Find minimum value of the algebraic expression

Question

With a,b,c>0 and a+b+c=3 find the minimum value of
$$P = \frac{a}{b^2+1} + \frac{b}{c^2+1} + \frac{c}{a^2+1}$$ This problem is the last part of my second midterm exam which i couldn't do since it is the hardest one.
For me, i think that the minimum value is
P $\ge \frac{a}{2} + \frac{b}{2} + \frac{c}{2} = \frac{a+b+c}{2} = \frac{3}{2}$
The equality happens when $a=b=c=1$
However i don't know how to prove it so i hope you guys can help me with this problem