Difference between revisions of "2021 WSMO Speed Round Problems/Problem 3"
(Created page with "==Problem== Let <math>a@b=\frac{a^2-b^2}{a+b}</math>. Find the value of <math>1@(2@(\dots(2020@2021)\dots)</math>. ==Solution== Note that <math>a^2-b^2=(a-b)(a+b).</math> Thi...") |
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==Solution== | ==Solution== | ||
Note that <math>a^2-b^2=(a-b)(a+b).</math> This means that <math>a@b=a-b.</math> Now, <math>a@(b@c)=a-(b-c)=a-b+c</math> and <math>a@(b@(c@d))=a-(b-(c-d))=a-b+c-d.</math> Using this pattern, we find that <cmath>1@(2@(\dots(2020@2021)\dots)=1-2+3-4+\ldots-2020+2021=</cmath><cmath>(1-2)+(3-4)+\ldots+(2019-2020)+2021=-1010+2021=\boxed{1011}.</cmath> | Note that <math>a^2-b^2=(a-b)(a+b).</math> This means that <math>a@b=a-b.</math> Now, <math>a@(b@c)=a-(b-c)=a-b+c</math> and <math>a@(b@(c@d))=a-(b-(c-d))=a-b+c-d.</math> Using this pattern, we find that <cmath>1@(2@(\dots(2020@2021)\dots)=1-2+3-4+\ldots-2020+2021=</cmath><cmath>(1-2)+(3-4)+\ldots+(2019-2020)+2021=-1010+2021=\boxed{1011}.</cmath> | ||
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Latest revision as of 10:20, 23 December 2021
Problem
Let 
. Find the value of 
.
Solution
Note that 
This means that 
Now, 
and 
Using this pattern, we find that ![\[1@(2@(\dots(2020@2021)\dots)=1-2+3-4+\ldots-2020+2021=\]](http://latex.artofproblemsolving.com/f/2/7/f2790eeaf2d5b8bf2e6023dd723715710b519022.png)
![\[(1-2)+(3-4)+\ldots+(2019-2020)+2021=-1010+2021=\boxed{1011}.\]](http://latex.artofproblemsolving.com/1/7/5/175fbded15deb838ca12f0de85dd811156698d76.png)
~pinkpig
