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Part of 2013 Putnam

Problems(2)

Putnam 2013 A2

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12/9/2013
Let SS be the set of all positive integers that are not perfect squares. For nn in S,S, consider choices of integers a1,a2,…,ara_1,a_2,\dots, a_r such that n<a1<a2<⋯<arn<a_1<a_2<\cdots<a_r and n⋅a1⋅a2⋯arn\cdot a_1\cdot a_2\cdots a_r is a perfect square, and let f(n)f(n) be the minimum of ara_r over all such choices. For example, 2⋅3⋅62\cdot 3\cdot 6 is a perfect square, while 2⋅3,2⋅4,2⋅5,2⋅3⋅4,2\cdot 3,2\cdot 4, 2\cdot 5, 2\cdot 3\cdot 4, 2⋅3⋅5,2⋅4⋅5,2\cdot 3\cdot 5, 2\cdot 4\cdot 5, and 2⋅3⋅4⋅52\cdot 3\cdot 4\cdot 5 are not, and so f(2)=6.f(2)=6. Show that the function ff from SS to the integers is one-to-one.
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Putnam 2013 B2

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12/9/2013
Let C=⋃N=1∞CN,C=\bigcup_{N=1}^{\infty}C_N, where CNC_N denotes the set of 'cosine polynomials' of the form f(x)=1+∑n=1Nancos⁡(2πnx)f(x)=1+\sum_{n=1}^Na_n\cos(2\pi nx) for which:
(i) f(x)≥0f(x)\ge 0 for all real x,x, and (ii) an=0a_n=0 whenever nn is a multiple of 3.3.
Determine the maximum value of f(0)f(0) as ff ranges through C,C, and prove that this maximum is attained.
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