PRMO 2019 Question 15 number of diagonals in a regular 10-gon

In how many ways can a pair of parallel diagonals of a regular polygon of 10 sides be selected?
Hint: First try for an octagon.

Visual solution.

Solution:

If the decagon is P1⋯P10P_1 \cdots P_{10} then there are two types of sets of parallel diagonals. One is typified by
P3P10, P4P9, P5P8P_3P_{10},\ P_4P_9,\ P_5P_8 (parallel to the side P1P2P_1P_2) and the other by P1P3, P10P4, P9P5P_1P_3,\ P_{10}P_4,\ P_9P_5 and P8P6P_8P_6.
There are five sets of each type.

If you are counting unordered sets of parallel diagonals, then the answer is

5(32)+5(42)=45.







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