# Inequalities with strange equality conditions

This page serves to debunk the myth that

An expression attains its maximum or minimum when all (some) of the variables are equal.

## If \(a, b\) are real numbers such that \( a + b = 4 \), what is the minimum value of \[\]

\[\big[(a-1) ( b-1)\big]^2?\]

Clearly, since the expression is a perfect square, the minimum that it can attain is 0. This can indeed be attained, say with \( a = 1, b=3 \). The minimum does not occur when \( a = b = \frac{4}{2} \), even though the expression is symmetric. \(_\square\)

Some inequalities are less obvious and they do not have equality case when all variables are equal.

If \(x,y,z\ge 0\) satisfy \(x+y+z=3\), find the maximum value of \(x^2y+y^2z+z^2x\).

Setting \(x=y=z=1\) gives \(x^2y+y^2z+z^2x=3\). However, is this the maximum value?

In fact, setting \(x=0, y=2, z=1\) gives \(x^2y+y^2z+z^2x=4,\) which is the actual maximum value. \(_\square\)

Try your hand at the list of problems below. Be warned that these problems are hard to (properly) solve, and could require a lot of ingenuity.

**Cite as:**Inequalities with strange equality conditions.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/inequalities-with-strange-equality-conditions/