Algorithm for finding 2 missing items in a stream of integers

Question

I saw this post and wondered why the approach described in the accepted answer works. The same problem and solution is described a bit nicer here.

So let's say we receive a stream of $n-2$ pairwise different numbers $a_1 + a_2 + \cdots + a_{n-2}$, from the set $\left\{1,\dots,n\right\}$. What I learned from the above post is that for the problem of finding 2 missing numbers $x$ and $y$, given a stream as described above, can be solved by finding the solution to the following system of equations:

\begin{alignat*}{4} x+y & {}={} & \dfrac{n(n+1)}{2} - (a_1 + a_2 + \cdots + a_{n-2}) \\ x^2 + y^2 & {}={} & \dfrac{n(n+1)(2n+1)}{6} - (a_1^2 + a_2^2 + \cdots + a_{n-2}^2) \end{alignat*}

Now it is easy to convert this into a single pass algorithm which supposedly returns the two missing numbers for a given stream.

How would one prove this algorithm's correctness? I think showing that the system must have exactly two solutions would be enough to prove correctness but I'm not sure how to approach this?

How can I convince myself (and others) that the algorithm is correct?

Answer 1

Suppose that $x + y = A$ and $x^2 + y^2 = B$. Then $$ xy = \frac{(x+y)^2 - (x^2+y^2)}{2} = \frac{A^2-B}{2}. $$ Consider now the polynomial $$ (t-x)(t-y) = t^2 - (x+y)t + xy = t^2-At + \frac{A^2-B}{2}. $$ The polynomial on the right has the two roots $x,y$.

If $x + y = z + w$ and $x^2 + y^2 = z^2 + w^2$ then both $x,y$ and $z,w$ are the two roots of the same polynomial, and so $\{x,y\} = \{z,w\}$.