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Lists with given mean and median

Solve the first problem 18 problems

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1. Smallest of five consecutive integers ●○○○○ 10 ptsThe mean (average) of 5 consecutive integers is 9. What is the smallest of these 5 integers?
2. Greatest integer in a set of five ●○○○○ 10 ptsFive positive integers have an average of 15. What is the greatest possible value of one of these integers?
3. Smallest average of distinct even integers ●○○○○ 10 ptsWhat is the smallest possible average of four distinct positive even integers?
4. Largest of four integers from triple averages ●●○○○ 20 ptsDewa writes down a list of four integers. He calculates the average of each group of three of the four integers. These averages are $32, 39, 40, 44$. What is the largest of the four integers?
5. Range of a five-integer set ●●○○○ 20 ptsA set of five positive integers has mean 5, median 5, and 8 as its only mode. What is the difference between the largest and smallest integers in the set?
6. Largest integer in a five-number list ●●○○○ 20 ptsA list of five positive integers has all of the following properties: - The only integer in the list that occurs more than once is $8$. - Its median is $9$. - Its average (mean) is $10$. What is the largest possible integer that could appear in the list?
7. Maximum size of a special set ●●●○○ 50 ptsA set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S}$, the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that $1$ belongs to $\mathcal{S}$ and that $2002$ is the largest element of $\mathcal{S}$, what is the greatest number of elements that $\mathcal{S}$ can have?
8. Maximum size of a set with integer means ●●●○○ 50 ptsLet $S$ be a set of positive integers with the property that for any $x \in S$, the arithmetic mean of the remaining numbers in $S$ after removing $x$ is a positive integer. Suppose $1 \in S$ and $2016$ is the largest element of $S$. Find the maximum value of $|S|$.
9. Greatest integer after adding four ●●●○○ 50 ptsFour integers are added to the group of integers 3, 4, 5, 5 and 8. As a result, the mean, the median and the mode of the data each increase by 1. What is the greatest integer in the new group of integers?
10. Possible values of the largest removed integer ●●●●○ 100 ptsYann writes down the first $n$ consecutive positive integers, $1,2,3,4,\ldots,n-1,n$. He removes four different integers $p,q,r,s$ from the list. At least three of $p,q,r,s$ are consecutive and $100<p<q<r<s$. The average of the integers remaining in the list is $89.5625$. What is the number of possible values of $s$?
11. Maximum smallest ball in bowl Jia ●●●●○ 100 ptsPlace the 27 balls numbered $1$ to $27$ into three bowls, Jia, Yi and Bing, so that the average ball number in bowl Jia is $15$, in bowl Yi is $3$, and in bowl Bing is $18$, and each bowl contains at least $4$ balls. Find the maximum possible value of the smallest ball number in bowl Jia.

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