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 real
,
, and
. There exists some minimum integer
such that the equation
has some solution in
,
, and
when
and no solutions when
. This is the case if and only if
has y-intercept
, where
is some constant. What is the value 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)