MathDB

Putnam 2012 A3

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December 3, 2012

Putnamfunctionlimitalgebrafunctional equationtopologycollege contests

Problem Statement

Let

f:[-1,1]\to\mathbb{R}

be a continuous function such that
(i)

f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right)

for every

x

in

[-1,1],

(ii)

f(0)=1,

and
(iii)

\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}}

exists and is finite.
Prove that

f

is unique, and express

f(x)

in closed form.

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