Nemotron-IMO-Bench 2026 A32

Prove that if real numbers $a,b,c,d$ add up to $2$, then $$\frac{a}{a^2-a+1}+ \frac{b}{b^2-b+1}+ \frac{c}{c^2-c+1}+ \frac{d}{d^2-d+1}\leq \frac{8}{3}.$$