Difference between revisions of "2024 USAMO Problems/Problem 4"

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Let <math>m</math> and <math>n</math> be positive integers. A circular necklace contains <math>m n</math> beads, each either red or blue. It turned out that no matter how the necklace was cut into <math>m</math> blocks of <math>n</math> consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair <math>(m, n)</math>.
 
Let <math>m</math> and <math>n</math> be positive integers. A circular necklace contains <math>m n</math> beads, each either red or blue. It turned out that no matter how the necklace was cut into <math>m</math> blocks of <math>n</math> consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair <math>(m, n)</math>.
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==Solution 1==
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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:
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- **No matter how the necklace is cut into m blocks of n consecutive beads, each block has a distinct number of red beads.**
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Necessary Condition:
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1. **Maximum Possible Distinct Counts:**
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  - In a block of n beads, the number of red beads can range from 0 to n.
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  - Therefore, there are n+1 possible distinct counts of red beads in a block.
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  - Since we have m blocks, the maximum number of distinct counts must be at least m.
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  - **Thus, we must have:** 
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    <cmath> m \leq n + 1 </cmath>
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Sufficient Construction:
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We will show that for all positive integers m and n satisfying mn+1, such a necklace exists.
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1. **Construct Blocks:**
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  - Create m blocks, each containing n beads.
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  - Assign to each block a unique number of red beads, ranging from 0 to m1.
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2. **Design the Necklace:**
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  - Arrange these m blocks in a fixed order to form the necklace.
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  - Since the necklace is circular, cutting it at different points results in cyclic permutations of the blocks.
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3. **Verification:**
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  - In any cut, the sequence of blocks (and thus the counts of red beads) is a cyclic shift of the original sequence.
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  - Therefore, in each partition, the blocks will have distinct numbers of red beads.
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Example:
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Let's construct a necklace for m=3 and n=2:
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- **Blocks:**
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  - Block 1: 0 red beads (BB)
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  - Block 2: 1 red bead (RB)
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  - Block 3: 2 red beads (RR)
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- **Necklace Arrangement:**
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  - Place the blocks in order: **BB-RB-RR**
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- **Verification:**
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  - Any cut of the necklace into 3 blocks of 2 beads will have blocks with red bead counts of 0, 1, and 2.
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Conclusion:
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- **All ordered pairs (m,n) where mn+1 satisfy the condition.**
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- **Therefore, the possible values of (m,n) are all positive integers such that mn+1.**
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Final Answer:
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**Exactly all positive integers m and n with mn+1; these are all possible ordered pairs (m,n).**
  
 
==Video Solution==
 
==Video Solution==
 
https://youtu.be/ZcdBpaLC5p0 [video contains problem 1 and problem 4]
 
https://youtu.be/ZcdBpaLC5p0 [video contains problem 1 and problem 4]

Revision as of 01:40, 15 November 2024

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

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:**  
    \[m \leq n + 1\]

Sufficient Construction:

We will show that for all positive integers m and n satisfying mn+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 mn+1 satisfy the condition.** - **Therefore, the possible values of (m,n) are all positive integers such that mn+1.**

Final Answer:

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

Video Solution

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