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Part of 2014 Putnam

Problems(2)

Putnam 2014 A3

Source:

12/7/2014
Let a0=5/2a_0=5/2 and ak=ak−12−2a_k=a_{k-1}^2-2 for k≥1.k\ge 1. Compute ∏k=0∞(1−1ak)\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right) in closed form.
Putnamlimitinductioninequalitiesalgebradifference of squaresPutnam 2014
Putnam 2014 B3

Source:

12/8/2014
Let AA be an m×nm\times n matrix with rational entries. Suppose that there are at least m+nm+n distinct prime numbers among the absolute values of the entries of A.A. Show that the rank of AA is at least 2.2.
Putnampigeonhole principlevectorPutnam matricesPutnam 2014