Suppose we have a quadratic function $y = a x ^ { 2 } + b x + c$ in $x$ which satisfies the following conditions 【＊】：
【＊】 When $x = - 1$ ，then $y = - 8$ and when $x = 3$ ，then $y = 16$ ．Further，in the interval $- 1 \leqq x \leqq 3$ ，the value of $y$ increases with the increase of the value of $x$ ．
We are to find the conditions which $a , b$ and $c$ must satisfy．
From【＊】，it follows that $b$ and $c$ can be expressed in terms of $a$ as
$$\begin{aligned} & b = \mathbf { A B } a + \mathbf { A } \\ & c = \mathbf { D E } a - \mathbf { F } . \end{aligned}$$
Hence，the axis of symmetry of the graph of this quadratic function has the equation
$$x = \mathbf { G } - \frac { \mathbf { H } } { a } .$$
Thus $a , b$ and $c$ must satisfy the relationships（1）and（2），and furthermore
$$0 < a \leqq \frac { \mathbf { I } } { \mathbf { J } } \quad \text { or } \frac { \mathbf { K L } } { \mathbf { M } } \leqq a < 0 .$$

Let $a , b , c$ and $d$ be real numbers satisfying $a < b < c < d$. Suppose that the two subsets of real numbers
$$A = \{ x \mid a \leqq x \leqq c \} , \quad B = \{ x \mid b \leqq x \leqq d \}$$
satisfy
$$A \cap B = \left\{ x \mid x ^ { 2 } - 4 x + 3 \leqq 0 \right\} .$$
Then, answer the questions for cases (1) and (2).
(1) Let the union of $A$ and $B$ be
$$A \cup B = \left\{ x \mid x ^ { 2 } - 5 x - 24 \leqq 0 \right\} .$$
Then the values of $a , b , c$ and $d$ are
$$a = \mathbf { \text { NO } } , \quad b = \mathbf { P } , \quad c = \mathbf { Q } , \quad d = \mathbf { Q } .$$
(2) Let the intersection of $A$ and the complement $\bar { B }$ of $B$ be
$$A \cap \bar { B } = \left\{ x \mid x ^ { 2 } + 5 x - 6 \leqq 0 \text { and } x \neq 1 \right\} ,$$
and let the intersection of the complement $\bar { A }$ of $A$ and $B$ be
$$\bar { A } \cap B = \left\{ x \mid x ^ { 2 } - 9 x + 18 \leqq 0 \text { and } x \neq 3 \right\} .$$
Then the values of $a , b , c$ and $d$ are
$$a = \mathbf { S T } , \quad b = \mathbf { U } , \quad c = \mathbf { V } , \quad d = \mathbf { W } .$$

Consider a polynomial in $x$ and $y$
$$P = ( 3 x + 4 y + 1 ) ^ { 5 } .$$
Let us denote the coefficient of $x ^ { n } y$ in the expansion of $P$ by $a _ { n }$, where $n$ is an integer. Note that $x ^ { 0 } = y ^ { 0 } = 1$.
(1) Let us find the value of the coefficient $a _ { 1 }$. First, we note that
$$P = \{ ( 3 x + 1 ) + 4 y \} ^ { 5 }$$
and use the binomial theorem to expand $P$. Then $x y$ appears when we expand the term AB $( 3 x + 1 ) ^ { \text {C} } y$. Further, the coefficient for $x$ in the expansion of $( 3 x + 1 ) ^ { \text {C} }$ is DE. It follows that
$$a _ { 1 } = \mathbf { F G H } .$$
(2) The number of values which $n$ can take is $\square$ in all. Also, the value of $a _ { n }$ is maximized at $n =$ $\square$ J .

Consider the integral expression
$$P = ( x - 1 ) ^ { 2 } ( y + 5 ) + ( 2 x - 3 ) ( y + 4 ) - ( x - 1 ) ^ { 2 } .$$
(1) $P$ can be transformed into
$$P = \left( x ^ { 2 } - \mathbf { K } \right) ( y + \mathbf { L } ) .$$
(2) The pairs $( x , y )$ of integers $x$ and $y$ which give $P = 7$ are
$$( \pm \mathbf { M } , \mathbf { N O P } ) , \quad ( \pm \mathbf { Q } , \mathbf { R S } ) .$$
(3) Let $a$ be a rational number. If $x = \sqrt { 2 } + 2 \sqrt { 3 }$ and $y = a + \sqrt { 6 }$, then the value of $a$ such that the value of $P$ is a rational number is $\mathbf { T U }$.

For each of A $\sim$ D in questions (1)$\sim$(4) below, choose the appropriate answer from among (0) $\sim$ (3) of each question. For $\mathbf { E } \sim \mathbf { G }$ in question (5), put the correct number.
Suppose that $a , b$ and $c$ are integers, and $a > 0$. Also, suppose that the graph of a quadratic function $y = a x ^ { 2 } - 2 b x + c$ intersects the $x$-axis and all points of intersection are in the interval $0 < x < 1$.
(1) The relationship between $a$ and $b$ is A. (0) $a > b$
(1) $a < b$
(2) $a = b$
(3) indeterminate
(2) The conditions on $b$ and $c$ are $\mathbf { B }$. (0) $b < 0 , c < 0$
(1) $b < 0 , c > 0$
(2) $b > 0 , c < 0$
(3) $b > 0 , c > 0$
(3) The relationship between $2 b$ and $a + c$ is $\mathbf { C }$. (0) $2 b > a + c$
(1) $2 b < a + c$
(2) $2 b = a + c$
(3) indeterminate
(4) The relationship between $b$ and $c$ is $\mathbf { D }$. (0) $b > c$
(1) $b < c$
(2) $b = c$
(3) indeterminate
(5) The smallest integer which $a$ can take is $\mathbf { E }$. In this case, the value of $b$ is $\mathbf { F }$, and the value of $c$ is $\mathbf { G }$.

Let $O$ be the vertex (apex) of a right circular cone such that the radius of the base is 1 and the slant height is 3.
(1) Consider the net of the cone, which consists of a sector and a circle. (The net of a solid is a 2-dimensional shape that can be folded to form that solid.) The central angle of the sector is $\square$ ABC , and the area of the sector is $\square$ D $\pi$.
(2) Take two points A and B on the circumference of the base such that the line segment AB is a diameter. Take a point P on the segment OB and consider a path on the side of this circular cone which starts from the point A , passes through the point P and returns to A . Denote the length of the path by $\ell$.
(i) If $\mathrm { OP } = 2$, then the smallest value of $\ell$ is $\mathbf { E }$. $\mathbf { F }$.
(ii) Let point P be any point on the line segment OB . When $\ell$ is minimized, then $\mathrm { OP } = \frac { \mathbf { G } } { \mathbf { G } }$, and the value of $\ell$ is $\square \sqrt { } \square$.

Suppose we have a quadratic function $y = a x ^ { 2 } + b x + c$ in $x$ which satisfies the following conditions 【＊】：
【＊】 When $x = - 1$ ，then $y = - 8$ and when $x = 3$ ，then $y = 16$ ．Further，in the interval $- 1 \leqq x \leqq 3$ ，the value of $y$ increases with the increase of the value of $x$ ．
We are to find the conditions which $a , b$ and $c$ must satisfy．
From【＊】，it follows that $b$ and $c$ can be expressed in terms of $a$ as
$$\begin{aligned} & b = \mathbf { A B } a + \mathbf { C } \\ & c = \mathbf { D E } a - \mathbf { F } . \end{aligned}$$
Hence，the axis of symmetry of the graph of this quadratic function has the equation
$$x = \mathbf { G } - \frac { \mathbf { H } } { a } .$$
Thus $a , b$ and $c$ must satisfy the relationships（1）and（2），and furthermore
$$0 < a \leqq \frac { \mathbf { I } } { \mathbf { J } } \quad \text { or } \frac { \mathbf { K L } } { \mathbf { M } } \leqq a < 0 .$$

Let $a , b , c$ and $d$ be real numbers satisfying $a < b < c < d$. Suppose that the two subsets of real numbers
$$A = \{ x \mid a \leqq x \leqq c \} , \quad B = \{ x \mid b \leqq x \leqq d \}$$
satisfy
$$A \cap B = \left\{ x \mid x ^ { 2 } - 4 x + 3 \leqq 0 \right\} .$$
Then, answer the questions for cases (1) and (2).
(1) Let the union of $A$ and $B$ be
$$A \cup B = \left\{ x \mid x ^ { 2 } - 5 x - 24 \leqq 0 \right\} .$$
Then the values of $a , b , c$ and $d$ are
$$a = \mathbf { \text { NO } } , \quad b = \mathbf { P } , \quad c = \mathbf { Q } , \quad d = \mathbf { Q } .$$
(2) Let the intersection of $A$ and the complement $\bar { B }$ of $B$ be
$$A \cap \bar { B } = \left\{ x \mid x ^ { 2 } + 5 x - 6 \leqq 0 \text { and } x \neq 1 \right\} ,$$
and let the intersection of the complement $\bar { A }$ of $A$ and $B$ be
$$\bar { A } \cap B = \left\{ x \mid x ^ { 2 } - 9 x + 18 \leqq 0 \text { and } x \neq 3 \right\} .$$
Then the values of $a , b , c$ and $d$ are
$$a = \mathbf { S T } , \quad b = \mathbf { U } , \quad c = \mathbf { V } , \quad d = \mathbf { W } .$$

Given a sphere $S$ whose center is at O and whose radius is 1 , take three points A , B and C on $S$ such that
$$\overrightarrow { \mathrm { OA } } \cdot \overrightarrow { \mathrm { OB } } = \overrightarrow { \mathrm { OB } } \cdot \overrightarrow { \mathrm { OC } } = \overrightarrow { \mathrm { OC } } \cdot \overrightarrow { \mathrm { OA } } = 0 .$$
Note that $\overrightarrow { \mathrm { OA } } \cdot \overrightarrow { \mathrm { OB } }$, etc., refers to the inner product of the two vectors.
(1) It follows that $\overrightarrow { \mathrm { AB } } \cdot \overrightarrow { \mathrm { AC } } = \square \mathbf { A } , | \overrightarrow { \mathrm { AB } } | = \sqrt { \mathbf { B } } , \cos \angle \mathrm { BAC } = \frac { \square \mathbf { C } } { \square }$ and the area of the triangle ABC is $\frac { \sqrt { \mathbf { E } } } { \mathbf { F } }$.
(2) Let G be the center of gravity of triangle ABC and P be the intersection point of the ray (half line) OG and sphere $S$.
Since $\overrightarrow { \mathrm { OG } } = \frac { \mathbf { G } } { \mathbf { G } } ( \overrightarrow { \mathrm { OA } } + \overrightarrow { \mathrm { OB } } + \overrightarrow { \mathrm { OC } } )$, we have
$$\begin{gathered} | \overrightarrow { \mathrm { OG } } | = \frac { \sqrt { \mathbf { I } } } { \sqrt { \mathbf { J } } } , \quad | \overrightarrow { \mathrm { PG } } | = \frac { \square \mathbf { K } - \sqrt { \square \mathbf { L } } } { \square \mathbf { M } } \\ \overrightarrow { \mathrm { AG } } \cdot \overrightarrow { \mathrm { PG } } = \square \mathbf { N } . \end{gathered}$$
Hence the volume of the tetrahedron PABC is $\frac { \sqrt { \mathbf { Q } } - \mathbf { P } } { \mathbf { Q } }$.

Given real numbers $x$ and $y$ that satisfy
$$\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 4 } = 1 , \quad x \geqq 0 , \quad y \geqq 0$$
we are to find the maximum value of
$$P = x ^ { 2 } + x y + y ^ { 2 } .$$
Let $x$ and $y$ satisfy the conditions. When we set $x = \sqrt { 2 } \cos \theta \left( 0 \leqq \theta \leqq \frac { \pi } { 2 } \right)$, we have
$$y = \mathbf { A } \sin \theta .$$
Thus $P$ can be represented as
$$\begin{aligned} P & = \sqrt { \mathbf { B } } \sin 2 \theta - \cos 2 \theta + \mathbf { C } \\ & = \sqrt { \mathbf { D } } \sin ( 2 \theta - \alpha ) + \mathbf { E } \end{aligned}$$
where
$$\sin \alpha = \frac { \sqrt { \mathbf { F } } } { \mathbf { G } } , \quad \cos \alpha = \frac { \sqrt { \mathbf { H } } } { \mathbf { I } } \quad \left( 0 < \alpha < \frac { \pi } { 2 } \right) .$$
Hence the maximum value of $P$ is $\sqrt { \square \mathbf { J } } + \mathbf { K }$. Let us denote the $\theta$ at which the value of $P$ is maximized by $\theta _ { 0 }$. Then we have
$$2 \theta _ { 0 } = \alpha + \frac { \pi } { \square } ,$$
and hence
$$\sin 2 \theta _ { 0 } = \frac { \sqrt { \mathbf { M } } } { \mathbf { N } } , \quad \cos 2 \theta _ { 0 } = - \frac { \sqrt { \mathbf { O } } } { \mathbf { N } } .$$

Let us define a sequence $\left\{ S _ { n } \right\}$ as
$$S _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { \sqrt { k } } \quad ( n = 1,2,3 , \cdots ) .$$
We are to find the following two limits:
$$\begin{aligned} & \lim _ { n \rightarrow \infty } S _ { n } , \\ & \lim _ { n \rightarrow \infty } \frac { S _ { 2 n } - S _ { n } } { \sqrt { n } } . \end{aligned}$$
(1) For each of A $\sim$ I in the following sentences, choose the appropriate answer from among (0) $\sim$ (9) at the bottom of this page.
Let us find $\lim _ { n \rightarrow \infty } S _ { n }$. Look at the function $y = \frac { 1 } { \sqrt { x } }$. We have
$$y ^ { \prime } = - \frac { \mathbf { A } } { 2 \sqrt { x ^ { \mathbf { B} } } } ,$$
and hence this function $y$ is $\square$ C . So, considering each interval $k \leqq x \leqq k + 1 ( k = 1,2 , \cdots , n )$, we obtain
$$\frac { 1 } { \sqrt { k } } \mathbf { D } \int _ { k } ^ { k + 1 } \frac { 1 } { \sqrt { x } } d x .$$
When we separately add the left-hand sides and the right-hand sides of this expression from $k = 1$ to $k = n$, we have
$$S _ { n } \mathbf { E } \int _ { \mathbf { F } } ^ { \mathbf { G } } \frac { 1 } { \sqrt { x } } d x = \mathbf { H } ( \sqrt { \square \mathbf { G } } - 1 )$$
and finally
$$\lim _ { n \rightarrow \infty } S _ { n } = \infty .$$
Choices: (0) $\infty$
(1) 1
(2) 2
(3) 3
(4) $n$
(5) $n + 1$ (6) $<$ (7) $>$ (8) monotonically increasing (9) monotonically decreasing
(2) For each of $\square$ J $\sim$ $\square$ P in the following, choose the appropriate answer from among (0) $\sim$ (9) below.
Let us find $\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } - S _ { n } } { \sqrt { n } }$. Since
$$S _ { 2 n } - S _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { \sqrt { \mathbf { J } } } ,$$
we have from quadrature (mensuration) by parts that
$$\begin{aligned} \lim _ { n \rightarrow \infty } \frac { S _ { 2 n } - S _ { n } } { \sqrt { n } } & = \lim _ { n \rightarrow \infty } \frac { 1 } { \mathbf { K } } \sum _ { k = 1 } ^ { n } \frac { 1 } { \sqrt { \mathbf { L } + \frac { k } { n } } } \\ & = \int _ { \mathbf { M } } ^ { \mathbf { N } } \frac { 1 } { \sqrt { 1 + x } } d x \\ & = \mathbf { O } ( \sqrt { \mathbf { P } } - 1 ) . \end{aligned}$$
Choices: (0) 0
(1) 1
(2) 2
(3) $n - 1$
(4) $n$
(5) $n + 1$ (6) $n - k$ (7) $n + k$ (8) $n + k - 1$ (9) $n + k + 1$

For each of Q , S , V in the following sentences, choose the appropriate expression from among (0) $\sim$ (7) at the bottom of this page. For the other $\square$, enter the correct number.
Suppose we have a differentiable function $f ( x )$ which satisfies the equation
$$\int _ { 0 } ^ { x } f ( t ) d t = \left( 1 + e ^ { - x } \right) f ( x ) + 2 x - 4 \log 2 \tag{1}$$
We are to find $f ( x )$ and the value of $\lim _ { x \rightarrow \infty } f ( x )$.
When we differentiate each side of (1) with respect to $x$ and transform the equation, we have
$$\left( 1 + e ^ { - x } \right) ( \mathbf { Q } ) = \mathbf { R } . \tag{2}$$
Next we set $f ( x ) = e ^ { x } g ( x )$, and using (2), we obtain
$$g ^ { \prime } ( x ) = \frac { \mathbf { S } } { 1 + e ^ { - x } }$$
and hence
$$g ( x ) = \mathbf { T } \log \left( 1 + e ^ { - x } \right) + C ,$$
where $C$ is an integral constant. Furthermore, since $g ( 0 ) = f ( 0 )$, we see that $C = \mathbf { U }$. Thus we obtain $g ( x )$ and from that,
$$f ( x ) = \mathbf { V } \log \left( 1 + e ^ { - x } \right) .$$
Finally, we set $e ^ { - x } = t$ and obtain
$$f ( x ) = \mathbf { W } \log ( 1 + t ) ^ { \frac { 1 } { t } }$$
and hence
$$\lim _ { x \rightarrow \infty } f ( x ) = \lim _ { t \rightarrow \mathbf { X } } \mathbf { W } \log ( 1 + t ) ^ { \frac { 1 } { t } } = \mathbf { Y }$$
Choices: (0) $f ^ { \prime } ( x ) - f ( x )$
(1) $f ( x ) - f ^ { \prime } ( x )$
(2) $f ^ { \prime } ( x ) - 2 f ( x )$
(3) $f ( x ) - 2 f ^ { \prime } ( x )$
(4) $2 e ^ { x }$
(5) $- 2 e ^ { x }$ (6) $2 e ^ { - x }$ (7) $- 2 e ^ { - x }$