AMC 10B/12B 2024 (Problem 5/5)
In the following expression, Melanie changed some of the plus signs to minus signs:
When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?
When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?
To minimize the number of sign changes, change the plus signs before the largest numbers to minus signs.
Since
we have
Changing the plus sign before to a minus sign gives
Similarly, changing the signs before and gives
In general, if we change the signs before all the terms from through , the new expression becomes
we have
Changing the plus sign before to a minus sign gives
Similarly, changing the signs before and gives
In general, if we change the signs before all the terms from through , the new expression becomes
Using the arithmetic series formula,
Since is an integer and , the greatest possible value of is
For , the terms whose signs are changed are
or
The number of these terms is
Therefore, the least number of sign changes is .
or
The number of these terms is
Therefore, the least number of sign changes is .
(B)