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Putnam 2017 A3

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December 3, 2017
PutnamPutnam 2017Putnam calculusPutnam analysis

Problem Statement

Let aa and bb be real numbers with a<b,a<b, and let ff and gg be continuous functions from [a,b][a,b] to (0,∞)(0,\infty) such that ∫abf(x) dx=∫abg(x) dx\int_a^b f(x)\,dx=\int_a^b g(x)\,dx but f≠g.f\ne g. For every positive integer n,n, define In=∫ab(f(x))n+1(g(x))n dx.I_n=\int_a^b\frac{(f(x))^{n+1}}{(g(x))^n}\,dx. Show that I1,I2,I3,…I_1,I_2,I_3,\dots is an increasing sequence with lim⁡n→∞In=∞.\displaystyle\lim_{n\to\infty}I_n=\infty.
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