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Challenge 308: Nearby or Not?

(i) 13 points are picked at random inside a regular hexagon with side length 1 cm.

Show that it is always possible to draw an equilateral triangle of side length 1 cm so that at least 3 of these 13 points lie inside or on the perimeter of the triangle.

(ii) 15 points are picked at random inside a circle of radius 2 cm.

Is it always possible to draw a circle of radius 1 cm so that at least 3 of these 15 points lie inside or on the circumference of the circle? Justify your answer clearly!

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