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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