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Let be $a, b, c$ real numbers such that: $$a+b+c \neq 0 \mbox{ and } ab+bc+ca \mbox{ is rational. }$$

Give a counterexample to prove that the property is not true if $a+b+c = 0$.

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Where is the problem from? I do not understand what it means. – Jonas Meyer Feb 10 at 23:53
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I think you forgot to mention what the "property" is... – David Mitra Feb 11 at 0:04

closed as not a real question by Andres Caicedo, Jonas Meyer, Henry T. Horton, 5PM, tomasz Feb 11 at 3:48

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, see the FAQ.

1 Answer

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Let $a=0$, $b=\pi$ and $c=-\pi$

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If this is an answer to the question, could you please share what the question is? I mean, I see that you've chosen $a,b,c$ such that $a+b+c=0$, and I see that $-\pi^2$ is not rational, but I don't know what the question is. – Jonas Meyer Feb 11 at 0:10
I believe that the poster was trying to identify why the statement required $a+b+c \neq 0$ – rckrd Feb 11 at 0:29
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But, Matt, $a+b+c\ne0$ certainly doesn't imply $ab+bc+ca$ is rational. So, what question are we actually answering here? – Gerry Myerson Feb 11 at 0:54
Yeah, you're right. There really is no property at all. If we let $a,b,c$ be irrational, can we say that $ab+bc+ca$ is irrational if $a+b+c\neq 0$? – rckrd Feb 11 at 1:15
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@Matt: No; e.g., $a=b=c=\sqrt 2$. – Jonas Meyer Feb 11 at 1:38

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