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Part of 2012 Putnam

Problems(2)

Putnam 2012 A1

Source:

12/3/2012
Let d1,d2,…,d12d_1,d_2,\dots,d_{12} be real numbers in the open interval (1,12).(1,12). Show that there exist distinct indices i,j,ki,j,k such that di,dj,dkd_i,d_j,d_k are the side lengths of an acute triangle.
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Putnam 2012 B1

Source:

12/3/2012
Let SS be a class of functions from [0,∞)[0,\infty) to [0,∞)[0,\infty) that satisfies:
(i) The functions f1(x)=ex−1f_1(x)=e^x-1 and f2(x)=ln⁡(x+1)f_2(x)=\ln(x+1) are in S;S;
(ii) If f(x)f(x) and g(x)g(x) are in S,S, the functions f(x)+g(x)f(x)+g(x) and f(g(x))f(g(x)) are in S;S;
(iii) If f(x)f(x) and g(x)g(x) are in SS and f(x)≥g(x)f(x)\ge g(x) for all x≥0,x\ge 0, then the function f(x)−g(x)f(x)-g(x) is in S.S.
Prove that if f(x)f(x) and g(x)g(x) are in S,S, then the function f(x)g(x)f(x)g(x) is also in S.S.
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