1. [5] What is the greatest rational root that the equation of the form $a x^{2}+b x+c=0$ can have, where $a, b$ and $c$ are natural numbers not exceeding $100 ?$
Answer: - $\frac{1}{\text {. }}$ Solution Note that $-\frac{1}{2}$ is a root of the equation $99 x^{2}+1$ Mikhail Evdokimov 99 . Solution. Note that - $\frac{1}{99}$ is a root of the equation $99 x^{2}+100 x+1=0$.

Let's prove the maximality. It is clear that the root $x$ of such an equation, as in the condition, is negative. Suppose $|x|0-100 \cdot 0.01+1=0$. Contradiction.