Scribd
Upload
72 views1 page

Practice Problems

The document introduces several olympiad-style inequalities involving three positive variables a, b, and c. It provides examples of proofs for inequalities of this form, including using Holder's inequality. It then lists several practice problems involving similar inequalities for readers to attempt, along with references to their origins. The problems range from simpler cases involving sums of terms to more complex expressions involving roots and fractional exponents.

Uploaded by

p001
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
72 views1 page

Practice Problems

The document introduces several olympiad-style inequalities involving three positive variables a, b, and c. It provides examples of proofs for inequalities of this form, including using Holder's inequality. It then lists several practice problems involving similar inequalities for readers to attempt, along with references to their origins. The problems range from simpler cases involving sums of terms to more complex expressions involving roots and fractional exponents.

Uploaded by

p001
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 1

Evan Chen (April 30, 2014) A Brief Introduction to Olympiad Inequalities

Example 3.7 (USA TST 2010)


1 1 1 1
If abc = 1, prove a5 (b+2c)2
+ b5 (c+2a)2
+ c5 (a+2b)2 3.

Proof. We can use Hölder to eliminate the square roots in the denominator:
!2 ! !3
X X 1 X1
ab + 2ac 3(ab + bc + ca)2 .
cyc cyc
a (b + 2c)2
5
cyc
a

§3.3 Practice Problems


p p p p p p
1. If a + b + c = 1, then ab + c + bc + a + ca + b 1 + ab + bc + ca.
p p p
2. If a2 + b2 + c2 = 12, then a · 3 b2 + c2 + b · 3 c2 + a2 + c · 3 a2 + b2  12.
q q q
3. (ISL 2004) If ab + bc + ca = 1, prove 3 a1 + 6b + 3 1b + 6c + 3 1c + 6a  abc 1
.
p p p p
4. (MOP 2011) a2 ab + b2 + b2 bc + c2 + c2 ca + a2 + 9 3 abc  4(a + b + c).

5. (Evan Chen) If a3 + b3 + c3 + abc = 4, prove

(5a2 + bc)2 (5b2 + ca)2 (5c2 + ab)2 (10 abc)2


+ + .
(a + b)(a + c) (b + c)(b + a) (c + a)(c + b) a+b+c

When does equality hold?

§4 Problems
1. (MOP 2013) If a + b + c = 3, then
p p p p
a2 + ab + b2 + b2 + bc + c2 + c2 + ca + a2 3.

1 1 1 3
2. (IMO 1995) If abc = 1, then a3 (b+c)
+ b3 (c+a)
+ c3 (a+b) 2.

P (2a+b+c)2
3. (USA 2003) Prove cyc 2a2 +(b+c)2  8.

4. (Romania)
Pn Let x1 , x2 , . . . , xn be positive reals with x1 x2 . . . xn = 1. Prove that
1
i=1 n 1+xi  1.

5. (USA 2004) Let a, b, c be positive reals. Prove that

a5 a2 + 3 b5 b2 + 3 c5 c2 + 3 (a + b + c)3 .

p p
7 p
6. (Evan Chen) Let a, b, c be positive reals satisfying a + b + c = 7
a+ b+ 7
c.
Prove aa bb cc 1.

10

You might also like