# 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/