Today (11/08/19) was PRMO 2019.
Q.15 asks
In how many ways can a pair of parallel diagonals of a regular polygon of 10 sides be selected?
I could not visualise the polygon during the test, so I thought only 5 pairs are there, that are the pairs of lines connecting the vertices of opposite parallel sides of the polygon which form a rectangle.
But now after looking at image of the polygon, I think there are 30 pairs, as there seem to be 4 parallel diagonals between any two parallel sides, so 4C2 * 5.
What is the correct answer?
Thank you.
Note- 4C2 means ways to choose 2 objects out of 4.
Best Answer
If the decagon is $P_1\cdots P_{10}$ then there are two types of sets of parallel diagonals. One is typified by $P_3P_{10}$, $P_4P_9$, $P_5P_8$ (parallel to the side $P_1P_2$) and the other by $P_1P_3$, $P_{10}P_4$, $P_9 P_5$ and $P_8P_6$. There are five sets of each type.
If you are counting unordered sets of parallel diagonals, then the answer is $$5\binom32+5\binom42=45.$$