For a): construction of the points: Take $M$ on $|BC|$ so that $\angle BAM\leq 22.5^\circ$, $E$ on $|BC|$ so that $AE$ is the reflection of $AB$ in $AM$, take $N$ on $|BC|$ so that $MN\bot AE$ and $P$ on $|AD|$ so that $NP$ is the reflection of $AM$ in perpendicular bisector of $MN$. $M, N, P$ are the required points and pt a) is then obvious. Best regards, sunken rock

And the fact that, other than $x_1 = \pi/8$ which clearly is a solution, there exists another solution $0< x_2 < \pi/8$ (quite close to $\pi/8$), is short of a miracle ...