MathDB
Maximum of Minimum Nonzero Sum

Source: 2021 Taiwan TST Round 1 Mock Day 2 P4

March 20, 2021
InequalityalgebraTaiwan

Problem Statement

Let nn be a positive integer. For each 4n4n-tuple of nonnegative real numbers a1,…,a2na_1,\ldots,a_{2n}, b1,…,b2nb_1,\ldots,b_{2n} that satisfy ∑i=12nai=∑j=12nbj=n\sum_{i=1}^{2n}a_i=\sum_{j=1}^{2n}b_j=n, define the sets A:={∑j=12naibjaibj+1:i∈{1,…,2n} s.t. ∑j=12naibjaibj+1≠0},A:=\left\{\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:i\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\}, B:={∑i=12naibjaibj+1:j∈{1,…,2n} s.t. ∑i=12naibjaibj+1≠0}.B:=\left\{\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:j\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\}. Let mm be the minimum element of A∪BA\cup B. Determine the maximum value of mm among those derived from all such 4n4n-tuples a1,…,a2n,b1,…,b2na_1,\ldots,a_{2n},b_1,\ldots,b_{2n}.
[I]Proposed by usjl.
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