New Year 2023. No.1

[rsoldo]

Quadrilateral $ABCD$ is inscribed in a circle. Prove that there is a unique pair of points $M,N$ who lying on the sides $AB$ and $CD$ for which the line $MN$ is the bisector of the angles $CMD$ and $ANB.$

[Dendr]

It's easy to show that there must be at least one pair $M, N$.

We just start with first point $P\in AB$, then built the bisector $PQ$ of $\angle CPD$, and after that the director $QR$ of $\angle AQB$.
Since $ABCD$ is inscribed , thus convex, $R$ is always lying between $A$ and $B$. It means that if $P=A$, the point $R$ is between $P$ and $B$, and if $P=B$ -- between $P$ and $A$.

While $P$ is moving continuously, so is $R$, and at some point they cross, which means that $P=R=N, Q=M$. So existence is proven.

But.

This approach doesn't help us to prove uniqueness. Surely, we can introduce a variable $x=\frac{AP}{AB}, y=\frac{AR}{AB}$, define closed form for a function $f(x)=y(x)-x$ and prove that $f(x)$ is monotonous, or $f'(x) \ne0$ if $0<x<1$. Though, it's incredibly hard.

Next attempt is geometry...

[TOTAL]

Продлим $AB$ и $CD$ до пересечения в точке $P$. Из точки $P$ как из центра рисуем вторую окружность, радиус которой равен длине касательной из точки $P$ к первой окружности. Таким образом на $AB$ и $CD$ получаем две искомых насечки $M$ и $N$.