Counting Chains with Casework in Combinatorics: A Problem from the RMO 2024

In this exploration, we dive into a combinatorial problem from the 2024 Regional Math Olympiad (RMO) in India, centered on counting specific number sequences, called "chains." Using a function \ (f(n) \), defined as the number of chains that start at 1 and end at \( n \) with each previous number dividing the next, the problem applies strategic casework to calculate \( f(2^m \cdot 3) \).

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Problem Overview

We want to determine:
\(f(2^m \cdot 3)\)
where each chain is a sequence that:

1. Begins at 1 and ends at \( 2^m \cdot 3 \),
2. Has each term dividing the next.

Concepts Used:

1. Combinatorial Casework: Breaking down problems by considering specific scenarios helps in counting complex structures systematically.
2. Binomial Theorem: Key in calculating possible combinations, where the sum of binomial coefficients up to \( n \) equals \( 2^n \).

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Solution Outline:

The solution involves structured casework using the position of the first appearance of 3 as a "switch" to organize sub-cases.

Understanding Chains

• For example, \( f(4) \) is the number of "4-chains" starting at 1 and ending at 4, where each term divides the next, such as \( [1, 4] \) and \( [1, 2, 4] \).

Casework on Position of 3:

• Identify when 3 first appears in the sequence, which can happen at positions like \( 3, 2^j \cdot 3, \ldots, 2^m \cdot 3 \).
• Divide cases based on different powers of 2, allowing systematic counting of sequences in each case.

Applying Binomial Coefficients:

• Use binomial coefficients to count the number of valid choices for sequences involving powers of 2 on either side of the appearance of 3.
• This yields:
  \[2^{m-1} \times m + 2^m\]

Final Answer:

By organizing cases and summing possibilities, we obtain the final count:
\[2^{m-1} \cdot (m + 2)\]

This problem exemplifies the effectiveness of casework in combinatorics, teaching a methodical approach to counting sequences. Through strategic splitting and summing, it provides a beautiful solution to a challenging problem in combinatorial mathematics.