When　establishing　an　effective　treatment　with　binary　data　in　a　two-stage　design,　one-sided　tests　for　a　proportion　p　are　employed.　Researchers　use　the　parameter　configuration　at　the　boundary　of　the　null　hypothesis　space　to　determine　a　rejection　region　and　an　optimal　design.　However,　it　is　unclear　whether　the　(maximum)　Type　I　error　rate　is　achieved　at　the　boundary　especially　when　the　sample　size　in　stage　2　varies.　In　this　paper,　we　first　prove　that　this　is　true　for　a　large　family　of　tests　in　adaptive　two-stage　designs　by　showing　that　any　test　in　the　family　has　a　nondecreasing　power　in　p　and　then　derive　optimal　designs.　Second,　similar　results　are　established　for　m-stage　designs　with　m　＞　2.　Supplementary　materials　for　this　article　are　available　online.