mateforum.ro Forum Index mateforum.ro

 
 FAQFAQ   SearchSearch   MemberlistMemberlist   UsergroupsUsergroups   RegisterRegister 
 ProfileProfile   Log in to check your private messagesLog in to check your private messages   Log inLog in 

Numere cu suma 53

 
Post new topic   Reply to topic    mateforum.ro Forum Index -> Juniori -> Teoria Numerelor
View previous topic :: View next topic  
Author Message
Marius Mainea
Gauss


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

PostPosted: Tue Dec 01, 2009 9:58 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.
Back to top
View user's profile Send private message Send e-mail
Andi Brojbeanu
Newton


Joined: 22 Mar 2009
Posts: 383
Location: Targoviste (Dambovita)

PostPosted: Sat Dec 19, 2009 4:53 pm    Post subject: Reply with quote

Presupunem ca dintre cele 53 de numere nu putem alege doua numere cu suma 53.
Atunci va trebui sa selectam 53 de numere naturale distincte cat mai mici, astfel incat suma acestora sa fie cat mai mica. Suma celor 53 de numere trebuie sa aiba valoarea maxima 2009.
Pentru ca suma sa ia cea mai mica valoare, alegem urmatoarele numere naturale consecutive: 0, 1, 2, 3, 4, ........, 25, 26 pentru a face parte din sirul de 53 de numere. Observam ca nu putem selecta in continuarea sirului numerele 27, 28, 29, ...., 53 deoarece suma  0+53=53; 1+52=53; 2+51=53; ...... 25+28=53; 26+27=53, ceea ce ar duce la contradictie cu presupunerea facuta.
Atunci, cele 53 numere vor fi 0, 1, 2, 3, ....., 25, 26, 54, 55, ......, 78, 79 si suma acestora va fi egala cu
1+2+3+4+....+26+54+55+....+78+79=
\frac{26\cdot 27}{2}+53+1+53+2+ .....+ 53+25+ 53+26=
\frac{26\cdot 27}{2}+53\cdot 26+ (1+2+3+...+26)=
\frac{26\cdot 27}{2}+53\cdot 26+ \frac{26\cdot 27}{2}=
26\cdot 27+ 53\cdot 26=26\cdot(53+27)=26\cdot 80= 2080>2009, contradictie cu ipoteza.
Back to top
View user's profile Send private message Send e-mail Yahoo Messenger
Display posts from previous:   
Post new topic   Reply to topic    mateforum.ro Forum Index -> Juniori -> Teoria Numerelor All times are GMT + 2 Hours
Page 1 of 1

 
Jump to:  
You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot vote in polls in this forum



Powered by phpBB © 2001, 2005 phpBB Group