Show that for any bounded shape, you can make a straight cut in any direction to create halves of equal area On an infinite frying pan, there is a bounded pancake. Prove that one can make a straight cut in any given direction (that is, parallel to a given line) that splits the pancake in halves of equal area.
I think that you can use intermediate value theorem, but I'm not quite sure where to start or how to apply the theorem

Here's the outline of a solution. For every nonzero vector $v \in \mathbb{R}^n$ and $\lambda \in \mathbb{R}$ consider the half-space defined as $H_{\lambda} = x \cdot v \geq \lambda$. For any bounded region $B$, let $B_{\lambda} = H_{\lambda} \cap B$. This corresponds to the part of $B$ on side of a cut that runs parallel to any vector perpendicular to $v$. Fix the vector $v$ and define the function $f(\lambda) = \mbox{Volume}(B_\lambda)$. Show that $f(\lambda)$ is continuous, that $\lim_{\lambda \to \infty} f(\lambda) = 0$, and that $\lim_{\lambda \to -\infty} f(\lambda) = \mbox{Volume}(B)$. Finally, apply the intermediate value theorem.
Perhaps this picture might help:

