Friday, September 9, 2022

Vietnamese Olympiad 2022: Find the value of 1a2023+1b2023+1c2023

     If a+b+c=2022 and 1a+1b+1c=12022 Find the value of 1a2023+1b2023+1c2023

Solution

We had: a+b+c=2022 and 1a+1b+1c=12022

Then, 1a+b+c=12022 So that, 1a+b+c=1a+1b+1c

           1a+b+c-1a=1b+1c

            a-a-b-ca(a+b+c)=b+cbc

            -b+ca(a+b+c)=b+cbc

            b+ca(a+b+c)+b+cbc

            (b+c)(1a(a+b+c)+1bc)=0

            (b+c)(a(a+b+c)+bcabc(a+b+c))=0

Therefore, (b+c)(bc+a2+ab+ac)=0

            (b+c)[c(b+a)+a(a+b)]=0

            (a+b)(b+c)(a+c)=0

There are 3 cases of above equatoin:

+ Case 01: a+b=0 Then a=-b It is easy to see that 1a2023=-1b2023

Also from this a+b+c=2022 We can see that c=2022

Then,    1c2023=120222023

Hence, 1a2023+1b2023+1c2023=120222023

+ For other 2 cases b+c=0 and a+c=0 We do the same and we will get the answer is 1a2023+1b2023+1c2023=120222023

Solution by: Thin Sokkean

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