2023 AIME II Problems/Problem 14

==Solution (3-d vector analysis, analytic geometry)

We introduce a Cartesian coordinate system to the diagram. We put the origin at $A$ . We let the $z$ -components of $B$ , $C$ , $D$ be positive. We set the $x$ -axis in a direction such that $B$ is on the $x-O-z$ plane.

The coordinates of $A$ , $B$ , $C$ are $A = \left( 0, 0, 0 \right)$ , $B = \left( x_B, 0 , 2 \right)$ , $C = \left( x_C, y_C, 8 \right)$ .

Because $AB \perp AC$ , $\overrightarrow{AB} \cdot \overrightarrow{AC} = 0$ . Thus, $x_B x_C + 16 = 0 . \hspace{1cm} (1)$

Because $AC$ is a diagonal of a face, $AC^2 = 2 AB^2$ . Thus, $x_C^2 + y_C^2 + 8^2 = 2 \left( x_B^2 + 2^2 \right) . \hspace{1cm} (2)$

Because plane $ABCD$ is perpendicular to plan $P$ , $\hat z \cdot \left( \overrightarrow{AB} \times \overrightarrow{AC} \right) = 0$ . Thus, $\begin{vmatrix} 0 & 0 & 1 \\ x_B & 0 & 2 \\ x_C & y_C & 8 \end{vmatrix} = 0 . \hspace{1cm} (3)$

Jointly solving (1), (2), (3), we get one solution $x_B = 4 \sqrt{2}$ , $x_C = - 2 \sqrt{2}$ , $y_C = 0$ . Thus, the side length of the cube is $|AB| = \sqrt{x_B^2 + 2^2} = 6$ .

Denote by $P$ and $Q$ two vertices such that $AP$ and $AQ$ are two edges, and satisfy the right-hand rule that $\widehat{AB} \times \widehat{AP} = \widehat{AQ}$ . Now, we compute the coordinates of $P$ and $Q$ .

Because $|AB| = 6$ , we have $\overrightarrow{AP} \times \overrightarrow{AQ} = 6 \overrightarrow{AB}$ , $\overrightarrow{AQ} \times \overrightarrow{AB} = 6 \overrightarrow{AP}$ , $\overrightarrow{AB} \times \overrightarrow{AP} = 6 \overrightarrow{AQ}$ .

Hence, $\begin{align*} \begin{bmatrix} \hat i & \hat j & \hat k \\ x_P & y_P & z_P \\ x_Q & y_Q & z_Q \end{bmatrix} & = 6 \left( 4 \sqrt{2} \hat i + 2 \hat k \right) , \\ \begin{vmatrix} \hat i & \hat j & \hat k \\ x_Q & y_Q & z_Q \\ 4 \sqrt{2} & 0 & 2 \end{vmatrix} & = 6 \left( x_P \hat i + y_P \hat j + z_P \hat k \right) , \\ \begin{vmatrix} \hat i & \hat j & \hat k \\ 4 \sqrt{2} & 0 & 2 \\ x_P & y_P & z_P \end{vmatrix} & = 6 \left( x_Q \hat i + y_Q \hat j + z_Q \hat k \right) . \end{align*}$

By solving these equations, we get \[ y_P^2 + y_Q^2 = 36 . \]

In addition, we have $\overrightarrow{AC} = \overrightarrow{AP} + \overrightarrow{AQ}$ . Thus, $P = \left( - \sqrt{2} , 3 \sqrt{2} , 4 \right)$ , $Q = \left( - \sqrt{2} , - 3 \sqrt{2} , 4 \right)$ .

Therefore, the volume of the water is $\begin{align*} V = & 6^3 \int_{u=0}^1 \int_{v=0}^1 \int_{w=0}^1 \mathbf 1 \left\{ z_B u + z_P v + z_Q w \leq 7 \right\} dw dv du \\ & = 6^3 \int_{u=0}^1 \int_{v=0}^1 \int_{w=0}^1 \mathbf 1 \left\{ 2 u + 4 v + 4 w \leq 7 \right\} dw dv du \\ & = 6^3 - 6^3 \int_{u=0}^1 \int_{v=0}^1 \int_{w=0}^1 \mathbf 1 \left\{ 2 u + 4 v + 4 w > 7 \right\} dw dv du . \end{align*}$

Define $u' = 1 - u$ , $v' = 1 - v$ , $w' = 1 - w$ . Thus, $\begin{align*} V & = 6^3 - 6^3 \int_{u=0}^1 \int_{v=0}^1 \int_{w=0}^1 \mathbf 1 \left\{ 2 u' + 4 v' + 4 w' < 3 \right\} dw dv du \\ & = 6^3 - 6^3 \int_{u'=0}^1 \left( \int_{v'=0}^1 \int_{w'=0}^1 \mathbf 1 \left\{ v' + w' < \frac{3}{4} - \frac{u'}{2} \right\} dw' dv' \right) du' \\ & = 6^3 - 6^3 \int_{u'=0}^1 \frac{1}{2} \left( \frac{3}{4} - \frac{u'}{2} \right)^2 du' . \end{align*}$

Define $u'' = \frac{3}{4} - \frac{u'}{2}$ . Thus, $\begin{align*} V & = 6^3 - 6^3 \int_{u'' = 1/4}^{3/4} \left( u'' \right)^2 du'' \\ & = 6^3 - 6^3 \frac{1}{3} \left( \left(\frac{3}{4}\right)^3 - \left(\frac{1}{4}\right)^3 \right) \\ & = 216 - \frac{117}{4} \\ & = \frac{747}{4} . \end{align*}$

Therefore, the answer is $747 + 4 = \boxed{\textbf{(751) }}$ .

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2023 AIME II Problems/Problem 14