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\listoverridecount0\ls2}{\listoverride\listid1897084486\listoverridecount0\ls3}}{\info{\title A m\'fbszaki f\'f5iskol\'e1k 2004}{\author  }{\operator Kis G\'e1bor}{\creatim\yr2006\mo2\dy25\hr7\min31}{\revtim\yr2006\mo2\dy25\hr8\min7}{\version5}{\edmins21}
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\pnlcrm\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}\pard\plain \qc\nowidctlpar\widctlpar\adjustright \fs20\lang1038\cgrid {\b\i\fs28\lang1033 A }{\b\i\fs28 m\'fbszaki f\'f5iskol\'e1k 1997. \'e9vi Haj\'f3s Gy\'f6rgy
\par matematika verseny\'e9nek feladatai
\par }\pard \nowidctlpar\widctlpar\adjustright {\fs24 
\par 
\par 
\par 
\par 
\par 
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\fs24 , }{\i\fs24 x}{\fs24\sub 2}{\fs24 , }{\i\fs24 x}{\fs24\sub 3}{\fs24 , \'85}{\i\fs24  x}{\fs24\sub 1997}{\fs24 , sorozat tagjai az 1, 2, 3, \'85, 1997 sz\'e1mok egy prmut\'e1ci\'f3j\'e1t alkotj\'e1k. Igaz-e, hogy ekkor az
\par }\pard \qc\li426\nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright {\fs24 (}{\i\fs24 x}{\fs24\sub 1}{\fs24 +1)}{\fs24\super .}{\fs24 (}{\i\fs24 x}{\fs24\sub 2}{\fs24 +2)}{\fs24\super .}{\fs24 (}{\i\fs24 x}{\fs24\sub 3}{\fs24 +3)
}{\fs24\super  .}{\fs24 ...}{\fs24\super  .}{\fs24  (}{\i\fs24 x}{\fs24\sub 1997}{\fs24 +1997)
\par }\pard \qj\li426\nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright {\fs24 Szorzat mindig oszthat\'f3 3-mal?
\par }\pard \nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright {\fs24 
\par 
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mok halmaz\'e1n az al\'e1bbi egyenletet:
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\fs24 
\par }\pard \qj\li426\nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright {\fs24 (\{}{\i\fs24 y}{\fs24 \}=}{\i\fs24 y\_}{\fs24 [}{\i\fs24 y}{\fs24 ], ahol [}{\i\fs24 y}{\fs24 ] az }{\i\fs24 y}{\fs24  sz\'e1m eg\'e9sz r\'e9sz\'e9
t jelenti.)
\par }\pard \nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright {\fs24 
\par 
\par {\pntext\pard\plain\cgrid \hich\af0\dbch\af0\loch\f0 3.\tab}}\pard \qj\fi-360\li360\nowidctlpar\widctlpar\jclisttab\tx360{\*\pn \pnlvlbody\ilvl0\ls1\pnrnot0\pndec\pnstart1\pnindent360\pnhang{\pntxta .}}\ls1\adjustright {\fs24 Egy egys\'e9gnyi oldalhossz
\'fas\'e1g\'fa n\'e9gyzet tetsz\'f5leges }{\i\fs24 P}{\fs24  bels\'f5 pontj\'e1n \'e1t az oldalakkal p\'e1rhuzamosan h\'fazott egyenesek a n\'e9gyzetet n\'e9gy t\'e9glalapra osztj\'e1k.
\par }\pard \qj\li426\nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright {\fs24 a) Bizony\'edtsa be, hogy b\'e1rmelyik k\'e9t, \'e1tl\'f3san elhelyezked\'f5 (teh\'e1t csak egy k\'f6z\'f6s cs\'faccsal rendelkez\'f5) t\'e9glalapot v\'e9
ve, nem fordulhat el\'f5, hogy mindkett\'f5 ter\'fclete nagyobb 1/4-n\'e9l!
\par b) \'c1ltal\'e1nos\'edtsa a fenti \'e1ll\'edt\'e1st n\'e9gyzet helyett m\'e1s s\'edk- \'e9s t\'e9rbeli alakzatokra.
\par }\pard \nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright {\fs24 
\par 
\par {\pntext\pard\plain\cgrid \hich\af0\dbch\af0\loch\f0 4.\tab}}\pard \fi-360\li360\nowidctlpar\widctlpar\jclisttab\tx360{\*\pn \pnlvlbody\ilvl0\ls1\pnrnot0\pndec\pnstart1\pnindent360\pnhang{\pntxta .}}\ls1\adjustright {\fs24 Hat\'e1rozza meg az \'f6
sszes olyan polinomot, amelynek a 2}{\i\fs24 x}{\fs24  helyen vett helyettes\'edt\'e9si \'e9rt\'e9ke minden }{\i\fs24 x}{\fs24 -re megegyezik az els\'f5 \'e9s m\'e1sodik deriv\'e1ltja }{\i\fs24 x}{\fs24  helyen vett helyettes\'edt\'e9si \'e9rt\'e9k\'e9kei
nek szorzat\'e1val!
\par }\pard \nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright {\fs24 
\par 
\par {\pntext\pard\plain\cgrid \hich\af0\dbch\af0\loch\f0 5.\tab}}\pard \qj\fi-360\li360\nowidctlpar\widctlpar\jclisttab\tx360{\*\pn \pnlvlbody\ilvl0\ls1\pnrnot0\pndec\pnstart1\pnindent360\pnhang{\pntxta .}}\ls1\adjustright {\fs24 Az }{\i\fs24 ABCD}{\fs24 
 tetra\'e9der }{\i\fs24 AB}{\fs24  \'e9l\'e9nek egy pontj\'e1n kereszt\'fcl a tetra\'e9der }{\i\fs24 ACD}{\fs24 , illetve }{\i\fs24 BCD}{\fs24  oldallapj\'e1val p\'e1rhuzamos s\'edkokat vett\'fcnk fel. Az \'edgy keletkezett k\'e9t kis tetra\'e9der t\'e9
rfogata 8 illetve 1 cm}{\fs24\super 3}{\fs24 . Mekkora az }{\i\fs24 ABCD}{\fs24  tetra\'e9der t\'e9rfogata?
\par }}