1. ## Tetrahedron problem

"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.

2. Originally Posted by amoeba
"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 $\displaystyle T_1,\ T_2,\ T_3,\ T_4$, with $\displaystyle T_1\leqslant T_2\leqslant T_3\leqslant T_4$. If $\displaystyle T_1<T_2$ then (by the intermediate value theorem) there are points on the edges $\displaystyle T_1T_3$ and $\displaystyle T_1T_4$ where the temperature is $\displaystyle T_2$. If $\displaystyle T_1 = T_2 = T_3$ then we already have three points where the temperature is the same.

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