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September 7th, 2017, 10:01 AM  #1 
Newbie Joined: Sep 2017 From: London Posts: 3 Thanks: 0  Equation system
Plis, I need help. The order is: "Solve in real numbers" a^2 * b + 2 = a + 2*b*c b^2 * c + 2 = b + 2*a*c c^2 * a + 2 = c + 2*a*b 
September 7th, 2017, 10:29 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,720 Thanks: 1536 
If you try letting a = b = c, you get a³  2a²  a + 2 = 0, i.e. (a + 1)(a  1)(a  2) = 0.

September 7th, 2017, 10:35 AM  #3 
Newbie Joined: Sep 2017 From: London Posts: 3 Thanks: 0 
Yeah but how to prove that a=b=c and there are no more roots?

September 7th, 2017, 01:15 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,454 Thanks: 567 
Skipjack doesn't need to prove anything. He has demonstrated that there is a particular solution when a=b=c.

September 7th, 2017, 10:23 PM  #5 
Newbie Joined: Sep 2017 From: London Posts: 3 Thanks: 0 
Yeah, but I cannot write "If a=b=c there are this 3 solutions1,1,2" the profesor give me 0 points and write "how you know it, and why you think that there arent more solution"

September 8th, 2017, 04:09 AM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,976 Thanks: 807 
Yes, so you need to work out whether there are other solutions before you turn it in. Did you notice that all three equations are the same except that "a", "b", and "c" are rearranged?

September 8th, 2017, 05:42 AM  #7 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 11,865 Thanks: 768  
September 8th, 2017, 06:00 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 18,720 Thanks: 1536 
Hint: rearrange each equation.

September 8th, 2017, 08:00 PM  #9 
Global Moderator Joined: Dec 2006 Posts: 18,720 Thanks: 1536 
This type of problem occurs from time to time in mathematics competitions, some of which are open competitions where the candidates are asked not to obtain help from internet forums. I haven't found the above problem on any other website, but I think it would be best not to provide further help other than general advice until some weeks have elapsed or it can be shown that the problem isn't from a current competition (for example, by showing that it appears in a published book).


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