I found this question some time ago in an Elementary Olympiad book:  If $a, b, c, d$ are positive integers such that $abcd =1$, then prove that $(1+a)(1+b)(1+c)(1+d) \geq 16$.  Evidently this was a direct consequence of Hölder's inequality, so I merely gave the following proof. $$(1^4+(a^{\frac{1}{4}})^4)(1^4+(b^{\frac{1}{4}})^4)(1^4+(c^{\frac{1}{4}})^4)(1^4+(d^{\frac{1}{4}})^4) \geq (1+(abcd)^{\frac{1}{4}})^4 $$ $$(1+a)(1+b)(1+c)(1+d) \geq (1+1)^4$$ $$(1+a)(1+b)(1+c)(1+d) \geq 16$$ QED.  However, I was requested by my instructor to try the problem again using only the AM-GM inequality or the Cauchy-Schwarz inequality, because it is supposedly solvable using only those two. At this point I expanded the expression believing that all the terms so formed would be of the degree $4$, and would, when taken together in groups, yield a power of $abcd$ on the GM side, which could then be added. Doing so, however, I only get the weaker result that $(1+a)(1+b)(1+c)(1+d) \geq 14$. I would appreciate a hint for this tantalizing problem.
Suppose $a_1, a_2,\dots , a_n$ are $n$ positive real numbers with $a_1a_2 \dots a_n = 1$.   Then what is the minimum value of $(1 + a_1)(1 + a_2). . .(1 + a_n)$ ?    I think $(1 + a_1)(1 + a_2). . .(1 + a_n)$ takes its minimum value when $a_1=a_2=\dots=a_n=1$ and thus the minimum value is $2^n$.  I don't know how to prove it. Please help.
Can I see a list of gyms I left a pokemon at? I captured a gym today just before the servers went down. I am wondering if I lost my progress. Is there a way to see if I own a gym or not? How does Earth's interior dynamo work? I'm interested in getting a basic physical understanding of how Earth's magnetic field is generated. I understand that it's a "dynamo" type of effect, driven by convection currents in the molten outer core. These currents cause charges to move, and this generates the field.  However, what I can't find a good explanation of is why there is a separation of charges in the first place. Presumably, moving neutrally charged molten iron would have the same effect as moving any other neutrally charged thing, i.e. it wouldn't create a field.  And presumably, if the fluid wasn't moving then it would become neutral pretty quickly, since molten iron is a good electrical conductor.  So am I right in thinking that the charge separation is the result of positive feedback, in that an intial deviation from neutrality would generate a field, and this would (somehow) cause a greater separation of charges, resulting in a kind of self-maintaining charge separation?  Or is there another explanation?  In either case, does anyone know of a good resource that explains the basic principles in physical terms?  I know that the interior dynamo is a very complex phenomenon, but I'd like something that gives a good physical picture of how the electromagnetic and fluid dynamical phenomena interact, rather than diving straight into partial differential equations. Is there a single word for "moving unhinderedly"? How do I install an older version of an app on an iPhone? I am working on plugin to Drupal Commerce, and I started getting this error:   DatabaseTransactionOutOfOrderException: in DatabaseConnection-&gt;rollback() (line 1058 of /Library/WebServer/Documents/includes/database/database.inc).   Can anyone shed some light on what this error means? If I install a fresh drupal and reinstall my plugin, is that likely to clear things up?   Any advice?   Thanks! Wifi losing connection, weak signal, Intel 7260 adapter I yesterday upgraded from Windows to Ubuntu 15.04 but I have problem with my wireless adapter Intel Corporation Wireless 7260 (rev 73). It randomly loses internet connection. Restarting adapter always helps. (same it did with Windows, except that sometimes restarting adapter didn't help).   Additionally, I have weaker connection then I had with Windows.  Thank you! I have a sword that currently rotates around a pivot point thats off of the actual object. Im new to blender and im not sure how i would go about changing it so that the sword rotates around the handle. Any help would be appreciated! Ubuntu 18.04 error on waking up from sleep : Read-error on swap device After the laptop has been in sleep mode for a couple of hours,when I try and resume my session I get the following error:     Read-error on swap device    It takes about 30 seconds for the login screen to load post this happening. Once I log in, the screen blanks out once or twice for a second, and none of my programs are open anymore. I get a "System problem detected" prompt. When I click on "Send Report", another notification pops up saying:     Sorry the program "Xorg" closed unexpectedly. Your computeer does not have enough free memory to automatically analyze the problem and send a report to the developers.   What I have tried so far is to increase the swap space available. It was around 2 GB initially and I created another swap file of 9 GB. This hasn't helped. The occupied swap space ( as per the swapon command ) after the crash is always around 170MB.  The DMESG for when I resume my session, up to the read error on swap device is as follows:       [64046.474054] ACPI: Low-level resume complete [64046.474162] ACPI: EC: EC started [64046.474162] PM: Restoring platform NVS memory [64046.475139] Enabling non-boot CPUs ... [64046.475196] x86: Booting SMP configuration: [64046.475196] smpboot: Booting Node 0 Processor 1 APIC 0x2 [64046.475663]  cache: parent cpu1 should not be sleeping [64046.475859] CPU1 is up [64046.475910] smpboot: Booting Node 0 Processor 2 APIC 0x4 [64046.476330]  cache: parent cpu2 should not be sleeping [64046.476506] CPU2 is up [64046.476539] smpboot: Booting Node 0 Processor 3 APIC 0x6 [64046.477071]  cache: parent cpu3 should not be sleeping [64046.477255] CPU3 is up [64046.477274] smpboot: Booting Node 0 Processor 4 APIC 0x1 [64046.477721]  cache: parent cpu4 should not be sleeping [64046.477922] CPU4 is up [64046.477947] smpboot: Booting Node 0 Processor 5 APIC 0x3 [64046.478371]  cache: parent cpu5 should not be sleeping [64046.478571] CPU5 is up [64046.478591] smpboot: Booting Node 0 Processor 6 APIC 0x5 [64046.479018]  cache: parent cpu6 should not be sleeping [64046.479229] CPU6 is up [64046.479247] smpboot: Booting Node 0 Processor 7 APIC 0x7 [64046.479675]  cache: parent cpu7 should not be sleeping [64046.479899] CPU7 is up [64046.485913] ACPI: Waking up from system sleep state S3 [64046.639206] ACPI: EC: event unblocked [64046.639711] sd 2:0:0:0: [sda] Starting disk [64046.873289] usb 1-11: reset full-speed USB device number 2 using xhci_hcd [64046.976869] ata4: SATA link down (SStatus 4 SControl 300) [64046.976892] ata2: SATA link down (SStatus 4 SControl 300) [64047.149289] usb 1-6: reset high-speed USB device number 40 using xhci_hcd [64047.437370] psmouse serio1: synaptics: queried max coordinates: x [..5660], y [..4570] [64047.476302] psmouse serio1: synaptics: queried min coordinates: x [1364..], y [1284..] [64047.922603] OOM killer enabled. [64047.922605] Restarting tasks ... done. [64047.928727] thermal thermal_zone1: failed to read out thermal zone (-61) [64047.930036] Bluetooth: hci0: Bootloader revision 0.0 build 2 week 52 2014 [64047.935036] Bluetooth: hci0: Device revision is 5 [64047.935037] Bluetooth: hci0: Secure boot is enabled [64047.935038] Bluetooth: hci0: OTP lock is enabled [64047.935038] Bluetooth: hci0: API lock is enabled [64047.935039] Bluetooth: hci0: Debug lock is disabled [64047.935040] Bluetooth: hci0: Minimum firmware build 1 week 10 2014 [64047.935042] Bluetooth: hci0: Found device firmware: intel/ibt-11-5.sfi [64047.944372] PM: suspend exit [64048.050329] Read-error on swap-device (8:0:1543400288) [64048.460888] [drm] RC6 on   Please let me know if any other information is needed. How to add a confirm dialog to a command button? I'd like to install a  client for Xubuntu (12.04).  I'm getting non-English Google results, and I didn't find  in the Xubuntu repositories.   Does anyone know of a client that works well for Ubuntu or it's supported derivates (or installation instructions for grive?) Foreign key constraint on array member? How do you use a gray card? There are already   on how to make a gray card, but unfortunately my ignorance goes deeper than that.  How do you use a gray card? Are bicategories and lax 2-categories the same? Non in front of hyphenated adjective Sending signals to USB port using C I want to send a simple 5V signal to the usb port. The USB cabel's insulation will be removed and connected to a relay which is connected to an electric door system. I think I should write this code in C. Can somebody give me a hint? I need to knock out a quick animation in C#/Windows Forms for a Halloween display.  Just some 2D shapes moving about on a solid background.  Since this is just a quick one-off project I really don't want to install and learn an entire new set of tools for this.  (DirectX dev kits, Silverlight, Flash, etc..)  I also have to install this on multiple computers so anything beyond the basic .Net framework (2.0) would be a pain in the arse.  For tools I've got VS2k8, 25 years of development experience, a wheelbarrow, holocaust cloak, and about 2 days to knock this out.  I haven't done animation since using assembler on my Atari 130XE (hooray for page flipping and player/missile graphics!)  Advice?  Here's some of the things I'd like to know:   I can draw on any empty widget (like a panel) by fiddling with it's OnPaint handler, right?  That's how I'd draw a custom widget.  Is there a better technique than this? Is there a page-flipping technique for this kind of thing in Windows Forms?  I'm not looking for a high frame rate, just as little flicker/drawing as necessary.   Thanks.  Post Mortem Edit ... "a couple of coding days later"  Well, the project is done.  The links below came in handy although a couple of them were 404.  (I wish SO would allow more than one reply to be marked "correct").  The biggest problem I had to overcome was flickering, and a persistent bug when I tried to draw on the form directly.   Using the OnPaint event for the Form: bad idea.  I never got that to work; lots of mysterious errors (stack overflows, or ArgumentNullExceptions).  I wound up using a panel sized to fill the form and that worked fine. Using the OnPaint method is slow anyway.  Somewhere online I read that building the PaintEventArgs was slow, and they weren't kidding.  Lots of flickering went away when I abandoned this.  Skip the OnPaint/Invalidate() and just paint it yourself.  Setting all of the "double buffering" options on the form still left some flicker that had to be fixed.  (And I found conflicting docs that said "set them on the control" and "set them on the form".  Well controls don't have a .SetStyle() method.)  I haven't tested without them, so they might be doing something (this is the form):      this.SetStyle(ControlStyles.UserPaint, true);     this.SetStyle(ControlStyles.OptimizedDoubleBuffer, true);     this.SetStyle(ControlStyles.AllPaintingInWmPaint, true);    So the workhorse of the code wound up looking like (pf is the panel control):      void PaintPlayField()     {         Bitmap bufl = new Bitmap(pf.Width, pf.Height);         using (Graphics g = Graphics.FromImage(bufl))         {             g.FillRectangle(Brushes.Black, new Rectangle(0, 0, pf.Width, pf.Height));             DrawItems(g);             DrawMoreItems(g);             pf.CreateGraphics().DrawImageUnscaled(bufl, 0, 0);         }     }   And I just called PaintPlayField from the inside of my Timer loop.  No flicker at all. Formatting a bootable USB that thinks it's a CD-ROM After upgrading to Windows 10, I remember that I had an old bootable recovery usb drive from the manufacturer that I no longer needed. I want to format it but it's showing up as a CD-ROM and format options are not available.  I also tried formatting it in a live linux mint environment, but linux cannot even mount the cd-rom.  How can I format this recovery usb that is showing up as a cd-rom? What is the 64-bit Firefox Beta PPA? Date Not Saving in Mysql from Php Registration Form Uneven movement on camera following a path. I render a short movie. I set the camera to the circle curve to make it fly in circle. The problem is after render, the camera is a little shaky. Can anyone help me? Thanks first.