Iranian Mathematical Olympiad

First of all consider $4$ points $A$, $B$, $C$ and $D$.

i) If convex hull of these points is a triangle (without loss of generality let $D$ be the inner point)
$$S_{\triangle ABC} - S_{\triangle ABD} - S_{\triangle ACD} - S_{\triangle BCD} = 0$$

ii) If convex hull of these points is a quadrilateral
$$S_{\triangle ABC} - S_{\triangle ABD} + S_{\triangle ACD} - S_{\triangle BCD} = 0$$

Let $A_1$, $A_2$, $A_3$, $\dots$, $A_8$ be $8$ problem's $8$ points. Now consider these sets of $4$ points, it is easy to see that each triangle of $56$ triangle is exactly in one of these sets, and from above it is possible to put $-$ and $+$ between areas in every set such that the result becomes zero, hence we are done.