MMM2020 Solutions.pdf
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… −p. Therefore, the slope equals −p 1−p . It now easily follows that this line is described by: y = − p 1−p x + p 1−p + 1− p For the line with parameter q we have likewise y = − q 1−q x + q 1−q + 1− q. At the intersection point of these two lines we find that the same y-value occurs if − p 1−p x + p 1−p + 1− p = − q 1−q x + q 1−q + 1− q which is equivalent to [ q 1−q − p 1−p ]x = q 1−q + 1− q − [ p 1−p + 1− p] or [q(1− p)− p(1− q)]x = q(1− p) + (1− q)2(1− p)− p(1− q)− (1− p)2(1− q) This can be … y = p(1 − q) + 1 − p = 1 − pq. Summarizing, the intersection point has coordinates: (1− (1− p)(1− q), 1− pq) or (p + q − pq, 1− pq) (c) It’s a parabola. See after part (d). (d) Consider the intersection point for the two lines with parameters p and q, computed under (b). If we keep one line (with parameter p) fixed and we 1 vary the other parameter q, we see that their intersection point converges to the tangent point of the fixed line at C, if q converges to p. So, if we choose p = q, we find that … xF = 4 and xP = 6. By similar logic, the expected number of cards one has to by in order to get pictures of Finn and Poe, xFP , satisfies the equation xFP = 1/4 · (1 + xP ) + 1/6 · (1 + xF ) + 7/12 · (1 + xFP ) which implies after a straightforward computation that xFP = 73 5 . Similarly, xRP = 7 and xRF = 52 7 . Finally, the desired number xRFP satisfies the equation xRFP = 1/3 · (1 + xFP ) + 1/4 · (1 + xRP ) + 1/6 · (1 + xRF ) + 1/4 · (1 + xRFP ) with solution xRFP ≈ 8.22. Answer to Problem 4: …