Inequalities with Strange Equality Conditions

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 $ab+ac+bd+cd$?