Let $M$ and $N$ be two points on the Thales circle of segment $AB$, different from $A$ and $B$. Let $C$ be the midpoint of segment $NA$, and $D$ be the midpoint of segment $NB$. The circle is intersected by line $MC$ at point $E$ (second intersection), and by line $MD$ at point $F$ (second intersection). What is the value of

$$
M C \cdot C E + M D \cdot D F
$$

if $AB = 2$ units?

Imre Merényi, Budapest
It is known that if the intersection point of chords $P Q$ and $R S$ of a circle is $T$, then $P T \cdot T Q = R T \cdot T S$ (Figure 1). Therefore, $M C \cdot C E = A C \cdot C N = \frac{1}{4} A N^{2}$ and $M D \cdot D F = B D \cdot D N = \frac{1}{4} B N^{2}$. The triangle $A B N$ is right-angled, so $A N^{2} + B N^{2} = A B^{2} = 4$. Based on these, $M C \cdot C E + M D \cdot D F = \frac{1}{4} \cdot 4 = 1$.

Therefore, the sought value is 1.

Based on the work of Tóth Gábor Zsolt (Budapest, Árpád Gymnasium, 2nd year).