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Putnam 1959 B7

Source: Putnam 1959

June 16, 2022
Putnamfunctionarithmetic meansymmetry

Problem Statement

For each positive integer nn, let fnf_n be a real-valued symmetric function of nn real variables. Suppose that for all nn and all real numbers x1,…,xn,xn+1,yx_1,\ldots,x_n, x_{n+1},y it is true that
  (1)  fn(x1+y,…,xn+y)=fn(x1,…,xn)+y,\;(1)\; f_{n}(x_1 +y ,\ldots, x_n +y) = f_{n}(x_1 ,\ldots, x_n) +y,   (2)  fn(−x1,…,−xn)=−fn(x1,…,xn),\;(2)\;f_{n}(-x_1 ,\ldots, -x_n) =-f_{n}(x_1 ,\ldots, x_n),   (3)  fn+1(fn(x1,…,xn),…,fn(x1,…,xn),xn+1)=fn+1(x1,…,xn).\;(3)\; f_{n+1}(f_{n}(x_1,\ldots, x_n),\ldots, f_{n}(x_1,\ldots, x_n), x_{n+1}) =f_{n+1}(x_1 ,\ldots, x_{n}).
Prove that fn(x1,…,xn)=x1+⋯+xnn.f_{n}(x_{1},\ldots, x_n) =\frac{x_{1}+\cdots +x_{n}}{n}.
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