True or False: Inequality

Algebra Level 2

We can find many triplets of positive real numbers $a, b, c$ satisfying these inequalities below: $a+\frac{1}{b} < 2$ $b+\frac{1}{c} < 2$ $c+\frac{1}{a} < 2$

Is the statement above true or false ?

False Paradox True

1 solution

Chris Lewis
Jul 31, 2019

Note that for all positive $x$ we have $x+\frac1x \ge 2$ (easily proved by, for example, AM-GM).

Now add the three inequalities and reorder the terms to get $a+\frac1a+b+\frac1b+c+\frac1c<6$ . But the left-hand side is clearly $\ge6$ ; contradiction. So no such triples exist.

I've used the AM-HM inequality :)

A Former Brilliant Member - 1 year, 10 months ago

True or False: Inequality

1 solution

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