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21 problems match “arithmetic mean”. Clear

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?
Arithmetic mean from geometric and harmonic means ●●○○○ 20 ptsFind the arithmetic mean of two positive numbers whose geometric mean is 4 and whose harmonic mean is $\frac{16}{5}$ .
Harmonic mean from the other means ●●○○○ 20 ptsFind the harmonic mean of two positive numbers whose arithmetic mean is $\frac{15}{2}$ and whose geometric mean is $6$ .
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?
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 $|S|$ .
Two numbers from mean conditions ●●●○○ 50 ptsFind two numbers whose arithmetic mean exceeds their geometric mean by $\frac{1}{2}$ and whose harmonic mean is $\frac{25}{26}$ less than their arithmetic mean.
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?
Two numbers from their means ●●●○○ 50 ptsTwo positive numbers have an arithmetic mean that exceeds their geometric mean by $2$ , and their geometric mean exceeds their harmonic mean by $1.6$ . Find the two numbers.
Two numbers from three means ●●●○○ 50 ptsFind two positive numbers whose arithmetic mean exceeds their geometric mean by $2$ and whose harmonic mean is $3.6$ less than their arithmetic mean. Give them in increasing order.
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 $s$ ?