Find five positive integers whose reciprocals sum to $1$ Find a positive integer solution $(x,y,z,a,b)$ for which
$$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$
Is your answer the only solution? If so, show why. 
I was surprised that a teacher would assign this kind of problem to a 5th grade child. (I'm a college student tutor) This girl goes to a private school in a wealthy neighborhood.
Please avoid the trivial $x=y=z=a=b=5$. Try looking for a solution where $ x \neq y \neq z \neq a \neq b$ or if not, look for one where one variable equals to another, but explain your reasoning. The girl was covering "unit fractions" in her class. 

I am posting another solution since one of my students found this method.
(He is 16 years old but as you see he is gifted)
Start with $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}$. No write $\frac{1}{6}=\frac{3}{18}=\frac{1}{18}+\frac{2}{18}=\frac{1}{9}+\frac{1}{18}$ and you get $\frac{1}{2}+\frac{1}{3}+\frac{1}{9}+\frac{1}{18}=1$. Proceed like this: Write the last fraction (which will have an even denominator) of the sum as $\frac{1}{2n}=\frac{1}{6n}+\frac{2}{6n}=\frac{1}{3n}+\frac{1}{6n}$ and so on!
