ࡱ> OQNV7 ~[bjbjUU 7|7|+|0l8H@>40 ;02=K9 2>64L ?;52 @5H8; ?>AB@>8BL =0 A2>59 75@>40 4;O >1>@>=K >B A>A54=8E ?;53>, GB>1K >1>@>=0 1K;0 :@5?:>9, =0720=8O 3>@>4>2 4>;6=K 1KBL =5?CABK4AB@>:03> A;>20 W. >4AB@>:>9 =07K205BAO ?>A;54>20B5;L=>ABL ?>4@O4 84CI8E A8;>2, 2E>4OI8E 2 7040==CN AB@>:C, =0?@84AB@>:0 AB@>:8  ababbaca , 0  bab  =5 O2;O5BAO ?>4AB@>:>9 A;>20  baobaab . 07C@>40 >4=8< 8 B5< 65 8<5=564L E>G5B ?>AB@>8BL :0: 6=> 1>;LH5 3>@>4>2. 0H0 7040G0  C7=0BL, A:>;L:> 3>@>4>2 A65B ?>AB@>8BL 2>64L. E>4=K5 40==K5  ?5@2>9 AB@>:5 2E>4=>3> D09;0 70?8A0=> F5;>5 G8A;> N (1 d" N d" 1000)  4;8=0 A;>20 W. > 2B>@>9 AB@>:5 70?8A0=> A0 A;>2> W, A>AB>OI55 87 AB@>G=KE 1C:2 ;0B8=A:>3> 0;D028B0. KE>4=K5 40==K5  548=AB25==CN AB@>:C 2KE>4=>3> D09;0 =C6=> 2K25AB8 >4=> F5;>5 G8A;>  :>;8G5AB2> 3>@>4>2, :>B>@>5 A65B ?>AB@>8BL 2>64L D;0B;0=4>2. @84=>9 D09;: input.txtKE>4=>9 D09;: output.txt7 baobaab23 040G0 2. @C3;0O >1>;>G:0 0 ?;>A:>AB8 7040=> 65AB2> ?>?0@=> =5A>2?040NI8E >:@C6=>AB59. "@51C5BAO =09B8 ?5@8I04L 2K?C:;>9 D83C@K A <8=8<0;L=K< ?5@8B>@0O A>45@68B 2A5 MB8 >:@C6=>AB8.  E>4=K5 40==K5  ?5@2CN AB@>:C 2E>4=>3> D09;0 70?8A0=> >4=> F5;>5 G8A;> 1 ( N d"100  :>;8G5AB2> >:@C6=>AB59. !;54CNI85 N AB@>: A>45@60B ?> B@8 F5;KE G8A;0, @0745;5==KE ?@>15;0>@48=0BK 8 @048CA A>>B25BAB2CNI59 >:@C6=>AB8. 064>5 G8A;> ?> 4C;N =5 ?@52>AE>48B 104, :@>3>, @048CA =5>B@8F0B5;5=. KE>4=K5 40==K5  2KE>4=>9 D09; =5>1E>48 2K25AB8 G5@57 ?@>15; 420 25I5AB25==KE G8A;0. 5@2>5 G8A;>  MB> ?5@8@>5  ?;>I04L ?>;CG5==>9 D83C@K.  G8A;0E 4>?CA:05BAO ?>3@5H=>ABL 510-4. @84=>9 D09;: input.txtKE>4=>9 D09;: output.txt1 0 0 16.2832 3.1416 4 2 2 1 2 -2 1 -2 2 1 -2 -2 122.2832 35.1416 040G0 3. "@5B89 9 ?;0=5B5 2 B@5C3>;L=KE 75<;OE 682CB ACI5AB20 A 9 ?0;LF0 B@8 =0 :064>9 @C:5). !59G0A, ?>A;5 A>740=8O :>@01;59, C =8E 2>7=8:;0 ?@>1;58A:0 >?B83> ?CB8 <564C 42C<O @A:8@B0;L=K5 8 O2;ONBAO @0725@B:>9 B5B@0M4@0. 4=0 87 B0:8E :0@B ?>:070=0 =0 @8AC=:5 (@51@0 B5B@0M4@0 =0@8A>20=K 68@=K<8 ;8=8O<8). 0 =59 15;K;L=8:01>7=0G5=0 ACH0. AB0;L=>5  70=840, ?> :>B>@>9 65B ?@>E>48BL ?CBL. >@BK @0A?>;030NBAO =0 2>45 8 >1O70B5;L=> ?@8G:8 =0 @8AC=:5). 8DK 3CB =0E>48BLAO B>;L:> 2 25@H8=0E B@5C3>;L=8:>2. > B@5C3>;L=8:0<, ?@8;530NI8< : @8D0E>48BL =5 65B. 725AB=>, GB> ?>@BK =5 ?@8;530NB : @8D0<. !8AB5>@48=0B C ACI5AB2 A;54CNI0O: 4;O ;N1>3> B@5C3>;L=8:0 53> ?5@2>9 :>>@48=0B>9 O2;O5BAO G8A;> B@5C3>;L=8:>2, @0A?>;>65==KE A;520 >B =53>, 0 2B>@>9  =>B>@>48BAO 40==K9 B@5C3>;L=8:. C2 ?@>872>48BAO A=87C 225@E 8 =0G8=05BAO A =C;O. 8D 70405BAO :>>@48=0B0;L=8:0, 2 25@H8=5 :>B>@>3> >= =0E>48BAO. A;8 B@5C3>;L=8: =0?@02;5= 225@E, B> @8D 2 25@E=59 25@H8=5, 5A;8 2=87  B> 2 =86=59. ;8=>9 ?CB8 =07K205BAO G8A;> B@5C3>;L=8:>2, ?> :>B>@KE>48B ?CBL, 2:;NG0O =0G0;L=K9 8 :>=5G=K9 B@5C3>;L=8:8. CBL 4>;65= ?5@5A5:0BL B>;L:> AB>@>=K A>A54=8E B@5C3>;L=8:>2, => =5 25@H8=K. 0H0 7040G0: =0?8A0BL ?@>3@0<B>@0O >?@545;O5B 4;8=C :@0BG09H53> ?CB8 <564C 42C<O 7040==K@B0<8. 48= 87 :@0BG09H8E ?CB59 @B0:070= =0 :0@B5 ?C=:B8@=>9 ;8=859. E>4=K5 40==K5  ?5@2>9 AB@>:5 2E>4=>3> D09;0 70?8A0=> G8A;> N (1 ( N ( 50)  4;8=0 @51@0 B5B@0M4@0. > 2B>@>9  425 ?0@K G8A5;  :>>@48=0BK ?>@B>2. "@5BLO AB@>:0 A>45@68B G8A;> K  :>;8G5AB2> :;5B>: ACH8.  A;54CNI85 K AB@>: 70?8A0=> ?> 420 G8A;0 =0 AB@>:C  :>>@48=0BK :;5B>: ACH8. 0;55 845B AB@>:0, 2 :>B>@>9 70405BAO G8A;> @8D>2 L, 0 70B5: >?8AK20NB :>>@48=0BK @8D>2. KE>4=K5 40==K5  2KE>4=>9 D09; =C6=> 2K40BL F5;>5 G8A;>  4;8=C :@0BG09H53> ?CB8 87 >4=>3> ?>@B0 2 4@C3>9. A;8 B0:>3> ?CB8 =5 ACI5AB2C5B, 8;8 4;8=0 ?CB8 1>;LH5 103, B> 2K40BL 0. !!! A5 2E>4=K5 8 2KE>4=K5 40==K5 ?@54AB02;5=K 2 B@>8G=>9 A8AB5<5 AG8A;5=8O . @84=>9 D09;: input.txtKE>4=>9 D09;: output.txt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d" N d" 100). KE>4=K5 40==K5  548=AB25==CN AB@>:C 2KE>4=>3> D09;0 =5>1E>48 2K25AB8 >4=> F5;>5 G8A;>  8A:>5 :>;8G5AB2> A?>A>1>2 157 ;848@CNI8E =C;59. @84=>9 D09;: input.txtKE>4=>9 D09;: output.txt3220895014631192902121 040G0 5. 5AQ;0O 83@0 K A 4@C7LOA:5 ?>4@O4 N =0BC@0;L=KE G8A5;. 0H0 7040G0  =09B8 :0: 6=> 1>;LH5 ?>4@O4 84CI8E G8A5;, :>B>@K5 1K 45;8;8AL =0 >4=> 8 B> 65 G8A;>, 1>;LH55 1. "0: :0: 2@CG=CN 8A:0BL >B25B A;>6=>, 2K @5H8;8 =0?8A0BL ?@>3@0<B>@0O A45;05B @01>BC 70 20A. E>4=K5 40==K5  ?5@2>9 AB@>:5 2E>4=>3> D09;0 7040=> G8A;> N (1 d" N d" 10000). > 2B>@>9 AB@>:5 70?8A0=> G5@57 ?@>15; N F5;KE G8A5; A1 ... AN (1 d" Ai d" 10000, 1 d" i d" N). -B> B5 A0B>@K5 =0?8A0;8 20H8 4@C7LO. =8 40=K 2 B>@O4:5, 2 :>B>@>=8 @0A?>;>65=K =0 4>A:5. KE>4=K5 40==K5 0H0 ?@>3@0<;6=0 2K25AB8 2 2KE>4=>9 D09; >4=> F5;>5 G8A;>  =081>;LH55 :>;8G5AB2> ?>4@O4 84CI8E G8A5; 7040==>9 ?>A;54>20B5;L=>AB8, :>B>@K5 1K 45;8;8AL =0 >4=> 8 B> 65 =0BC@0;L=>5 G8A;>, 1>;LH55 1. @84=>9 D09;: input.txtKE>4=>9 D09;: output.txt3 6 10 152 040G0 6. 0@0 A>AB>8B 87 ?;>I04>:, A>548=5==KE 1>9 C7:8E>409 ?;>I04:5 ;568B :0:>5-B> :>;8G5AB2> :0I04>: @07=K9 @59B8=3, 7028AOI89 >B 2KA>BK ?;>I04:8  G5I04:0, B5;LH5 55 @59B8=3 A@548 :0, GB> =0 ?;>I04:0E A 1>;55 2KA>:8;LH55 :>;8G5AB2> :048= A>;=5G=K9 45=L 3;02=K9 :04=>9 87 ?;>I04>: >1@0B8; 2=8 2>:@C3 AB0;> >G5=L B5A=>. > :0<=8 =5 3CB :0B8BLAO 225@E, 0 =0 =86=85 ?;>I04:8 >=8 :0B8BLAO =5 E>BOB 87-70 ?;>E>3> @59B8=30. >MB><C 3;02=K9 :0H5; =0 E8B@>ABL. = 2K1@0; M :0 345-B> 2=87C ?>O28;0AL 70I04:0 A A0<K:8@5. ;O ?>;=>9 ?@024>?>4>1=>AB8 >= A>AB028; :0@BC ?;>I04>:, =0 :>B>@K5 6=> A:0B8BLAO A 53> ?;>I04:8. A53> (2:;NG0O ?5@5?>;=5==CN ?;>I04:C) >:070;>AL N ?;>I04>:. >A:>;L:C 3;02=K9 :0= =5 AB0; @8A>20BL 2A5 ?CB8 I04:0AB> =0@8A>20; <8=85 :>;8G5AB2> ?@>E>4>2 I04:01K 87 53> ?;>I04:8 6=> 1K;> 4>1@0BLAO 2 ;N1CN, =0E>4OICNAO =865. = ?>:070; :064><C 87 M :01JOA=8;, 345 =0E>48BAO 70I04:0 (GB>1K :0<=8; ?CBL). @8G51K 3;02=K5 :0I04>: =5 8<5;8 : =53> =0H5AB28O, =5:>B>@K< :0<=O= ?>:070; =0 4@C385 ?;>I04:8, G5AB0;L=K1K =5 A>740BL ;028=C, >= A:070; :0<=O;LH8< 1>9. 53> ?>4C= @5H8;, GB> 6=> ?CA:0BL :0<=8 ?0@01K ?@>AB@0=AB2> 1KAB@55 >A2>1>48;>AL. 0<=8 525@8;8 8 AB0;8 A>18@0BLAO 2 4>@>3C. 0MB>AG8B0BL, A:>;L:> 2@5=8 1C4CB :0B8BLAO 2<5AB5, 8, 65B 1KBL, ?5@5AB@>8BL ?0@K, GB>1K C25;8G8BL MB> 2@5, GB>1K A:0B8BLAO ?> ?@>E>4C A >4=>9 ?;>I04:8 2 A;54CNICN, =5 70E>4O ?> ?CB8 =0 4@C385 ?;>I04:8, :04=C  A:;04K20BL G8A;0 :0<=8 =5 C< 8IL. E>4=K5 40==K5  ?5@2>9 AB@>:5 2E>4=>3> D09;0 70?8A0=K 420 F5;KE G8A;0  N 8 K, 345 1 d" N d" 25000, K= EMBED Equation.3 , 0 d" M d" 2 000.  A;54CNI8E (N 1) AB@>:0E =0E>4OBAO ?0@K G8A5; i 8 j, >7=0G0NI85, GB> 87 i-9 ?;>I04:8 6=> =0?@OI04:8 70=C20=K G8A;0B 1 4> N, 25@E=OO ?;>I04:0, >B:C40 :0BOBAO :0<=8, 65B 89 =>2 ?;>I04>: 4>AB0B>G=>, GB>1K >?@545;8BL ?CBL 87 A09 25@E=59 ?;>I04:8 4> ;N1>9 4@C3>9. 0;LH5 84CB K AB@>:, =0 :064>9 AB@>:5 G5@57 ?@>15; 70?8A0=0 ?0@0 G8A5; a 8 b (1d"a, bd"N), >1>7=0G0NI0O ?C=:BK =07=0G5=8O 4;O >G5@54=>9 ?0@K :0<=59: ?5@2K9 :0I04:C a, 0 2B>@>9  =0 ?;>I04:C b. KE>4=K5 40==K5  2KE>4=>9 D09; =C6=> 2K25AB8 K AB@>:, =0 :064>9 AB@>:5 4>;6=> =0E>48BLAO ?> B@8 F5;KE G8A;0 a, b 8 c, 70?8A0==KE G5@57 ?@>15;. -B8 G8A;0 >7=0G0NB, GB> ?0@0 :0I04:8 a 8 b, 1C45B :0B8BLAO 2<5AB5 c 9:8 G8A5; =C6=> 2K4020BL 2 B>@O4:5, 2 :>B>@>>B25BAB2CNI85 8 2E>4=>< D09;5. @84=>9 D09;: input.txtKE>4=>9 D09;: output.txt6 2 1 2 1 3 1 4 3 5 3 6 2 4 5 62 4 0 5 6 1 040G0 7. 5@5A5G5=85 65AB2 0=K 420 =5C?>@O4>G5==KE =01>@0 =0BC@0;L=KE G8A5; (65B 1KBL, A ?>2B>@5=8O2B>@5=89 2 ?>@O4:5 2>7@0AB0=8O 2A5 B5 G8A;0, :>B>@K5 2AB@5G0NBAO 2 >1>8E =01>@0E. E>4=K5 40==K5  ?5@2>9 AB@>:5 2E>4=>3> D09;0 70?8A0=> G5@57 ?@>15; 420 F5;KE G8A;0 N 8  (1(N,(106)  :>;8G5AB2> M;52 ?5@2>3> 8 2B>@>3> =01>@>2, A>>B25BAB25==>.  A;54CNI8E AB@>:0E 70?8A0=> A=0G0;0 N G8A5; ?5@2>3> =01>@0, 0 70B5@>3> =01>@0. '8A;0 @0745;5=K ?@>15;0<8 8;8 A8;0=F0 AB@>:8. 064>5 87 MB8E G8A5; ?>?0405B 2 ?@>: >B 1 4> 105. KE>4=K5 40==K5  2KE>4=>9 D09; =C6=> 70?8A0BL 2 2>7@0AB0NI5@O4:5 157 ?>2B>@5=89 2A5 G8A;0, :>B>@K5 2E>4OB :0: 2 ?5@2K9, B0: 8 2> 2B>@>9 =01>@. '8A;0 @0745;OBL >4=815;> 2KE>4=>9 D09; 4>;65= >AB020BLAO ?CABK<. @84=>9 D09;: input.txtKE>4=>9 D09;: output.txt11 6 2 4 6 8 10 12 10 8 6 4 2 3 6 9 12 15 186 12 040G0 8. !C?5@A204L10 !J52L ;N18B5;59 ;N128 ?>4E>4OB : :>=FC, 8 AF5=0@8ABK @5H8;8 ?>=5F A5@80;0 AC?5@A204L1C. >4 AC?5@A204L1>9 ?>=84=>2@55 ?@>2545=85 N A20451 <564C N <C6G8=0<8 8 N 65=I8=03> 9 65=I8=K 8725AB=>, 3CB ;8 >=8 ?>65=8BLAO. @>4NA5@0 A5@80;0 708=B5@5A>20; 2>?@>A: G5B=> ;8 G8A;> 2>76=KE AC?5@A20451? 0?8H8B5 ?@>3@0<B>@0O >B25B8B =0 MB>B 2>?@>A. E>4=K5 40==K5  ?5@2>9 AB@>:5 2E>4=>3> D09;0 70?8A0=> G8A;> F5;>5 N (1 d" N d" 100).  :064>9 87 A;54CNI8E N AB@>: G5@57 ?@>15; 70?8A0=> ?> N G8A5;. -B8 G8A;0 7040NB B01;8FC @075 G8A;> 2 i-9 AB@>:5 @02=> 1 B>340 8 B>;L:> B>340, :>340 <5@ j 3CB ?>65=8BLAO 2 :>=F5 A5@80;0, 2 ?@>B82=>< 8B 0. 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