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Math Help - Tetrahedron problem

  1. #1
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    Tetrahedron problem

    The question is asking,
    "Prove that the temperature of a tetrahedron must have at least three
    distinct points on the edges or vertices of the tetrahedron with the
    same value. Assume the temperature is a continuous function."

    My approach would be somehow to use the mean value theorem and represent the edges as lines, but I honestly don't even know how to start this problem.
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  2. #2
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    Quote Originally Posted by amoeba View Post
    The question is asking,
    "Prove that the temperature of a tetrahedron must have at least three
    distinct points on the edges or vertices of the tetrahedron with the
    same value. Assume the temperature is a continuous function."

    My approach would be somehow to use the mean value theorem and represent the edges as lines, but I honestly don't even know how to start this problem.
    I think you mean "intermediate value theorem" rather than "mean value theorem".

    Suppose that the temperatures at the four vertices are T_1,\ T_2,\ T_3,\ T_4, with T_1\leqslant T_2\leqslant T_3\leqslant T_4. If T_1<T_2 then (by the intermediate value theorem) there are points on the edges T_1T_3 and T_1T_4 where the temperature is T_2. If T_1 = T_2 = T_3 then we already have three points where the temperature is the same.

    That leaves us with the case T_1=T_2<T_3. In that case, choose T_0 with T_1<T_0<T_3, and check that the edges T_1T_3,\ T_2T_3,\ T_1T_4,\ T_2T_4 each have a point where the temperature is T_0.
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