IMO Inequalities 2

Question 1

For positive reals x and y with x + y = 10, what is the maximum value of xy?

Accepted Answer

25

Answer

By AM-GM, xy ≤ ((x+y)/2)² = 25, with equality when x = y = 5.

Question 2

What is the correct ordering of the Harmonic Mean (HM), Geometric Mean (GM), Arithmetic Mean (AM), and Quadratic Mean (QM) for positive reals?

Accepted Answer

HM ≤ GM ≤ AM ≤ QM

Answer

The power mean inequality gives HM ≤ GM ≤ AM ≤ QM for positive reals, with equality when all values are equal.

Question 3

For positive reals a, b, c, what is the minimum value of (a+b)(b+c)(c+a) given abc = 1?

Accepted Answer

8

Answer

By AM-GM, a+b ≥ 2√(ab), b+c ≥ 2√(bc), c+a ≥ 2√(ca), so (a+b)(b+c)(c+a) ≥ 8abc = 8.

Question 4

The AM-GM inequality a + b ≥ 2√(ab) for positive reals a, b holds with equality when:

Accepted Answer

a = b

Answer

AM-GM becomes an equality if and only if all terms are equal, so here when a = b.

Question 5

Hölder's inequality is a generalization of which of the following classical inequalities?

Accepted Answer

Cauchy-Schwarz inequality

Answer

Cauchy-Schwarz is the special case of Hölder's inequality with exponents p = q = 2.

IMO - International Mathematics Olympiad Practice Test