Question 1042421
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Eighteen points are equally spaced on a circle, from which you will choose a certain number at random. 
How many do you need to choose to guarantee that you will have the four corners of at least one rectangle?
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You need to have 2 (two) pairs of diametrically opposite points. Two different pairs, I mean.


Among any of 10 of the given points, there are at least two (one pair) of diametrically opposite points 
(Dirichlet's principle // pigeonhole principle).


If you randomly choose 10 = {{{18/2 + 1}}} points, you guarantee at least 1 pair of diametrically opposite points.


If you randomly choose 11 = 10 + 1 points, you guarantee one pair of diametrically opposite points; 
and after removing/withdrawing these two points, still there are at least one pair of diametrically opposite points 
among the remaining 9 = 11-2 points.


<U>Answer</U>. You need to choose 11 points randomly.