11. In a drawer there are $x$ white gloves and $y$ red gloves with $x>y$ and $x+y \leq 2008$. When two gloves are drawn at random, the probability that the two gloves are of the same colour is exactly one-half. Find the largest possible value of $x$.
(2 marks)
一個抽幄裡有 $x$ 隻白色手套和 $y$ 隻紅色手套, 其中 $x>y$ 而 $x+y \leq 2008$ 。當隨意抽出兩隻手套時, 兩隻手套顏色相同的概率剛好是二分之一。求 $x$ 的最大可能值。
11. 990
11. The number of ways of selecting two gloves of the same colour is $C_{2}^{x}+C_{2}^{y}$ while that of of selecting two gloves of different colours is $x y$. We thus have $\frac{x(x-1)}{2}+\frac{y(y-1)}{2}=x y$, which simplifies to $(x-y)^{2}=x+y$. Since $x=\frac{(x-y)+(x+y)}{2}=\frac{(x-y)+\sqrt{x-y}}{2}$, maximising $x$ is equivalent to maximising $x-y$. As $(x-y)^{2}=x+y \leq 2008$, the maximum possible value of $x-y$ is 44 . This corresponds to $x+y=44^{2}=1936$ and the maximum possible value of $x$ being $\frac{44+1936}{2}=990$.