We are given a convex quadrilateral $ABCD$ in the plane.
([i]i[/i]) If there exists a point $P$ in the plane such that the areas of $\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP$ are equal, what condition must be satisfied by the quadrilateral $ABCD$?
([i]ii[/i]) Find (with proof) the maximum possible number of such point $P$ which satisfies the condition in ([i]i[/i]).
1. **Given a convex quadrilateral \(ABCD\) and a point \(P\) such that the areas of \(\triangle ABP\), \(\triangle BCP\), \(\triangle CDP\), and \(\triangle DAP\) are equal, we need to determine the condition that must be satisfied by \(ABCD\).**

   Let's denote the area of each of these triangles as \(A\). Therefore, we have:
   \[
   [ABP] = [BCP] = [CDP] = [DAP] = A
   \]
   Since the sum of the areas of these four triangles equals the area of the quadrilateral \(ABCD\), we have:
   \[
   [ABCD] = [ABP] + [BCP] + [CDP] + [DAP] = 4A
   \]

2. **Suppose \(BP\) intersects \(AC\) at \(M\). Since \([ABP] = [BCP]\), point \(M\) must be the midpoint of \(AC\).**

   Similarly, if \(DP\) intersects \(AC\) at \(N\), then \(N\) must also be the midpoint of \(AC\). Therefore, \(P\) must lie on the line through \(B\) parallel to \(AC\) and on the line through \(D\) parallel to \(AC\).

3. **Let \(N\) be the midpoint of \(BD\). If \(B, M, D\) lie on a straight line, then \(P\) must lie on \(BD\).**

   In this case, \(P\) must also lie on \(AN\) and \(CN\). The only possible point \(P\) is \(N\), and the condition for such a quadrilateral is:
   \[
   [ABD] = [BCD]
   \]

4. **If \(B, M, D\) and \(A, N, C\) are not collinear, then \(P\) cannot lie on both \(BM\) and \(DM\).**

   Without loss of generality, assume \(P\) lies on the line through \(D\) parallel to \(AC\) and on \(BM\). Similarly, \(P\) lies on the line through \(C\) parallel to \(BD\) and on \(AN\). If \(AC\) intersects \(BD\) at \(E\), then \(DECP\) forms a parallelogram. The condition for such a quadrilateral is:
   \[
   [ABCD] = 2[DEC]
   \]

5. **To find the maximum possible number of such points \(P\), we consider the cases:**

   - If \(B, M, D\) are collinear, there is only one such point \(P\), which is \(N\).
   - If \(B, M, D\) and \(A, N, C\) are not collinear, there is still only one such point \(P\) that satisfies the condition.

   Therefore, the maximum number of such points \(P\) is 1.

The final answer is \( \boxed{ 1 } \).