Consider ten concentric circles and ten rays as in the following figure.
At the points where the inner circle is intersected by the rays write successively, in direction clockwise, the numbers 1,2,3,4,5,6,7,8,9,10. In the next circle we write the numbers 11,12,13,14,15,16,17,18,19,20 successively, and so on successively until the last round were we write the numbers 91,92,93,94,95,96,97,98,99,100 successively. In this orde, the numbers 1,11,21,31,41,51,61,71,81,91 are in the same ray, and similarly for the other rays. In front of 50 of those 100 numbers, we use the sign ''−'' such as:
a) in each of the ten rays, exist exactly 5 signs ''−'' , and also
b) in each of the ten concentric circles, to be exactly 5 signs ''−''.
Prove that the sum of the 100 signed numbers that occur, equals zero.
https://cdn.artofproblemsolving.com/attachments/9/d/ffee6518fcd1b996c31cf06d0ce484a821b4ae.gif algebracombinatoricsSumcircle