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Prove that number of numbers t in I is less than 1000

Source: Iran Third Round Problems 1993 – Poblem 6

July 29, 2011
algebra proposedalgebra

Problem Statement

Let x1,x2,…,x12x_1, x_2, \ldots, x_{12} be twelve real numbers such that for each 1≤i≤121 \leq i \leq 12, we have ∣xi∣≥1|x_i| \geq 1. Let I=[a,b]I=[a,b] be an interval such that b−a≤2b-a \leq 2. Prove that number of the numbers of the form t=∑i=112rixit= \sum_{i=1}^{12} r_ix_i, where ri=±1r_i=\pm 1, which lie inside the interval II, is less than 10001000.
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