Question 1

There are $2n$ positive integers written on a whiteboard. Here is a one-player game with these numbers: You start with a score of 0 . You will increase your score by performing the following move exactly $n$ times: - Choose two integers $x$ and $y$ that are written on the whiteboard. - Add $\min(x, y)$ to your score. - Erase $x$ and $y$ from the whiteboard. Note that after performing the move $n$ times, there will be no more integers written on the whiteboard. In general, our goal is to find the maximum final score you can achieve if you optimally perform the $n$ moves. We will refer to this value as the *answer*. Based on the above data answer the given subquestions. What is the answer if the input is 1 1 2 1? There are $2n$ positive integers written on a whiteboard. Here is a one-player game with these numbers: You start with a score of 0 . You will increase your score by performing the following move exactly $n$ times: - Choose two integers $x$ and $y$ that are written on the whiteboard. - Add $\min(x, y)$ to your score. - Erase $x$ and $y$ from the whiteboard. Note that after performing the move $n$ times, there will be no more integers written on the whiteboard. In general, our goal is to find the maximum final score you can achieve if you optimally perform the $n$ moves. We will refer to this value as the *answer*. Based on the above data answer the given subquestions. What is the answer if the input is 1 2 3 4 5 6 7 8 9 10? There are $2n$ positive integers written on a whiteboard. Here is a one-player game with these numbers: You start with a score of 0 . You will increase your score by performing the following move exactly $n$ times: - Choose two integers $x$ and $y$ that are written on the whiteboard. - Add $\min(x, y)$ to your score. - Erase $x$ and $y$ from the whiteboard. Note that after performing the move $n$ times, there will be no more integers written on the whiteboard. In general, our goal is to find the maximum final score you can achieve if you optimally perform the $n$ moves. We will refer to this value as the *answer*. Based on the above data answer the given subquestions. What is the answer if 2*n* = 50 and the input is the set of all even numbers between 1 and 100? (Hint: you might want to use the fact that the sum of the first *n* odd numbers is *n*².)