IMO Inequalities

Question 1

Which classical inequality states that for positive reals a and b, (a + b) / 2 ≥ √(ab)?

Accepted Answer

AM-GM inequality

Answer

Cauchy-Schwarz inequality

Answer

Jensen's inequality

Answer

Chebyshev's inequality

Question 2

For positive reals a and b, what is the minimum value of a/b + b/a?

Accepted Answer

2

Answer

0

Answer

1

Answer

4

Question 3

The Cauchy-Schwarz inequality for real sequences a₁, …, aₙ and b₁, …, bₙ states:

Accepted Answer

(Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²)

Answer

(Σaᵢbᵢ)² ≥ (Σaᵢ²)(Σbᵢ²)

Answer

Σaᵢbᵢ ≤ n·max(aᵢbᵢ)

Answer

Σaᵢbᵢ = √(Σaᵢ² · Σbᵢ²)

Question 4

For a convex function f and real numbers x, y, Jensen's inequality gives:

Accepted Answer

f((x+y)/2) ≤ (f(x)+f(y))/2

Answer

f((x+y)/2) ≥ (f(x)+f(y))/2

Answer

f(x+y) = f(x)+f(y)

Answer

f(x+y) ≤ f(x)·f(y)

Question 5

The inequality a/(b+c) + b/(a+c) + c/(a+b) ≥ 3/2 for positive reals a, b, c is known as:

Accepted Answer

Nesbitt's inequality

Answer

Schur's inequality

Answer

Markov's inequality

Answer

Hölder's inequality

Question 6

If x > 0, the minimum value of x + 1/x is:

Accepted Answer

2

Answer

0

Answer

1

Answer

4

Question 7

For positive reals a, b, c with a + b + c = 3, what is the minimum value of a² + b² + c²?

Accepted Answer

3

Answer

1

Answer

6

Answer

9