Problem 1 If u is harmonic in the connected domain Ω and is not constant, then u(Ω) is open in \R. Suppose Ω is bounded and that its boundary ∂Ω is connected. If u is harmonic in Ω, then u(Ω)⊂u(∂Ω). Problem 2 A radial harmonic function on B is constant. A positive harmonic function on \Rd is constant. Problem 3 Suppose u(x) is harmonic in some domain in \Rn. Then v(x)=|x|2−nu(x|x|2) is also harmonic in a suitable domain. Problem 4 For n=2, find the Green's function for the Laplace operator on the first quadrant.