Question

даны 3 числа 100. 2017. 2019. любое из чисел можно заменить разностью между его квадратом и утроенным произведением 2 других чисел. с новой тройкой можно проделать то же самое и т.д. можно ли в результате получить тройку чисел сумма которых равна нулю?

Answer 1

Problem Analysis

We are given three numbers: 100, 2017, and 2019. We can replace any of these numbers with the difference between its square and the triple product of the other two numbers. We need to determine if it is possible to obtain a triple of numbers whose sum is zero.

Solution

To solve this problem, we can iterate through all possible combinations of the three given numbers and apply the replacement operation. We will check if any of the resulting triples have a sum of zero.

Let's go through the steps to find out if it is possible to obtain a triple of numbers whose sum is zero:

1. Start with the given numbers: 100, 2017, and 2019. 2. Replace the first number (100) with the difference between its square and the triple product of the other two numbers (2017 and 2019). 3. Replace the second number (2017) with the difference between its square and the triple product of the other two numbers (100 and 2019). 4. Replace the third number (2019) with the difference between its square and the triple product of the other two numbers (100 and 2017). 5. Check if the sum of the resulting triple is zero. If it is, we have found a solution. If not, continue to the next iteration.

Let's perform the calculations step by step:

1. Replace the first number (100) with the difference between its square and the triple product of the other two numbers (2017 and 2019): - New first number = 100^2 - 3 * 2017 * 2019 = -244,160,683. - New triple: (-244,160,683, 2017, 2019). - The sum of the new triple is not zero.

2. Replace the second number (2017) with the difference between its square and the triple product of the other two numbers (100 and 2019): - New second number = 2017^2 - 3 * 100 * 2019 = -12,186,077. - New triple: (-244,160,683, -12,186,077, 2019). - The sum of the new triple is not zero.

3. Replace the third number (2019) with the difference between its square and the triple product of the other two numbers (100 and 2017): - New third number = 2019^2 - 3 * 100 * 2017 = -12,186,077. - New triple: (-244,160,683, -12,186,077, -12,186,077). - The sum of the new triple is not zero.

Since none of the resulting triples have a sum of zero, it is not possible to obtain a triple of numbers whose sum is zero using the given replacement operation.

Conclusion

In conclusion, it is not possible to obtain a triple of numbers whose sum is zero by repeatedly applying the given replacement operation to the numbers 100, 2017, and 2019.