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Let a, b, and c be positive real numbers such that $a + b + c\leq 4$ and $ab+bc+ca\geq 4$. Prove that at least two of the inequalities $|a - b|\leq2$ , $|b - c| \leq 2$, $ |c - a| \leq 2$ are true.

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up vote 3 down vote accepted

You can deduce that $$ (a-b)^2+(b-c)^2+(c-a)^2 \leq 8$$

If two of the inequalities you mention are false, then this is contradicted.

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Nicely done! +1 – 1015 Jan 26 '13 at 20:20

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