Question 9
Let $ n \geq 3 $ be a given positive integer. We wish to label each side and each diagonal of a regular $ n $-gon $ P_1 \ldots P_n $ with a positive integer less than or equal to r so that:
(i) every integer between 1 and r occurs as a label;
(ii) in each triangle $P_{i}P_{j}P_{k}$ two of the labels are equal and greater than the third. Given these conditions:
(a) Determine the largest positive integer r for which this can be done.
(b) For that value of r, how many such labellings are there?
(i) every integer between 1 and r occurs as a label;
(ii) in each triangle $P_{i}P_{j}P_{k}$ two of the labels are equal and greater than the third. Given these conditions:
(a) Determine the largest positive integer r for which this can be done.
(b) For that value of r, how many such labellings are there?