2024 USAMO Problems/Problem 4

Let and be positive integers. A circular necklace contains beads, each either red or blue. It turned out that no matter how the necklace was cut into blocks of consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair .

Solution 1

To solve this problem, we need to determine all possible positive integer pairs \((m, n)\) such that there exists a circular necklace of \(mn\) beads, each colored red or blue, satisfying the following condition:

- **No matter how the necklace is cut into \(m\) blocks of \(n\) consecutive beads, each block has a distinct number of red beads.**

Necessary Condition:

1. **Maximum Possible Distinct Counts:**

  - In a block of \(n\) beads, the number of red beads can range from \(0\) to \(n\).
  - Therefore, there are \(n + 1\) possible distinct counts of red beads in a block.
  - Since we have \(m\) blocks, the maximum number of distinct counts must be at least \(m\).
  - **Thus, we must have:**  
    

Sufficient Construction:

We will show that for all positive integers \(m\) and \(n\) satisfying \(m \leq n + 1\), such a necklace exists.

1. **Construct Blocks:**

  - Create \(m\) blocks, each containing \(n\) beads.
  - Assign to each block a unique number of red beads, ranging from \(0\) to \(m - 1\).

2. **Design the Necklace:**

  - Arrange these \(m\) blocks in a fixed order to form the necklace.
  - Since the necklace is circular, cutting it at different points results in cyclic permutations of the blocks.

3. **Verification:**

  - In any cut, the sequence of blocks (and thus the counts of red beads) is a cyclic shift of the original sequence.
  - Therefore, in each partition, the blocks will have distinct numbers of red beads.

Example:

Let's construct a necklace for \(m = 3\) and \(n = 2\):

- **Blocks:**

 - Block 1: \(0\) red beads (BB)
 - Block 2: \(1\) red bead (RB)
 - Block 3: \(2\) red beads (RR)

- **Necklace Arrangement:**

 - Place the blocks in order: **BB-RB-RR**

- **Verification:**

 - Any cut of the necklace into \(3\) blocks of \(2\) beads will have blocks with red bead counts of \(0\), \(1\), and \(2\).

Conclusion:

- **All ordered pairs \((m, n)\) where \(m \leq n + 1\) satisfy the condition.** - **Therefore, the possible values of \((m, n)\) are all positive integers such that \(m \leq n + 1\).**

Final Answer:

• • Exactly all positive integers \(m\) and \(n\) with \(m \leq n + 1\); these are all possible ordered pairs \((m, n)\).**

Video Solution

https://youtu.be/ZcdBpaLC5p0 [video contains problem 1 and problem 4]