Prove inequality $(1+x+y)(1+y+z)(1+z+x) \ge 9(xy+xz+yz)$ I need to prove inequality $$(1+x+y)(1+y+z)(1+z+x) \ge 9(xy+xz+yz)$$ for nonnegative $x,y,z \ge 0$  I can do it with partitional derivates $$f(x,y,z)=((1+x+y)(1+y+z)(1+z+x)-9(xy+xz+yz))$$ and find $$ \frac{\partial f}{\partial x}=2xy + 2xz + 2yz + z^2 + y^2 + 2x - 6y - 6z + 2$$ $$ \frac{\partial f}{\partial y}=2xy + 2xz + 2yz + x^2 + z^2 + 2y - 6c - 6x + 2$$ $$ \frac{\partial f}{\partial c}=2xy + 2xz + 2yz + x^2 + y^2 + 2z - 6x - 6y + 2$$ minimum value of function $f$ is $f(1,1,1)=0$ that's why $(1+x+y)(1+y+z)(1+z+x) \ge 9(xy+xz+yz)$.  But I need the prove without partitional derivates.  Any standart inequality (Inequality of arithmetic and geometric means, Cauchy–Schwarz inequality) don't help me.
Prove $(1+a+b)(1+b+c)(1+c+a)\ge 9(ab+bc+ca)$ How one can prove the following.  Let $a$, $b$ and $c$ be non-negative real numbers. Then the inequality holds:  $(1+a+b)(1+b+c)(1+c+a)\ge9(ab+bc+ca).$  WLOG one can assume that $0\le a\le b\le c$.  It is not difficult to prove the statement when $0\le a\le b\le c\le 1$ or $1\le a\le b\le c$.
How to compute $k$'th roots $\!\bmod n$? I have the following equation: $$ n \equiv M^k\mod(b) $$ where n, a, b are integers, and M is unknown. How do I solve this equation to find the M value? Those a and b are public keys of the RSA algorithm and n is the result of encryption. I need to find the encrypted message. Does time freeze at Absolute Zero? Time has many definitions per se, but the basic idea being it's "the measurement of change" so as we know, all matter looses it's ability of changing with the loss of kinetic energy. and the where it becomes zero is the absolute zero (−273.15°C). so my question would be, will time stop/freeze at Absolute zero? and does time get slow while reaching there? Vectorization operator Notification that ‘someone might be watching what you browse’ Convert from MySQL datetime to another format with PHP I think this is a duplicate of an old question without a good answer:  My child was doing schoolwork on her school-issued Macbook Air. Suddenly lighter greys are totally white. This generally gives the images on the screen a vague feeling of being &quot;off&quot;, and pixellates all curved edges. The biggest problem is that text on some of her sites is unreadable. I've checked the following:  Color profiles - it looked like it was using the default, but a cycled through a few built-ins. I could see noticeable changes to color balance, but the light greys remained the same as white. Also deleted contents of ~/Library/ColorSync/. Accessibility - my child has accidentally inverted colors before, this isn't that. No other accessibility settings seemed related to color. Display settings - Modifying brightness was also fruitless. The screen overall got brighter or darker, but still couldn't distinguish between light grey and white Rebooting - colors look fine until shortly after logging in - seems connected to user profile somehow.  Here are some images to demonstrate the issue. This is a generic color test image:  On the Mac, it looks like this (simulated with photoshop):  Any solution would be appreciated, but if there is a keyboard shortcut that can cause this, it would be the most likely explanation (8 year old says they &quot;might have rested their arms on the keyboard&quot; during a class meeting) No Sound (Dummy Output) error Ubuntu 19.10 When is tension constant in a rope? Consider a massless rope with pulling forces applied at each end.  How do we decide if tension is constant or not in a rope? Consider a few example scenarios:  For example, if there is a knot in the rope the tension is not constant throughout (why?. Similarly, if the rope is hung over a cylindrical pulley of a non-negligible radius, the tension is not constant (why?). If there isn't anything touching the rope, for example, 2 people tugging on the rope at each end, the tension is constant (why?). Android Respond To URL in Intent I want my intent to be launched when the user goes to a certain url: for example, the android market does this with  urls. so does youtube. I want mine to do that too. Multiple letters without spacing in Math Boot stuck at logo screen after upgrading to 14.10 $(xy + 1)(yz + 1)(zx + 1) $ is a perfect square if and only if each factor is A singularity, by the definition I know, is a point in space with infinite of a property such as density.  Density is Mass/Volume.  Since the volume of a singularity is 0, then the density will thus become infinite because Mass/0 = undefined  However, is it possible to have a singularity with a mass of 0? 0/0 is indeterminate, but would it be possible for a singularity to exist even if its mass and volume were zero? Lost access to sudo sudo: unable to resolve host   I think I need to edit the /etc/hosts file to make an entry for 127.0.1.1 PC name.  But how can I do that if I can't access sudo? 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You could implement (rather UGLY) auto_ptrs using #defines, if you really wanted to.  This way of thinking about templates helps me understand how my code will actually work, but I feel that I must be missing the point somehow. Macros are meant evil incarnate, yet "template metaprogramming" is all the rage.  So, what ARE the real distinctions? and how can templates avoid the dangers that #define leads you into, like   Inscrutable compiler errors in places where you don't expect them? Code bloat?  Difficulty in tracing code? Setting Debugger Breakpoints?