BALTIC-WAY 2025 Problem 16

Let $a, b, c, n$ be positive integers satisfying $abc = 10^n - 1$ . If $S(x)$ denotes the sum of digits of $x$ in decimal notation, prove that

$S(a) + S(b^2) + S(c^4) \geq \sqrt[3]{243n}.$

Official text Baltic Way 2025 official paper, page 2. Quoted with source.

A model read this text from the scan of the paper by OCR, with vision/local (GLM-OCR-8bit). No person has checked it against the paper yet.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.