Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $$  \frac a{a^2+b^3+c^3} + \frac b{a^3+b^2+c^3} + \frac c{a^3+b^3+c^2}\le\frac1{5abc}.$$   I tried using AM-GM-HM-QM inequality but had no luck.
Let $a,b,c&gt; 0$ be three real numbers such that $a+b+c=1$. I want to prove that   $$\frac{a}{a^2+b^3+c^3}+\frac{b}{b^2+a^3+c^3}+\frac{c}{c^2+a^3+b^3}\le\frac{1}{5abc}.$$  My attempt: Using AM-GM on each denominator gives (here, LHS denotes the left-hand side) $$LHS\le\frac{1}{3a^{\frac12}bc}+\frac{1}{3b^{\frac12}ac}+\frac{1}{3c^{\frac12}ab}.$$  However, I think that my attempt doesn't work because the last expression can get larger than $\frac{1}{5abc}$. In fact, if we multiply with $3abc$ then the original inequality is: $$\sqrt a+\sqrt b+\sqrt c\le\frac35$$  However, as $a,b,c&lt;1$, $$\sqrt a +\sqrt b + \sqrt c&gt; a+b+c=1&gt;\frac35.$$ So my upper bound is always larger than the given one.
Update to latest version of java after ppa is discontinued I have been trying to install the latest version of java-8. However, I found out that the  due to changes in the java licence.  How should I update my java to 8.0_201 or 8.0_211 now?  Thanks! What is the difference between <%# and <%= in ASP.NET MVC? I frequently create labels that follow the same format: white text on a 75% opaque black background, with 10px of margin around the text. Currently, I have to manually build these labels, which takes a lot of work.  In one method, I write the text, figure out its height and width, then create a separate rectangle and place it behind it. This takes a lot of clicking around for every label.  In the other method, I draw out a text box, select it directly, give it the correct background and margins, and then insert the text. However, I must then resize the box manually, meaning more clicking around and greater imprecision. 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I ask this because I am designing a simple bench power supply using a 125VA transformer with a 24Vrms secondary and am trying to figure out how much power I can draw before the current spikes into the inductor are too much for the transformer. I also don't want to have to resort to active PFC as I don't want to risk messing up any HVAC PCB design.  Thank you DRAM Frequency showing 800 instead of 1600 I am running a Core i7 CPU originally at 3Ghz overclocked to 4Ghz using a multiplier of x20 and a Bus Speed of 200 Mhz  My memory is 1600 Mhz OCZ.  I installed CPU-Z to see what is going on - but it shows me (under the "Memory" Tab):  DRAM Frequency 800 Mhz  Channel Triple - NB Frequency 3800.2 Mhz  So what is the actual frequency its running on then? I am slightly confused. Is it OK to leave "What have you tried?" comments? Why does the area of a bar in a histogram represent the frequency of that particular class In a bar-graph, the height of the bars represent the frequency of a particular category but it is counter-intuitive that in a histogram, instead of the height of the bar(which represents the frequency density), the area of the bar represents the frequency of the particular class. Why is this so? What is the difference between "till" and "until"? What is the difference between till and until?    When to use till or until? Please explain with examples. Does "until [date]" mean "before that date"? What does until mean in the following?     You need to deliver this product within 2 days (until August 18, 2011)   to meet your deadline and get paid.   Does this mean that I have to deliver the product by 23:59 on 17 August 2011 or 23:59 on 18 August 2011? The order time is 21:04 on August 16 2011. How to setup OpenVPN Client I get the following error in my log files every time I try to upload a large file.  a client request body is buffered to a temporary file /var/lib/nginx/body/0000000001   Although the file uploads successfully, I always get the above error.  I increased the client_body_buffer_size to 1000m which is what I expect the largest file uploaded to be. However, this is was just a guess and although I don't get that error anymore I am wondering if this is an appropriate value to set for the client_body_buffer_size?  I would appreciate it if anyone can shed some light on this directive and how it should be used. How do I create an .exe for a Java program? What can I do if I forgot my Windows password? How do I check if a string contains a specific word? Live USB for installation does not boot, black error screen followed by visual artifacts on a purple screen I'm trying to install Ubuntu 18.04 desktop with Rufus on a computer with the following specs:  CPU: AMD Ryzen 5 2600 MOBO: ASUS Prime B450M-A/CSM RAM: G-Skill Ripjaws DDR4 F4-2400C15D-16GVB GPU: Nvidia GeForce GTX750 PSU: Apexgaming AG-650M SSHD: Seagate Firecuda Compute mATX: Rosewill SCM-01  Basically it boots from the USB (FAT32) and brings up a black error screen with words and numbers. Then it transitions to the purple splash screen with &quot;ubuntu&quot; and 5 dots, however the screen shows various visual glitches depending on which USB port I plug into. When trying to boot the computer, it boots to a black screen with the following error message: ACPI Error: Needed [Integer/String/Buffer], found [Region]    that shows what I mean. Why are we limited to six questions per day?