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Value of integer sums

Source: USAMO 1992

October 27, 2005
inductionnumber theory unsolvednumber theory

Problem Statement

For a nonempty set  S \, S \, of integers, let  σ(S) \, \sigma(S) \, be the sum of the elements of  S\, S. Suppose that  A={a1,a2,…,a11} \, A = \{a_1, a_2, \ldots, a_{11} \} \, is a set of positive integers with  a1<a2<⋯<a11 \, a_1 < a_2 < \cdots < a_{11} \, and that, for each positive integer  n≤1500, \, n\leq 1500, \, there is a subset  S \, S \, of  A \, A \, for which  σ(S)=n\, \sigma(S) = n. What is the smallest possible value of  a10\, a_{10}?
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