2008年全国高中数学联赛(吉林赛区)预赛试题与简答

200820082008525830—11001903661sin(2)6yxπ=−cos2yx=(A)6π(B)3π(C)6π(D)3π220085105A160B320C32D1203ABCOABCP1112322OPOAOBOC⎛⎞=++⎜⎝⎠⎟PABCAABBABCDAB4α22sin3coslog(2)xxαα−=−+x(A){1012xxx−≤≤}(B){1012xxx}−(C){01}xx≤(D){12xx}−5MN{|,}MNxxMxN−=∈∉()()MNMNNM⊕=−−∪2{|3,},{|2,}xAyyxxxRByyxR==−∈==−∈AB⊕=(A)9(,04−](B)9[,04−)(C)9(,)[0,4)−∞−+∞∪(D)9(,)(0,4)−∞−+∞∪6{}na11221,33nnna−−⎡⎤⎛⎞⎛⎞=⎢⎜⎟⎜⎟⎝⎠⎝⎠⎢⎥⎣⎦−⎥AB1,a4a1,aCD3a1,a3a5497()()()()2320121111nnnxxxxbbxbxb++++++++=++++xn1226nbbb+++=81111ABCDABCD−A”111AAAD→→⋅⋅⋅⋅⋅⋅1ABBB→→⋅⋅⋅⋅⋅⋅2n+2008n()fx9R1()fx2xR∈(4)()fxf+=x[0,2]x∈()2fxx=+4y=()fx103a2aa11()2fxxa=+x{}{}()0,R(())0,Rxfxxxffxx=∈==∈≠∅a1212,AA12AAA∪=12[,]AA[,A12]AA][,21AAA{1,2,3}A=A520100QPBDCA13αABCD1BAD=60αP-ABDQ-CBDAPB=90ABDCBDP-BD-QPQBD146CDMCD16,ACDBCD∆∆90AMB∠=°AMB∆152()fxxaxb=++a|(|,b)||2430fxxx≤+−2x{{}na}nb112a=12()1nnafa−5=+2,3,4n=12nnba=+n{ST1,2,3,=}nbnnnn12a=15b=−2n12nnTS+n+161234(,,,)Aaaaa1234(,,,)Bbbbb421()iiiABab==−∑1234{(,,,)|01,1,2,3,4}iIPccccci===In12{,,,}nQPPP=,,ijkPPPQ∈ijkPPP∆ijjkkiPPPPPP==n17,,abc2472abcab++≤cabc++BABACD748294106311{|04}aa≤121413(2)1cos3α=2(3)31440041152169n1753,,22abc===abc++152