Correct solutions were submitted by Bader Al-Kandari.  Further correct solutions were received from Christophe Carde, Agustin Murillo, A. Pozharskiy, Ross Millikan, Lou Cairoli, Juan Carrara, Burkart Venzke, Alejandro Vellano, Ron Welch, Nancy Schwarzkopf, Aaron Kahn, Jens Vob, Juan Marivela.

Solvers used a variety of methods, but all required some trigonometric identity.  Here is a simple one.  

Place a coordinate system on figure with the origin at O, the positive vertical axis along BO and the positive horizontal axis along OC.  With the point C at (1,0), the line along AX has equation  y  = tan(a)×(x + 1), while the line along BC has equation y  = tan(2a)×(1 - x).   Equate the two expressions and solve for the x-coordinate of X; convert the tangents to sines and cosines, and simplify to get the last expression in the first row below, then use the angle-sum and difference formula to get:

An application of L'Hôpital's Rule gives that x ® 1/3 as a ®  0.  

This situation leads to other questions:
1.  What is limiting value of x as a ® p/4?
2.  What is the locus of points traced out by the point X as a ranges from 0 to p/4?

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