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.
Let \(a, b, c, d \in \left[\frac{1}{2}, 2\right]\). Suppose \(abcd = 1\). Find the maximum possible value of \(\left(a + \dfrac{1}{b}\right)\left(b + \dfrac{1}{c}\right)\left(c + \dfrac{1}{d}\right)\left(d + \dfrac{1}{a}\right)\).
Let \(a, b, c, d\) be positive integers such that \(a+b+c+d=120,\) and define \[S=ab+bd+da+bc+cd.\] Find the number of ordered quadruples \((a,b,c,d)\) for which \(S\) attains its maximum.
Let \( x, y, z\) be non-negative real numbers satisfying the condition \( x+y+z = 1\). The maximum possible value of
\[ x^3y^3 + y^3z^3 + z^3x^3 \]
has the form \( \frac {a} {b} ,\) where \(a\) and \(b\) are positive, coprime integers. What is the value of \(a+b\)?
Let \(x,y,z\) be real numbers such that \(x^2+y^2+z^2+(x+y+z)^2=9\) and \(xyz\leq \frac{15}{32}\). To 2 decimal places, what is the greatest possible value of \(x\)?
Suppose \(a,\) \(b,\) and \(c\) are non-negative real numbers with \(a+b+c=1.\) The largest possible value of the expression \(ab^2+bc^2+ca^2\) can be written as \(\frac{n}{m},\) where \(n\) and \(m\) are coprime positive integers. What is the value of \(n+m\)?
What is the smallest real number \(k\) (to 3 decimal places), such that for all ordered triples of non-negative reals \( (a,b,c) \) which satisfy \( a + b + c = 1 \), we have
\[ \frac{ a}{ \sqrt{1-c} } + \frac{b} { \sqrt{1-a} } + \frac{ c} { \sqrt{1-b} } \leq 1 + k ? \]
If \(a, b, c \) are non-negative real numbers, what is the maximum value of
\[ \frac{ ab^2 + bc^2 + ca^2 } { ( a + b + c) ^ 3 } ? \]
Consider all pairs of real numbers such that \( a + b = 3 \).
What is the minimum value of
\[ a^2 b^2 - 2a^2 b - 2 a b^2 + a^2 + 4ab + b^2 - 2a - 2b + 1? \]
Let a triangle with sides of length \(a,b,c\) have perimeter \(2\). What is the maximum value of \(k\) such that \[\dfrac{1-a}{b}+\dfrac{1-b}{c}+\dfrac{1-c}{a}\ge k\] is always true? Prove your claim.
Over all positive triples of real numbers, what is the largest value of \(k\) (to 2 decimal places) such that
\[ \sqrt{ \frac{a} { b+c} } + \sqrt{ \frac{b}{ c + a }} + \sqrt{ \frac{ c}{a+b} } \geq k ?\]
For real numbers \(a\) and \(b\) such that \(a>b>0\), what is the minimum value of
\[ a+\frac{1}{b(a-b)} ?\]
Suppose that \(a, b, c, d\) are positive real numbers such that
\[ ab \leq 1000, ~~ac \leq 1000, ~~bd \leq 1000, ~~cd \leq 500. \]
What is the maximum possible value of \( ab+ac+bd+cd\)?