MathDB
Joints type inequality

Source: 2023 Taiwan TST Round 2 Independent Study 2-A

April 6, 2023
Inequalityanalytic geometryinequalitiesTaiwan

Problem Statement

For each positive integer kk greater than 11, find the largest real number tt such that the following hold: Given nn distinct points a(1)=(a1(1),…,ak(1))a^{(1)}=(a^{(1)}_1,\ldots, a^{(1)}_k), …\ldots, a(n)=(a1(n),…,ak(n))a^{(n)}=(a^{(n)}_1,\ldots, a^{(n)}_k) in Rk\mathbb{R}^k, we define the score of the tuple a(i)a^{(i)} as ∏j=1k#{1≤i′≤n such that πj(a(i′))=πj(a(i))}\prod_{j=1}^{k}\#\{1\leq i'\leq n\textup{ such that }\pi_j(a^{(i')})=\pi_j(a^{(i)})\} where #S\#S is the number of elements in set SS, and πj\pi_j is the projection Rk→Rk−1\mathbb{R}^k\to \mathbb{R}^{k-1} omitting the jj-th coordinate. Then the tt-th power mean of the scores of all a(i)a^{(i)}'s is at most nn.
Note: The tt-th power mean of positive real numbers x1,…,xnx_1,\ldots,x_n is defined as (x1t+⋯+xntn)1/t\left(\frac{x_1^t+\cdots+x_n^t}{n}\right)^{1/t} when t≠0t\neq 0, and it is x1⋯xnn\sqrt[n]{x_1\cdots x_n} when t=0t=0.
Proposed by Cheng-Ying Chang and usjl
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