Question

设 $A$ 、 $B$ 、 $C$ 、 $D$ 是双曲线 $\frac{x^2}{20} - \frac{y^2}{24} = 1$ 上的点，使得 $ABCD$ 是一个菱形，且它的对角线在原点相交。求一个最大数，使它对所有这样的菱形 $ABCD$ 都小于 $BD^2$ 。

Let $A,$ $B,$ $C,$ and $D$ be points on the hyperbola $\frac{x^2}{20} - \frac{y^2}{24} = 1$ such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the largest number less than $BD^2$ for all rhombuses $ABCD.$

答案

答案：480

知识点：双曲线菱形最优化

难度评级：2710

解答：

菱形的对角线互相垂直且互相平分，所以 $C = -A$ 、 $D = -B$ ，并且 $OA \perp OB$ 。设直线 $BD$ 的斜率为 $m$ ，于是 $B = (x, mx)$ ，并满足 $x^2\left(\frac{1}{20} - \frac{m^2}{24}\right) = 1,$ $x^2 = \frac{120}{6 - 5m^2},$ 即这要求 $m^2 \lt \frac{6}{5}$ 。于是 $BD^2 = 4(x^2 + m^2x^2)$ $= \frac{480(1 + m^2)}{6 - 5m^2}$ 。直线 $AC$ 的斜率为 $-\frac{1}{m}$ ，所以它与双曲线相交仅当 $\frac{1}{m^2} \lt \frac{6}{5}$ ，也就是 $m^2 \gt \frac{5}{6}$ 。

在区间 $\frac{5}{6} \lt m^2 \lt \frac{6}{5}$ 上，量 $\frac{480(1 + m^2)}{6 - 5m^2}$ 随 $m^2$ 严格递增：当 $m^2 \to \frac{5}{6}$ 时，它趋近于 $480 \cdot \frac{11/6}{11/6} = 480$ ，而当 $m^2 \to \frac{6}{5}$ 时，它无界增长。因此 $BD^2$ 正好取到 $(480, \infty)$ 中的值，并且永不等于 $480$ 。

对每一个这样的菱形都小于 $BD^2$ 的最大数因此是 $480$ 。

The diagonals of a rhombus are perpendicular bisectors of each other, so $C = -A,$ $D = -B,$ and $OA \perp OB.$ Let line $BD$ have slope $m,$ so $B = (x, mx)$ with $x^2\left(\frac{1}{20} - \frac{m^2}{24}\right) = 1,$ i.e. $x^2 = \frac{120}{6 - 5m^2},$ which requires $m^2 \lt \frac{6}{5}.$ Then $BD^2 = 4(x^2 + m^2x^2)$ $= \frac{480(1 + m^2)}{6 - 5m^2}.$ Line $AC$ has slope $-\frac{1}{m},$ so it meets the hyperbola only when $\frac{1}{m^2} \lt \frac{6}{5},$ that is $m^2 \gt \frac{5}{6}.$

On the interval $\frac{5}{6} \lt m^2 \lt \frac{6}{5},$ the quantity $\frac{480(1 + m^2)}{6 - 5m^2}$ is strictly increasing in $m^2:$ as $m^2 \to \frac{5}{6}$ it tends to $480 \cdot \frac{11/6}{11/6} = 480,$ and as $m^2 \to \frac{6}{5}$ it grows without bound. Hence $BD^2$ takes exactly the values in $(480, \infty)$ and never equals $480.$

The largest number that is less than $BD^2$ for every such rhombus is therefore $480.$

2024 AIME I 第 9 题

答案：480

难度评级：2710

解答：

其他年份的第 9 题