Problem 245

Let's suppose that at least one of $a$, $b$, or $c$ is not equal to $0$. We will check each assumption to see if the equation holds:

1. If $a=b=0$ and $c\neq 0$, the equation becomes $c\sqrt{34} = 0$. Since $\sqrt{34}$ is not $0$, $c$ must be $0$ to satisfy the equation, which is contradictory to the assumption.

2. If $a\neq 0$ and $b=0$, the equation becomes $a + c\sqrt{34} = 0$. This implies $c\sqrt{34} = -a$. Since $a$ and $c$ are rational numbers, and the square root of a non-square integer is an irrational number, $\sqrt{34}$ is irrational. Therefore, the product of $c$ (a rational number) and $\sqrt{34}$ (an irrational number) cannot equal $a$ (a rational number). This leads to a contradiction.

In both cases, we arrive at a contradiction, and therefore, $a$, $b$, and $c$ must all be equal to $0$ to satisfy the equation.

$$\boxed{a = b = c = 0}$$