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one of ab(1-c)^2 , bc(1-a)^2 , ca(1- b)^2 <= 1/16 if 0<a,b,c<=1

Source: Chile Finals 2014 L2 p1

October 5, 2022

algebrainequalities

Problem Statement

Let

a, b,c

real numbers that are greater than

0

and less than

1

. Show that there is at least one of these three values

ab(1-c)^2

bc(1-a)^2

ca(1- b)^2

which is less than or equal to

\frac{1}{16}