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

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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:
 
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 <math>m</math> blocks of <math>n</math> 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 <math>n</math> beads, the number of red beads can range from <math>0</math> to <math>n</math>.
+
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 <math>n + 1</math> 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 <math>m</math> blocks, the maximum number of distinct counts must be at least <math>m</math>.
+
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:
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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.
 
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.
  
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1. **Construct Blocks:**
+
1. Construct Blocks:
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  - Create <math>m</math> blocks, each containing <math>n</math> beads.
+
Create <math>m</math> blocks, each containing <math>n</math> beads. Assign to each block a unique number of red beads, ranging from <math>0</math> to <math>m - 1</math>.
−
  - Assign to each block a unique number of red beads, ranging from <math>0</math> to <math>m - 1</math>.
 
  
−
2. **Design the Necklace:**
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2. Design the Necklace:
−
  - Arrange these <math>m</math> 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.
 
  
−
3. **Verification:**
+
3. Verification:
−
  - In any cut, the sequence of blocks (and thus the counts of red beads) is a cyclic shift of the original sequence.
+
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.
−
  - Therefore, in each partition, the blocks will have distinct numbers of red beads.
 
  
 
Example:
 
Example:
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Let's construct a necklace for <math>m = 3</math> and <math>n = 2</math>:
 
Let's construct a necklace for <math>m = 3</math> and <math>n = 2</math>:
  
−
- **Blocks:**
+
Blocks:
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  - Block 1: <math>0</math> red beads (BB)
+
Block 1: <math>0</math> red beads (BB)
−
  - Block 2: <math>1</math> red bead (RB)
+
Block 2: <math>1</math> red bead (RB)
−
  - Block 3: <math>2</math> red beads (RR)
+
Block 3: <math>2</math> red beads (RR)
  
−
- **Necklace Arrangement:**
+
Necklace Arrangement:
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  - Place the blocks in order: **BB-RB-RR**
+
Place the blocks in order: BB-RB-RR.
  
−
- **Verification:**
+
Verification:
−
  - 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>.
+
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:
  
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- **All ordered pairs <math>(m, n)</math> where <math>m \leq n + 1</math> 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 <math>(m, n)</math> are all positive integers such that <math>m \leq n + 1</math>.**
+
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 <math>m</math> and <math>n</math> with <math>m \leq n + 1</math>; these are all possible ordered pairs <math>(m, n)</math>.**
+
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>.
 +
 
 +
~[https://artofproblemsolving.com/wiki/index.php/User:Athmyx Athmyx]
  
 
==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]

Latest revision as of 01:50, 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 $(m, n)$.

~Athmyx

Video Solution

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

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