[The 1985 solutions hosted here were prepared by someone else, not rusin@math.niu.edu ; evidently there are some errors.] ============================================================================== From: "Raymond Cassella" To: Subject: Solution to Putnam B4 1985 Date: Wed, 7 Nov 2001 22:26:13 -0500 The solution to problem B4 located at http://www.math.niu.edu/~rusin/problems-math/1985.pdf seems rather wrong. First of all, wouldn't the area of the rectangle be 4|sin(theta)*cos(theta)|, not 2|sin(theta)*cos(theta)|? And also, the second expression of the probability as an integral is wrong. Neither the last nor next to last equalities hold. My answer to this problem was 4/Pi^2. ============================================================================== Date: 11 Jun 2002 13:40:37 -0500 From: "micah mcdaniel" To: rusin@math.niu.edu Subject: putnam exam 1985 problem A-4 your solution to problem A-4 is incorrect, I think. In particular: all numbers on the right hand side are mod 100 a(1) = 3 a(2) = 3^3 = 27 a(3) = 3^27 = 87 , not 29 a(4) = 3^87 = 87 a(n) = a(n-1) so the only numbers that appear are 03, 27, and 87. Micah McDaniel mcdaniel@parthus.com 512-249-2330x215 ============================================================================== Date: Wed, 19 Dec 2001 17:57:05 +0100 From: "Gerald Petit" To: rusin@math.niu.edu CC: rcassell@ic.sunysb.edu, petitge@yahoo.com Subject: Calculation mistakes in my Putnam 1985 solution Dear Prof Rusin, I just had a new look at your Putnam page which now contains more info (excellent). I thank the reader who detected the calculation mistake in my proposed solution of B4 for 1985. I had noticed it before when visiting John Schole's site. There are two other mistakes. I didn't sent you corrections because I cannot work in TeX anymore (I don't trust my equipment to support it) and I could not alter my original file: - A4 was also wrong since 3^27 = 87 and not 29 modulo 100. Also 3^87 = 87 modulo 100 - thus the only integer solution to the problem is 87. - For B5 the approach for calculating the integral is right, but I have screwed the last line: I = 2 * f(1) and not 2 * f(m) (thus I = ((pi/1985)^(1/2)) * exp (-2 * 1985)). Ge'rald Petit