Problem
ALG-B2-M01-P022 Differences as bounds
A sequence \(a_1,\ldots,a_{150}\) satisfies \(a_n-a_k\ge n^3-k^3\) for all \(n,k\). Given \(a_{75}=0\), find \(a_{150}\).
Hint 1. Write the condition for the pair \((150,75)\).
Hint 2. Then swap \(n\) and \(k\).
For \(n=150,k=75\), we get \(a_{150}-a_{75}\ge150^3-75^3\), so \(a_{150}\ge150^3-75^3\). For \(n=75,k=150\), we get \(a_{75}-a_{150}\ge75^3-150^3\), so \(a_{150}\le150^3-75^3\). Therefore \[a_{150}=150^3-75^3=7\cdot75^3.\]
A. Source analysis. Main objects: inequalities, order, an extremal element, or an invariant. The obvious first move usually gives only a local estimate. The hidden observation is to choose the right nondecreasing quantity, or to add/multiply inequalities only after signs are controlled. The needed step is an ordering, an invariant, a product transformation, or a boundary case.
F. Difficulty justification. Regional level 6: the idea is short, but the condition must be used in both directions.
G. Why this is not a one-step exercise. One substitution gives only the lower bound.