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

Schur's inequality with t = 1 states that a(a−b)(a−c) + b(b−a)(b−c) + c(c−a)(c−b) ≥ 0 for all non-negative reals a, b, c.

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

From (a+b+c)² = a²+b²+c²+2(ab+bc+ca) and a²+b²+c² ≥ (a+b+c)²/3, we get ab+bc+ca ≤ 1/3, achieved at a=b=c=1/3.

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

The Rearrangement Inequality states the sum Σaᵢbᵢ is maximized when both sequences are sorted in the same order and minimized when sorted in opposite orders.

Question 4

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

Accepted Answer

The geometric mean

Answer

As r → 0, the power mean M_r converges to the geometric mean (a₁a₂···aₙ)^(1/n), which can be shown via L'Hôpital's rule applied to the logarithm.

Question 5

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

Accepted Answer

n · Σaᵢbᵢ ≥ (Σaᵢ)(Σbᵢ)

Answer

Chebyshev's sum inequality states n · Σaᵢbᵢ ≥ (Σaᵢ)(Σbᵢ) when both sequences are similarly ordered (and reverses for oppositely ordered sequences).

IMO - International Mathematics Olympiad Practice Test

IMO - International Mathematics Olympiad Practice Test

IMO Inequalities 3