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4

Part of 2012 Putnam

Problems(2)

Putnam 2012 A4

Source:

12/3/2012
Let qq and rr be integers with q>0,q>0, and let AA and BB be intervals on the real line. Let TT be the set of all b+mqb+mq where bb and mm are integers with bb in B,B, and let SS be the set of all integers aa in AA such that rara is in T.T. Show that if the product of the lengths of AA and BB is less than q,q, then SS is the intersection of AA with some arithmetic progression.
Putnaminequalitiesmodular arithmeticarithmetic sequencecollege contests
Putnam 2012 B4

Source:

12/3/2012
Suppose that a0=1a_0=1 and that an+1=an+e−ana_{n+1}=a_n+e^{-a_n} for n=0,1,2,… .n=0,1,2,\dots. Does an−log⁡na_n-\log n have a finite limit as n→∞?n\to\infty? (Here log⁡n=log⁡en=ln⁡n.\log n=\log_en=\ln n.)
Putnamlogarithmslimitinequalitiesinductioncalculusintegration