14. A regular 201-sided polygon is inscribed inside a circle of center $C$. Triangles are drawn by connecting any three of the 201 vertices of the polygon. How many of these triangles have the point $C$ lying inside the triangle?
14. 一個正 201 邊形內接於一個以 $C$ 為圓心的圓形內。把多邊形 201 個頂點中的任意三個頂點連起, 可以組成不同的三角形。有多少個這樣的三角形, 將 $C$ 點包含在其內部?
(2 分)
14. 338350
14. Fix a vertex $A$ and draw a diameter from $A$. This diameter splits the remaining 200 vertices into two groups of 100 each. Number them as $P_{1}, P_{2}, \ldots, P_{100}$ and $Q_{1}, Q_{2}, \ldots, Q_{100}$ according to the distance of each vertex from $A$. Note that a triangle with $A$ as vertex contains point $C$ if and only if it is of the form $A P_{i} Q_{j}$ where $i+j>100$. Hence there are $1+2+\cdots+100=5050$ triangles with $A$ as a vertex which contains $C$. Since $A$ is arbitrary we have $5050 \times 201=1015050$ such triangles. However, in this way each triangle is counted three times. So the answer is $1015050 \div 3=338350$.