<s>[INST] Given positives $x$ and $y$ such that $x^2 + y^2 = xy + 1$, prove that $\frac{x}{x^2 + y} + \frac{y}{y^2 + x} \le 1$. 
Given positives $x$ and $y$ such that $x^2 + y^2 = xy + 1$, prove that $$\large \frac{x}{x^2 + y} + \frac{y}{y^2 + x} \le 1$$

Well, I have provided my solution below, which is clearly not the most straightforward and convenient one. I would be greatly appreciated if you could come up with any other solutions.
 [/INST] We need to prove that $$(x+y)(x-y)^2\geq1-x^2y^2$$ or $$(x+y)(x-y)^2\geq(x-y)^2(x^2+y^2)$$ or $$(x+y)^2(x^2-xy+y^2)\geq(x^2+y^2)^2$$ or $$(x-y)^2xy\geq0.$$
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