Math Problem Statement
each student participates in exactly two lessons and any pair of students participates in exactly one lesson together. Show that there is a course in which at least 2/3 of the students participate.
Solution
To solve this, let’s use a combinatorial approach. We assume we have students, each participating in exactly two lessons, and each pair of students appears together in exactly one lesson. We want to show that there exists a course in which at least of the students participate.
Step 1: Analyzing Participation and Pairs
- Let be the set of students, and be the set of lessons.
- Since each student participates in exactly two lessons, each student appears in two distinct lessons from .
- For each pair of students, there is exactly one lesson they participate in together. This constraint implies a unique pairing structure across lessons, similar to the properties of a projective plane configuration or graph-theoretic design.
Step 2: Counting Total Lesson Slots
If each student is in exactly two lessons, the total "student-lesson slots" (or incidences) is .
If each lesson contains students, we get: Therefore, the total number of students across all lessons is twice the number of students in .
Step 3: Using the Pigeonhole Principle
Now, let's consider the average number of students per lesson. Since there are lessons and student-lesson incidences, the average number of students per lesson is: To ensure that at least one lesson contains at least of the students, we apply the pigeonhole principle.
Suppose each lesson contained fewer than students. Then: which contradicts the fact that the sum of all students in lessons is exactly .
Conclusion
Thus, there must be at least one lesson containing at least of the students, completing the proof.
Would you like a more detailed explanation of any part?
Related Questions
- What is the role of projective planes in combinatorial design problems like this one?
- How does the pigeonhole principle apply in combinatorial proofs?
- What are some real-world applications of combinatorial design principles?
- Could a similar principle be used if each student attended three lessons instead?
- How does this problem change if we relax the “exactly one lesson together” condition?
Tip
When tackling combinatorial proofs, break down the problem into manageable parts, like counting incidences or using averages, before applying general principles like pigeonhole
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Math Problem Analysis
Mathematical Concepts
Combinatorics
Graph Theory
Projective Planes
Pigeonhole Principle
Formulas
Total student-lesson slots: 2n
Sum of students in all lessons: ∑ |L_i| = 2n
Average number of students per lesson: 2n / k
Theorems
Pigeonhole Principle
Combinatorial Design Theory
Suitable Grade Level
Undergraduate Level (Discrete Mathematics/Combinatorics)
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