IMO Inequalities 3

Question 1

For positive reals a, b, c, Schur's inequality with parameter t = 1 states:

Accepted Answer

a(a−b)(a−c) + b(b−a)(b−c) + c(c−a)(c−b) ≥ 0

Answer

a²(b+c) + b²(a+c) + c²(a+b) ≤ abc

Answer

(a+b+c)³ ≥ 27abc

Answer

a/b + b/c + c/a ≥ 3

Question 2

For positive reals a, b, c with a + b + c = 1, what is the maximum value of ab + bc + ca?

Accepted Answer

1/3

Answer

1/9

Answer

1/4

Answer

1/2

Question 3

The Rearrangement Inequality states that for sequences a₁ ≤ ··· ≤ aₙ and b₁ ≤ ··· ≤ bₙ, the sum Σaᵢbᵢ is:

Accepted Answer

Maximized when both sequences are sorted in the same order

Answer

Minimized when both sequences are sorted in the same order

Answer

Independent of how the sequences are paired

Answer

Equal to (Σaᵢ)(Σbᵢ)/n for any pairing

Question 4

The power mean M_r of positive reals a₁, …, aₙ as r → 0 converges to:

Accepted Answer

The geometric mean

Answer

The arithmetic mean

Answer

The harmonic mean

Answer

Zero

Question 5

The Chebyshev sum inequality for similarly ordered sequences a₁ ≤ ··· ≤ aₙ and b₁ ≤ ··· ≤ bₙ states:

Accepted Answer

n · Σaᵢbᵢ ≥ (Σaᵢ)(Σbᵢ)

Answer

n · Σaᵢbᵢ ≤ (Σaᵢ)(Σbᵢ)

Answer

Σaᵢbᵢ = (Σaᵢ)(Σbᵢ) / n²

Answer

Σaᵢbᵢ ≤ max(aᵢ) · Σbᵢ

Question 6

For positive reals a, b, c, which inequality directly implies a³ + b³ + c³ ≥ a²b + b²c + c²a?

Accepted Answer

Rearrangement inequality

Answer

AM-GM inequality applied to triples

Answer

Jensen's inequality for f(x) = x³

Answer

Cauchy-Schwarz in Engel (Titu) form

Question 7

In the Engel form (Titu's Lemma) of Cauchy-Schwarz, the inequality Σ(aᵢ²/bᵢ) ≥ (Σaᵢ)²/Σbᵢ is most directly useful for:

Accepted Answer

Establishing lower bounds on sums of fractions

Answer

Proving products of sums are bounded

Answer

Computing exact values of symmetric sums

Answer

Deriving upper bounds on geometric means