[1] Let $x$ be a positive real number satisfying $x ^ { 2 } + \frac { 4 } { x ^ { 2 } } = 9$. Then
$$\left( x + \frac { 2 } { x } \right) ^ { 2 } = \text { A B }$$
Therefore, $x + \frac { 2 } { x } = \sqrt { \text { A B } }$. Furthermore,
$$\begin{aligned} x ^ { 3 } + \frac { 8 } { x ^ { 3 } } & = \left( x + \frac { 2 } { x } \right) \left( x ^ { 2 } + \frac { 4 } { x ^ { 2 } } - \right. \\ & = \text { D } \sqrt { \text { E F } } \end{aligned}$$
□ C
Also,
$$x ^ { 4 } + \frac { 16 } { x ^ { 4 } } = \text { G H }$$
[2] Let $p$ and $q$ be two conditions regarding the real number $x$:
$$\begin{array} { l l } p : & x = 1 \\ q : & x ^ { 2 } = 1 \end{array}$$
Also, let $\bar { p }$ and $\bar { q }$ denote the negations of conditions $p$ and $q$, respectively.
(1) Choose one from the options (0)–(3) below for each of the blanks K, L, M, N. You may select the same option more than once. $q$ is a \text{____} condition for $p$ (K) $\bar { p }$ is a \text{____} condition for $q$ (L) ($p$ or $\bar { q }$) is a \text{____} condition for $q$ (M) ($\bar { p }$ and $q$) is a \text{____} condition for $q$ (N) (0) necessary but not sufficient
(1) sufficient but not necessary
(2) necessary and sufficient
(3) neither necessary nor sufficient
(2) Let $r$ be a condition regarding the real number $x$:
$$r : x > 0$$
Choose one from options ⓪–⑦ below for the blank O.
Consider the three propositions:
A: ``($p$ and $q$) $\Longrightarrow r$'' B: ``$q \Longrightarrow r$'' C: ``$\bar { q } \Longrightarrow \bar { p }$''
The correct statement about the truth values of these propositions is □ O. (0) A is true, B is true, C is true
(1) A is true, B is true, C is false
(2) A is true, B is false, C is true
(3) A is true, B is false, C is false
(4) A is false, B is true, C is true
(5) A is false, B is true, C is false (6) A is false, B is false, C is true (7) A is false, B is false, C is false
[3] Let $a$ be a constant, and let $g ( x ) = x ^ { 2 } - 2 \left( 3 a ^ { 2 } + 5 a \right) x + 18 a ^ { 4 } + 30 a ^ { 3 } + 49 a ^ { 2 } + 16$. The vertex of the parabola $y = g ( x )$ is
$$\text { ( P } a ^ { 2 } + \text { Q } a , \text { R } a ^ { 4 } + \text { S T } a ^ { 2 } + \text { U V } \text { ) }$$
When $a$ varies over all real numbers, the minimum value of the $x$-coordinate of the vertex is $- \frac { \text { W X } } { \text { Y Z } }$.
Next, let $t = a ^ { 2 }$. Then the $y$-coordinate of the vertex can be expressed as
$$\text { R } t ^ { 2 } + \text { S T } t + \text { U V }$$
Therefore, when $a$ varies over all real numbers, the minimum value of the $y$-coordinate of the vertex is AA.

[1] In $\triangle \mathrm { ABC }$, $\mathrm { AB } = \sqrt { 3 } - 1$, $\mathrm { BC } = \sqrt { 3 } + 1$, and $\angle \mathrm { ABC } = 60 ^ { \circ }$.
(1) $\mathrm { AC } = \sqrt { \text { A } }$, so the circumradius of $\triangle \mathrm { ABC }$ is $\sqrt { \text { B } }$, and
$$\sin \angle \mathrm { BAC } = \frac { \sqrt { \square } + \sqrt { \square } } { \square }$$
Here, the order of answers for □ C and □ D does not matter.
(2) Let D be a point on side AC such that the area of $\triangle \mathrm { ABD }$ is $\frac { \sqrt { 2 } } { 6 }$. Then
$$\mathrm { AB } \cdot \mathrm { AD } = \frac { \square \sqrt { * } - \square } { \square }$$
Therefore, $\mathrm { AD } = \frac { \square } { \square }$. □ E F
[2] Ski jumping is a competition where athletes compete on both jump distance and the beauty of their aerial posture. Athletes slide down a slope and take off from the edge of the slope into the air. A score $X$ is determined from the jump distance $D$ (in meters), and a score $Y$ is determined from the aerial posture. Consider 58 jumps at a certain competition.
(1) For the score $X$, the score $Y$, and the takeoff velocity $V$ (in km/h), three scatter plots in Figure 1 were obtained.
Choose one from options (0)–(6) below for each of G, H, and I. The order of answers does not matter.
The correct statements that can be read from Figure 1 are □ G, □ H, and □ I. (0) The correlation between $X$ and $V$ is stronger than the correlation between $X$ and $Y$.
(1) There is a positive correlation between $X$ and $Y$.
(2) The jump with maximum $V$ also has maximum $X$.
(3) The jump with maximum $V$ also has maximum $Y$.
(4) The jump with minimum $Y$ does not have minimum $X$.
(5) All jumps with $X \geq 80$ have $V \geq 93$. (6) There is no jump with $Y \geq 55$ and $V \geq 94$.
(2) The score $X$ is calculated from the jump distance $D$ using the following formula:
$$X = 1.80 \times ( D - 125.0 ) + 60.0$$
Choose one from options (0)–(6) below for each of J, K, and L. You may select the same option more than once. – The variance of $X$ is □ J times the variance of $D$. – The covariance of $X$ and $Y$ is □ K times the covariance of $D$ and $Y$. Here, covariance is defined as the average of the products of deviations from the mean for each of the two variables. – The correlation coefficient of $X$ and $Y$ is □ L times the correlation coefficient of $D$ and $Y$. (0) $- 125$
(1) $- 1.80$
(2) $1$
(3) $1.80$
(4) $3.24$
(5) $3.60$ (6) $60.0$
(3) The 58 jumps were performed by 29 athletes, each performing 2 jumps. A histogram for the values of $X + Y$ (sum of scores $X$ and $Y$) for the first jump and a histogram for the values of $X + Y$ for the second jump are either A or B in Figure 2. Also, a box plot for the values of $X + Y$ for the first jump and a box plot for the values of $X + Y$ for the second jump are either a or b in Figure 3. The minimum value of $X + Y$ for the first jump was 108.0.
Choose one from options ⓪–③ in the table below for the blank M.
The correct combination of histogram and box plot for the values of $X + Y$ for the first jump is □ M.

\cline { 2 - 5 } \multicolumn{1}{c|}{}	(0)	(1)	(2)	(3)
Histogram	A	A	B	B
Box plot	a	b	a	b

Choose one from options (0)–(3) below for the blank N.
The correct statement that can be read from Figure 3 is □ N. (0) The interquartile range of $X + Y$ for the first jump is larger than the interquartile range of $X + Y$ for the second jump.
(1) The median of $X + Y$ for the first jump is larger than the median of $X + Y$ for the second jump.
(2) The maximum value of $X + Y$ for the first jump is smaller than the maximum value of $X + Y$ for the second jump.
(3) The minimum value of $X + Y$ for the first jump is smaller than the minimum value of $X + Y$ for the second jump.

There is a lottery consisting of 4 tickets in total: 2 winning tickets and 2 losing tickets. Three people A, B, and C draw one ticket each in that order. Once a ticket is drawn, it is not returned.