Which of the following inequalities are NOT correct?

(A) If $a > 1, b > 1,$ then $\log_b a + \log_a b \leq 2$

(B) For any real number $x, (9^x + 9^{1-x}) \geq 9$

(C) If $a,b,c$ are non-zero real numbers of the same sign, then $\left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right) \leq 3$

(D) If $a,b,c$ are three distinct real numbers, then $(a + b)(b + c)(c + a) \geq 8abc$

Choose the correct answer from the options given below:

For option (A), let $t = \log_b a$. Since $a > 1$ and $b > 1$, we know $t > 0$.

Also, $\log_a b = \dfrac{1}{t}$, so the expression becomes $t + \dfrac{1}{t}$.

By AM-GM inequality, for any $t > 0$:

$t + \dfrac{1}{t} \geq 2\sqrt{t \cdot \dfrac{1}{t}}$

$= 2$

This means $\log_b a + \log_a b \geq 2$, not $\leq 2$.

Inequality (A) is NOT correct.

---

For option (B), apply AM-GM on $9^x$ and $9^{1-x}$ (both always positive):

$9^x + 9^{1-x} \geq 2\sqrt{9^x \cdot 9^{1-x}}$

$= 2\sqrt{9^{x+1-x}}$

$= 2\sqrt{9}$

$= 6$

The minimum value is $6$ (at $x = \frac{1}{2}$, giving $3 + 3 = 6 < 9$).

The expression is always $\geq 6$, but NOT always $\geq 9$.

Inequality (B) is NOT correct.

---

For option (C), if $a, b, c$ are all positive, each ratio is positive.

By AM-GM inequality:

$\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} \geq 3\sqrt[3]{\dfrac{a}{b} \cdot \dfrac{b}{c} \cdot \dfrac{c}{a}}$

$= 3\sqrt[3]{1}$

$= 3$

If $a, b, c$ are all negative, let $a = -p,\ b = -q,\ c = -r$ where $p, q, r > 0$:

$\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} = \dfrac{p}{q} + \dfrac{q}{r} + \dfrac{r}{p} \geq 3$

In both cases, the sum is $\geq 3$, not $\leq 3$.

Inequality (C) is NOT correct.

---

For option (D), apply AM-GM to each pair of positive reals:

$a + b \geq 2\sqrt{ab}$

$b + c \geq 2\sqrt{bc}$

$c + a \geq 2\sqrt{ca}$

$(a+b)(b+c)(c+a) \geq 8\sqrt{a^2b^2c^2} = 8abc$

Since $a, b, c$ are distinct, equality (which requires $a = b = c$) can never hold, so the strict inequality holds.

Inequality (D) is correct.

---

The inequalities that are NOT correct are (A), (B) and (C) only.

$\boxed{\textbf{(A), (B) and (C) only}}$