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Example 1.2: Evan Chen (April 30, 2014) A Brief Introduction To Olympiad Inequalities

This document provides an introduction to Olympiad inequalities through examples and proofs. It proves that a2 + b2 + c2 ≤ ab + bc + ca and a4 + b4 + c4 ≤ a2bc + b2ca + c2ab using the AM-GM inequality. It also states exercises to prove that a3 + b3 + c3 ≤ a2b + b2c + c2a and a5 + b5 + c5 ≤ a3bc + b3ca + c3ab ≤ abc(ab + bc + ca). The document then introduces Muirhead's inequality, which states that if sequences x1x2...xn and y1y2...yn

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60 views1 page

Example 1.2: Evan Chen (April 30, 2014) A Brief Introduction To Olympiad Inequalities

This document provides an introduction to Olympiad inequalities through examples and proofs. It proves that a2 + b2 + c2 ≤ ab + bc + ca and a4 + b4 + c4 ≤ a2bc + b2ca + c2ab using the AM-GM inequality. It also states exercises to prove that a3 + b3 + c3 ≤ a2b + b2c + c2a and a5 + b5 + c5 ≤ a3bc + b3ca + c3ab ≤ abc(ab + bc + ca). The document then introduces Muirhead's inequality, which states that if sequences x1x2...xn and y1y2...yn

Uploaded by

p001
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© © All Rights Reserved
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Evan Chen (April 30, 2014) A Brief Introduction to Olympiad Inequalities

Example 1.2
Prove that a2 + b2 + c2 ab + bc + ca and a4 + b4 + c4 a2 bc + b2 ca + c2 ab.

Proof. By AM-GM,

a 2 + b2 2a4 + b4 + c4
ab and a2 bc.
2 4
Similarly,
b2 + c 2 2b4 + c4 + a4
bc and b2 ca.
2 4
c2 + a2 2c4 + a4 + b4
ca and c2 ab.
2 4
Summing the above statements gives

a 2 + b2 + c 2 ab + bc + ca and a4 + b4 + c4 a2 bc + b2 ca + c2 ab.

Exercise 1.3. Prove that a3 + b3 + c3 a2 b + b2 c + c2 a.

Exercise 1.4. Prove that a5 + b5 + c5 a3 bc + b3 ca + c3 ab abc(ab + bc + ca).

The fundamental intuition is being able to decide which symmetric polynomials of a


given degree are bigger. For example, for degree 3, the polynomial a3 + b3 + c3 is biggest
and abc is the smallest. Roughly, the more “mixed” polynomials are the smaller. From
this, for example, one can immediately see that the inequality

(a + b + c)3 a3 + b3 + c3 + 24abc

must be true, since upon expanding the LHS and cancelling a3 + b3 + c3 , we find that the
RHS contains only the piddling term 24abc. That means a straight AM-GM will suffice.
A useful formalization of this is Muirhead’s Inequality. Suppose we have two sequences
x1 x2 · · · xn and y1 y2 · · · yn such that

x 1 + x 2 + · · · + x n = y1 + y2 + · · · + yn ,

and for k = 1, 2, . . . , n 1

x1 + x2 + · · · + xk y1 + y2 + · · · + yk ,

Then we say that (xn ) majorizes (yn ), written (xn ) (yn ).


Using the above, we have the following theorem.

Theorem 1.5 (Muirhead’s Inequality)


Ifa1 , a2 , . . . , an are positive reals, and (xn ) majorizes (yn ) then we have the inequality.
X X y y
ax1 1 ax2 2 . . . axnn a11 a22 . . . aynn .
sym sym

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