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JBTST IV 2007, Problema 4

 
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Laurian Filip
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Joined: 25 Nov 2007
Posts: 473
Location: Bucuresti

PostPosted: Sun Apr 20, 2008 8:21 am    Post subject: JBTST IV 2007, Problema 4 Reply with quote

Sa se determine toate numerele naturale nenule n care se pot scrie sub forma

n=[a,b]+[b,c]+[c,a]

cu a,b,c naturale nenule.
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Aelius Pop
Euclid


Joined: 08 Nov 2008
Posts: 22
Location: Arad

PostPosted: Thu Apr 16, 2009 2:20 am    Post subject: Reply with quote

Fie a=2^xm ;m impar
b=2^yn ;n impar
c=2^zp ;p impar
Deoarece relatia este simetrica putem presupune fara a reduce din generalitatea problemei ca x \leq y \leq z

N=2^y[m;n]+2^z[n;p]+2^z[m;p]

[m;n];[n;p];[m;p] impare
N=2^d+2^ze+2^zf; d,e,f impare
N=2^y[d+2^{z-y}(e+f )]
d impar
2^{z-y}(e+f) par

=>[d+2^{z-y}(e+f )] este impar
=>N=2^y(2k+1) k\in \mathbb{N^*}
Deci N \in \mathbb{N}^*-\lbrace2^q|q\in \mathbb{N}\rbrace
Luam z=y si gasim m;n;p astfel incat N sa fie orice nr care nu e putere a lui 2.
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