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{SECT 0 {SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Hatlap\372 toroidok" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart;with(plots):" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Eszk\366zt\341r" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 366 "a:=array[1..3]:\nb:=array[1..3]:\nc:=array
[1..3]:\nvegyesszorzat:= (a,b,c) -> a[1]*b[2]*c[3]\n                  \+
        +a[2]*b[3]*c[1]\n                          +a[3]*b[1]*c[2]\n  \+
                        -a[3]*b[2]*c[1]\n                          -a[
2]*b[1]*c[3]\n                          -a[1]*b[3]*c[2]:\nsik:= (a,b,c
) -> vegyesszorzat(b-a,c-a,[x-a[1],y-a[2],z-a[3]])=0:" }{TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 
0 "" {TEXT 258 47 "P\351lda  h\341rom s\355k metsz\351spontj\341nak ki
sz\341m\355t\341s\341ra" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "
A1:=[ 5,-2,10]: A2:=[-1,-3, 8]: A3:=[ 3, 4, 5]: h1:=[A1,A2,A3];\nB1:=[
 1,10,-4]: B2:=[-3,12,-7]: B3:=[ 2,12, 3]: h2:=[B1,B2,B3];\nC1:=[10, 5
, 0]: C2:=[ 6,-4,-3]: C3:=[ 4,-2, 4]: h3:=[C1,C2,C3];\npolygonplot3d([
h1,h2,h3],scaling= constrained);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 280 "e:=sik(A1,A2,A3);\nf:=sik(B1,B2,B3);\ng:=sik(C1,C2,C
3);\nm:=solve(\{e,f,g\}):\nassign(m):\nP:=[x,y,z]:\nunassign('x','y','
z'):\nprint(` x= `,P[1],`   y = `, P[2],`   z = `,P[3]);\nn1:=[P,A1,A2
,A3];\nn2:=[P,B1,B2,B3];\nn3:=[P,C1,C2,C3];\npolygonplot3d([n1,n2,n3,h
1,h2,h3],scaling= constrained);" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Sz\341m\355t
\341sok" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "A t\363rusz tengely
\351t" }{TEXT 259 1 " " }{TEXT 260 6 "metsz\365" }{TEXT 256 1 " " }
{TEXT -1 28 "oldal\372 hatsz\366gek el\365\341ll\355t\341sa" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 34 "El\365sz\366r megadjuk a t\363rusz adatai
t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 255 "d:=12:   # gy\373r
\373 sugara\nr:= 4:   # keresztmetszet sugara\np:= 7;   # gy\373r\373 \+
lapjainak sz\341ma             index: i=0..p-1 \nq:= 3;   # keresztmet
szet lapjainak sz\341ma    index: j=0..q-1 \nf:=Pi/p:  g:=Pi/q:   # sz
\366gek\n         dg:= g/2:   # kezd\365 \351rt\351k egy k\366r\366n" 
}{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Kisz\341m\355tju
k a n\351gysz\366gekb\365l fel\351p\355tett toroid cs\372csait." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "for i from 0 to 2*p-1  do\n
 for j from 0 to 2*q-1  do\nn[i,j]:=[evalhf((d+r*cos(j*g+dg))*cos(i*f)
),evalhf((d+r*cos(j*g+dg))*sin(i*f)),evalhf(r*sin(j*g+dg))]:\n  od;\no
d;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Kisz\341m\355tjuk a hatsz
\366gek azon  cs\372csait, melyek s\355kok mettsz\351spontjaik\351nt \+
\341llnak el\365." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2045 "H:=a
rray(0..p-1,0..q-1):\nK:=array(0..p-1,0..q-1):\nR:=array(0..p-1,0..q-1
):\nS:=array(0..p-1,0..q-1):\n\n for i from 0 to p-1 do\n   for j from
 0 to q-1 do\n     k:=2*i-1; l:= 2*j;\n\ns1:=sik(n[(k-1) mod 2*p, l   \+
 mod 2*q],\n        n[ k    mod 2*p, l    mod 2*q],\n        n[ k    m
od 2*p,(l+1) mod 2*q]):\ns2:=sik(n[ k    mod 2*p,(l-1) mod 2*q],\n    \+
    n[(k+1) mod 2*p,(l-1) mod 2*q],\n        n[(k+1) mod 2*p,(l  ) mod
 2*q]):\ns3:=sik(n[(k+1) mod 2*p,(l  ) mod 2*q],\n        n[(k+2) mod \+
2*p,(l  ) mod 2*q],\n        n[(k+2) mod 2*p,(l+1) mod 2*q]):\n     m \+
:=solve(\{s1,s2,s3\}):\n     assign(m):\n     H[i,j]:=[x,y,z];\n     u
nassign('x','y','z');\n\ns1:=sik(n[(k-1) mod 2*p, l    mod 2*q],\n    \+
    n[ k    mod 2*p, l    mod 2*q],\n        n[ k    mod 2*p,(l+1) mod
 2*q]):\ns2:=sik(n[ k    mod 2*p,(l+1) mod 2*q],\n        n[(k+1) mod \+
2*p,(l+1) mod 2*q],\n        n[(k+1) mod 2*p,(l+2) mod 2*q]):\ns3:=sik
(n[(k+1) mod 2*p,(l  ) mod 2*q],\n        n[(k+2) mod 2*p,(l  ) mod 2*
q],\n        n[(k+2) mod 2*p,(l+1) mod 2*q]):\n     m :=solve(\{s1,s2,
s3\});\n     assign(m):\n     K[i,j]:=[x,y,z];\n     unassign('x','y',
'z'):\n\n    k:=2*i; l:= 2*j+1;\n\ns1:=sik(n[(k-1) mod 2*p, l    mod 2
*q],\n        n[ k    mod 2*p, l    mod 2*q],\n        n[ k    mod 2*p
,(l+1) mod 2*q]):\ns2:=sik(n[ k    mod 2*p,(l-1) mod 2*q],\n        n[
(k+1) mod 2*p,(l-1) mod 2*q],\n        n[(k+1) mod 2*p,(l  ) mod 2*q])
:\ns3:=sik(n[(k+1) mod 2*p,(l  ) mod 2*q],\n        n[(k+2) mod 2*p,(l
  ) mod 2*q],\n        n[(k+2) mod 2*p,(l+1) mod 2*q]):\n     m :=solv
e(\{s1,s2,s3\}):\n     assign(m):\n     R[i,j]:=[x,y,z];\n     unassig
n('x','y','z');\n\ns1:=sik(n[(k-1) mod 2*p, l    mod 2*q],\n        n[
 k    mod 2*p, l    mod 2*q],\n        n[ k    mod 2*p,(l+1) mod 2*q])
:\ns2:=sik(n[ k    mod 2*p,(l+1) mod 2*q],\n        n[(k+1) mod 2*p,(l
+1) mod 2*q],\n        n[(k+1) mod 2*p,(l+2) mod 2*q]):\ns3:=sik(n[(k+
1) mod 2*p,(l  ) mod 2*q],\n        n[(k+2) mod 2*p,(l  ) mod 2*q],\n \+
       n[(k+2) mod 2*p,(l+1) mod 2*q]):\n        m :=solve(\{s1,s2,s3
\});\n     assign(m):\n     S[i,j]:=[x,y,z];\n     unassign('x','y','z
'):\n   od;\n od;" }}}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 12 "Kontroll 2
: " }{TEXT 265 71 "\nMegvizsg\341ljuk, hogy a sz\341m\355t\341sainkhoz
 j\363l szervezt\374k-e az adatainakat." }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 118 "A t\363ruszt rajzol\363 utas\355t\341ssal nem tudjuk bef
oy\341solni, hogy hol kezd\365dj\366n az egy oszlopba rajzolt pontokat
 meghat\341roz\363 " }{TEXT 266 2 "2q" }{TEXT -1 16 " oldal\372 soksz
\366g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "tubeplot([d*sin(t
),d*cos(t),0],t=0..2*Pi,radius=r,grid=[2*p+1,2*q+1],scaling=constraine
d);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "unas
sign('i','j'):\nplot3d([(d+r*cos(j))*cos(i),(d+r*cos(j))*sin(i),r*sin(
j)],i=0..2*Pi,j=0..2*Pi,scaling= constrained,grid=[2*p+1,2*q+1]);" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "A fenti k\351t ut
as\355t\341snak ugyanolyan rajzot kell eredm\351nyeznie. " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Az al\341bbiak is \+
ilyen alakzatot \341ll\355tan\341nak el\365 " }{TEXT 268 4 "dg=0" }
{TEXT -1 61 " esetben, de ez nem lenne alkalmas a hatsz\366gek el\365
\341ll\355t\341s\341ra." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "unassign('i','j'):\nplot3d([(d+r*c
os(j+dg))*cos(i),(d+r*cos(j+dg))*sin(i),r*sin(j+dg)],i=0..2*Pi,j=0..2*
Pi,scaling= constrained,grid=[2*p+1,2*q+1]);" }}{PARA 13 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "negyzet:=[
seq(seq(\n   [n[i             , j           ],\n    n[i             ,(
j+1) mod 2*q],\n    n[(i+1) mod 2*p ,(j+1) mod 2*q],\n    n[(i+1) mod \+
2*p ,j            ]]\n,i=0..2*p-1),j=0..2*q-1)]:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 43 "polygonplot3d(negyzet,scaling=constrained);" 
}}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
38 "Az ut\363bbi k\351t rajzon m\341r \351rv\351nyes\374l a " }{TEXT 
269 3 "dg " }{TEXT -1 20 "nagys\341g\372 elfordul\341s." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 238 "fele:=[seq(seq(\n   [n[i          \+
  ,(2*j  +(i mod 2)) mod 2*q ],\n    n[i            ,(2*j+1+(i mod 2))
 mod 2*q ],\n    n[(i+1) mod 2*p,(2*j+1+(i mod 2)) mod 2*q ],\n    n[(
i+1) mod 2*p,(2*j  +(i mod 2)) mod 2*q ]]\n  ,i=0..2*p-1),j=0..q-1)]:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "racs:=polygonplot3d(fel
e,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r
acs;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "HK:
=polygonplot3d([seq(seq(\n[ H[i,j],K[i,j]]    \n,i=0..p-1),j=0..q-1)])
:\nRS:=polygonplot3d([seq(seq(\n[ R[i,j],S[i,j]]    \n,i=0..p-1),j=0..
q-1)]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "display([racs,HK
,RS],scaling= constrained);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Itt m\341r berajzoltuk az \372jonn
an keletkezett hatsz\366g\351leket." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 0 {PARA 20 "" 0 "" {TEXT -1 13 "T(6,3) toroid" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 217 "hatszog:=[seq(seq(\n   [ [R[i,j],K[i,j],H[
i,j], \n      S[i,(j-1) mod q],H[(i+1) mod p,j],K[(i+1) mod p,j]],\n  \+
   [H[i,(j+1) mod q],S[(i-1) mod p,j],R[(i-1) mod p,j],\n      K[i,j],
R[i,j],S[i,j]]]\n ,i=0..p-1),j=0..q-1)]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "polygonplot3d(hatszog,scaling= constrained):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 214 "ketsor:=[seq(seq(\n[ [R[i,j],K[i,j],H[i,j], \n    \+
  S[i,(j-1) mod q],H[(i+1) mod p,j],K[(i+1) mod p,j]],\n     [H[i,(j+1
) mod q],S[(i-1) mod p,j],R[(i-1) mod p,j],\n      K[i,j],R[i,j],S[i,j
]]]    \n,i=0..0),j=0..q-1)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 43 "polygonplot3d(ketsor,scaling= constrained);" }}{PARA 13 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 521 "szin:=[red,green, blue,yellow, pin
k,aquamarine, cyan, grey,white,black]:\n for s from 1 to 7 do\negyszin
:=[seq(seq(\n[ [R[(i-j) mod p,j],K[(i-j) mod p,j],H[(i-j) mod p,j], \n
      S[(i-j) mod p,(j-1) mod q],H[(i+1-j) mod p,j],K[(i+1-j) mod p,j]
],\n     [H[(i-j) mod p,(j+1) mod q],S[(i-1-j) mod p,j],R[(i-1-j) mod \+
p,j],\n      K[(i-j) mod p,j],R[(i-j) mod p,j],S[(i-j) mod p,j]]]    \+
\n,i=s..s),j=0..q-1)]:\n\nt[s]:=polygonplot3d(egyszin,color=szin[s],sc
aling=constrained):\nod:\nszines:=display3d(seq(t[i],i=1..7),insequenc
e=false):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "szines;" }{TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "t[1];" }{TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 " VRML f
\341jl" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "with(plottools):\n
vrml(szines,`63T7_3.wrl`,\n     background_color=COLOR(RGB,0,1,1));" }
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 
"" {TEXT 264 28 "Egy eredm\351nytelen k\355s\351rlet: " }{TEXT 262 21 
"\nA t\363rusz tengely\351re " }{TEXT 261 9 "mer\365leges" }{TEXT 263 
29 " oldal\372 hatsz\366gek el\365\341ll\355t\341sa" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 255 "d:=10:   # gy\373r\373 sugara\nr:= 3:   # \+
keresztmetszet sugara\np:= 3;   # gy\373r\373 lapjainak sz\341ma      \+
       index: i=0..p-1 \nq:= 5;   # keresztmetszet lapjainak sz\341ma \+
   index: j=0..q-1 \nf:=Pi/p:   g:=Pi/q:  # sz\366gek\n           dg:=
 g/2: # kezd\365 \351rt\351k egy k\366r\366n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"pG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG\"
\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2387 "for i from 0 to 2*
p-1  do\n for j from 0 to 2*q-1  do\nn[i,j]:=[evalhf((d+r*cos(j*g+dg))
*cos(i*f)),evalhf((d+r*cos(j*g+dg))*sin(i*f)),evalhf(r*sin(j*g+dg))]:
\n  od;\nod;\nH:=array(0..p-1,0..q-1):\nK:=array(0..p-1,0..q-1):\nR:=a
rray(0..p-1,0..q-1):\nS:=array(0..p-1,0..q-1):\n\n for i from 0 to p-1
 do\n   for j from 0 to q-1 do\n\n    k:=2*i-1; l:= 2*j;\n\ns1:=sik(n[
(k-1) mod 2*p, l    mod 2*q],\n        n[ k    mod 2*p, l    mod 2*q],
\n        n[ k    mod 2*p,(l+1) mod 2*q]):              \ns2:=sik(n[ k
    mod 2*p,(l-1) mod 2*q],\n        n[(k+1) mod 2*p,(l-1) mod 2*q],\n
        n[(k+1) mod 2*p, l    mod 2*q]):\ns3:=sik(n[ k    mod 2*p,(l+1
) mod 2*q],\n        n[(k+1) mod 2*p,(l+1) mod 2*q],\n        n[(k+1) \+
mod 2*p,(l+2) mod 2*q]):               \n               \n     m :=sol
ve(\{s1,s2,s3\}):\n     assign(m):\n     H[i,j]:=[x,y,z];\n     unassi
gn('x','y','z');\n               \ns1:=sik(n[(k+1) mod 2*p, l    mod 2
*q],\n        n[(k+2) mod 2*p, l    mod 2*q],\n        n[(k+2) mod 2*p
,(l+1) mod 2*q]):              \ns2:=sik(n[ k    mod 2*p,(l-1) mod 2*q
],\n        n[(k+1) mod 2*p,(l-1) mod 2*q],\n        n[(k+1) mod 2*p, \+
l    mod 2*q]):\ns3:=sik(n[ k    mod 2*p,(l+1) mod 2*q],\n        n[(k
+1) mod 2*p,(l+1) mod 2*q],\n        n[(k+1) mod 2*p,(l+2) mod 2*q]): \+
              \n     m :=solve(\{s1,s2,s3\});\n     assign(m):\n     K
[i,j]:=[x,y,z];\n     unassign('x','y','z'):\n\n     k:=2*i; l:= 2*j+1
;\n\ns1:=sik(n[(k-1) mod 2*p, l    mod 2*q],\n        n[ k    mod 2*p,
 l    mod 2*q],\n        n[ k    mod 2*p,(l+1) mod 2*q]):             \+
 \ns2:=sik(n[ k    mod 2*p,(l-1) mod 2*q],\n        n[(k+1) mod 2*p,(l
-1) mod 2*q],\n        n[(k+1) mod 2*p, l    mod 2*q]):\ns3:=sik(n[ k \+
   mod 2*p,(l+1) mod 2*q],\n        n[(k+1) mod 2*p,(l+1) mod 2*q],\n \+
       n[(k+1) mod 2*p,(l+2) mod 2*q]):               \n              \+
 \n     m :=solve(\{s1,s2,s3\}):\n     assign(m):\n     R[i,j]:=[x,y,z
];\n     unassign('x','y','z');\n   \ns1:=sik(n[(k+1) mod 2*p, l    mo
d 2*q],\n        n[(k+2) mod 2*p, l    mod 2*q],\n        n[(k+2) mod \+
2*p,(l+1) mod 2*q]):              \ns2:=sik(n[ k    mod 2*p,(l-1) mod \+
2*q],\n        n[(k+1) mod 2*p,(l-1) mod 2*q],\n        n[(k+1) mod 2*
p, l    mod 2*q]):\ns3:=sik(n[ k    mod 2*p,(l+1) mod 2*q],\n        n
[(k+1) mod 2*p,(l+1) mod 2*q],\n        n[(k+1) mod 2*p,(l+2) mod 2*q]
):               \n     m :=solve(\{s1,s2,s3\});\n     assign(m):\n   \+
  S[i,j]:=[x,y,z];\n     unassign('x','y','z'):            \n   od;\n \+
od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "hatszog:=[seq(seq(
\n   [ [S[(i-1) mod p,j],H[i,j],K[i,j], \n      R[i,j],K[i,(j+1) mod q
],H[i,(j+1) mod q]],\n     [K[i,(j+1) mod q],R[i,j],S[i,j],\n      H[(
i+1) mod p,(j+1) mod q],S[i,(j+1) mod q],\n      R[i,(j+1) mod q]]]\n \+
,i=0..p-1),j=0..q-1)]:\npolygonplot3d(hatszog,scaling= constrained);" 
}}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "62-%)POLYGONSG6$
7(7%$\"+2$z_O$!\"*$!+k/1/'*F*$\"+l>5aLF*7%$\"+/m&G%oF*$!+Rn,T5!\")$\"+
;!\\H2#F*7%$\"+s(*oV7F4$!+&fR+@(!#5F57%$\"+\"*********F*$\"+jGh()=F*$
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{TEXT -1 141 "Ez a megold\341s nem megfelel\365, mert vannak olyan lap
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\351rt\351kek mellett - \366n\341tmetsz\365v\351 v\341lnak." }{TEXT 
267 69 "Vajon lehet-e ilyen elrendez\351s\373 lapokb\363l szab\341lyos
 toroidot \351p\355teni? " }{TEXT -1 182 "\nSejthet\365, hogy nem, ugy
anis a felhaszn\341lt n\351gyzetr\341csnak vagy a lapjai, vagy a s\355
kjai illeszkednek a t\363rusz tengely\351re mer\365leges s\355kra, \+
\351s ez okozza az eml\355tett jelens\351gek egyik\351t. " }}}}{EXCHG 
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