parmenides51 wrote: Given $a,b,c$ ,$(a, b,c \in (0,2)$), with $a + b + c = ab+bc+ca$, prove that $$\frac{a^2}{a^2-a+1}+\frac{b^2}{b^2-b+1}+\frac{c^2}{c^2-c+1} \le 3$$(D. Pirshtuk) It's true for any reals $a$, $b$ and $c$ such that $a+b+c=ab+ac+bc.$ There is very nice solution without $uvw$.

Any solution please ??

Tricky as hell: substitute $a’=\frac{1}{a}-\frac12>0$ etc. Then homogenization and AM-GM.

parmenides51 wrote: Given $a,b,c$ ,$(a, b,c \in (0,2)$), with $a + b + c = ab+bc+ca$, prove that $$\frac{a^2}{a^2-a+1}+\frac{b^2}{b^2-b+1}+\frac{c^2}{c^2-c+1} \le 3$$(D. Pirshtuk) $$ LHS=\frac{a^2}{a^2-a+1}+\frac{b^2}{b^2-b+1}+\frac{(a+b-ab)^2}{(a+b-ab)^2-(a+b-ab)(a+b-1)+(a+b-1)^2}=3-\frac{(a-b)^2+3(a+b-2)^2}{4(a^2-a+1)(b^2-b+1)}$$