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Putnam 1939 A7

Source:

August 20, 2021
Putnam

Problem Statement

Do either (1)(1) or (2)(2):
(1)(1) Let CaC_a be the curve (y−a2)2=x2(a2−x2).(y - a^2)^2 = x^2(a^2 - x^2). Find the curve which touches all CaC_a for a>0.a > 0. Sketch the solution and at least two of the Ca.C_a.
(2)(2) Given that (1−hx)−1(1−kx)−1=∑i≥0aixi,(1 - hx)^{-1}(1 - kx)^{-1} = \sum_{i\geq0}a_i x^i, prove that (1+hkx)(1−hkx)−1(1−h2x)−1(1−k2x)−1=∑i≥0ai2xi.(1 + hkx)(1 - hkx)^{-1}(1 - h^2x)^{-1}(1 - k^2x)^{-1} = \sum_{i\geq0} a_i^2 x^i.
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