Let $a, b, c$ be three positive real numbers such that $a + b + c = 1$. Prove that among the three numbers $a − ab, b − bc, c − ca$ there is one which is at most $1/4$ and there is one which is at least $2/9$

I have proceeded by the AM-GM inequality to prove that \begin{align*}a(1 − a) \leq \frac 14, \qquad b(1 − b) \leq \frac 14, \qquad c(1 − c) \leq \frac 14.\end{align*} Multiplying these we obtain $$abc(1 − a)(1 − b)(1 − c) \leq \frac{1}{4^3}.$$ After that, I could not do much.

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    $\begingroup$ Show us your effort. $\endgroup$ – S.C.B. May 31 '16 at 16:14
  • $\begingroup$ I have proceeded by AM GM inequality to prove a(1 − a) ≤1/4 b(1 − b) ≤1/4 c(1 − c) ≤1/4 Multiplying these we obtain abc(1 − a)(1 − b)(1 − c) ≤1/4^3 after that i could not do much . $\endgroup$ – starunique2016 May 31 '16 at 16:19
  • $\begingroup$ I mean you should add that information to the question, not in a comment. $\endgroup$ – S.C.B. May 31 '16 at 16:20

Hint 1: For the first inequality, without loss of generality let $a$ be the smallest of the three numbers. Then by the AM-GM, we have $$a(1-b) \le \left(\frac{a+(1-b)}2\right)^2.$$

Hint 2: For the second inequality, let $a$ be the largest number and $b$ the smallest. Then $a \ge \frac13$ and $1-b \ge \frac23$.


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