Hamilton 2015 Q5
5. Some boys and girls are standing in a row, in some order, about to be photographed. All of them are facing the photographer. Each girl counts the number of boys to her left, and each boy counts the number of girls to his right.
Let the sum of the numbers counted by the girls be \(G\), and the sum of the numbers counted by the boys be \(B\).
Prove that \(G = B\).
Hamilton Olympiad 2015, Question 5. © UK Mathematics Trust. Original paper on ukmt.org.uk
Consider one of the girls. For each boy to her left, draw an arc connecting the two, as indicated in the following figure (which is shown from the photographer's point of view).
Thus one arc is drawn for each boy that she counts. We repeat this for every girl, and as a result \(G\) arcs are drawn altogether.
Now consider adopting the same procedure for the boys. The arcs that would be drawn for the boys are exactly the same as the arcs drawn for the girls. Indeed, in the above figure, the arc corresponds either to a girl with a boy to her left, or to a boy with a girl to his right.
It follows that \(G = B\) since both count the total number of arcs.
Solution © UK Mathematics Trust, from the Hamilton Olympiad 2015 solutions.