1,令An=a^n+b^n,则易知A(n+2)=A(n+1)+An.
事实上,由a^2=a+1得a^(n+2)=a^(n+1)+a^n,同理b^(n+2)=b^(n+1)+b^n,相加即得A(n+2)=A(n+1)+An
于是题目等价于求数列{An}的第21项.
因为a+b=1,a^2+b^2=(a+1)+(b+1)=3,固前21项为1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778,9349,15127,从而a^21+b^21=15127
2,abcba=10000a+1000b+100c+10b+a=10101a+1110b=111(91a+10b)=3*37(91a+10b),于是abcba是3,37的倍数,从而abcba是3^2,37^2的倍数.
于是abcba=111^2或222^2(333^2是六位数不满足条件)
经检验,abcba=111^2=12321满足条件,从而abba=1221
3,设角ACD=x(度),则
角BCE=x,角DCE=2x,角CAD=90-x,角CBD=90-3x.
在三角形ACE中,由正弦定理得AE/CE=sin3x/sin(90-x)
在三角形BCE中,由正弦定理得BE/CE=sinx/sin(90-3x)
又因为AE=BE,固sin3x/sin(90-x)=sinx/sin(90-3x),即sin3xcos3x=sinxcosx
于是sin6x=sin2x
又因为角CBD=90-3x>0,所以0