Turkey — Team Selection Test

Note that
$$\frac{a+b+c}{2}r = \frac{b+c-a}{2}r_a = \frac{a+b+c}{2}r_a - ar_a = \frac{a+b+c}{2}r_b - br_b = \frac{a+b+c}{2}r_c - cr_c.$$
Therefore, we have $\frac{3r(a+b+c)}{2} = \frac{a+b+c}{2}(r_a + r_b + r_c) - ar_a - br_b - cr_c$ and hence
$$ar_a + br_b + cr_c = \frac{a+b+c}{2}(r_a + r_b + r_c - 3r).$$
On the other hand, the Cauchy-Schwarz inequality implies
$$ar_a + br_b + cr_c \le \sqrt{a^2 + b^2 + c^2} \cdot \sqrt{r_a^2 + r_b^2 + r_c^2}$$
and the result follows.