, 2012

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.)