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

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?

Accepted Answer

2

Answer

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:

Accepted Answer

(Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²)

Answer

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:

Accepted Answer

f((x+y)/2) ≤ (f(x)+f(y))/2

Answer

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:

Accepted Answer

Nesbitt's inequality

Answer

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.

IMO - International Mathematics Olympiad Practice Test

IMO - International Mathematics Olympiad Practice Test

IMO Inequalities