MathLabs

Problem 1

Let a1,a2,…,ana_1,a_2,\dots,a_n be a sequence of non-negative integers, and let An=(a1+a2+⋯+an)/nA_n=(a_1+a_2+\cdots+a_n)/n. Prove that a1!a2!⋯an!≥(⌊An⌋!)na_1!a_2!\cdots a_n!\ge (\lfloor A_n\rfloor!)^n, where a!=1⋅2⋯aa!=1\cdot2\cdots a for a≥1a\ge1 and 0!=10!=1. When does equality hold?
Step 5 of 6: First equality case
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A=B=0  ⟺  a1=a2=⋯=an.A=B=0\iff a_1=a_2=\cdots=a_n.
Detailed analysis

If there are no factors above or below s, every entry equals s, so all entries are equal. Conversely, an equal sequence clearly gives equality.