Let $T = TNFTPP$.  $x$ and $y$ are nonzero real numbers such that \[18x - 4x^2 + 2x^3 - 9y - 10xy - x^2y + Ty^2 + 2xy^2 - y^3 = 0.\] The smallest possible value of $\tfrac{y}{x}$ is equal to $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers.  Find $m+n$.

[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question.  For those who wanted to try the problem separately, [hide=here's the value of T]$T=6$[/hide].
Given the equation:
\[ 18x - 4x^2 + 2x^3 - 9y - 10xy - x^2y + 6y^2 + 2xy^2 - y^3 = 0 \]

We are given that \( T = 6 \). Substituting \( T \) into the equation, we get:
\[ 18x - 4x^2 + 2x^3 - 9y - 10xy - x^2y + 6y^2 + 2xy^2 - y^3 = 0 \]

Let \( \frac{y}{x} = k \). Then \( y = kx \). Substituting \( y = kx \) into the equation, we get:
\[ 18x - 4x^2 + 2x^3 - 9kx - 10xkx - x^2kx + 6(kx)^2 + 2x(kx)^2 - (kx)^3 = 0 \]

Simplifying, we get:
\[ 18x - 4x^2 + 2x^3 - 9kx - 10kx^2 - kx^3 + 6k^2x^2 + 2k^2x^3 - k^3x^3 = 0 \]

Combining like terms, we get:
\[ (18 - 9k)x + (-4 - 10k + 6k^2)x^2 + (2 - k + 2k^2 - k^3)x^3 = 0 \]

Factoring out \( x \), we get:
\[ x \left( (18 - 9k) + (-4 - 10k + 6k^2)x + (2 - k + 2k^2 - k^3)x^2 \right) = 0 \]

Since \( x \neq 0 \), we have:
\[ (18 - 9k) + (-4 - 10k + 6k^2)x + (2 - k + 2k^2 - k^3)x^2 = 0 \]

This can be rewritten as:
\[ (2 - k)x \left( (k^2 + 1)x^2 - (6k + 2)x + 9 \right) = 0 \]

We can see that one solution is \( k = 2 \). Now, suppose that \( k \neq 2 \). We need to solve the quadratic equation:
\[ (k^2 + 1)x^2 - (6k + 2)x + 9 = 0 \]

For this quadratic equation to have a real solution, the discriminant must be nonnegative:
\[ (6k + 2)^2 - 4(k^2 + 1) \cdot 9 \geq 0 \]

Simplifying the discriminant, we get:
\[ (6k + 2)^2 - 36(k^2 + 1) \geq 0 \]
\[ 36k^2 + 24k + 4 - 36k^2 - 36 \geq 0 \]
\[ 24k - 32 \geq 0 \]
\[ 24k \geq 32 \]
\[ k \geq \frac{32}{24} \]
\[ k \geq \frac{4}{3} \]

Therefore, the minimum value of \( k \) is \( \frac{4}{3} \). The smallest possible value of \( \frac{y}{x} \) is \( \frac{4}{3} \). 

Thus, \( m = 4 \) and \( n = 3 \), and \( m + n = 4 + 3 = 7 \).

The final answer is \( \boxed{ 7 } \)