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Part of 2000 Putnam

Problems(2)

Putnam 2000 A3

Source:

9/6/2011
The octagon P1P2P3P4P5P6P7P8P_1P_2P_3P_4P_5P_6P_7P_8 is inscribed in a circle with the vertices around the circumference in the given order. Given that the polygon P1P3P5P7P_1P_3P_5P_7 is a square of area 55, and the polygon P2P4P6P8P_2P_4P_6P_8 is a rectangle of area 44, find the maximum possible area of the octagon.
Putnamgeometryrectangleanalytic geometryquadraticsgeometric transformationreflection
Putnam 2000 B3

Source:

9/6/2011
Let f(t)=∑j=1Najsin⁡(2πjt)f(t) = \displaystyle\sum_{j=1}^{N} a_j \sin (2\pi jt), where each aja_j is areal and aNa_N is not equal to 00. Let NkN_k denote the number of zeroes (including multiplicites) of dkfdtk\dfrac{d^k f}{dt^k}. Prove that N0≤N1≤N2≤⋯ and lim⁡k→∞Nk=2N. N_0 \le N_1 \le N_2 \le \cdots \text { and } \lim_{k \rightarrow \infty} N_k = 2N. [Only zeroes in [0, 1) should be counted.]
Putnamtrigonometrylimitcalculusderivativealgebrapolynomial