1. **State the problem:** Given that $x + \frac{1}{x} = 99$, find the value of $$\frac{100x}{2x^2 + 102x + 2}$$. 2. **Rewrite the denominator:** Notice the denominator can be factored or simplified. First, write it as: $$2x^2 + 102x + 2 = 2(x^2 + 51x + 1)$$ 3. **Use the given relation:** From $x + \frac{1}{x} = 99$, multiply both sides by $x$ (assuming $x \neq 0$): $$x^2 + 1 = 99x$$ 4. **Express $x^2 + 51x + 1$ in terms of $x + \frac{1}{x}$:** Rewrite $x^2 + 51x + 1$ as: $$x^2 + 51x + 1 = (x^2 + 1) + 51x$$ Using step 3, substitute $x^2 + 1 = 99x$: $$= 99x + 51x = 150x$$ 5. **Substitute back into the denominator:** $$2(x^2 + 51x + 1) = 2(150x) = 300x$$ 6. **Simplify the original expression:** $$\frac{100x}{2x^2 + 102x + 2} = \frac{100x}{300x}$$ Since $x \neq 0$, cancel $x$: $$= \frac{100}{300} = \frac{1}{3}$$ **Final answer:** $$\boxed{\frac{1}{3}}$$