# Titu's Lemma

Titu's lemma states that:

It is a direct consequence of Cauchy-Schwarz inequality.

Equality holds when for .

Titu's lemma is named after Titu Andreescu and is also known as T2 lemma, Engel's form, or Sedrakyan's inequality.

## Contents

## Examples

### Example 1

Given that positive reals , , and are subject to , find the minimum value of . (Source: cxsmi)

#### Solution

This is a somewhat standard application of Titu's lemma. Notice that When solving problems with Titu's lemma, the goal is to get perfect squares in the numerator. Now, we can apply the lemma.

### Example 2

Prove Nesbitt's Inequality.

#### Solution

For reference, Nesbitt's Inequality states that for positive reals , , and , We rewrite as follows. This is the application of Titu's lemma. This step follows from .

### Example 3

Let , , , , , , , be positive real numbers such that . Show that (Source)

#### Solution

By Titu's Lemma, This is valid because (from the problem statement).

## Problems

### Introductory

- There exists a smallest possible integer such that for all real sequences . Find the sum of the digits of . (Source)

### Intermediate

- Let positive real numbers , , and be roots of the polynomial for some fixed integer . For this , there exists an integer such that, as varies through the reals, . What is the sum of all possible values of ? (Source: cxsmi)

- Prove that, for all positive real numbers (Source)

### Olympiad

- Let be positive real numbers such that . Prove that (Source)

- Let be positive real numbers such that . Prove that

(Source)