AMC 12 Mock Exam 4 Problem 16

(Every lattice point strictly between the origin and a tree has both coordinates between $1$ and $10$, so it is itself a tree; “blocked by a tree” and “contains another lattice point” mean the same thing.) The segment from $(0,0)$ to $(x,y)$ passes through the lattice points $\left(\frac{kx}{g},\frac{ky}{g}\right)$ for $k=0,\dots,g$, where $g=\gcd(x,y)$, and through no others. So $(x,y)$ is visible exactly when $\gcd(x,y)=1$.

Count pairs with $x<y$: for each $y$, there are $\varphi(y)$ values $x<y$ coprime to $y$ (for $y\ge2$), and none for $y=1$. So these number $\varphi(2)+\cdots+\varphi(10)=1+2+2+4+2+6+4+6+4=31$. By symmetry, $31$ pairs have $x>y$, and the only coprime pair with $x=y$ is $(1,1)$. Total: $31+31+1=\boxed{63}$.