Ta có:
[imath]\dfrac{ab}{a+b}=\dfrac{bc}{b+c}=\dfrac{ca}{c+a}\\\Rightarrow \dfrac{a+b}{ab}=\dfrac{b+c}{bc}=\dfrac{c+a}{ca}\\\Rightarrow \dfrac{a}{ab}+\dfrac{b}{ab}=\dfrac{b}{bc}+\dfrac{c}{bc}=\dfrac{c}{ca}+\dfrac{a}{ca}\\\Rightarrow \dfrac1b+\dfrac1a=\dfrac1c+\dfrac1b=\dfrac1a+\dfrac1c[/imath]
Từ [imath]\dfrac1b+\dfrac1a=\dfrac1c+\dfrac1b[/imath] suy ra [imath]\dfrac1a=\dfrac1c\Rightarrow a=c[/imath]
Từ [imath]\dfrac1c+\dfrac1b=\dfrac1a+\dfrac1c[/imath] suy ra [imath]\dfrac1b=\dfrac1a\Rightarrow b=a[/imath]
Suy ra [imath]a=b=c[/imath]
Do đó
[imath]C=\dfrac{(ab+bc+ca)^{1011}}{a^{2022}+b^{2022}+c^{2022}}=\dfrac{(a^2+a^2+a^2)^{1011}}{a^{2022}+a^{2022}+a^{2022}}=\dfrac{(3a^2)^{1011}}{3a^{2022}}=\dfrac{3^{1011}.a^{2022}}{3a^{2022}}=3^{1010}[/imath]
Nếu bạn có thắc mắc cứ hỏi tại đây, tụi mình sẽ hỗ trợ.
Ngoài ra, bạn tham khảo kiến thức tại topic này nha
https://diendan.hocmai.vn/threads/t...c-mon-danh-cho-ban-hoan-toan-mien-phi.827998/