1985 IMO SL #11

Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+  a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.

Let $-x_1, \cdots, -x_6$ be the roots of $P(x)$. Let $s_{k,i}$, where $k \le i \le 6$, denote the sum of all products of $k$ of the numbers $x_1, \cdots, x_i$. Using Vieta's Formulas, note that $a_k=s_{k,6}$. Also, notice that $$s_{k,i}=s_{k-1,i-1} x_i+s_{k,i-1}.$$ Using this, we can compute all $a_k$ as shown in the below diagram, where horizontal arrow represents multiplication and vertical arrow represents addition. $$\begin{array}{ccccccccccc}{x}_1& \to <!--\; \to \; -->& s_{2,2}& \to <!--\; \to \; -->& s_{3,3}& \to <!--\; \to \; -->& s_{4,4}& \to <!--\; \to \; -->& s_{5,5}& \to <!--\; \to \; -->& a_6\\ \downarrow <!--\; \downarrow \; -->& & \downarrow <!--\; \downarrow \; -->& & \downarrow <!--\; \downarrow \; -->& & \downarrow <!--\; \downarrow \; -->& & \downarrow <!--\; \downarrow \; -->& \\ s_{1,2}& \to <!--\; \to \; -->& s_{2,3}& \to <!--\; \to \; -->& s_{3,4}& \to <!--\; \to \; -->& s_{4,5}& \to <!--\; \to \; -->& a_5\\ \downarrow <!--\; \downarrow \; -->& & \downarrow <!--\; \downarrow \; -->& & \downarrow <!--\; \downarrow \; -->& & \downarrow <!--\; \downarrow \; -->& \\ s_{1,3}& \to <!--\; \to \; -->& s_{2,4}& \to <!--\; \to \; -->& s_{3,5}& \to <!--\; \to \; -->& a_4& & & \\ \downarrow <!--\; \downarrow \; -->& & \downarrow <!--\; \downarrow \; -->& & \downarrow <!--\; \downarrow \; -->& & & & \\ s_{1,4}& \to <!--\; \to \; -->& s_{2,5}& \to <!--\; \to \; -->& a_3& & & & \\ \downarrow <!--\; \downarrow \; -->& & \downarrow <!--\; \downarrow \; -->& & & & & \\ s_{1,5}& \to <!--\; \to \; -->& a_2& & & & & \\ \downarrow <!--\; \downarrow \; -->& & & & & & & \\ a_1& & & & & \end{array}$$ We are done. $\square$