$x$,$y$ and $z$ are positive real numbers.  We have to show that $$\frac{y}{xy+2y+1} + \frac{z}{yz+2z+1} + \frac{x}{xz+2x+1}\leq\frac{3}{4}.$$ I am the equivalent of a senior in high school  (not a US citizen) and we didn't take a multi-variable calculus course. Is there a way this could be solved without it?   (It's a problem I found in a math olympiad somewhere,but I can't remember which one.)
Let $x,y,z\ge 0$, show that $$\dfrac{y}{xy+2y+1}+\dfrac{z}{yz+2z+1}+\dfrac{x}{zx+2x+1}\le\dfrac{3}{4}$$  I had solve $$\sum_{cyc}\dfrac{y}{xy+y+1}\le 1$$ becasuse After some simple computations, it is equivalent to  $$(1-xyz)^2\ge 0$$
The dock doesn't work correctly after I enter my password, either. Who else knew that Lord Voldemort was Tom Marvolo Riddle? We know that some people were aware of the fact that Lord Voldemort was Tom Marvolo Riddle prior to his return in Goblet of Fire; at the very least his close soon-to-be-Death-Eater &quot;friends&quot; and Albus Dumbledore:  “Very few people know that Lord Voldemort was once called Tom Riddle. I taught him myself, fifty years ago, at Hogwarts...&quot; ... underwent so many dangerous, magical transformations, that when he resurfaced as Lord Voldemort, he was barely recognizable. Hardly anyone connected Lord Voldemort with the clever, handsome boy who was once Head Boy here.” (Professor Dumbledore to Weasleys, Chamber of Secrets, CHAPTER EIGHTEEN, Dobby’s Reward)  and  “Voldemort,” said Riddle softly, “is my past, present, and future, Harry Potter…” He pulled Harry’s wand from his pocket and began to trace it through the air, writing three shimmering words: TOM MARVOLO RIDDLE Then he waved the wand once, and the letters of his name rearranged themselves: I AM LORD VOLDEMORT “You see?” he whispered. “It was a name I was already using at Hogwarts, to my most intimate friends only, of course. ...  (Tom's Diary manifestation in the Chamber of Secrets).  QUESTION: Who - based on canon (books/interviews/pottermore) - knew about the fact that Lord Voldemort was Tom Marvolo Riddle prior to Ginny being taken to the Chamber of Secrets? Obviously, Albus Dumbledore and Tom himself. Who else? How do I find the caller of a method using stacktrace or reflection? About the anime version of Air:  Some sources on the net state that Kanna's curses were broken, and her next reincarnation would be a happy one.  Some say that the pair of children playing on the beach were the reincarnations of Misuzu-Kanna / Yukito-Ryūya.  What is the canonical explanation of this ending? Did the curses really get lifted? Please include sources in your answer. Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$ Hi I don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please.   The ring that I am talking about is $\mathbb{Z}[x]$ so $\langle 2,x \rangle$ refers to $2g(x) + xf(x)$ where $g(x)$, $f(x)$ belongs to $\mathbb{Z}[x]$. To which extent is general relativity a gauge theory? Does mathematics become circular at the bottom? What is at the bottom of mathematics? I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign), functions and relations.  Now is the "=" taken as undefined? I have seen it been defined in terms of the identity relation.  But in order to talk about functions and relations you need set theory.  However,  seems to be a part of mathematical logic.  Does this mean that (naive) set theory comes before sentential and predicate logic? Is (naive)set-theory at the absolute bottom, where we can define relations and functions and the eqality relation. And then comes sentential logic, and then predicate logic?  I am a little confused because when I took an introductory course, we had a little logic before set-theory. But now I see in another book on introduction to proofs that set-theory is in a chapter before logic. So what is at the bottom/start of mathematics, logic or set theory?, or is it circular at the bottom?  Can this be how it is at the bottom?  naive set-theory $\rightarrow$ sentential logic $\rightarrow $ predicate logic $\rightarrow$ axiomatic set-theory(ZFC) $\rightarrow$ mathematics  (But the problem with this explanation is that it seems that some naive-set theory proofs use logic...)  (The arrows are of course not "logical" arrows.)  simple explanation of the problem:  a book on logic uses at the start: functions, relations, sets, ordered pairs, "="  a book on set theory uses at the start: logical deductions like this: "$B \subseteq A$", means every element in B is in A, so if $C \subseteq B, B \subseteq A$, a proof can be "since every element in C is in B, and every element in B is in A, every element of C is in A: $C \subseteq A$". But this is first order logic? ($(c \rightarrow b \wedge b \rightarrow a)\rightarrow (c\rightarrow a)$).  Hence, both started from each other? Restoring open software after a restart event in windows I find that at the end of a long day, I sometimes have a large number of programs running. All which I will need to use tomorrow. Normally, this isn't an issue, I can simply lock the machine and come back tomorrow. My problem arrises when windows update launches in the middle of the night and force restarts my computer. That in turns closes all my open software. I of course save everything regularly so I don't loose anything, but I waste time reopening all of those resources whenever there is a restart.   [EDIT]  I should clarify that I still want to be able to restart my computer when an update comes down. Preventing the restart only delays the problem until later. I should have been more specific in that I want to be able to recover my working environment after a restart for any reason. Things like scheduled maintence, power loss, updates, and software installs.  [EDIT]  I can't simply have them setup to launch at startup becasuse those files change from week to week. So I need something that monitors what I have open, and gives me the option to "recover" those software sessions when I log back in.  Anyone have any suggestions on what I can do? I'd even be willing to purchase software to do this for me if that is the only option.  Thanks Image uploads not working on Chrome When I try to upload an image, I'm presented with the following dialog box,    The dialog box does not allow uploading either from my computer or by posting a link.  This happens only on Chrome browser (running on Windows). Clearing cache and cookies, and restarting computer do not resolve.  Is this just my computer? Or is this a site-wide bug?  Additional info:  Clicking on 'click' to upload does nothing. Dragging an image onto dialog redirects me to the file location on my computer. Clicking 'Add picture' redirects me to http://[SE-domain].com/upload/image?https=true.  UPDATE: A Norton extension for Chrome was the source of the problem. At  there is this "only in ubuntu" that says "Built in virus protection":  What is the Ubuntu built it protection? What is the program in charge of this and how does it work? What is a more efficient way to draw this tree? I have drawn the following tree in TikZ. What I don't like about it is   that I have to manually adjust the sibling distance for each level; and the syntax is too cumbersome.   It has been suggested to me by @percusse and @AlanMunn that using tikz-qtree and/or forest may be a more efficient way to produce the same result. But I don't have experience in using either package.   It would be nice if someone can reproduce my example using either of these packages, so that I can have quick example to learn from instead of reading a 150-page manual trying to find an answer.    \documentclass{standalone} \usepackage{tikz} \usepackage{amsmath}  \begin{document}  \tikzset{   % Two node styles for game trees: solid and hollow   solid node/.style={circle,draw,inner sep=1.2,fill=black},   hollow node/.style={circle,draw,inner sep=1.2}, }  % macro for entering payoffs \newcommand\payoff[1]{   $\begin{pmatrix} #1 \end{pmatrix}$ }  \begin{tikzpicture}[font=\footnotesize]   \tikzset{     level 1/.style={level distance=15mm,sibling distance=65mm},     level 2/.style={level distance=15mm,sibling distance=30mm},     level 3/.style={level distance=15mm,sibling distance=15mm},     level 4/.style={level distance=15mm,sibling distance=10mm},   }    \node[hollow node,label=above:{Nature}]{}     child{node[solid node,label=left:{P1}]{}       child{node(l1)[solid node]{}         child{node[solid node,label=below:{\payoff{1\\-1}}]{}edge from parent node[left]{L}}         child{node[solid node,label=below:{\payoff{0\\0}}]{}edge from parent node[right]{R}}         edge from parent node[left]{U}       }       child{node(l2)[solid node]{}         child{node[solid node,label=below:{\payoff{0\\0}}]{}edge from parent node[left]{L}}         child{node[solid node,label=below:{\payoff{2\\-2}}]{}edge from parent node[right]{R}}         edge from parent node[right]{D}       }       edge from parent node[left,xshift=-10]{A}     }     child{node[solid node,label=right:{P1}]{}       child{node(r1)[solid node]{}         child{node[solid node,label=below:{\payoff{-2\\2}}]{}edge from parent node[left]{L}}         child{node[solid node,label=below:{\payoff{0\\0}}]{}edge from parent node[right]{R}}         edge from parent node[left]{U}       }       child{node(r2)[solid node]{}         child{node[solid node,label=below:{\payoff{0\\0}}]{}edge from parent node[left]{L}}         child{node[solid node,label=below:{\payoff{-1\\1}}]{}edge from parent node[right]{R}}         edge from parent node[right]{D}       }       edge from parent node[right,xshift=10]{B}     }   ;    \draw[dashed](l1)--(l2)--node[midway,above]{P2}(r1)--(r2); \end{tikzpicture}  \end{document} How can I run a program on startup, minimized? I just want Telegram to be run and I have added it to startup apps. The point is that I need it to be minimized. Any commands? I read this sentence today:  If I were five years old again, I would sleep in every Saturday while I still had the chance!  I am confused why is had used in the last part. When the first part is referring to a hypothetical present/future then how can the last part be in past tense. The answer I received from a native speaker is that :  in type 2 and type 3 conditional sentences, a subordinate clause within the protasis or apodosis goes in the past (sometimes past perfect) tense.  Is this true? If yes, where can I find this information ? These are very advanced grammar concepts for me and any resource suggestion can be very helpful for me. I built a desktop computer a couple years back with the following specs:  CPU: Intel  Yorkfield 2.5GHz 6 MB L2 Cache LGA 775 95W Quad-Core Processor BX80580Q9300 Motherboard:  LGA 775 NVIDIA nForce 680i SLI ATX Intel Motherboard Video Card: Two  GeForce 8600GT SCC 256 MB 128-bit GDDR3 PCI Express x16 SLI Supported Video Card PSU: SeaSonic S12 Energy Plus  ATX12V V2.3 / EPS12V V2.91 SLI Certified CrossFire Ready 80 PLUS Certified Active PFC Power Supply Memory: Two G.SKILL 4 GB (2 x 2 GB) 240-Pin DDR2 SDRAM DDR2 800 (PC2 6400) Dual Channel Kit Desktop Memory   Since its inception, the machine has periodically locked up, the regularity having varied over the years from once a day to once a month. Typically, lockups happen once every few days. By &quot;lockup&quot; I mean my computer just freezes. The screen locks up, I can't move the mouse. Hitting keys on my keyboard that normally turn LEDs on or off on the keyboard (such as Caps Lock) no longer turn the LEDs on or off. If there was music playing at the time of the lockup, noise keeps coming out of the speakers, but it's just the current frequency/note that plays indefinitely. There is no BSOD. When such a lockup occurs I have to do a hard reboot by either turning off the computer or hitting the reset button. I have the most recent version of the NVIDIA hardware drivers, and update them semi-regularly, but that hasn't seemed to help. I am currently using Windows 7 x64, but was previously using Windows Server 2003 x64 and having the same lockup issues. My guess is that it's somehow video driver or motherboard related, but I don't know how to go about diagnosing this problem to narrow down which of the two is the culprit.  Additional information re: cooling Regarding cooling... I've not installed any after-market cooling systems aside from two regular fans I scavenged from an older computer. The fan atop the CPU is the one that shipped with it. One of the two scavenged fans I added it located at the bottom tower of the corner, in an attempt to create some airflow from front to back. The second fan is pointed directly at the two video cards.  SpeedFan installation and readings Per studiohack's suggestion, I installed , which provided the following temperature readings:  GPU: 63C GPU: 65C System: 76C CPU: 64C AUX: 36C Core 0: 78C Core 1: 76C Core 2: 79C Core 3: 79C   Update #3: Another Lockup :-( Well, I had another lockup last night. :-( SpeedFan reported the CPU temp at 38 C when it happened, and there was no spike in temperature leading up to the freeze. One thing I notice is that the freeze seems more likely to happen if I am watching a video. In fact, of the last 5 freezes over the past month, 4 of them have been while watching a video on Flickr. Not necessarily the same video, but a video nevertheless. I don't know if this is just coincidence or if it means anything. (As an aside, each night before bedtime my 2 year old daughter sits on my lap and watches some home videos on Flickr and, in the last month, has learned the phrase, &quot;Uh oh, computer broke.&quot;)  Update #4: MemTest86 and 3DMark06 Test Results: Per suggestions in the comments, I ran the MemTest86 overnight and it cycled through the 8 GB of memory 5 times without error. I also ran the 3DMark06 test without a problem (see my scores at ). So... what now? :-) Any further suggestions on what to check? Is there some way to get a stack trace or something when the computer locks like that? Resolution I have never did figure out the particular problems, but based on the suggestions here and elsewhere, I'm presuming it was a motherboard issue. In any event, I recently upgraded my system, buying a new motherbeard, PSU, CPU, and RAM, and that new rig has been working splendidly the past several weeks. I am using the same graphic cards as in the old setup, so I think it's safe to reason that they weren't the cause of the problem. I have been using LaTeX for a long time now, and I find myself often going to this website to solve my problems. This often includes copy pasting codes which include \newcommand, \newenvironment and \def. I think it is time I start to learn how to use them so I can solve my problems myself! It is always better if you actually know what is going on inside your code.  I am wondering if there is any turtorial or book out there that handles how to use those commands. Or is the best way to learn to just mess with a random file?   Thanks in advance!  (Also this is my first time I post here after using this website for so long, I really appreciate everyone who posts and tries to help other people here!) Normalization of basis vectors with a continuous index? I have an infinite basis which associates with each point, $x$, on the $x$-axis, a basis vector $|x\rangle$ such that the matrix of $|x\rangle$ is full of zeroes and a one by the $x^{\mathrm{th}}$ element. The book on Quantum Mechanics by Shankar says that the inner product between a basis vector and itself is not one, why not? Why can't these basis vectors be normalized to one, only to the ? How does the current limiting resistor for an LED affect current and voltage drops? I'm having a bit of trouble understanding currrent limiting resistors in simple LED circuits. I know that I can determine the optimal resistor like so:  \$\displaystyle R=\frac{V_{s}-V_{f}}{I_{f}}\$  But I'm having a hard time understanding how this one value modifies the voltage and the current to the correct values for the LED. For example, if my calcuations for a super bright blue LED (with \$V_{f}\$ being 3.0-3.4&nbsp;V and \$I_{f}\$ being 80&nbsp;mA, and a voltage source of 5&nbsp;V) gives me 25&nbsp;ohms (using the lower bound of the forward voltage), that's fine. So the current throughout should be 80&nbsp;mA and the voltage drop for the resistor and LED should be 2 and 3 volts, respectively.  But what if I used a 100&nbsp;ohm resistor instead? Or any other value—how would I calculate the voltage drops and current? Would I assume one of them stays the same? How do I debug when the system freezes or when it crashes back to login? I am fairly new to Ubuntu and I was wonder what a good way is to debug when something crashes?   For example: I installed 11.04 and the default media player banshee makes my laptop crash after a few minutes of playing music. Since the system completely freezes and there is not error message it's hard to find out what is wrong.  Second problem I have is that sometimes the system crashes to login. I am suspecting it to be a Chromium or Flash but I am not 100% sure.  So anybody got some tips on how to debug stuff like this? Are there strange attractors whose coordinates are "almost integers"? "ping: unknown host google.com" but IPs works fine