The quadratic equation $x^2-5x+t =-x^2+3x$ has only positive integer roots. Find the average of all distinct possible values of $t$.
Rearranging the equation, we have \[2x^2-8x + t = 0.\]By Vieta's formulas, the sum of the possible roots of this equation is $$\frac 82 = 4$$Since the quadratic factors into $(2x-a)(x-b)$ for some positive integers $a,b,$ we have $ab = 2t$, and $a + b = 4$, which is possible only for $\{a,b\} = \{1,8\}$ or $\{2,4\}$. In either case, $t=ab/2=4$, so the average of all possible $t$ is $\boxed{4}$. (Though, to be thorough, we could verify that both of these cases actually do produce integer roots of the original quadratic.)
To find the average of all distinct possible values of $t$, we need to find the sum of all distinct possible values of $t$ and then divide it by the number of distinct possible values of $t$.
First, let's rewrite the given quadratic equation:
$x^2-5x+t = -x^2+3x$
$x^2+8x+t = 0$
Since the equation has only positive integer roots, we know that it can be factored into the form $(x-a)(x-b) = 0$, where $a$ and $b$ are positive integers.
Expanding the factored form, we get:
$x^2 - (a+b)x + ab = 0$
Comparing this with the given equation, we can see that $a+b = -8$ and $ab = t$.
To find all the possible values of $t$, we need to find all the pairs of positive integers $(a, b)$ that satisfy the conditions above.
We can start by listing all the pairs of positive integers $a$ and $b$ such that their sum is $-8$:
$(1, -9), (2, -10), (3, -11), \ldots, (8, 0), (9, -1), (10, -2), \ldots$
Since $a$ and $b$ must be positive, we can remove all the pairs where either $a$ or $b$ is negative:
$(1, 8), (2, 6), (3, 4), (4, 3), (6, 2), (8, 1)$
For each pair, the corresponding value of $t$ is given by $ab$:
$t_1 = 1 \cdot 8 = 8$
$t_2 = 2 \cdot 6 = 12$
$t_3 = 3 \cdot 4 = 12$
$t_4 = 4 \cdot 3 = 12$
$t_5 = 6 \cdot 2 = 12$
$t_6 = 8 \cdot 1 = 8$
The distinct possible values of $t$ are $8$ and $12$. Therefore, the sum of all distinct possible values of $t$ is $8 + 12 = 20$.
Since there are $2$ distinct possible values of $t$, the average can be calculated as $\frac{20}{2} = \boxed{10}$.