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JBTST VI 2007, Problema 4

 
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Laurian Filip
Site Admin


Joined: 25 Nov 2007
Posts: 473
Location: Bucuresti

PostPosted: Sun Apr 20, 2008 8:27 am    Post subject: JBTST VI 2007, Problema 4 Reply with quote

Un numar real x cu 0<x<1 se numeste interesant daca este irational si, in scrierea sa zecimala, primele patru zecimale sunt egale. Sa se determine cel mai mic numar natural n cu proprietatea ca orice numar real t cu 0<t<1 se poate scrie ca suma de n numere interesante distincte doua cate doua.
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Aelius Pop
Euclid


Joined: 08 Nov 2008
Posts: 22
Location: Arad

PostPosted: Mon Apr 20, 2009 4:01 pm    Post subject: Reply with quote

Un numar de forma \overline{0,aaaa} necesita cele mai multe numere.
Indubitabil ca 1112 numere sunt suficiente pentru a scrie 0.1111 si inseamna ca 1112 numere sunt suficiente pt a scrie orice numar. Deci n=1112.
Configuratia de numere cu suma 0.1111: pt. 1\leq k \leq 1111, fie a_k=0,0001 -  10^{-100} \cdot k \pi si a_{1112}= (1+2+...+1111) \cdot 10^{-100} \cdot  \pi
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