Absolute 2

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.

#Algebra #Sharky

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

For 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.

Eddie The Head - 7 years, 2 months ago

its not true

Mayank Singh - 7 years, 1 month ago

It actually is true.

Sharky Kesa - 7 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$