A most interesting problem

by djmathman, Jun 20, 2013, 11:35 PM

Problem wrote:
Let $a_1,a_2,\ldots,a_{100}$ and $b_1,b_2,\ldots,b_{100}$ be $200$ distinct real numbers. Consider an $100\times 100$ table and put the number $a_i+b_j$ in the $(i,j)$ position. Suppose that the product of the entries in each column is $1$. Prove that the product of the entries in each row is $-1$.

Solution

Not difficult, but still very clever.
This post has been edited 1 time. Last edited by djmathman, Sep 5, 2013, 1:30 AM

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A blog documenting a (no longer) high school youth and his struggles with advancing his mathematical skill.

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djmathman
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  • dj so orz :omighty:

    by Yiyj1, Mar 29, 2025, 1:42 AM

  • legendary problem writer

    by Clew28, Jul 29, 2024, 7:20 PM

  • orz $$\,$$

    by balllightning37, Jul 26, 2024, 1:05 AM

  • hi dj $ $ $ $

    by OronSH, Jul 23, 2024, 2:14 AM

  • i wanna submit my own problems lol

    by ethanzhang1001, Jul 20, 2024, 9:54 PM

  • hi dj, may i have the role of contributer? :D

    by lpieleanu, Feb 23, 2024, 1:31 AM

  • This was helpful!

    by YIYI-JP, Nov 23, 2023, 12:42 PM

  • waiting for a recap of your amc proposals for this year :D

    by ihatemath123, Feb 17, 2023, 3:18 PM

  • also happy late bday man! i missed it by 2 days but hope you are enjoyed it

    by ab456, Dec 30, 2022, 10:58 AM

  • Contrib? :D

    by MC413551, Nov 20, 2022, 10:48 PM

  • :love: tfw kakuro appears on amc :love:

    by bissue, Aug 18, 2022, 4:32 PM

  • Hi dj :)

    by 799786, Aug 10, 2022, 1:44 AM

  • Roses are red,
    Wolfram is banned,
    The best problem writer is
    Djmathman

    by ihatemath123, Aug 6, 2022, 12:19 AM

  • hello :)

    by aidan0626, Jul 26, 2022, 5:49 PM

  • Do you have a link to your main blog that you started after graduating from high school, I couldn't find it. @dj I met you IRL at Awesome Math summer Program several years ago.

    by First, Mar 1, 2022, 5:18 PM

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