### 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. $\begin{array}{ccccccccccc}{x}_{1}& \to & {s}_{2,2}& \to & {s}_{3,3}& \to & {s}_{4,4}& \to & {s}_{5,5}& \to & {a}_{6}\\ ↓& & ↓& & ↓& & ↓& & ↓& \\ {s}_{1,2}& \to & {s}_{2,3}& \to & {s}_{3,4}& \to & {s}_{4,5}& \to & {a}_{5}\\ ↓& & ↓& & ↓& & ↓& \\ {s}_{1,3}& \to & {s}_{2,4}& \to & {s}_{3,5}& \to & {a}_{4}& & & \\ ↓& & ↓& & ↓& & & & \\ {s}_{1,4}& \to & {s}_{2,5}& \to & {a}_{3}& & & & \\ ↓& & ↓& & & & & \\ {s}_{1,5}& \to & {a}_{2}& & & & & \\ ↓& & & & & & & \\ {a}_{1}& & & & & \end{array}$ We are done. $\square$

1. Anonymous1/10/2022

beautifully explained

2. Anonymous1/10/2022

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ur blog has blown up so much
it deserves a custom domain

3. schukkayapally1/10/2022

i agree man
get a custom domain
make it math4l.net

your blog is literally famous in the math olympiad community at this point

if u only had like 500 views i wouldnt tell u to get custom domain
but now that ur blog is literally so famous, you need it to look more professional

4. schukkayapally1/10/2022

tbh i wouldnt be surprised if u surpass all math olympiad organizations including aops in like 2 months

i highly highly recommend u to get custom domain

bcuz if u think about it
15,000 is a HUGE HUGE number of views
and so many people love ur blog, so they keep revisiting

get a custom domain bro

5. Anonymous1/11/2022

Orzzzzz

I agree with you all

If this helpful and amazing guy doesn't get a custom domain, idek why it exists

Get custom domain
If I were u
I would've gotten it
At only like 5k view
U have 16k view now

आप बहुत प्रो हैं।