IMO Shortlist 1971 problem 9


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Dodao/la: arhiva
2. travnja 2012.
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Let T_k = k - 1 for k = 1, 2, 3,4 and
T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k \qquad  (k \geq 3).
Show that for all k,
1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right] \quad \text{and} \quad 1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right],
where [x] denotes the greatest integer not exceeding x.
Izvor: Međunarodna matematička olimpijada, shortlist 1971