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:**
![\[m \leq n + 1\]](//latex.artofproblemsolving.com/8/d/d/8dd4d9b222a26993bff86efe292ce7589b923967.png)
![\[m \leq n + 1\]](http://latex.artofproblemsolving.com/8/d/d/8dd4d9b222a26993bff86efe292ce7589b923967.png)
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]
