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The sum is less than 2n+1 [Iran Second Round 1990]

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December 1, 2010
number theory proposednumber theory

Problem Statement

(a) For every positive integer nn prove that 1+122+132+⋯+1n2<21+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} <2
(b) Let X={1,2,3,…,n} (n≥1)X=\{1, 2, 3 ,\ldots, n\} \ ( n \geq 1) and let AkA_k be non-empty subsets of X (k=1,2,3,…,2n−1).X \ (k=1,2,3, \ldots , 2^n -1). If aka_k be the product of all elements of the set Ak,A_k, prove that ∑i=1m∑j=1m1ai⋅j2<2n+1\sum_{i=1}^{m} \sum_{j=1}^m \frac{1}{a_i \cdot j^2} <2n+1
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