Problem
ALG-B1-M02-P026 Increasing a Product After a Shift
In a product of three positive integers, each factor was decreased by \(4\). Could the new product be exactly \(2020\) greater than the original one? If yes, give an example and explain how to find it.
Try making two factors equal to \(1\), so that after the decrease they create a positive coefficient in front of the third factor.
The obvious attempt to take all factors large is not helpful: the product tends to decrease. The hidden idea is that small factors may become negative, and two negative factors give a positive product.
Let the original numbers be \(1,1,a\). The original product is \(a\), while the new product is \((-3)(-3)(a-4)=9a-36\).
We need the new product to be \(2020\) greater:
\[9a-36=a+2020.\]
Thus \(8a=2056\), so \(a=257\). Check: the original product is \(1\cdot1\cdot257=257\), the new product is \((-3)(-3)\cdot253=2277\), and \(2277-257=2020\). Therefore it is possible.
Rewritten olympiad idea: instead of searching blindly, choose small factors constructively and solve a linear equation for the last factor.