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Arithmetic, geometric and harmonic means

Solve the first problem 17 problems

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1. Sum of squares from means ●○○○○ 10 ptsReal numbers $x$ and $y$ have an arithmetic mean of 18 and a geometric mean of $\sqrt{92}$. Find $x^2+y^2$.
2. Average of integers between two fractions ●○○○○ 10 ptsWhat is the average of all the integer values of $M$ such that $\frac{M}{56}$ is strictly between $\frac{3}{7}$ and $\frac{1}{4}$?
3. Geometric mean of three numbers ●○○○○ 10 ptsFind the geometric mean of $3$, $6$ and $12$.
4. Median of consecutive integer sets ●●○○○ 20 ptsLet $a$, $b$, and $c$ be integers with $a < b < c$. Let $S$ be the set of all integers from $a$ to $b$, inclusive, and $Q$ the set of all integers from $b$ to $c$, inclusive. The median of $S$ is $\frac{3}{4}b$ and the median of $Q$ is $\frac{7}{8}c$. Let $R$ be the set of all integers from $a$ to $c$, inclusive. What fraction of $c$ is the median of $R$? Give your answer as a decimal.
5. 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 $\frac{16}{5}$.
6. 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 $6$.
7. 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.
8. 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.
9. 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.

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