Pigeonhole principle in geometry

stan

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Dec 4, 2018
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More than half of the sphere is filled with color. Show that if we color the area in whatever way there can always be drawn a line in a way that the line goes through the center of the sphere and both points at the end of the line are on colored area.

I came across this problem and don't know how to explain it.
 
Looks like a good candidate for a proof by contradiction.
 
More than half of the sphere is filled with color. Show that if we color the area in whatever way there can always be drawn a line in a way that the line goes through the center of the sphere and both points at the end of the line are on colored area.

I came across this problem and don't know how to explain it.
Pick a painted point and assume that the point opposite is not painted. keep assuming this. What can't be true???
 
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