Question 45
Let $n$ be a positive integer not divisible by $3$. A triangular array of order $n$ is obtained by dividing a large equilateral triangle into $n^2$ congruent small equilateral triangles by the lines through all the points dividing each side into $n$ equal parts, parallel to the other sides. Put an orange at every vertex of the small triangles of the triangular array of order $n$; there are
$$
\frac{(n+1)(n+2)}{2}
$$
oranges in total. A set of three oranges $A, B, C$ is called **good** if and only if $\overline{AB}$ and $\overline{AC}$ are each a side of some small triangle and $\angle BAC = 120^{\circ}$.
Each time, **Yanyan** may choose a good set of three oranges and take them away. Find the maximum number of oranges that **Yanyan** can take away.
$$
\frac{(n+1)(n+2)}{2}
$$
oranges in total. A set of three oranges $A, B, C$ is called **good** if and only if $\overline{AB}$ and $\overline{AC}$ are each a side of some small triangle and $\angle BAC = 120^{\circ}$.
Each time, **Yanyan** may choose a good set of three oranges and take them away. Find the maximum number of oranges that **Yanyan** can take away.