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Problem Set J

The problem below are from the 1995 Iowa Collegiate Mathematics Competition

  1. Let A be the integer whose base 10 representation consists of 119 ones

    A = \underbrace{1111 \ldots 1}_{119}
    Prove that A is not prime.

  2. Let a_1, a_2, \ldots a_n be real numbers with

    a_1^2 + a_2^2 + \cdots + a_n^2 =1
    Prove that
    \displaystyle \frac{-1}{2} \leq \sum_{1\leq j < k\leq n } a_ja_k \leq \frac{n-1}{2}
    Note: On the original 1995 Iowa Collegiate Mathematics Competition this problem was stated as
    \displaystyle \frac{-1}{2} \leq \sum_{1\leq j < k\leq n } a_ja_k \leq \frac{n-2}{2}
    which is not true if all the a_i = a_1 for all i

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