IT-CESENATICO 2016 Problem 6

A mysterious machine contains a secret combination of $2016$ integer numbers $x_1,x_2,\ldots,x_{2016}$ . It is known that all the numbers in the combination are equal but one. One may ask questions to the machine by giving to it a sequence of $2016$ integer numbers $y_1,\ldots,y_{2016}$ , and the machine answers by telling the value of the sum $$x_1y_1+\dots+x_{2016}y_{2016}.$$ After answering the first question, the machine accepts a second question and then a third one, and so on.

Determine how many questions are necessary to determine the combination: (a) knowing that the number which is different from the others is equal to zero; (b) not knowing what the number different from the others is.