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Putnam 2012 B1

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December 3, 2012

Putnamfunctionlogarithmsalgebrapolynomialcollege contests

Problem Statement

Let

S

be a class of functions from

[0,\infty)

to

[0,\infty)

that satisfies:
(i) The functions

f_1(x)=e^x-1

and

f_2(x)=\ln(x+1)

are in

S;

(ii) If

f(x)

and

g(x)

are in

S,

the functions

f(x)+g(x)

and

f(g(x))

are in

S;

(iii) If

f(x)

and

g(x)

are in

S

and

f(x)\ge g(x)

for all

x\ge 0,

then the function

S,

is in

S.

Prove that if

f(x)

and

g(x)

are in

S,

then the function

f(x)g(x)

is also in

S.

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