Subcontests
(14)Putnam 1959 B7
For each positive integer n, let fn be a real-valued symmetric function of n real variables. Suppose that for all n and all real numbers x1,…,xn,xn+1,y it is true that (1)fn(x1+y,…,xn+y)=fn(x1,…,xn)+y,
(2)fn(−x1,…,−xn)=−fn(x1,…,xn),
(3)fn+1(fn(x1,…,xn),…,fn(x1,…,xn),xn+1)=fn+1(x1,…,xn).Prove that fn(x1,…,xn)=nx1+⋯+xn. Putnam 1959 A7
If f is a real-valued function of one real variable which has a continuous derivative on the closed interval [a,b] and for which there is no x∈[a,b] such that f(x)=f′(x)=0, then prove that there is a function g with continuous first derivative on [a,b] such that fg′−f′g is positive on [a,b]. Putnam 1959 A6
Let m and n be integers greater than 1 and a1,a2,…,am+1 be real numbers. Prove that there exist real n×n matrices A1,A2,…,Am such that
(i) det(Aj)=aj for j=1,2,…,m and
(ii) det(A1+A2+…+Am)=am+1. Old Seemingly Simple Putnam Problem
"Let ω3=1,ω=1. Show thatz1,z2,−ωz1,−ω2z2 are the vertices of an equilateral triangle."