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 \(a+b\)?

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

\[ a^2 b^2 - 2a^2 b - 2 a b^2 + a^2 + 4ab + b^2 - 2a - 2b + 1? \]

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 \( ab+ac+bd+cd\)?