48th Putnam 1987 Problem B1 Evaluate ∫ 2 4 ln 1/2 (9 - x)/( ln 1/2 (9 - x) + ln 1/2 (x + 3) ) dx. Solution
: 1. Offputting, but easy! The integrand looks horrible. A little playing around, differentiating things like ln 1/2 (9 - x), soon confirms this impression. But we know the first question in each Putnam session is always easy, so there must be some trick! We might notice that as x goes from 2 to 4, 9 - x goes from 7 to 5, and x + 3 goes from 5 to 7. That might suggest substituting y such that 9 - y = x + 3 (and hence also y + 3 = 9 - x). dy = - dx, the limits interchange and we get ∫ 2 4 ln 1/2 (y + 3)/( ln 1/2 (9 - y) + ln 1/2 (y + 3) ) dy. We are home, because we can add the two integrals to get ∫ 2 4 1 dx = 2. 48th Putnam 1987 © John Scholes jscholes@kalva.demon.co.uk 7 Oct 1999