Question 15
A student wrote the following 99 numbers on the board:
$$1,-2,3,-4,5,-6,\ldots,-98,99.$$
She arbitrarily picked two of these numbers, say $a$ and $b$, and replaced them by the sum $a+b$. For example, if she had picked 37 and $-24$, she would erase those two numbers and add the number 13 to the board. Suppose that she repeated the process, and after 98 replacements, there was one number left on the board. What is the largest possible value of this number?
$$1,-2,3,-4,5,-6,\ldots,-98,99.$$
She arbitrarily picked two of these numbers, say $a$ and $b$, and replaced them by the sum $a+b$. For example, if she had picked 37 and $-24$, she would erase those two numbers and add the number 13 to the board. Suppose that she repeated the process, and after 98 replacements, there was one number left on the board. What is the largest possible value of this number?