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Part of 2003 Polish MO Finals

Problems(1)

Inequality with strictly increasing sequences - Poland 2003

Source:

11/1/2010
Let 0<a<10 < a < 1 be a real number. Prove that for all finite, strictly increasing sequences k1,k2,…,knk_1, k_2, \ldots , k_n of non-negative integers we have the inequality (∑i=1naki)2<1+a1−a∑i=1na2ki.\biggl( \sum_{i=1}^n a^{k_i} \biggr)^2 < \frac{1+a}{1-a} \sum_{i=1}^n a^{2k_i}.
inequalitiesinductioninequalities unsolved