CF 1736A - Make A Equal to B
This is a Type B (prove) problem. The statement requires a universal claim for all natural numbers $n$, namely that at least one of $S(7n)$ or $S(9n)$ exceeds $S(n)/5$.
Rating: 800
Tags: brute force, greedy, sortings
Solve time: 1m 57s
Verified: no
Solution
Problem-Type Check
This is a Type B (prove) problem. The statement requires a universal claim for all natural numbers $n$, namely that at least one of $S(7n)$ or $S(9n)$ exceeds $S(n)/5$. A valid solution must establish the inequality rigorously for all $n$, not just for small examples or via intuition. Stepwise logical deductions with proper justifications are required.
Step-by-Step Verification
Step 1: Exploration with small values of $n$ - VALID. The computations for $n=1,2,10,111$ correctly illustrate the plausibility of the inequality, but they do not constitute a proof.
Step 2: Lemma 1, claiming $S(9k) \ge S(k)$ for all $k$