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Difference between revisions of "2024 USAMO Problems/Problem 4"
(→‎Solution 1)
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==Solution 1==
 
==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:
+
We need to determine all possible positive integer pairs <math>(m, n)</math> such that there exists a circular necklace of <math>mn</math> 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.**
+
- **No matter how the necklace is cut into <math>m</math> blocks of <math>n</math> consecutive beads, each block has a distinct number of red beads.**
  
 
Necessary Condition:
 
Necessary Condition:
  
 
1. **Maximum Possible Distinct Counts:**
 
1. **Maximum Possible Distinct Counts:**
−
   - In a block of \(n\) beads, the number of red beads can range from \(0\) to \(n\).
+
   - In a block of <math>n</math> beads, the number of red beads can range from <math>0</math> to <math>n</math>.
−
   - Therefore, there are \(n + 1\) possible distinct counts of red beads in a block.
+
   - Therefore, there are <math>n + 1</math> possible distinct counts of red beads in a block.
−
   - Since we have \(m\) blocks, the maximum number of distinct counts must be at least \(m\).
+
   - Since we have <math>m</math> blocks, the maximum number of distinct counts must be at least <math>m</math>.
 
   - **Thus, we must have:**   
 
   - **Thus, we must have:**   
−
     <cmath> m \leq n + 1 </cmath>
+
     <cmath>m \leq n + 1</cmath>
  
 
Sufficient Construction:
 
Sufficient Construction:
  
−
We will show that for all positive integers \(m\) and \(n\) satisfying \(m \leq n + 1\), such a necklace exists.
+
We will show that for all positive integers <math>m</math> and <math>n</math> satisfying <math>m \leq n + 1</math>, such a necklace exists.
  
 
1. **Construct Blocks:**
 
1. **Construct Blocks:**
−
   - Create \(m\) blocks, each containing \(n\) beads.
+
   - Create <math>m</math> blocks, each containing <math>n</math> beads.
−
   - Assign to each block a unique number of red beads, ranging from \(0\) to \(m - 1\).
+
   - Assign to each block a unique number of red beads, ranging from <math>0</math> to <math>m - 1</math>.
  
 
2. **Design the Necklace:**
 
2. **Design the Necklace:**
−
   - Arrange these \(m\) blocks in a fixed order to form the necklace.
+
   - Arrange these <math>m</math> 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.
 
   - Since the necklace is circular, cutting it at different points results in cyclic permutations of the blocks.
  
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Example:
 
Example:
  
−
Let's construct a necklace for \(m = 3\) and \(n = 2\):
+
Let's construct a necklace for <math>m = 3</math> and <math>n = 2</math>:
  
 
- **Blocks:**
 
- **Blocks:**
−
   - Block 1: \(0\) red beads (BB)
+
   - Block 1: <math>0</math> red beads (BB)
−
   - Block 2: \(1\) red bead (RB)
+
   - Block 2: <math>1</math> red bead (RB)
−
   - Block 3: \(2\) red beads (RR)
+
   - Block 3: <math>2</math> red beads (RR)
  
 
- **Necklace Arrangement:**
 
- **Necklace Arrangement:**
Line 45: Line 45:
  
 
- **Verification:**
 
- **Verification:**
−
   - Any cut of the necklace into \(3\) blocks of \(2\) beads will have blocks with red bead counts of \(0\), \(1\), and \(2\).
+
   - Any cut of the necklace into <math>3</math> blocks of <math>2</math> beads will have blocks with red bead counts of <math>0</math>, <math>1</math>, and <math>2</math>.
  
 
Conclusion:
 
Conclusion:
  
−
- **All ordered pairs \((m, n)\) where \(m \leq n + 1\) satisfy the condition.**
+
- **All ordered pairs <math>(m, n)</math> where <math>m \leq n + 1</math> satisfy the condition.**
−
- **Therefore, the possible values of \((m, n)\) are all positive integers such that \(m \leq n + 1\).**
+
- **Therefore, the possible values of <math>(m, n)</math> are all positive integers such that <math>m \leq n + 1</math>.**
  
 
Final Answer:
 
Final Answer:
  
−
**Exactly all positive integers \(m\) and \(n\) with \(m \leq n + 1\); these are all possible ordered pairs \((m, n)\).**
+
**Exactly all positive integers <math>m</math> and <math>n</math> with <math>m \leq n + 1</math>; these are all possible ordered pairs <math>(m, n)</math>.**
  
 
==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 01:43, 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

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 $m \leq n + 1$, 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]

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