Problem 265

The average of any two numbers is found by adding the two numbers and dividing by two.

Thus, the three averages when the numbers are taken in pairs are $$\text{25+5 2 = 30 2 =15, , , 5+29 2 = 34 2 =17, and 25+29 2 = 54 2 =27.}$$
The average of 2 and 6 is $\dfrac{2+6}{2}=4$.

Since 6 is greater than 2, then the average of 6 and $n$ is greater than the average of 2 and $n$.

Therefore, the average of 6 and $n$ is 13 and the average of 2 and $n$ is 11.

Since the average of 6 and $n$ is 13, then $\dfrac{6+n}{2}=13$ or $6+n=26$ and so $n=20$.

We can check that $n=20$ is correct by recognizing that the average of 2 and 20 is indeed 11.
When each of the three numbers is added to the average of the other two, the resulting three expressions are $$2+\frac{a+b}{2},\,\,a+\frac{2+b}{2},\,\,b+\frac{2+a}{2}.$$ To determine which of these expressions is equal to which of the results, 14, 17, 21, we must order the three expressions from smallest to largest.

Since $2<a<b$, then $2+(2+a+b)<a+(2+a+b)<b+(2+a+b)$

or $4+a+b<2a+2+b<2b+2+a$.

Dividing by 2, $\dfrac{4+a+b}{2}<\dfrac{2a+2+b}{2}<\dfrac{2b+2+a}{2}$ or $\dfrac{4}{2}+\dfrac{a+b}{2}<\dfrac{2a}{2}+\dfrac{2+b}{2}<\dfrac{2b}{2}+\dfrac{2+a}{2}$

and so $2+\dfrac{a+b}{2}<a+\dfrac{2+b}{2}<b+\dfrac{2+a}{2}$.

Since $2+\dfrac{a+b}{2}$ is the smallest of the three expressions, then it must equal the smallest of the three results, 14.

Since $b+\dfrac{2+a}{2}$ is the largest of the three expressions, then it must equal the largest of the three results, 21.

We now solve the following system of two equations and two unknowns. $$\begin{align*} 2+\dfrac{a+b}{2} & = 14 \tag{1} \\ b+\dfrac{2+a}{2} & = 21 \tag{2} \end{align*}$$ Multiplying each equation by 2, $$\begin{align*} 4+a+b & = 28 \tag{3} \\ 2b+2+a & = 42 \tag{4} \end{align*}$$ Thus, $$\begin{align*} a+b & = 24 \tag{5}\\ a+2b & = 40 \tag{6} \end{align*}$$ Subtracting equation $(5)$ from equation $(6)$, we get $b=16$.

Substituting $b=16$ into equation $(5)$, $a+16=24$, and so $a=8$.

(We may check that our solution is correct by substituting $a=8$ and $b=16$ into the third expression $a+\dfrac{2+b}{2}$ to get the third result, 17.)