Welcome to three new puzzles. Solutions to the first two will be published next month; the third is (as yet) unsolved. In each, the issue is how your intuition matches up with the mathematics.
Puzzled: Variations on the Ham Sandwich Theorum
I think the truth of 2 and 3 may depend on how you define (closed) curve, so please give your precise definitions. For instance, there are unbounded, space filling curves for which 2 would not hold, and one can make such a curve 'closed' by self-crossing. So, maybe 2 should be restricted to simple (Jordan) curves?
A continuous tangent should be sufficient. Are weaker conditions sufficient ?
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