Chord Lines

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Let AB be a chord of a circle with center O and radius R. M is the midpoint of AB and line OM intersects the circle at point C. Solve each of the following in terms of R.
  1. Find AM and CM if AM equals 3*CM, 2*CM, CM, CM/2, or CM/3. Can you solve the general case for AM=X*CM?
  2. Line segments are drawn from C to point Q on AM, from Q to point P on the circle, and from P to A. Find CQ if CQ=QP=PA. I believe there are multiple answers, so solve for CQ in terms of CM (or AB). What is CQ if CM=R/2?

Source: Original. Based upon a design on the side of a building in Rocklin, CA on I-80.

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