15. $A B C D$ is a convex quadrilateral and $E, F$ are the mid-points of $B C$ and $C D$ respectively. The line segments $A E, A F$ and $E F$ divide $A B C D$ into four triangles, whose areas are four consecutive integers. Find the greatest possible area of $\triangle A B D$.
(2 marks) $A B C D$ 是凸四邊形, $E 、 F$ 分別是 $B C$ 和 $C D$ 的中點。線段 $A E 、 A F$ 和 $E F$ 把 $A B C D$ 分成四個三角形, 它們的面積分別是四個連續整數。求 $\triangle A B D$ 的面積的最大可能值。
(2 分)
15. 6
15. Let the areas of the four triangles be $n, n+1$, $n+2$ and $n+3$, where $n$ is a positive integer. The area of the quadrilateral $A B C D$ is thus $4 n+6$. Note that the area of $\triangle B C D$ is four times that of $\triangle E C F$, which is at least $4 n$. Hence the area of $\triangle A B D$ is at most 6 .

Equality can be attained when $A B C D$ is an isosceles trapezium with parallel sides $A D=6$ and $B C=4$, and height 2 . (We can check in this case that the areas of $\triangle C E F, \triangle A B E, \triangle A D F$ and $\triangle A E F$ are 1, 2, 3 and 4 respectively, and $\triangle A B D$ has area 6). The answer is thus 6 .