The quadratic equation $x^2-5x+t =3x$ has only positive integer roots. Find the average of all distinct possible values of $t$.
Rearranging and factoring, we have that $x^2 - 8x + t = 0$. By Vieta's formulas or otherwise, we know the sum of the roots is $8$. Hence the distinct values of $t$ are of the form $k(1+n)$ where $k$ ranges over the distinct prime factors of $8$, $k=2, 3$, and $n$ ranges over proper divisors of $8/k$. These values of $n$ are $1, 2, 4$ for $k=2$ and $1$ for $k=3$. Hence $t$ ranges over $3, 6, 9, 12$. The average of these is $\boxed{7.5}$.
We can start by simplifying the given equation $x^2-5x+t = 3x$ by combining like terms: $x^2-8x+t=0$.
Since we are looking for quadratic equations with positive integer roots, we know that the sum of the roots of the quadratic equation $x^2-8x+t=0$ is equal to $8$.
Also, the average of the roots can be found by dividing the sum of the roots by the number of roots.
Let's denote the two roots of the quadratic equation as $a$ and $b$. We have:
$a+b = 8$ (1)
The average of the roots is then:
$\frac{a+b}{2}$.
We can substitute the value of $a+b$ from equation (1) into the average expression:
$\frac{8}{2} = 4$.
Therefore, the average of all distinct possible values of $t$ is $4$.