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Enigma 1465.We can consider the vote counting as path going from

(0, 0)to(a, b)(whereaandbare the number of votes for each candidate).This Python program constructively counts the possible paths. It runs in 389ms.

Solution:150 membersdid notvote in the election.The only possible scenario is that George received 101 votes, and his opponent received 99 votes. So 200 of 350 members voted.

Analytically, if George got

avotes and his opponent gotbvotes then the number of different ways that the votes can be counted is:And the number of ways they can be counted such that George is always ahead is:

(See: [ https://en.wikipedia.org/wiki/Catalan%27s_triangle ]).

We are interested in when the ratio of these two terms is 100:1. i.e. when:

From which we see the solution is

a= 101,b= 99 asa + b< 350.In this case the numbers involved are quite large:

This solution is essentially the same as that given above by Jim. I could have combined the two path functions (as Jim does) but I found that it ran faster when they were kept separate. I also think it is a bit easier to understand these functions when they are kept separate.