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[obm-l] Putnam Problems

Um da Putnam...



  ai and bi are constants. Let A be the (n+1) x
(n+1) matrix Aij, defined as follows: Ai1 = 1;
A1j = xj-1 for j ≤ n; A1 (n+1) = p(x); Aij
= ai-1j-1 for i > 1, j ≤ n; Ai (n+1) = bi-1
for i > 1. We use the identity det A = 0 to
define the polynomial p(x). Now given any
polynomial f(x), replace bi by f(bi) and p(x) by
q(x), so that det A = 0 now defines a polynomial
q(x). Prove that f( p(x) ) is a multiple of
∏ (x - ai) plus q(x). 


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TRANSIRE SVVM PECTVS MVNDOQVE POTIRI

CONGREGATI EX TOTO ORBE MATHEMATICI OB SCRIPTA INSIGNIA TRIBVERE

Fields Medal(John Charles Fields)




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