A list of positive integers has the following properties: $\bullet$ The sum of the items in the list is $30$. $\bullet$ The unique mode of the list is $9$. $\bullet$ The median of the list is a positive integer that does not appear in the list itself. Find the sum of the squares of all the items in the list.

Starting Point: We know that the sum of the items in the list is 30. This is the starting point for our calculations.Unique Mode: The unique mode of the list is 9, which means that 9 appears more frequently than any other number in the list. To ensure that 9 is the mode, it must appear at least twice.Median Calculation: The median of the list is a positive integer that does not appear in the list itself. Since the list is made of positive integers, the median must be at least 1. Given that the median is not in the list, the list must have an even number of elements, because if it had an odd number, the median would be one of the elements of the list.Increasing Elements: Let's assume the list has the smallest even number of elements, which is 2. However, with only two elements, we cannot have a median that is not in the list. Therefore, we need to increase the number of elements. The next even number is 4.Distribution of Numbers: If we have four elements and 9 is the mode, we must have at least two 9s. Let's assume we have exactly two 9s. The sum of these two 9s is 18, leaving us with 12 to distribute among the remaining two numbers.Adjusting List for Median: If we distribute the remaining 12 equally among the two numbers, we get two 6s. However, this would make 6 the median, which contradicts the condition that the median is not in the list. Therefore, we need to distribute the 12 in a way that the two numbers are not equal and do not include 9.Final List Formation: One way to distribute the 12 is to have one element be 5 and the other be 7. This gives us a list of [5, 9, 9, 7]. The median of this list would be the average of 9 and 9, which is 9. However, this violates the condition that the median is not in the list. Therefore, we need to add more elements to the list.Sum of Squares Calculation: Let's add two more elements to the list, making it six elements long. We still have two 9s, and we need to distribute the remaining 12 among four numbers. If we add two 1s, we have a list of [1, 1, 9, 9, x, y] with x + y = 10. The median would be the average of the third and fourth elements, which are both 9, so this still violates the condition.We need to adjust the list so that the median is not 9. If we place the two 1s at the ends and have two 4s in the middle, we get a list of [1, 1, 4, 4, 9, 9]. The median of this list is the average of 4 and 4, which is 4. This satisfies the condition that the median is not in the list.Now we have a list that satisfies all conditions: [1, 1, 4, 4, 9, 9]. The sum of the squares of all the items in the list is 1^2 + 1^2 + 4^2 + 4^2 + 9^2 + 9^2.Calculating the sum of the squares: 1 + 1 + 16 + 16 + 81 + 81 = 196.

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A list of positive integers has the following properties: $\newline$ - The sum of the items in the list is $30$ . $\newline$ - The unique mode of the list is $9$ . $\newline$ - The median of the list is a positive integer that does not appear in the list itself. $\newline$ Find the sum of the squares of all the items in the list.

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Q. A list of positive integers has the following properties:

\newline

- The sum of the items in the list is

30

\newline

- The unique mode of the list is

9

\newline

- The median of the list is a positive integer that does not appear in the list itself.

\newline

Find the sum of the squares of all the items in the list.

Starting Point: We know that the sum of the items in the list is $30$ . This is the starting point for our calculations.
Unique Mode: The unique mode of the list is $9$ , which means that $9$ appears more frequently than any other number in the list. To ensure that $9$ is the mode, it must appear at least twice.
Median Calculation: The median of the list is a positive integer that does not appear in the list itself. Since the list is made of positive integers, the median must be at least $1$ . Given that the median is not in the list, the list must have an even number of elements, because if it had an odd number, the median would be one of the elements of the list.
Increasing Elements: Let's assume the list has the smallest even number of elements, which is $2$ . However, with only two elements, we cannot have a median that is not in the list. Therefore, we need to increase the number of elements. The next even number is $4$ .
Distribution of Numbers: If we have four elements and $9$ is the mode, we must have at least two $9$ s. Let's assume we have exactly two $9$ s. The sum of these two $9$ s is $18$ , leaving us with $12$ to distribute among the remaining two numbers.
Adjusting List for Median: If we distribute the remaining $12$ equally among the two numbers, we get two $6$ s. However, this would make $6$ the median, which contradicts the condition that the median is not in the list. Therefore, we need to distribute the $12$ in a way that the two numbers are not equal and do not include $9$ .
Final List Formation: One way to distribute the $12$ is to have one element be $5$ and the other be $7$ . This gives us a list of $[5, 9, 9, 7]$ . The median of this list would be the average of $9$ and $9$ , which is $9$ . However, this violates the condition that the median is not in the list. Therefore, we need to add more elements to the list.
Sum of Squares Calculation: Let's add two more elements to the list, making it six elements long. We still have two $9$ s, and we need to distribute the remaining $12$ among four numbers. If we add two $1$ s, we have a list of $[1, 1, 9, 9, x, y]$ with $x + y = 10$ . The median would be the average of the third and fourth elements, which are both $9$ , so this still violates the condition.
Sum of Squares Calculation: Let's add two more elements to the list, making it six elements long. We still have two $9$ s, and we need to distribute the remaining $12$ among four numbers. If we add two $1$ s, we have a list of $[1, 1, 9, 9, x, y]$ with $x + y = 10$ . The median would be the average of the third and fourth elements, which are both $9$ , so this still violates the condition.We need to adjust the list so that the median is not $9$ . If we place the two $1$ s at the ends and have two $4$ s in the middle, we get a list of $[1, 1, 4, 4, 9, 9]$ . The median of this list is the average of $4$ and $4$ , which is $4$ . This satisfies the condition that the median is not in the list.
Sum of Squares Calculation: Let's add two more elements to the list, making it six elements long. We still have two $9$ s, and we need to distribute the remaining $12$ among four numbers. If we add two $1$ s, we have a list of $[1, 1, 9, 9, x, y]$ with $x + y = 10$ . The median would be the average of the third and fourth elements, which are both $9$ , so this still violates the condition.We need to adjust the list so that the median is not $9$ . If we place the two $1$ s at the ends and have two $4$ s in the middle, we get a list of $[1, 1, 4, 4, 9, 9]$ . The median of this list is the average of $4$ and $4$ , which is $4$ . This satisfies the condition that the median is not in the list.Now we have a list that satisfies all conditions: $[1, 1, 4, 4, 9, 9]$ . The sum of the squares of all the items in the list is $12$ $4$ .
Sum of Squares Calculation: Let's add two more elements to the list, making it six elements long. We still have two $9$ s, and we need to distribute the remaining $12$ among four numbers. If we add two $1$ s, we have a list of $[1, 1, 9, 9, x, y]$ with $x + y = 10$ . The median would be the average of the third and fourth elements, which are both $9$ , so this still violates the condition.We need to adjust the list so that the median is not $9$ . If we place the two $1$ s at the ends and have two $4$ s in the middle, we get a list of $[1, 1, 4, 4, 9, 9]$ . The median of this list is the average of $4$ and $4$ , which is $4$ . This satisfies the condition that the median is not in the list.Now we have a list that satisfies all conditions: $[1, 1, 4, 4, 9, 9]$ . The sum of the squares of all the items in the list is $12$ $4$ .Calculating the sum of the squares: $12$ $5$ .