I've been trying to help my son with what seems an innocent enough equation, but I have to admit, I'm struggling and walking in circles - the question is: If $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=4$, what is $abc$? First I multiply both sides with the denominators: $a(a+c)(a+b)+b(b+c)(a+b)+c(b+c)(a+c)=4(a+b)(a+b)(b+c)$ $\Updownarrow$ $a^3+b^3+c^3=3(abc+a^2b+ab^2+b^2c+a^2c+ac^2+bc^2)$ From here I begin to run in circles - substituting $x=abc$ and expressing $a$, $b$ and $c$ in terms of $x$, for example, ends up after a while back at the beginning. So what is the trick - or is there no simple solution? (Sorry about the tag - I couldn't find anything more suitable for such an elementary question) Edit The suggested duplicate is not a full answer, I think. My question concerns the same equation, but the question is different, and the answers are only partial. I will accept the answer given by Rhys Hughes, since it answers my question and gives a good explanation of how he reached it.
If I had a very shallow question, then I am sorry. $x,y,z\in\mathbb{N}^{+}$ and$$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=4$$find $x,y,z$.  I try with AM-GM, just get$$ \frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}\geq\frac{3}{2}$$  This means that the equation must have a real solution, but can not be sure there is an integer solution.  Let: $x=ay=abz$, then the equation becomes:$$\frac{ab}{a+b}+\frac{b}{ab+1}+\frac{1}{a^2+ab}=4$$  Which makes the problem become non-homogeneous, and seems to become more difficult. I have no more ideas. Could anyone help me? Thanks a lot.
How to duplicate sys.stdout to a log file? Edit: Since it appears that there's either no solution, or I'm doing something so non-standard that nobody knows - I'll revise my question to also ask: What is the best way to accomplish logging when a python app is making a lot of system calls?  My app has two modes.  In interactive mode, I want all output to go to the screen as well as to a log file, including output from any system calls.  In daemon mode, all output goes to the log.  Daemon mode works great using os.dup2().  I can't find a way to "tee" all output to a log in interactive mode, without modifying each and every system call.    In other words, I want the functionality of the command line 'tee' for any output generated by a python app, including system call output.  To clarify:   To redirect all output I do something like this, and it works great:  # open our log file so = se = open("%s.log" % self.name, 'w', 0)  # re-open stdout without buffering sys.stdout = os.fdopen(sys.stdout.fileno(), 'w', 0)  # redirect stdout and stderr to the log file opened above os.dup2(so.fileno(), sys.stdout.fileno()) os.dup2(se.fileno(), sys.stderr.fileno())   The nice thing about this is that it requires no special print calls from the rest of the code.  The code also runs some shell commands, so it's nice not having to deal with each of their output individually as well.  Simply, I want to do the same, except duplicating instead of redirecting.  At first thought, I thought that simply reversing the dup2's should work.  Why doesn't it?  Here's my test:   import os, sys  ### my broken solution: so = se = open("a.log", 'w', 0) sys.stdout = os.fdopen(sys.stdout.fileno(), 'w', 0)  os.dup2(sys.stdout.fileno(), so.fileno()) os.dup2(sys.stderr.fileno(), se.fileno()) ###  print("foo bar")  os.spawnve("P_WAIT", "/bin/ls", ["/bin/ls"], {}) os.execve("/bin/ls", ["/bin/ls"], os.environ)   The file "a.log" should be identical to what was displayed on the screen. preventDefault when clicking bounty "cancel" link When you click "start a bounty" then "cancel," href="#" exiles you to the top of the page and you lose your spot near the end of the question's comments. Clicking anywhere else on the page simply closes the bounty popup; clicking "cancel" ought to do the same.  In case that wasn't clear:    Dramatization aside, it's a nuisance. Can the click event on that anchor be preventDefaulted? Send an email as an attachment from iPhone How to show hidden files in Nautilus 3.20.3 Ubuntu 16.10? How to show hidden files in Nautilus 3.20.3 Ubuntu 16.10? Why is Modus Ponens valid? I am having trouble understanding what defines Entailment operator. On Mathoverflow I posted this question on what I perceive to be .  Consider: :     P therefore Q  P            Therefore, Q   My question is could we have arbitrarily defined it such as:     P therefore Q  P            Therefore, P   However, if an advanced civilization receives information in the form of following nested structures/containers or packets, how will they interpret Modus Ponens if in an alien land Modus Ponens is not defined?    Therefore, it raises the validity of MP. How do we define the implication or entailment operator? Because if we use an argument that relies on implication or entailment operator ipso facto, then it is trivial that logic is circular.  Is there a flaw in my rationale? 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Clarification 2: I'm doing this as I have private methods in a class that take said enum, and I do not want the enum's values exposed - so, for example, I do not want anyone to know that E is defined as  enum E {     FUNCTIONALITY_NORMAL, FUNCTIONALITY_RESTRICTED, FUNCTIONALITY_FOR_PROJECT_X }   as project X is not something I want my users to know about.  So, I wanted to forward declare the enum so I could put the private methods in the header file, declare the enum internally in the cpp, and distribute the built library file and header to people.  As for the compiler - it's GCC. Why did Kattappa kill Baahubali? Let's say that there is a question asked in a bad manner, in a way that the question is missing. For example the user shows the code and doesn't explain very well which is the problem.    Usually someone starts digging in the OP mind posting comments and asking him for details. It may happen that the question gets edited and the OP edits it to a valid question.    Do you usually give the OP time to edit the question or you immediately vote to close it? In other words, are $\emptyset$ and $\mathbb{R}$ the only open and closed sets in $\mathbb{R}$? Why/Why not?  I tried by assuming a set is equal to its interior points and contains its limit points.  A bounded set will not do since stuff like $[1,4]$ and $\{5\}$ will not work, though that is not really proof. Help please?  Anyway, it must then be unbounded.  If $a$ is a real number then $(a,\infty)$, $(-\infty,a)$, $[a,\infty)$ and $(-\infty,a]$ don't seem to cut it so it must be $\mathbb{R}$. Is there a C pre-processor which eliminates #ifdef blocks based on values defined/undefined? Unary operator expected #!/bin/bash  SUBJECT="WARNING CPU USAGE HIGH" TO=gmail id MESSAGE=/tmp/messages echo "#######################" &gt; $MESSAGE echo "CPU statistics as follows.." &gt;&gt; $MESSAGE mpstat &gt;&gt; $MESSAGE echo "#######################" &gt;&gt; $MESSAGE CPU_USAGE=$(top -b -n1 | awk '/^Cpu/ {print $2}' | cut -d. -f1) [ $CPU_USAGE -gt 85 ] &amp;&amp; mail -s "$SUBJECT" "$TO" &lt; $MESSAGE`   ./cpu.sh: line 11: [: -gt: unary operator expected What might be the reason I asked a  on stackoverflow at the end of last year for a beta or pre-release product and got the answer I wanted.  3 months later someone else answered the question with the correct answer for a newer release.  This answer got way more upvotes than the one I accepted.  I only noticed this today so changed the accepted answer.  Can I get a notification email when an answer gets say 10 more votes than the one I accepted.  Just to let me know something might be up. I know that nobody likes duplicate questions, but I haven't found my answer on any of the other questions with this topic.  I built a computer with a  (made by Gigabyte).  It supposedly has 512 CUDA cores.  I followed  tutorial, (which enabled the Nvidia proprietary drivers) and was able to enable dual monitors and all the desktop graphics that are needed.  Unfortunetly, the GPU processing option still doesn't show up in Blender.  I've tried installing nvidia-cuda-toolkit, and that didn't make a difference.  I also tried to update my drivers from nvidia-346 to nvidia-349, and this had no apparent effect either.  What can I do to enable GPU rendering?  (I've tried both blender installed and just downloaded from the site). Class imbalance strategies When dealing with the class imbalance problem in a binary classifier, there are three ways I know of to address it: over-sampling, under-sampling and using cost-sensitive methods.   Are there any guidelines, rules of thumb or general strategies to choose among these methods? A possible answer would be: over-sample when the positive class has more than 100 instances (I just made it up). There's no need to  the  that have led to a lack of trust between  of  and . Suffice it to say that astoundingly they continue to assert that  and, instead of apologizing and reinstating Monica have demanded that she go through a process with no transparency, due process, or involvement from the community.   Has the time has come for Stack Exchange, Inc to spin off the community and all the data needed to keep the sites running to a separate non-profit, who would then in turn license the software from the company?  What's in it for the community?   The community can practice self determination for issues from as basic as the number of moderators they have to as fundamental as the policies they enforce. If the company development continues to be unresponsive to the community's needs, the community would be able to develop its own open source software that would meet its needs. They can find their own funding model instead of having ads imposed on them.   What's in it for the company?   Community actions will no longer reflect on the company. Interaction with the community as a customer may come more naturally to current leadership than the present paradigm. They don't have to worry about paying for the community's operating expenses. They can focus on doing one thing well. What's the meaning of a set to the power of another set? Redirect all requests to HTTPS, except for one subdirectory