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Korea's unhealthy obession with inequality-combinatorics??

Source: 2024 KJMO P7

November 9, 2024
combinatoricsinequalities

Problem Statement

Let AkA_k be the number of pairs (a1,a2,...,a2k)(a_1, a_2, ..., a_{2k}) for k≤50k\leq 50, where a1,a2,...,a2ka_1, a_2, ..., a_{2k} are all different positive integers that satisfy the following.
⋅\cdot a1,a2,...,a2k≤100a_1, a_2, ..., a_{2k} \leq 100 ⋅\cdot For an odd number less or equal than 2k−12k-1, we have ai>ai+1a_i > a_{i+1} ⋅\cdot For an even number less or equal than 2k−22k-2, we have ai<ai+1a_i < a_{i+1}
Prove that A1≤A2≤⋯≤A49A_1 \leq A_2 \leq \cdots \leq A_{49}.
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