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This is a problem that comes from a 9th grade math tournament.

We have the following picture: -

The plane $\alpha$ and the distinct points M and N that do not lie on it are fixed. We want to find the point O on the plane $\alpha$ such that $\mid MO-NO \mid$ has maximum value, if such point exists. If not, we want to prove it doesn't exist.

I've tried solving a simpler variation of the problem in 2 dimensions where instead of a plane there is a line d and the points M, N & O are in the plane rather than in space, but I couldn't a solution for it either.

Another idea I had was to show that for any given point O a point P can be constructed for which $\mid MP-NP \mid$ > $\mid MO-NO\mid$, but I couldn't find any construction for such P.

Any idea is welcome. Thanks.

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1 Answer 1

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(Don't overthink it.)

Hint: Triangle inequality.

Thus $|MO -NO| \leq MN$, with equality when ...

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  • $\begingroup$ Thank you for the solution! In the case where MN is parallel to $\alpha$, I said there is no maximum value for |MO−NO|, as the difference tends to be equal to MN as the point O heads more and more away from MN, in the direction of (MN, but never actually reaches that value. A much simpler solution than I anticipated! $\endgroup$
    – Wiirexu
    Commented Dec 15, 2019 at 13:34
  • $\begingroup$ Right, in the case where MN is parallel to the plane, then the maximum does not exist. The supremum is still MN. I doubt the organizers expect a 9th grader to make a distinction between maximum and supremum though. $\endgroup$
    – Calvin Lin
    Commented Dec 15, 2019 at 19:22

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