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

\(|a - b| \leq 2, |b - c| \leq 2, |c - a| \leq 2\)

are true.

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TopNewestFor sake of contradiction let us suppose \(|a-b| > 2\), \(|b-c| > 2\),\(|c-a| > 2\). which implies \[(a-b)^{2} + (b-c)^{2} + (c-a)^{2} > 8\] ---(1)

Given \(a + b + c \le 4\) \[a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca \le 16\] \[a^{2} + b^{2} + c^{2} - ab - bc - ca \le 16 - 3ab -3bc - 3ca\] \[a^{2} + b^{2} + c^{2} - ab - bc - ca \le 4\] \[2a^{2} + 2b^{2} + 2c^{2} - 2ab - 2bc - 2ca \le 8\] \[(a-b)^{2} + (b-c)^{2} + (c-a)^{2} \le 8\].

This contradicts (1) and hence our desired proof follows.

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its not true

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It actually is true.

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