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sum of 100 signed numbers in 10 concentric circles equals zero

Source: 1997 Greece Junior p4

March 17, 2020
algebracombinatoricsSumcircle

Problem Statement

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 1,2,3,4,5,6,7,8,9,101, 2, 3, 4, 5, 6, 7, 8, 9, 10. In the next circle we write the numbers 11,12,13,14,15,16,17,18,19,2011, 12, 13, 14, 15, 16, 17, 18, 19,20 successively, and so on successively until the last round were we write the numbers 91,92,93,94,95,96,97,98,99,10091, 92, 93, 94, 95, 96, 97, 98, 99, 100 successively. In this orde, the numbers 1,11,21,31,41,51,61,71,81,911, 11, 21, 31, 41, 51, 61, 71, 81, 91 are in the same ray, and similarly for the other rays. In front of 5050 of those 100100 numbers, we use the sign ''−-'' such as: a) in each of the ten rays, exist exactly 55 signs ''−-'' , and also b) in each of the ten concentric circles, to be exactly 55 signs ''−-''. Prove that the sum of the 100100 signed numbers that occur, equals zero. https://cdn.artofproblemsolving.com/attachments/9/d/ffee6518fcd1b996c31cf06d0ce484a821b4ae.gif
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