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 $\mathbf { G }$.
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 $\frac { \sqrt { \mathbf { Q } } - \mathbf { P } } { \mathbf { Q } }$.
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) $- 2 e ^ { - x }$