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4

Part of 1999 Putnam

Problems(2)

Putnam 1999 A4

Source:

12/22/2012
Sum the series ∑m=1∞∑n=1∞m2n3m(n3m+m3n).\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.
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Putnam 1999 B4

Source:

12/22/2012
Let ff be a real function with a continuous third derivative such that f(x)f(x), f′(x)f^\prime(x), f′′(x)f^{\prime\prime}(x), f′′′(x)f^{\prime\prime\prime}(x) are positive for all xx. Suppose that f′′′(x)≤f(x)f^{\prime\prime\prime}(x)\leq f(x) for all xx. Show that f′(x)<2f(x)f^\prime(x)<2f(x) for all xx.
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