*Problem:*

Let be non-negative numbers, no two of them are zero. Prove that

.

*1st solution:*

Let . We have

from which the desired inequality follows. Equality occurs if and only if .

*2nd solution:*

Divide each fraction with respectively. Then we get that

.

Let us denote . Then the inequality transforms to

.

Let us now use the transformation which makes the inequality to

.

From Cauchy’s inequality now, we get that

.

So, we only need to prove that

,

which is obviously true. Equality holds only for , *Q.E.D.*

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