Question 1
A grasshopper starts at $(1,1)$ in the coordinate plane. It may jump from $A$ to $B$ when $A,B$ have positive integer coordinates and triangle $OAB$ has area $\tfrac12$, where $O$ is the origin.
1. Determine all points $(m,n)$ it can reach in finitely many jumps.
2. Prove that each such point can be reached in fewer than $|m-n|$ jumps.
Editorial notice: Part 2 above faithfully reproduces a false strict bound in the supplied source. For example, $(2,1)$ requires one jump, not fewer than one. The valid corrected bound is **at most** $|m-n|$ jumps; $(1,1)$ requires zero. The solution proves the corrected statement.
1. Determine all points $(m,n)$ it can reach in finitely many jumps.
2. Prove that each such point can be reached in fewer than $|m-n|$ jumps.
Editorial notice: Part 2 above faithfully reproduces a false strict bound in the supplied source. For example, $(2,1)$ requires one jump, not fewer than one. The valid corrected bound is **at most** $|m-n|$ jumps; $(1,1)$ requires zero. The solution proves the corrected statement.