Today I’m going to show two very nice inequalities to know, which really are useful in the mathematical olympiads.

### What They Say And Their Proof

The Cauchy-Schwarz inequality asserts that, for real tuples we have:

Here’s the proof:

consider the expression

It’s obviously true since each term of the sum is non negative. Hence we expand the sum to get:

Since it’s non negative, it has at most one real root, so its discriminant must be less than or equal to 0, which gives us the desired result:

This proof gives also the equality case: there must exist a real number a such that

While Cauchy-Schwarz works for all reals, Titu’s Lemma only works for positive reals, in fact it’s a direct consequence of Cauchy-Schwarz where the tuples are

Substituting these we get its final form:

### An Example For Cauchy-Schwarz

We have three real numbers such that .

Find the minimum of

By Cauchy-Schwarz Inequality we get

Hence the minimum is 74.

### An Example For Titu

Let be positive reals and

Prove that

By Titu’s Lemma on the tuples we get

And we are done.