41st Putnam 1980 Problem B1 For which real k do we have cosh x ≤ exp(k x 2 ) for all real x? Solution
: k ≥ 1/2. cosh x = 1 + x 2 /2! + x 4 /4! + ... + x 2n /(2n)! + ... and exp(x 2 /2) = 1 + x 2 /2 + x 4 /8 + ... + x 2n /(2 n n!). But 2 n n! = 2n.(2n - 2).(2n - 4) ... 2  1, so cosh x  exp(k x 2 ) for sufficiently small x. Thus the inequality does not hold for k < 1/2. 41st Putnam 1980 © John Scholes jscholes@kalva.demon.co.uk 10 Dec 1999 Last corrected/updated 21 Nov 02