = (x+y)(x^2-xy+y^2)$ and substituting in for $x+y = 2$. We then get $x^3y^3(x^2-xy+y^2) \leq 1$. How do I prove this is true?
Why does my minecraft server disconnect the internet when it crashes? As far as I see the "Blatantly off-topic" flag      Blatantly off-topic (this question has nothing to do with %site topic%)    is only available for low-rep users without the close vote privilege.  What is the reason to remove this option for high-rep users? I think it could be very usefull in some cases where now we must select      Other (add a comment explaining what is wrong)    option and waste time to type obvious message for a reason. Creating volumetric lighting (crepuscular rays) in Cycles (a.k.a. "god rays" and many other names) can be created in Blender Internal by checking the Halo checkbox in the lamp properties box.  How can I create a similar effect using Cycles? The documentation for the  function states that you pass it a number, and the positions past the decimal to round. 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In the proof of the lemma     For every limit ordinal $\alpha$, $\textrm{cf } \alpha$ is a regular cardinal    Jech says that     It is easy to see that if $\alpha$ is not a cardinal, then using a mapping of $|\alpha|$ onto $\alpha$ one can construct a cofinal sequence in $\alpha$ of length $\leq |\alpha|$, and therefore $\textrm{cf } \alpha &lt; \alpha$.    I've spent a while trying to construct this cofinal sequence and was wondering if I was on the right path:   From a bijection $f$ between $\kappa = |\alpha|$ and $\alpha$ take $S = \{\beta \in \kappa : \ f(\gamma) &lt; f(\beta)$ for all $\gamma &lt; \beta\}$ then if $\xi$ is the order type of $S$ and $g$ is the isomorphism from $\xi$ to $S$ the function $f\circ g$ is a cofinal sequence in $\alpha$ and $\xi &lt; \alpha$.   Is this on the right track, or is there an easier way of doing it? Explanation behind Second Derivative of a Parametric Equation Formula I am looking to find out how the second derivative formula works.  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We can address the problem in the following way: the sum of the two variables has a density $h(x)$ given by the convolution product of the densities of the two variables, say $f$ and $g$, so that   $$ h(x) = \int_{-\infty}^{\infty} f(x-y)g(y)dy. $$  However,  $$ f(y) = g(y) = \frac{1}{\sqrt {2\pi}}\exp(-y^2/2). $$  Hence now the integral becomes:  $$ h(x) =\frac{1}{2\pi}\int_{-\infty}^{\infty}\exp(-(x-y)^2/2)\exp(-y^2/2)dy. $$  However, I am puzzled here. How should we go on from here? I cannot see how to transform this integral into the integral of a normal r.v of type $N(0,2)$. Thanks in advance for your help. Shell script to send an alert mail if disk usage exceeds 90%