This follows since is compact and since the map x,y)\mapsto (x+y)^2+y^2" alt="\varphi:\mathbb{S}^1\to\mathbb{R}x,y)\mapsto (x+y)^2+y^2" /> is continuous it follows that assumes a maximum on

Similarly, x,y)\mapsto (x-1)^2+y^2-1" alt="\varphi:\mathbb{S}^1\to\mathbb{R}x,y)\mapsto (x-1)^2+y^2-1" /> is continuous and connected and so is an interval.2. same kind of problem: show (x-1)^2 + y^2=1 has a solution on the circle x^2+y^2=1