Let with . Prove that

Determine when equality holds.

Determine when equality holds.

Singapore Team Selection Test 2004

Inequalities

Medium

Inequalities by BJ Venkatachala

Do you really need a hint? Try it first!

Show that there exists a triangle such that and . The easiest way to prove is is to define , and show that must be .

Now the inequality becomes equivalent to . Take a look at this well-known inequality. Note that the tangent is a convex function in .

Note that, are less than . This means that and are all less than . Hence the triangle is acute.

Using Jensen’s inequality, we get . Equality holds for an equilateral triangle.

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