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Part of 2012 Putnam

Problems(2)

Putnam 2012 A3

Source:

12/3/2012
Let f:[−1,1]→Rf:[-1,1]\to\mathbb{R} be a continuous function such that
(i) f(x)=2−x22f(x22−x2)f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right) for every xx in [−1,1],[-1,1],
(ii) f(0)=1, f(0)=1, and
(iii) lim⁡x→1−f(x)1−x\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}} exists and is finite.
Prove that ff is unique, and express f(x)f(x) in closed form.
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Putnam 2012 B3

Source:

12/3/2012
A round-robin tournament among 2n2n teams lasted for 2n−12n-1 days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the nn games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one winning team from each day without choosing any team more than once?
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