Question 189
Let $ABCD$ be a square, and the points $M \in [BC]$, $N \in [CD]$, $P \in [DA]$, such that
$$
\angle(\overrightarrow{AB}, \overrightarrow{AM}) = x, \quad \angle(\overrightarrow{BC}, \overrightarrow{MN}) = 2x, \quad \angle(\overrightarrow{CD}, \overrightarrow{NP}) = 3x.
$$
i) Show that, for any $x \in [0, \pi/8]$, such a configuration uniquely exists, and $P$ ranges over the entire segment $[DA]$;
ii) Determine the number of angles $x \in [0, \pi/8]$ for which $\angle(\overrightarrow{DA}, \overrightarrow{PB}) = 4x$.
$$
\angle(\overrightarrow{AB}, \overrightarrow{AM}) = x, \quad \angle(\overrightarrow{BC}, \overrightarrow{MN}) = 2x, \quad \angle(\overrightarrow{CD}, \overrightarrow{NP}) = 3x.
$$
i) Show that, for any $x \in [0, \pi/8]$, such a configuration uniquely exists, and $P$ ranges over the entire segment $[DA]$;
ii) Determine the number of angles $x \in [0, \pi/8]$ for which $\angle(\overrightarrow{DA}, \overrightarrow{PB}) = 4x$.