Arhimede_1-2-2003

  • November 2019
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<math.h> <stdio.h>

void main(void)

{

int cif[9],j,c,cj,nrr; unsigned int limn; float tr,cmax,cmin;tr=1/3.; unsigned long c1,c2,ip,k,m,n,s,st,sz; unsigned long ist[100][2]; clrscr();



for (c=3;c<10;c++) { nri=0; cj=c/2; cmax=pow10(c);cmin=pow10(c-1); limn=floor(pow(cmax,tr));



$5+,0('(V1ULDQXDULHIHEUXDULH for (n=1;n<=limn;n++){ m=n; etm: m=m+1; st=m*n*(m-n)*(m+n); if (st>cmax) goto etn; k=0; etk: k=k+1; if (k==1) goto etc; if (sqrt(k)==floor(sqrt(k))) goto etk; etc: s=k*k*st; if (scmax) goto etm; sz=s; for (j=1;j<=c; j++){ cif[j]=floor(sz/pow10(c-j)); sz=sz-cif[j]*pow10(c-j);} for (j=1;j<=cj;j++){ if (cif[j]!=cif[c+1-j]) goto etk;} c1=k*(m*m-n*n);c2=2*k*m*n;ip=k*(m*m+n*n); for (j=1;j<=nri;j++){ if (ist[j][1]==c1 && ist[j][2]==c2) goto etk; if (ist[j][1]==c2 && ist[j][2]==c1) goto etk;} nri=nri+1;ist[nri][1]=c1;ist[nri][2]=c2; cout<
}   

 

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          +    VLQ $VLQ %VLQ &   +     $ % & VLQ  &  VLQ VLQ VLQ 

 

,QHJDOLWDWHD OXL 3 6FKZHLW]HU FX Q   DILUP  F  GDF  [ \ ] ∈ >P 0@ ⊂ 5 + 

      P + 0  [ + \ + ]  + +  ≤    P0 [ \ ] 'DF vQ  FRQVLGHU P[ VLQ$\ VLQ%] VLQ&DWXQFLP VLQ$0       + VLQ  $       úLGHFL VLQ$VLQ%VLQ&   + +  ≤    VLQ  $  VLQ $ VLQ % VLQ &  'LQ  úL  UH]XOW HQXQ XO  π π 2EVHUYD LH'DF  ≤$DWXQFLVLQ ≤VLQ$⇔ ≤VLQ$VLQ%VLQ&≤    &XDFHDVWDGLQ  úL  GHGXFHPF  DWXQFL

   +      ⋅      D  + E  + F    +  +   ≤  = =     E F  D ⋅  



3UREOHPD2/GLQQU )LU$Q {«Q}Q∈1 úLI $Q FHOPDLPLFQXP UFXSURSULHWDWHDF  RULFHVXEPXO LPHDOXL$QFXI $Q HOHPHQWHFRQ LQHFHOSX LQWHUPHQLGLVWLQF LDL úLUXOXLOXL)LERQDFFL ) ) )Q )Q)Q±∀Q≥ 'DF I $Q   V VHDIOHQ *DEULHO'RVSLQHVFX2QHúWL  

$5+,0('(V1ULDQXDULHIHEUXDULH 6ROX LH)LHQ∈1 SHQWUXFDUHI $Q  'HILQLPσ $Q FDILLQGQXP UXO WHUPHQLORU )Q Q≥GLVWLQF LGLQPXO LPHD$Q&XP) ) DYHP {)« )σ $ + }⊆{«Q}GHFL )σ $ + ≤Q    )LH|F|QXP UXOGHHOHPHQWHDOHPXO LPLL&úLPXO LPHD%Q {)« )σ $ 

+ }⊆

⊆$Q$WXQFL|$Q±%Q| Q±σ $Q 6 DU W PF I $Q  Q±σ $Q 'DF I $Q ≤ ≤Q±σ $Q ILH'QDVWIHOvQFkW'Q⊆$Q'Q∩%Q ∅|'Q| Q±σ $Q ±I $Q úL (Q $Q± 'Q∪%Q $WXQFL|(Q| Q±|'Q∪$Q| Q± |'Q||%Q|±|'Q∩%Q|   Q± σ $Q Q±σ $Q ±I $Q  I $Q 'DU(QQXFRQ LQHWHUPHQLDLúLUXOXL)Q 'DF I $Q  Q±σ $Q VDXI $Q  Q±σ $Q OX P )Q >$Q± 'Q∪%Q @∪{;};∉ 'Q∪%Q UHVSHFWLY*Q >$Q± 'Q∪%Q @∪{[ \}[\∉ 'Q∪%Q $WXQFL)Q*QDXFkWHI $Q HOHPHQWHVXQWVXEPXO LPLDOHOXL $QGDUDXFHOPXOWGRLWHUPHQL )Q Q≥$úDGDUI $Q ≥Q±σ $Q  )LHεQ⊆$QεQ {D« DQ − σ $ +  }'DF εQFRQ LQHFHOPXOWGRLWHUPHQLDL

úLUXOXL )Q Q≥DWXQFL$Q±εQFRQ LQHFHOSX LQσ $Q ±WHUPHQLDGLF |$Q±εQ|≥ ≥σ $Q ±'DU|$Q±εQ| Q± Q±σ $Q   σ $Q ±σ $Q ±FRQWUDGLF LH 'HFLI $Q  Q±σ $Q úLFXPI $Q  ⇒σ $Q  Q±   'LQ  úL  ⇒)Q±≤Q 3H GH DOW  SDUWH HVWH HYLGHQW F  Q ≥ I $Q     6H RE LQH SULQ FDOFXO F  ) ≤   ) 0DL REVHUY P F    HVWH VDWLVI FXW  GH Q   3HQWUX Q ∈ ∈ {«}DYHPσ $Q   úLGLQ  ⇒ Q QX3UHVXSXQHPF  Q≥úLDU W PSULQLQGXF LHF )Q±!Q3HQWUXQ ) ))   !3UHVXSXQHPF )N±!N∀N∈{«Q±Q} úLDU W PF )Q±!Q'DU)Q± )Q±)Q±!QQ±!Q 'HFLQ HVWHQXP UXOFHUXW 



3UREOHPD;,GLQQU ÌQLQWHULRUXOXQXLFHUF& 2U VHFRQVLGHU XQSXQFW$)LHWULXQJKLXOHFKLOD WHUDO$%&DVWIHOvQFkW$HVWHSXQFWXOIL[DWvQLQWHULRUXOFHUFXOXL%VHDIO SHFHUF LDU&HVWHDOWUHLOHDYkUIDOWULXQJKLXOXL6 VHDIOHORFXOJHRPHWULFGHVFULVGHSXQF WXO&FkQG%SDUFXUJHFHUFXO SURI7UDLDQ3UHGD 6ROX LH&RQVLGHU PXQVLVWHPGHD[HFXRULJLQHDvQSXQFWXO$úLDVWIHOvQFkW FHQWUXOFHUFXOXLV DLE FRRUGRQDWHOH2 DD  $YHPGRX SRVLELOLW LGHFRQVWUXF LHDWULXQJKLXOXLHFKLODWHUDO FDvQILJXU  ∆$%&úL∆$%&  6WXGLHPORFXOJHRPHWULFGHVFULVGH& 3XQFWXO%VHRE LQHGLQSXQFWXO&SULQWURURWD LHGHFHQWUX$úLFXXQXQJKLGH °1RW P% [\ úL& [ \ ⇒ 



$5+,0('(V1ULDQXDULHIHEUXDULH

   [ = [′ ⋅ − \′ ⋅  [ = [′ FRV ° − \′ VLQ °    ⇒  ⇒  ′ ′ \ = [ VLQ  ° + \ FRV  °      \ = [′ ⋅  + \′ ⋅  3XQFWXO%VHDIO SHFHUF⇒ [±D  \±D  U⇒

\ &

% °

°

2 DD 

  $ [  [′ \′    [′  \′  & ⇒  − − D +  + − D  = U  ⇒             ⇒ > [ ±\  ±D  [  \ ±D @ U⇒ [′ + \′ D±  [ \ ±  ±[ DD  \  [′  \′ D  [ \ ±D  [ ±D\  U⇒

⇒ [′  \′ ±D[   D\  ± D U

[′ ±





 D  +     D   \′  D \  ±   D  −   D± [                 

 D  +    D  −    ±   ±           

D  ±   U ⇒ 







 D  +    D  −   U ⇒  [′ − D± + \′ +           





     [′ − D  +   +  \′ + D  −    U ⇒ & GHVFULH FHUFXO GH            D  +  D  −    úLUD] U FHQWUX         $QDORJSXQFWXO%VHRE LQHGLQSXQFWXO& SULQWURURWD LHFX±°⇒    [ = [′ ⋅ + \′ ⋅  [ = [′ FRV −° − \′ VLQ −°    ⇒  ⇒ ⇒ ′ ′ \ = [ VLQ −  ° + \ FRV −  °      \ = − [′ ⋅  + \′ ⋅  



 [′ \′    [′  \′   − D + − + − D  = U  ⇒ > [′ + \′ D  [ \ ± ⇒  +            ±D[ ±D  \  [′  \′ D±  [ \ D  [ ±D\ @ U⇒ ⇒ [′  \′ ±D[ D  [ ±D  \ ±D\ D U⇒

⇒ [′  \′ ±D[ D  [ ±D  \ ±D\ D U⇒  

$5+,0('(V1ULDQXDULHIHEUXDULH ⇒ [′ ±D[ ±   \′ ±D\   D U⇒ ⇒ [′ 





 D  −     D  −    \′ ± D  +  \   D  +   D± [               

 D  −    D  +    ±   ±           







 D  −    D  +   U ⇒  [′ + + \′ − D±           



 D D  −    D  +   ± ±      U⇒  [′ + + \′ −             D  −  D  +    úLUD] U  ⇒& GHVFULHFHUFXOGHFHQWUX  −       'HFLORFXOJHRPHWULFF XWDWHVWHIRUPDWGLQUHXQLXQHDDGRX FHUFXUL   D  +  D  −         U  úL&   D  −   D  +    U    &                  



U



3UREOHPD;,GLQQU 6HG úLUXO [Q Q≥[Q[Q[Q±[Q [![!6 VHDUDWHF úLUXOHVWH FRQYHUJHQWúLV VHFDOFXOH]HOLPLWDVD SURI$XUHO'RERúDQ \Q + \Q + \Q +  \Q +  \ ⋅ ⋅ −  Q + =   6ROX LH 1RW P [Q   UHOD LD GDW  GHYLQH \Q \Q \Q + \Q +  \Q GHXQGH\Q±\Q±\Q FXHFXD LDFDUDFWHULVWLF U±U± FDUHVHPDL VFULH U U U  FXU G FLQLOHU ±U ±U  $ − Q + + % − Q + + & ⋅ Q +  'HFL\Q $ ± Q% ± Q&⋅Q$WXQFL [Q = $ − Q + % − Q + & ⋅ Q Q +     Q +      $ −  + % −  + &         & = =  GDF &≠FHHDFHHVWH úL OLP [Q = OLP Q Q Q →∞ Q →∞     &   Q   $ −  + % −  + &         Q +

DGHY UDWGHRDUHFH [ =

W LOH 

$ +  % + & − $ − % + & ! [ = !FXSRVLELOL − $ −  % + & $ +  % + &



$5+,0('(V1ULDQXDULHIHEUXDULH

− $ −  % + & >  L   úLGLQ[!UH]XOW ±$±%&!$GXQkQGSUL  $ +  % + & >  PHOHGRX LQHJDOLW LJ VLP%&!DGLF %&!LDUGLQDGRXDúLDWUH LDLQHJDOLWDWHSULQDGXQDUHJ VLP±%&!DGLF ±%&!FDUHDGXQDW  FXFHDDQWHULRDU G &!DGLF &! − $ −  % + & <  LL   úLGLQ[!UH]XOW ±$±%&3URFHGkQGFD  $ +  % + & <  PDLVXVJ VLP& 'HFLRULFXP&≠úLDWXQFLúLUXO [Q QHVWHFRQYHUJHQWúLDUHOLPLWD 

3UREOHPD;,GLQQU       6HG PDWULFHD $ =           

D 'HWHUPLQD LDEF∈5DVWIHOvQFkW$D$E$F, 2 E $U WD LF $± $±$, SURI$XUHO'RERúDQ/XJRM               6ROX LHD , $ =       $ =                  

$WXQFL$D$E$F, 2HVWHHFKLYDOHQW FXVLVWHPXO 

 D E  D E D E D E D E D E



F   F  F

     

FXVROX LDD ±E F ±$úDGDU$±$$±, 2 ∗  − λ   ,,$OWIHO 3$ λ =

 

−λ 



−λ

 ±λλ±λ'XS WHRUHPDOXL

&D\OH\+DPLOWRQUH]XOW 3$ $  DGLF ±$$±$, 2VDX $±$$±, 2GHXQGHD ±E F ± E 'LQ$±$$±, 2J VLP, $±$$ $ $±$,     $ ±$, $úLFRQIRUPGHILQL LHLPDWULFHLLQYHUVHUH]XOW   

$5+,0('(V1ULDQXDULHIHEUXDULH $± $±$, 

3UREOHPD;,,GLQQU 'DF  *⋅ HVWHXQJUXSDEHOLDQV VHGHWHUPLQHELMHF LLOHI*→*úLP∈* DúDvQFkWI I$I [ ⋅I [  P ∀ [∈* '0% WLQH X*LXUJLX 6ROX LH'HRDUHFHIHVWHELMHF LHUH]XOW F  I$I [ ⋅I [  I± P  ∀ [∈*   'DF vQ  OX PI [ vQORFGH[GHGXFHP I$I$I [ ⋅ I$I [  I± P    'LQ  úL  UH]XOW  I$I [ ⋅I [   I$I$I [ ⋅ I$I [  ∀ [∈*úL GDWRULW IDSWXOXLF JUXSXOHVWHFRPXWDWLYRE LQHPF  I$I [ ⋅I [   I$I [ ⋅ I$I$I [  ∀ [∈* GHXQGHSULQVLPSOLILFDUHFX I$I [ GHGXFHP I$I$I [  I [  ∀ [∈*⇔ ⇔I I$I [  I [  ∀ [∈*   'DWRULW IDSWXOXLF IHVWHLQMHFWLY GLQ  RE LQHP I$I [  [ ∀ [∈*   LQkQGVHDPDGH  UHOD LD  GHYLQH[⋅I [  I± P  ∀ [∈*⇔ I [  [±⋅I± P  ∀ [∈*   'DF vQ  SXQHP[ HGHGXFHP I H  I± P ⇒ I$I H  H  I$I± P  PDGLF P H 3ULQXUPDUHI [  [±⋅I± H  [±⋅D ∀ [∈*XQGHD I± H  

3UREOHPD;,,GLQQU 6 VHFDOFXOH]HLQWHJUDOD   [  +  [  +  [ +   [ +  [  +  [ +   [  +  [ +   G[XQGH[! OQ ∫ [  [ +   [ +     [  [ +   [ +     SURI0LKDLO%HQF]H%UDúRY     = − + 6ROX LH)LH \ = $WXQFL [ [ +  [ +   [ [ +   [ + 

   [  +  [ +     G[  −  − G\   −  + G['HFL    [ [ +   [ +   [ +   [ +    [

  [  +  [  +  [ +   [ +  [  +  [ +   [  +  [ +      OQ ∫ [  [ +   [ +     [  [ +   [ +      − ∫ OQ \  +  \ +  G\  − ∫ OQ \ +  G\ − ∫ OQ \ +  G\ 

± \ OQ \ ± \ OQ \ \&  



$5+,0('(V1ULDQXDULHIHEUXDULH

      +   OQ +   −   −    [ [ +  [ +    [ [ +  [ + 

       ±  +   OQ +   + &  [ [ +  [ +    [ [ +  [ +   [ [ +  [ + 

352%/(0(352386( &/$6(/(,±,9

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$5+,0('(V1ULDQXDULHIHEUXDULH SX LQGHFkWMXP WDWHGLQFDQWLWDWHDFXOHDV vQSULPD]L&kWHNJGHVWUXJXULDUHFRO WDWIDPLOLDvQILHFDUH]L" vQY1LFRODH,RUG QHVFX* HúWL 

&/$6$$9$

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&/$6$$9,$

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S  Q + −  T  −  = Q∈1  S − T − SURI'RULQD0LKDHOD%RJGDQ&RQVWDQ D 9,,)LH$%&'XQGUHSWXQJKLFX$%    úL%& 3HODWXUD%&VHLD SXQFWXO( (∈>%&@ DVWIHOvQFkW&( &DOFXOD LOXQJLPHDSHUSHQGLFXODUHLGXV  GLQ$SH'( SURI,RDQDúL,XOLDQ0D]LOX8U]LFHQL 9,,ÌQWULXQJKLXO$%& $0HVWHELVHFWRDUHDXQJKLXOXL%$&0∈ %& úL 'HVWHPLMORFXOOXL>%&@3ULQSXQFWHOH0úL'VHGXFSDUDOHOHOD$&FDUHLQWHUVHF WHD] $%vQSXQFWHOH4UHVSHFWLY3ùWLLQGF $%úL$&VXQWGLUHFWSURSRU LRQDOHFX 4% úLDIOD L  $3 SURI/LOLDQD7RGHULF D+E 9,,'DF DE∈1 GLVWLQFWHFX DE ∈1V VHDUDWHF  ∉1  DE SURI'DQ1HGHLDQX'UREHWD7U6HYHULQ 9,,* VL LQ∈1 DVWIHOvQFkWQXP UXOQQQQQQVHGLYL GHFX SURI*K&ULVWHVFX  

$5+,0('(V1ULDQXDULHIHEUXDULH 9,,)LHDEF∈  'HPRQVWUD LF 

E D

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