1. A circle contains the points $(0,11)$ and $(0,-11)$ on its circumference and contains all points $(x, y)$ with $x^{2}+y^{2}<1$ in its interior. Compute the largest possible radius of the circle.
Answer: 61

Solution: Such a circle will be centered at $(t, 0)$ for some $t$; without loss of generality, let $t>0$. Our conditions are that
$$
t^{2}+11^{2}=r^{2}
$$
and
$$
r \geq t+1
$$

So, $t^{2} \leq(r-1)^{2}$, which means
$$
(r-1)^{2}+11^{2} \geq r^{2} \Longrightarrow 122 \geq 2 r
$$
so our answer is $61(r=61$ is attainable with $t=60)$.