Difference between revisions of "Titu's Lemma"
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− | *Let positive real numbers <math>x</math>, <math>y</math>, and <math>z</math> be roots of the polynomial <math>p(x) = x^3 + bx^2 + cx + | + | *Let positive real numbers <math>x</math>, <math>y</math>, and <math>z</math> be roots of the polynomial <math>p(x) = x^3 + bx^2 + cx + 2024</math> for some fixed integer <math>b</math>. For this <math>b</math>, there exists an integer <math>k</math> such that, as <math>c</math> varies through the reals, <math>\sqrt{\frac{x^2yz + 3x^2y + 2x^2z + 6x^2 + xy^2z + 3xy^2 + y^2z + 3y^2 + xyz^2 + 2xz^2 + yz^2 + 2z^2}{xyz + 3xy + 2xz + 6x + yz + 3y + 2z + 6}} \geq k</math>. What is the sum of all possible values of <math>p(1)+p(-1)</math>? (Source: [https://artofproblemsolving.com/wiki/index.php/User:Cxsmi cxsmi]) |
*Prove that, for all positive real numbers <math>a, b, c,</math> <cmath>\frac{1}{a^3+b^3+abc}+\frac{1}{b^3+c^3+abc}+\frac{1}{a^3+c^3+abc} \le \frac{1}{abc}.</cmath> ([https://artofproblemsolving.com/wiki/index.php/1997_USAMO_Problems/Problem_5 Source]) | *Prove that, for all positive real numbers <math>a, b, c,</math> <cmath>\frac{1}{a^3+b^3+abc}+\frac{1}{b^3+c^3+abc}+\frac{1}{a^3+c^3+abc} \le \frac{1}{abc}.</cmath> ([https://artofproblemsolving.com/wiki/index.php/1997_USAMO_Problems/Problem_5 Source]) |
Latest revision as of 19:10, 29 April 2024
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
[hide]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)