46th Putnam 1985 Problem B4 Let C be the circle radius 1, center the origin. A point P is chosen at random on the circumference of C, and another point Q is chosen at random in the interior of C. What is the probability that the rectangle with diagonal PQ, and sides parallel to the x-axis and y-axis, lies entirely inside (or on) C? Solution
: 4/π 2 . Given P, let R be the rectangle inscribed in C with sides parallel to the axes. Then the rectangle with diagonal PQ lies entirely inside C iff Q lies inside R. Let O be the center of C. Let the diameter of C parallel to the y-axis make an angle θ with OP. Then the area of R is 4 sin θ cos θ. So the required probability is 2/π ∫ 0 π/2 (4 sin θ cos θ)/π dθ = 4/π 2 ∫ 0 π/2 sin 2θ dθ = 4/π 2 . 46th Putnam 1985 © John Scholes jscholes@kalva.demon.co.uk 7 Jan 2001