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Five points in the plane - Iran First Round 2018, P4

Source:

March 6, 2021
points

Problem Statement

There are 55 points in the plane no three of which are collinear. We draw all the segments whose vertices are these points. What is the minimum number of new points made by the intersection of the drawn segments?
<spanclass=latexbold>(A)</span> 0<spanclass=latexbold>(B)</span> 1<spanclass=latexbold>(C)</span> 2<spanclass=latexbold>(D)</span> 3<spanclass=latexbold>(E)</span> 5<span class='latex-bold'>(A)</span>\ 0\qquad<span class='latex-bold'>(B)</span>\ 1\qquad<span class='latex-bold'>(C)</span>\ 2\qquad<span class='latex-bold'>(D)</span>\ 3\qquad<span class='latex-bold'>(E)</span>\ 5