LIV OM - I - Task 7

At Aunt Reni's, $ n \geq 4 $ people met (including Aunt Reni). Each of the attendees gave at least one gift to at least one of the others. It turned out that everyone gave three times as many gifts as they received, with one exception: Aunt Reni gave only $ \frac{1}{6} $ of the number of gifts she received. Determine, depending on $ n $, the smallest possible number of gifts Aunt Reni could have received.
Let $ x_i $ ($ i = 1,2, \ldots ,n-1 $) denote the number of gifts received by the $ i $-th guest of Aunt Renia, and let $ x_R $ denote the number of gifts Aunt Renia received. Then the total number of gifts is

Therefore, the number $ x_R $ is divisible by $ 12 $, say $ x_R = 12k $, where $ k $ is some non-negative integer.
Since each person gave at least one gift, they must have also received at least one gift. Hence, we infer that $ x_i \geq 1 $ for $ i = 1,2,\ldots ,n-1 $, which, by the second equality (1), gives $ 5k \geq n-1 $. Since $ k $ is an integer, the last inequality is equivalent to the inequality $ k \geq [(n-1+4)/5] $, where $ [a] $ denotes the greatest integer not greater than $ a $. Therefore,

It remains to show that under the conditions of the problem, it is possible for Aunt Renia to receive exactly $ 12 [(n+3)/5] $ gifts.
Assume that $ n - 1 = 5k - r $, where $ r = 0,1,2,3,4 $, and $ k $ is some positive integer. Divide the $ n-1 $ guests of Aunt Renia into (at most) three disjoint groups:

(Since $ n \geq 4 $, then $ 2k-r \geq 0 $.) Next, seat all the people $ A_i $ and $ C_i $ around a round table and assume that:

Each person $ A_i $ and $ C_i $ gives 1 gift to the person sitting to their right;
Each person $ A_i $ gives 2 gifts to Aunt Renia;
Each person $ B_i $ gives 3 gifts to Aunt Renia;
Each person $ C_i $ gives 5 gifts to Aunt Renia;
Aunt Renia gives 1 gift to each person $ B_i $ and $ C_i $.

In this way, each person $ A_i $ and $ B_i $ gave 3 gifts and received one, while each person $ C_i $ gave 6 gifts and received two. Thus, each guest of Aunt Renia gave three times as many gifts as they received. On the other hand, Aunt Renia received

gifts, while giving $ 2k $ gifts, which is six times less.
Answer: Aunt Renia could receive at least $ 12[(n+3)/5] $ gifts.