IMO Inequalities 1 — Questions and Answers
Question 1: Which classical inequality states that for positive reals a and b, (a + b) / 2 ≥ √(ab)?
- Cauchy-Schwarz inequality
- AM-GM inequality (Correct answer)
- Jensen's inequality
- Chebyshev's inequality
Correct answer: AM-GM inequality
The AM-GM inequality (Arithmetic Mean – Geometric Mean) states that the arithmetic mean of non-negative reals is always at least their geometric mean.
Question 2: For positive reals a and b, what is the minimum value of a/b + b/a?
- 0
- 1
- 2 (Correct answer)
- 4
Correct answer: 2
By AM-GM, a/b + b/a ≥ 2√(a/b · b/a) = 2, with equality when a = b.
Question 3: The Cauchy-Schwarz inequality for real sequences a₁, …, aₙ and b₁, …, bₙ states:
- (Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²) (Correct answer)
- (Σaᵢbᵢ)² ≥ (Σaᵢ²)(Σbᵢ²)
- Σaᵢbᵢ ≤ n·max(aᵢbᵢ)
- Σaᵢbᵢ = √(Σaᵢ² · Σbᵢ²)
Correct answer: (Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²)
The Cauchy-Schwarz inequality states (Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²), with equality when the sequences are proportional.
Question 4: For a convex function f and real numbers x, y, Jensen's inequality gives:
- f((x+y)/2) ≥ (f(x)+f(y))/2
- f((x+y)/2) ≤ (f(x)+f(y))/2 (Correct answer)
- f(x+y) = f(x)+f(y)
- f(x+y) ≤ f(x)·f(y)
Correct answer: f((x+y)/2) ≤ (f(x)+f(y))/2
For a convex function, Jensen's inequality states f((x+y)/2) ≤ (f(x)+f(y))/2, reflecting that the function lies below the chord between two points.
Question 5: The inequality a/(b+c) + b/(a+c) + c/(a+b) ≥ 3/2 for positive reals a, b, c is known as:
- Schur's inequality
- Nesbitt's inequality (Correct answer)
- Markov's inequality
- Hölder's inequality
Correct answer: Nesbitt's inequality
Nesbitt's inequality states that for positive reals a, b, c, the cyclic sum a/(b+c) + b/(a+c) + c/(a+b) is always at least 3/2.
Question 6: If x > 0, the minimum value of x + 1/x is:
- 0
- 1
- 2 (Correct answer)
- 4
Correct answer: 2
By AM-GM, x + 1/x ≥ 2√(x · 1/x) = 2, with equality at x = 1.
Question 7: For positive reals a, b, c with a + b + c = 3, what is the minimum value of a² + b² + c²?
- 1
- 3 (Correct answer)
- 6
- 9
Correct answer: 3
By the Cauchy-Schwarz or QM-AM inequality, a² + b² + c² ≥ (a+b+c)²/3 = 3, with equality when a = b = c = 1.
IMO Inequalities 1
Which classical inequality states that for positive reals a and b, (a + b) / 2 ≥ √(ab)?