IMO Inequalities 2 — Questions and Answers
Question 1: For positive reals x and y with x + y = 10, what is the maximum value of xy?
- 10
- 20
- 25 (Correct answer)
- 100
Correct answer: 25
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?
- HM ≤ GM ≤ AM ≤ QM (Correct answer)
- AM ≤ GM ≤ HM ≤ QM
- QM ≤ AM ≤ GM ≤ HM
- GM ≤ HM ≤ AM ≤ QM
Correct answer: HM ≤ GM ≤ AM ≤ QM
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?
- 1
- 4
- 8 (Correct answer)
- 27
Correct answer: 8
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:
- a + b = 2
- a · b = 1
- a = b (Correct answer)
- a > b
Correct answer: a = b
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?
- Triangle inequality
- Cauchy-Schwarz inequality (Correct answer)
- AM-GM inequality
- Chebyshev's sum inequality
Correct answer: Cauchy-Schwarz inequality
Cauchy-Schwarz is the special case of Hölder's inequality with exponents p = q = 2.
Question 6: For positive reals x, y, z with xyz = 8, what is the minimum of x + y + z?
- 2
- 6 (Correct answer)
- 8
- 24
Correct answer: 6
By AM-GM, (x+y+z)/3 ≥ (xyz)^(1/3) = 2, so x+y+z ≥ 6, with equality when x = y = z = 2.
Question 7: Which statement correctly expresses the triangle inequality in a normed vector space?
- ‖u + v‖ ≤ ‖u‖ + ‖v‖ (Correct answer)
- ‖u + v‖ ≥ ‖u‖ + ‖v‖
- ‖u + v‖ = ‖u‖ · ‖v‖
- ‖u + v‖ ≤ ‖u‖ · ‖v‖
Correct answer: ‖u + v‖ ≤ ‖u‖ + ‖v‖
The triangle inequality states ‖u + v‖ ≤ ‖u‖ + ‖v‖, meaning the norm of a sum does not exceed the sum of the norms.
IMO Inequalities 2
For positive reals x and y with x + y = 10, what is the maximum value of xy?