Australian Mathematics Competition 2017 Intermediate Q29
Timmy has arranged his toy soldiers in a rectangle. He notices that if he removes 100 of the soldiers, he can rearrange the rest of the soldiers into a rectangle with 5 fewer rows and 5 more columns.
How many soldiers must he remove from the original pile so he can rearrange his soldiers into a rectangle with 11 fewer rows and 11 more columns (as compared to the original rectangle)?
The answer is 286.
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1 solution
Let the original rectangle have x rows, y columns and x y soldiers. We have
( x − 5 ) ( y + 5 ) x y + 5 ( x − y ) − 2 5 5 ( x − y ) y − x = x y − 1 0 0 = x y − 1 0 0 = − 7 5 = 1 5
Let the answer to this problem be t . Now, in a rectangle with x − 1 1 rows and y + 1 1 columns, we have
( x − 1 1 ) ( y + 1 1 ) x y − 1 1 ( y − x ) − 1 2 1 1 6 5 + 1 2 1 t = x y − t = x y − t = t = 2 8 6