2014年数学竞赛预赛(数学专业）答案

6¶:OyÒ:¤3:)Œ Ò:;’:     —µ‚‰KžØ‡‡Ld‚     183¥IŒÆ)êÆ¿mýmÁò (êÆa, 2014c10) Á/ª:4òÁžm: 150©¨÷©: 100© KÒ ˜  n o Ê 8 o© ÷© 15 15 15 15 20 20 100 © 5¿: 1.¤k‰KÑL3dÁò’—µ‚m>,3Ù§’þ˜ÆÃ. 2.—µ‚†>žÕ‰K,—µ‚ Øk6¶9ƒ'IP. 3.X‰K˜xØ,Œ3￾¡,¿I²KÒ. ˜!(K15©)®˜mü^†‚ l1: x4 1 = y3 2 = z8 1 ; l2: x+ 1 7 = y+ 1 6 = z+ 1 1 : 1¤y²l1Úl2É¡¶ 2¤¦l1Úl2úR‚IO￾§¶ 3¤¦ël1þ?˜:Úl2þ?˜:‚ã¥:;,˜„￾§" (1)yyy²²²µl1þk:r1= (4;3;8)§￾••þ￾v1= (1;2;1)" l2þk:r2= (1;1;1)§￾••þ￾v2= (7;6;1)" q (r1r2; v1; v2) =         5 4 9 12 1 76 1         6= 0; l1Úl2É¡" (3©) (2)l1þ?˜:P1=r1+t1v1†l2þ?˜:P2=r2+t2v2ë‚￾•• þ￾ ! P1P2=r2r1+t2v2t1v1 = (5 + 7t2t1;46t2+ 2t1;9 +t2t1): úR‚￾••þ￾ v=v1v2=         i j k 12 1 76 1         = (4;6;8): 11￾£
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n = Z 1 0 f(x)
n dx: 2¤y²µlimn!1xn= 1: yyy²²²:1¤ f(0) n  Z 1 0 f(x)
n dxf(1) n ; dëY¼ê0Š5Ÿxn35" (3©) duf´î‚üN¼ê§xn´￾˜" (5©) 2¤é?¿ >0,dfšK5ÚüN5§ f(xn)
n  Z 1 1 f(1)
n = f(1)
n ;  f(xn) n p f(1); l lim inf n!1 f(xn)f(1): 12￾£
5￾¤ 6¶:OyÒ:¤3:)Œ Ò:;’:     —µ‚‰KžØ‡‡Ld‚     dfüN5§ lim inf n!1 xn1: d?¿5, limxn= 1. (15 ©) n!(K15©)V￾4«m[0;1]þN¢¼ê¤¢•þ˜m§Ù¥ •þ\{†Xþ¦{þ￾Ï~"f1; : : : ; fn2V.y²±eü^dµ 1¤f1; : : : ; fn‚5Ã'¶ 2¤9a1; : : : ; an2[0;1]¦det(fi(aj))6= 0,ùpdetL1ª. yyy²²²2))1).Ä￾§1f1+


+nfn= 0.òa1; : : : ; an©O“\§￾§ | 8 > > < > > : 1f1(a1) +


+nfn(a1) = 0 . . . 1f1(an) +


+nfn(an) = 0 5¿þã￾§|XêÝ￾(fi(aj)) T ,Ïdddet(fi(aj))6= 0†1=


=n= 0. (6 ©) 1))2).^8B{.Äk§n= 1ž§(Øw," Ùg§n=kž(Øý"Kn=k+1ž§df1; : : : ; fk+1‚5Ã'§f1; : : : ; fk ‚5Ã'.Ïd9a1;