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Numere cu suma 53

 
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Marius Mainea
Gauss


Joined: 26 May 2008
Posts: 1099
Location: Gaesti (Dambovita)

PostPosted: Thu Nov 19, 2009 11:21 pm    Post subject: Numere cu suma 53 Reply with quote

Sa se demonstreze ca din 53 de numere distincte a caror suma nu depaseste 2009 , putem alege doua numere cu suma egala cu 53.
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anamariaradu



Joined: 27 Aug 2010
Posts: 1

PostPosted: Tue Aug 31, 2010 11:53 am    Post subject: Reply with quote

Presupunem contrariul. Observam ca printre primele 53 de numere naturale se pot gasi 2 a caror suma sa fie 53 ( iar suma acestora ,1+2+...53 este mai mica decat 2009), de exemplu: 0+53, 1+52..etc.. Astfel vom lua in considerare primele 27 numere naturale,iar pe urmatoarele pana la al 53-lea le vom alege mai mari decat 53. asa incat sa nu existe printre ele doua cu suma 53.
Dar in acest caz suma celor 53 de numere va fi
0+1+2+...+26 + 54+55+....79=2080
Contradictie!!! caci 2080 este mai mare decat 2009!!
Deci presupunerea facuta este falsa!
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Marius Mainea
Gauss


Joined: 26 May 2008
Posts: 1099
Location: Gaesti (Dambovita)

PostPosted: Tue Aug 31, 2010 7:47 pm    Post subject: Reply with quote

Smile Foarte bine Ana,


Bine ai venit pe forum Smile


O sa fac eu unele precizari

Fie multimile A=\{0,1,2...53\}siB=\{54,55,......\}

Presupunand prin absurd ca nu exista 2 cu suma 53 , rezulta ca in A avem maxim 27 de numere ( principiul cutiei (0,53);(1,52);...(26,27)) restul fiind in B.

Atunci suma celor 53 de numere este asa cum ai zis si tu cel putin egala cu 0+1+...+26+54+55+...+79=2080.

Contradictie.
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