<s>[INST] Olympiad Math Problem About Simultaneous Equations Given the question:

*

*$x + y = 1$

*$x^2 + y^2 = 2$

*$x^5 + y^5 = ?$
After a bunch of manipulation of the above equations we find the solution to be 19/4. Yet could the above be solved by simultaneous equations?
From $1)$ we can conclude that $x = 1 - y$.
Substituting this into $2)$ we get a quadratic $2y^2 - 2y - 1 = 0$.
However, when we solve for $y$ we get two possible solutions. Hence how do we proceed from here? Why doesn't simultaneous equations work to solve the above problem?
 [/INST] First note that we can get
$$2xy = (x+y)^2 - x^2-y^2 = 1 - 2 = -1$$
We can then apply binomial theorem again to obtain $x^3+y^3$ and $x^5+y^5$
$$
\begin{align*}
x^3 + y^3 &= (x+y)^3 - 3x^2y - 3xy^2 \\
&= 1 + \frac{3}{2}(x+y) = \frac{5}{2} \\
x^5 + y^5 &= (x+y)^5 - 5x^4y - 10x^3y^2 - 10x^2y^3 - 5xy^4 \\
&= 1 + \frac{5}{2}(x^3+y^3) - \frac{5}{2}(x+y) \\
&= 1 + \frac{25}{4} - \frac{5}{2} = \frac{19}{4}
\end{align*}
$$
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