Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.4 AIME, harder Prove it Russia

In a product of 3 positive integers each multiple was decreased by 3. Could it happen that the product increases by 2016 after this change?

В произведении трёх положительных целых чисел каждый сомножитель уменьшили на 3. Могло ли случиться, что произведение увеличилось на 2016 после этого изменения?

Solution

Yes. An example is $1 \cdot 1 \cdot 676$ .

After the operation, we have $(-2) \cdot (-2) \cdot 673 = 4 \cdot 673 = 2692 = 676 + 2016$ .

Note. The given example is unique. Here is how it can be found. Suppose two of the factors are $1$ , and the third is $a$ . Their product is $a$ , and after decreasing each by $3$ , the product becomes $(-2)^2(a-3) = 4a - 12$ . Thus, for $4a - 12 = a + 2016$ , the condition is satisfied. Solving this equation, we get $a = 676$ .

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