Question

A set \( S = \{1, 2, 3, \ldots, n\} \) is partitioned into \( n \) disjoint subsets \( A_1, A_2, \ldots, A_n \), each containing four elements. It is given that in each subset, one element is the arithmetic mean of the other three. Which of the following statements is true?

• A. \( n \neq 1 \) and \( n \neq 2 \)
• B. \( n \neq 1 \) but can be equal to 2
• C. \( n \neq 2 \) but can be equal to 1
• D. It is possible to satisfy for \( n = 1 \) as well as for \( n = 2 \)

Hint:

Try small values of \( n \) to verify if subsets can be constructed with one element as arithmetic mean.

Answer 1

The Correct Option is D

Each subset has 4 elements, and one of them is arithmetic mean of the other three.
Check for \( n = 1 \Rightarrow S = \{1, 2, 3, 4\} \). Can any one of these be average of the other three? Try: \[ \frac{1 + 2 + 3}{3} = 2 → okay
\frac{2 + 3 + 4}{3} = 3 → okay \] So \( n = 1 \) works. Check for \( n = 2 \Rightarrow S = \{1, 2, \ldots, 8\} \). Can we partition into two 4-element sets each with 1 arithmetic mean? Yes. Hence, \boxed{\text{possible for both } n = 1 \text{ and } n = 2}