48th Putnam 1987

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Problem B1

Evaluate ∫24 ln1/2(9 - x)/( ln1/2(9 - x) + ln1/2(x + 3) ) dx.

 

Solution

Answer: 1.

Offputting, but easy!

The integrand looks horrible. A little playing around, differentiating things like ln1/2(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 ∫24 ln1/2(y + 3)/( ln1/2(9 - y) + ln1/2(y + 3) ) dy. We are home, because we can add the two integrals to get ∫24 1 dx = 2.

 


 

48th Putnam 1987

© John Scholes
jscholes@kalva.demon.co.uk
7 Oct 1999