Subcontests
(8)Points, lines, circle
Points A,B,C,D are on circle S, such that AB is the diameter of S, but CD is not the diameter. Given also that C and D are on different sides of AB. The tangents of S at C and D intersect at P. Points Q and R are the intersections of line AC with line BD and line AD with line BC, respectively.
(a) Prove that P, Q, and R are collinear.
(b) Prove that QR is perpendicular to line AB. Beautiful arrangements
A 10-digit arrangement 0,1,2,3,4,5,6,7,8,9 is called beautiful if (i) when read left to right, 0,1,2,3,4 form an increasing sequence, and 5,6,7,8,9 form a decreasing sequence, and (ii) 0 is not the leftmost digit. For example, 9807123654 is a beautiful arrangement. Determine the number of beautiful arrangements.