I have a question about a simple proposition, I suppose that this is something well-known or a special case of something well-known:

Let $D\subset\mathbb{R}^{2}$ be the closed unit disk in the plane and $f:D\rightarrow\mathbb{R}$ be a smooth function. Suppose that the restriction $f|_{\mathbb{S}^{1}}$ has exactly $n$ local extrema, but none of them is a local extremum of $f$. Then the function $f$ has at least $\frac{n}{2}+1$ local extrema inside $D$.

Note that it follows from the condition that the level line of $f$ at a local extremum of $f|_{\mathbb{S}^{1}}$ touches the boundary from inside. Some examples are given below as level lines diagrams.

Here the crossing points of level lines correspond to saddles (possibly degenerated). As the condition is simple and clear and the proof is quite elementary, I intend to propose this at a student competition. However, if this is well-known or folklore, it might not be suitable to propose it, so any references are welcome.

Remarks. 1) This proposition may be completed by adding a second part: Let $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{R}$ be a smooth function with $n$ local extrema, then it may be extended to a smooth $f:D\rightarrow \mathbb{R}$ satisfying the above conditions with exactly $\frac{n}{2}+1$ local extrema (and thus lying in the interior of $D$). Presumably this is true, but I don't have a proof.

2. For a given even $n$, we may ask about the number of (topologically) different diagrams up to symmetry (rotation, reflection) with exactly $\frac{n}{2}+1$ local extrema and without other critical points; but it seems to be a very hard task.