Tricky, tricky

Discrete Mathematics Level 5

In a certain game, the first player secretly chooses an \(2015-\)dimensional vector \(a = (a_1, a_2, . . . , a_{2015})\) all of whose components are integers.

The second player is to determine \(a\) by choosing any \(2015-\)dimensional vectors \(x_i\), all of whose components are also integers.

For each \(x_i\) chosen, and before the next \(x_i\) is chosen, the first player tells the second player the value of the dot product \(x_i \cdot a\).

What is the least number of vectors \(x_i\) the second player has to choose in order to be able to determine \(a\)?


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