<s>[INST] 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. 
 [/INST] There are many ways to obtain many solutions. Here is one systematic way to obtain solutions.
First look at a class of solutions such that $x \leq y \leq z \leq a \leq b \leq 10$ (other solution can be obtained as permutation of these).
This implies that $2 \leq x \leq 5$. So now start with $x=2$. This now means that $3 \leq y \leq 8$. Choose $y=3$. This now means that $7 \leq z \leq 10$. You will quickly find that no solution exists such that $x \leq y \leq z \leq a \leq b \leq 10$. Then choose $y=4$. Again you will find that no solution exists such that $x \leq y \leq z \leq a \leq b \leq 10$.
Going through this you will find that if we want $x=2$, and $x \leq y \leq z \leq a \leq b \leq 10$, then $$[2, 5, 10, 10, 10]; 
[2, 6, 9, 9, 9]; 
[2, 8, 8, 8, 8]$$ are the only solutions with $x=2$ and $x \leq y \leq z \leq a \leq b \leq 10$.
Now set $x=3$ and get bounds for the remaining variables to see that $$ 
[3, 3, 9, 9, 9]; 
[3, 4, 6, 8, 8]; 
[3, 5, 5, 6, 10]; 
[3, 6, 6, 6, 6]; $$ are the only solutions with $x=3$ and $x \leq y \leq z \leq a \leq b \leq 10$.
Here are the solutions such that $x \leq y \leq z \leq a \leq b \leq 10$
$$[x,y,z,a,b] \in \{[2, 5, 10, 10, 10]; 
[2, 6, 9, 9, 9]; 
[2, 8, 8, 8, 8]; 
[3, 3, 9, 9, 9]; 
[3, 4, 6, 8, 8]; 
[3, 5, 5, 6, 10];\\ 
[3, 6, 6, 6, 6]; 
[4, 4, 4, 8, 8]; 
[4, 4, 5, 5, 10]; 
[4, 4, 6, 6, 6]; 
[5, 5, 5, 5, 5]; \}
$$
There are $114$ distinct solutions i.e. without permutations such that
$$x \leq y \leq z \leq a \leq b \leq 100$$ and can be found here.
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