Problem 1314

1. Let the equation of the line through the origin be $y = kx$, where $k$ is the slope of the line.
2. Substitute $y = kx$ into the given equation of the parabola $5y = 2x^2 - 9x + 10$:
$$5(kx) = 2x^2 - 9x + 10$$
Simplify this to:
$$5kx = 2x^2 - 9x + 10$$
3. Rearrange the equation to form a standard quadratic equation:
$$2x^2 - (9 + 5k)x + 10 = 0$$
4. By Vieta's formulas, the sum of the roots of the quadratic equation $ax^2 + bx + c = 0$ is given by $-\frac{b}{a}$. Here, $a = 2$ and $b = -(9 + 5k)$, so the sum of the roots is:
$$\frac{9 + 5k}{2}$$
5. We are given that the sum of the $x$-coordinates of the points where the line intersects the parabola is 77. Therefore, we set up the equation:
$$\frac{9 + 5k}{2} = 77$$
6. Solve for $k$:
$$9 + 5k = 154$$
$$5k = 145$$
$$k = 29$$

The final answer is $\boxed{29}$.