Practice Problems on Cauchy Schwarz Inequality

Below are some problems that I have culled from Prof. Titu Andreescu’s encyclopaedic literature on mathematics olympiads.

Use Cauchy Schwarz inequality to prove the following:

Problem 1:

Let x, y, z > 0. Prove that

\frac{2}{x+y}+\frac{2}{y+z}+\frac{2}{z+x} \geq \frac{9}{x+y+z}

Problem 2:

Let a, b, x, y, z be positive real numbers. Prove that

\frac{x}{ay+bz}+\frac{y}{az+bx}+\frac{z}{ax+by} \geq \frac{3}{a+b}

Problem 3:

Let a, b, c>0. Prove that

\frac{a^{2}+b^{2}}{a+b}+\frac{b^{2}+c^{2}}{b+c}+\frac{c^{2}+a^{2}}{a+c} \geq a+b+c.

Note: 

Perhaps, applying the Cauchy Schwarz inequality directly may be cumbersome, or even impossible. In that case, use the following equivalent lemma for Cauchy Schwarz Inequality:

Lemma:

If a, b, x, y are real numbers and x, y>0, then the following inequality holds:

\frac{a^{2}}{x}+\frac{b^{2}}{y} \geq \frac{(a+b)^{2}}{x+y}.

(I  have solved the above questions using lemma only!! Yet to check whether a direct application of Cauchy Schwarz inequality will work out 🙂 You can enlighten me on this)

Nalin Pithwa

 

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