1. In the Cartesian plane, a line segment with midpoint $(2020,11)$ has one endpoint at $(a, 0)$ and the other endpoint on the line $y=x$. Compute $a$.
Answer: 4018
Solution: Let the other endpoint be $(t, t)$. The midpoint of $(a, 0)$ and $(t, t)$ is $\left(\frac{a+t}{2}, \frac{t}{2}\right)$. So, we know that $\frac{a+t}{2}=2020$ and $\frac{t}{2}=11$. The second equation yields $t=22$. Substituting this into the first yields $a=2 \cdot 2020-22=4018$.