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. x 1 s 2 , 2 s 3 , 3 s 4 , 4 s 5 , 5 a 6 s 1 , 2 s 2 , 3 s 3 , 4 s 4 , 5 a 5 s 1 , 3 s 2 , 4 s 3 , 5 a 4 s 1 , 4 s 2 , 5 a 3 s 1 , 5 a 2 a 1 We are done. $\square$  


  1. Anonymous1/10/2022

    beautifully explained

  2. Anonymous1/10/2022

    bro u should get a custom domain
    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


    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।


Post a Comment

Popular posts from this blog

1995 IMO #2

Inequality problem I made