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one of u(1 -v), v(1 -w), w(1 - u) is <=1/4 if 0<u,w,v<1

Source: 1961-62 Germany R4 12.2

October 12, 2024

algebrainequalities

Problem Statement

Let

u, v

and

w

be any positive numbers smaller than

1

. Prove that among the numbers

u(1 -v)

v(1 -w)

w(1 - u)

there is always at least one value not greater than

\frac14