<s>[INST] Find $a,b,c \in \mathbb{R^+}$ such that $a^2+b^2+c^2+abc=4$ and $a+b+c=3$ 
Find  $a,b,c \in \mathbb{R^+}$ such that $$a^2+b^2+c^2+abc=4\;\;\;{\rm  and}\;\;\;a+b+c=3$$

Seeing $a^2+b^2+c^2+abc=4$,I substituted $a=2\cos{A}$,$b=2\cos{B},c=\cos{C}$ with $A+B+C=180°$.
Therefore,$cosA+cosB+cosC=\frac 32$ have thousands of solution. What to do now? Please help me.
 [/INST] It is $$\cos(C)+\cos(B)+\cos(A)\le \frac{3}{2}$$ and this Comes from
$$\cos(A)+\cos(B)+\cos(C)=1+\frac{r}{R}$$
</s>