If a, b, c are positive numbers, prove that it is not possible for the inequalitiesa(1−b)>1/4b(1−c)>1/4,c(1−a)>1/4,toholdatthesametime

To prove that the inequalities
\[ a(1 - b) > \frac{1}{4}, \]
\[ b(1 - c) > \frac{1}{4}, \]
\[ c(1 - a) > \frac{1}{4} \]
cannot hold simultaneously for positive numbers \(a\), \(b\), and \(c\), let's start by assuming the inequalities are true and derive a contradiction.

Consider the product of the left-hand sides of the inequalities:

\[ a(1 - b) \cdot b(1 - c) \cdot c(1 - a). \]

First, let's simplify this expression:
\[ \begin{aligned}
a(1 - b) \cdot b(1 - c) \cdot c(1 - a) &= abc (1 - b)(1 - c)(1 - a).
\end{aligned} \]

Now, consider the product of the right-hand sides of the inequalities:
\[ \left( \frac{1}{4} \right) \cdot \left( \frac{1}{4} \right) \cdot \left( \frac{1}{4} \right) = \frac{1}{64}. \]

According to the initial inequalities, we have:
\[ a(1 - b) > \frac{1}{4}, \]
\[ b(1 - c) > \frac{1}{4}, \]
\[ c(1 - a) > \frac{1}{4}. \]

Multiplying these inequalities together, we get:
\[ a(1 - b) \

Learn how to prove that certain inequalities involving positive numbers cannot hold simultaneously. Explore detailed steps and logical deductions using mathematical inequalities.

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