题1
∵a/sinA=b/sinB=c/sinC
∴b- (1/2) c= a cosC
∵sinB-(1/2)sinC=sinAcosC
∴sin(A+C)-(1/2)sinC=sinAcosC+sinCcosA-(1/2)sinC=sinAcosC+sinC(cosA-1/2)=sinAcosC
∴sinC(cosA-1/2)=0
∵sinC≠0 ∴cosA=1/2 ∴A=60°
题2
∵∠ ADB = 120°
∴∠ADC=60°
设BD=t ∴DC=2t
S△ADC=3-√3 =1/2AD•DC•sin∠ ADB =1/2﹡2﹡2t﹡√3/2
∴t=√3 -1 ∴cos∠DAC=(DA^2+DC^2-AC^2)/(2﹡AD•DC) =1/2解得AC=3√2-√6
同理AB=√6
cos∠ BAC=(AB^2+AC^2-BC^2)/(2﹡AB•AC)=1/2
∴∠ BAC=60°
题3
根据基本不等式：a^2+b^2≥2ab
∴4 x^ 2+ y^2≥4xy
4 x^ 2+ y^2 +xy≥5xy
∴5xy≤1 ∴xy≤1/5 "="有且仅当x=√10/10,y=√10/5时取到
2x + y =√（2x + y）^2=√(4 x^ 2+ y^2+4xy)=√(1+3xy)≤(2/5)√10
∴2x + y 的最大值是(2/5)√10