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

10

Answer

20

Answer

100

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

AM ≤ GM ≤ HM ≤ QM

Answer

QM ≤ AM ≤ GM ≤ HM

Answer

GM ≤ HM ≤ AM ≤ QM

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

1

Answer

4

Answer

27

Question 4

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

Accepted Answer

a = b

Answer

a + b = 2

Answer

a · b = 1

Answer

a > b

Question 5

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

Accepted Answer

Cauchy-Schwarz inequality

Answer

Triangle inequality

Answer

AM-GM inequality

Answer

Chebyshev's sum inequality

Question 6

For positive reals x, y, z with xyz = 8, what is the minimum of x + y + z?

Accepted Answer

6

Answer

2

Answer

8

Answer

24

Question 7

Which statement correctly expresses the triangle inequality in a normed vector space?

Accepted Answer

‖u + v‖ ≤ ‖u‖ + ‖v‖

Answer

‖u + v‖ ≥ ‖u‖ + ‖v‖

Answer

‖u + v‖ = ‖u‖ · ‖v‖

Answer

‖u + v‖ ≤ ‖u‖ · ‖v‖