INMO 2003 Problem 5

Let $A B C$ be a triangle with sides $a, b, c$. Consider a triangle $A_{1} B_{1} C_{1}$ with sides equal to $a+\frac{b}{2}, b+\frac{c}{2}, c+\frac{a}{2}$. Show that

$$\left[A_{1} B_{1} C_{1}\right] \geq \frac{9}{4}[A B C]$$

where $[X Y Z]$ denotes the area of the triangle $X Y Z$.