Let $a,b,c&gt;0$, prove that: $$\frac{1}{(2a+b)^2}+\frac{1}{(2b+c)^2}+\frac{1}{(2c+a)^2}\geq\frac{1}{ab+bc+ca}$$  I tried to use the inequality $\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\geq\frac{(a+b+c)^2}{x+y+z} \forall x,y,z&gt;0$ but that's all I can do. I think this inequality is too tight to use AM-GM or Cauchy Schwarz alone.
Let $x,y,z&gt;0$: Prove that:   $\frac{1}{(x+2y)^2}+\frac{1}{(y+2z)^2}+\frac{1}{(z+2x)^2} \geq\frac{1}{xy+yz+zx}$  I tried to apply Cauchy - Schwarz's inequality but I couldn't prove this inequality!
Batch-Resizing images in OS X to a certain height Me and my friends want to listen to music together. Two of us have Bluetooth Headphones (Beats Solo2 Wireless, Beats Studio Wireless) and one has a wired headset. Is there any way to use all three of these together? Short story about a woman that travels to other planets. Read online recently. She just manages to live long enough to embrace new technology How do I search for questions with specific tags?       For more information about searching, see &quot;&quot; in the . Have Chrome autodetect language before spelling checking? What is a rocket engine thrusting against in space? I know Newton's third law of motion might be the answer for this but still I am wondering how the rockets could thrust in the empty space and move in the opposite direction. I guess an astronaut wouldn't be able to push in the empty space with his hands or legs to move himself, but with a rocket engine it's possible. How? What might be the explanation for this in General Relativity? How to establish the following identities without the help of calculus:  For positive integer $n, $ $$\sum_{1\le r\le n}\frac{(-1)^{r-1}\binom nr}r=\sum_{1\le r\le n}\frac1r $$  and  $$\sum_{0\le r\le n}\frac{(-1)^r\binom nr}{4r+1}=\frac{4^n\cdot  n!}{1\cdot5\cdot9\cdots(4n+1)}$$ I will be writing scripts for ArcGIS Desktop in Python with ArcPy.   Is there a community or open source project where code and models can be shared? What constitutes an observation/measurement in QM? Merging two pictures Is there a tool in Ubuntu to automatically merge two images having a common strip? I need to scan a large image, which will not fit at once in the scanner. I was thinking about scanning it in two parts and hand-editing the result in Gimp to join them. But probably an automatic tool can do the same more quickly and better.  EDIT I think I may have not made my point clear. My difficulty is not aligning two images. My problem is that one has to compensate for the rotation or distortion introduced by the manual process of putting the sheet into the scanner, and an automatic tool should obtain more precise results at analyzing common patterns in the two images to rotate and align them seamlessly. Why is variance squared? The mean absolute deviation is:   $$\dfrac{\sum_{i=1}^{n}|x_i-\bar x|}{n}$$   The variance is: $$\dfrac{\sum_{i=1}^{n}(x_i-\bar x)^2}{n-1}$$   So the mean deviation and the variance are measuring the same thing, yet variance requires squaring the difference. Why? Squaring always gives a non-negative value, but the absolute value is also a non-negative value. Why isn't it $|x_i-\bar x|^2$, then? Squaring just enlarges, why do we need to do this?  A similar question is , but mine is a little different. Thanks. My directory (in Linux) looks like com/com/{various,files} where com and com are folders of the same name and the second com holds the various files. I want the first com removed, but the second and files in the second kept. How to filter by messages, not by conversation? I have been emailing my Gmail account error messages from a project I am working on.   I am trying to use Gmail filters/labels to sort out the class of errors.  But because the subject line is basically always the same even if I have a specific filter for 3 known error cases, the label is applied to the conversation, instead of the specific message.  Therefore, all the errors get all the filters.  I do not yet want to delete the errors, but would rather file them separately.   I have not yet used the NOT contains string yet, but that looks to be non-scalable as I get to the fourth or fifth error case I am done with handling.  Is there any option other than turning off Conversations for all my email?  (I guess I could use one of my spare accounts to turn off conversations and do the work there but would be nice to have such an option). List the files containing a particular word in their text How could it be that the entropy of gas is not infinite?  If we use the formula $S=k\ln(\Omega)$ we get infinity because there are an infinite number of possible situations (because there are infinite possibilities of position and momentum for each molecule). Creating Centrelines from Road Polygons/Casings using ArcGIS Desktop? I want to create center lines from road polygons.   Are there any automated tools in ArcGIS for Desktop to do this?  I am including a Screenshot of my vector file below. I need to find the closure of these sets on the ordered square: $$A = \{(1/n)\times 0 | n\in \mathbb{Z}_+\}$$ $$B = \{(1-1/n)\times \frac{1}{2}| n\in \mathbb{Z}_+\}$$ $$C = \{x\times 0 | 0&lt;x&lt;1\}$$ $$D = \{x\times \frac{1}{2}| 0&lt;x&lt;1\}$$ $$E = \{\frac{1}{2}\times y | 0&lt;y&lt;1\}$$ I know that the closure is just the set itself united with its limit points. For $A$, the set of limit points, I imagine, are $\{&lt;1,0&gt;\}$, that is, the $x$ coordinate is $1$ and the $y$ is $0$. But  is says the reverse. What am I getting wrong? Following the same reasoning, $B' = \{&lt;1,\frac{1}{2}&gt;\}$ $C' = \{&lt;0,0&gt;,&lt;0,1&gt;\}$ What am I doing wrong? Let a,b,c be positive real number, such that  $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=a+b+c$. Prove that :  $\frac{1}{(2a+b+c)^2}+\frac{1}{(2b+c+a)^2}+\frac{1}{(2c+a+b)^2} \leq \frac{3}{16}$  Can anyone help me how to deal with it? Two-Weapon Fighting & Bonus Actions During the latest gaming session our level 3 Rogue was Two-Weapon fighting with a Rapier in the main-hand and a scimitar in the off-hand.  At the completion of our gaming session I contested the application of the Two-Weapon fighting rules as it seemed unbalanced.  Once I got home I reviewed the PHB and various RPGStack and Reddit but couldn't find a concrete answer.  My interpretation of the PHB Reference rules below is that although any class/character can technically wield a weapon in their off-hand, only those who have been granted a Bonus Action for which no explicit bonus action have been defined would be allowed to actually attack.  So in the case of a Rogue the Bonus Actions granted under Cunning Action allow for specific actions (none of which are attack).  The alternate interpretation would be the rules under Two-Weapon Fighting, perhaps it should be written      you receive a bonus action to attack with a different light melee weapon that you’re holding in the other hand.   This would, in my opinion, explicitly grant the Bonus Action to any character/class who is Two-Weapon Fighting.  After thorough review does it boil down to application of the rules-as-written and determining a house-rule to apply in our campaign?    References  Page 189 of PHB : Bonus Actions  Various class features, spells, and other abilities let you take an additional action on your turn called a bonus action. The Cunning Action feature, for example, allows a rogue to take a bonus action.      You can take a bonus action only when a special ability, spell, or other feature of the game states that you can do something as a bonus   action. You otherwise don’t have a bonus action to take.   You can take only one bonus action on your turn, so you must choose which bonus action to use when you have more than one available. You choose when to take a bonus action during your turn, unless the bonus action’s timing is specified, and anything that deprives you of your ability to take actions also prevents you from taking a bonus action.  Page 96 of PHB  A Rogue receives cunning action at level 2, You can take a bonus action on each of your turns in combat.      This action can be used only to take the Dash, Disengage, or Hide action.   Page 195 of PHB  Two-Weapon Fighting (Combat Rules)  When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative. If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.  Page 165 of PHB  Dual Wielder  (Feat)  You master fighting with two weapons, gaining the following benefits:   You gain a +1 bonus to AC while you are wielding a separate melee weapon in each hand. You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren't light. You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one. Which package can be used to write BNF grammars?