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Subhimpunan loyal dan dua fungsi

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Subhimpunan bilangan asli $B$ dikatakan loyal jika terdapat bilangan asli $i\leq j$ yang memenuhi $B=\{i,i+1, ..., j\}$. Diketahui $Q$ koleksi subhimpunan-subhimpunan loyal dari bilangan asli. Untuk masing-masing $A=\{a_1,a_2,...,a_k \} \subseteq \{1,2,...,n \}$, didefinisikan $g(A)=max_{B\subset A, B\in Q} |B|$, $f(A)=max_{1\leq i\leq k-1}(a_{i+1}-a{i})$, $G(n)=\sum\nolimits_{A\subseteq\{1,2,...,n\}} g(A)$, dan $F(n)=\sum\nolimits_{A\subseteq\{1,2,...,n\}} f(A)$

Buktikan terdapat $m\in \mathbb{N}$ sehingga $G(n)<F(n)$ untuk semua bilangan asli $n>m$

 

Tes 2 Nomor 4 Pelatnas Tahap 3 IMO 2016

Edited by -_-

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