1. Positive integers $a, b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that the equation $a x^{2}-b x+c=0$ has exactly one real solution $r$, and this value $r$ is less than 100. Compute the maximum possible value of $r$.
Answer: $\square$
Solution: Note that for there to be exactly one solution, the discriminant must be 0 , so $b^{2}-4 a c=0$. Thus, $b$ is even, so $k=2$. Since $r=\frac{b}{2 a}$, then $r$ is also a power of 2 , and the largest power of 2 less than 100 is 64 . This is achieved by $(x-64)^{2}=x^{2}-128 x+4096$.