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Putnam 1941 B7

Source: Putnam 1941

February 23, 2022
Putnamalgebrafunctional equationcalculusderivativefunctiontrigonometry

Problem Statement

Do either (1) or (2): (1) Show that any solution f(t)f(t) of the functional equation f(x+y)f(x−y)=f(x)2+f(y)2−1f(x+y)f(x-y)=f(x)^{2} +f(y)^{2} -1 for x,y∈Rx,y\in \mathbb{R} satisfies f′′(t)=±c2f(t)f''(t)= \pm c^{2} f(t) for a constant cc, assuming the existence and continuity of the second derivative. Deduce that f(t)f(t) is one of the functions ±cos⁡ct,      ±cosh⁡ct. \pm \cos ct, \;\;\; \pm \cosh ct.
(2) Let (ai)i=1,...,n(a_{i})_{i=1,...,n} and (bi)i=1,...,n(b_{i})_{i=1,...,n} be real numbers. Define an (n+1)×(n+1)(n+1)\times (n+1)-matrix A=(cij)A=(c_{ij}) by ci1=1,    c1j=xj−1  for  j≤n,    c1n+1=p(x),    cij=ai−1j−1  for  i>1,j≤n,    cin+1=bi−1  for  i>1. c_{i1}=1, \; \; c_{1j}= x^{j-1} \; \text{for} \; j\leq n,\; \; c_{1n+1}=p(x), \;\; c_{ij}=a_{i-1}^{j-1} \; \text{for}\; i>1, j\leq n,\;\; c_{in+1}=b_{i-1}\; \text{for}\; i>1. The polynomial p(x)p(x) is defined by the equation det⁡A=0\det A=0. Let ff be a polynomial and replace (bi)(b_{i}) with (f(bi))(f(b_{i})). Then det⁡A=0\det A=0 defines another polynomial q(x)q(x). Prove that f(p(x))−q(x)f(p(x))-q(x) is a multiple of ∏i=1n(x−ai).\prod_{i=1}^{n} (x-a_{i}).
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