2003 USAMO Problems/Problem 6
At the vertices of a regular hexagon are written six nonnegative integers whose sum is 2003. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.
Assume the original numbers are . Since is odd, either or must be odd. WLOG let be odd and .
. Define Operation A as the sequence of moves from Step 1 to Step 3, shown below:
Notice that Operation A changes the numbers to and they are all nonnegative, since . Their sum changes from to ; it decreases as long as . If we repeat Operation A enough times, its sum will decrease and eventually we will arrive at a point where at least one of the numbers in the positions originally occupied by has become a 0.
and . Define Operation B as the sequence of moves from Step 1 to Step 3, shown below:
where in Step 3, we take the nonnegative choice of or . is changed to either or . If we have , their sum is and this is less than (the original sum) unless , but since the original sum is odd by assumption. If we have , their sum is , which is less than . Operation B applied repeatedly will cause either or to become .
and . Define Operation C as the sequence of moves from Step 1 to Step 4, shown below:
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