Consider the quadratic function in $x$
$$y = a x ^ { 2 } + b x + c .$$
The function (1) takes its maximum value 16 at $x = 1$, its graph intersects the $x$-axis at two points, and the length of the segment connecting those two points is 8. We are to find the values of $a$, $b$ and $c$.
From the conditions, (1) can be represented as
$$y = a ( x - \mathbf { A } ) ^ { 2 } + \mathbf { B } \mathbf { C }$$
and the coordinates of the two points at which the graph of (1) and the $x$-axis intersect are
$$( - \mathbf { D } , 0 ) , \quad ( \mathbf { E } , 0 ) .$$
Thus we obtain $a = \mathbf { F G }$. Hence we have
$$b = \mathbf { H } , \quad c = \mathbf { I J } .$$

In a box there are ten cards on which the numbers from 0 to 9 have been written successively. We take three cards out of the box using two methods and consider the probabilities.
(1) We take out three cards simultaneously.
(i) The probability that each number on the three cards is 2 or more and 6 or less is $\dfrac{\mathbf{KL}}{\mathbf{MN}}$.
(ii) The probability that the smallest number is 2 or less and the greatest number is 8 or more is $\dfrac { \mathbf { N O } } { \mathbf { P Q } }$.
(2) Three times we take out one card from the box, check its number, and then return it to the box. The probability that the smallest number is 2 or more and the greatest number is 6 or less is $\dfrac { \mathbf { R } } { \mathbf { S } }$.

Let $n$ be a natural number and $a$ be a real number, where $a \neq 0$. Suppose that the integral expression $x ^ { n } + y ^ { n } + z ^ { n } + a ( x y + y z + z x )$ can be expressed as the product of $x + y + z$ and an integral expression $P$ in $x$, $y$ and $z$, i.e.
$$x ^ { n } + y ^ { n } + z ^ { n } + a ( x y + y z + z x ) = ( x + y + z ) P .$$
We are to find the values of $n$ and $a$.
The equality (1) holds for all $x$, $y$ and $z$. So, consider for example, two triples of $x$, $y$ and $z$ that satisfy $x + y + z = 0$:
$$x = y = 1 , \quad z = - \mathbf { A }$$
and
$$x = y = - \frac { \mathbf { B } } { \mathbf { C } } , \quad z = 1 .$$
By substituting each triple in (1), we obtain the following two equations:
$$\left( - \mathbf { A } \right) ^ { n } = \mathbf { D } \text{ a} - \frac{\mathbf{E}}{\mathbf{F}}$$
$$\left( - \frac { \mathbf { B } } { \mathbf { C } } \right) ^ { n } = \frac { \mathbf{E} }{ \mathbf{F} } \text{ a} - \frac { \mathbf { H } } { \mathbf { I } } .$$
From these two equations, we get an equation in $a$. Solving this, we obtain $a = \mathbf { K }$ and hence by the first equation that $n = \mathbf { L }$.
Conversely, when $a = \mathbf{K}$ and $n = \mathbf{L}$, there exists a $P$ such that (1) holds, and hence these values of $a$ and $n$ are the solution.

Consider all segments PQ of length 2 such that the end points P and Q are on the parabola $y = x ^ { 2 }$. Denote the mid-point of the segment PQ by M. Among all M, we are to find the coordinates of the ones nearest to the $x$-axis.
Let us denote the coordinates of the end points of segment PQ by $\mathrm{ P }\left( p , p ^ { 2 } \right)$ and $\mathrm{ Q }\left( q , q ^ { 2 } \right)$. Then the $y$-coordinate $m$ of M is
$$m = \frac { p ^ { 2 } + q ^ { 2 } } { \mathbf { M } } .$$
Next, since $\mathrm{ PQ } = 2$, then
$$( p - q ) ^ { 2 } + \left( p ^ { 2 } - q ^ { 2 } \right) ^ { 2 } = \mathbf { N }$$
by the Pythagorean theorem.
Now, when we set $t = p q$, we obtain from (1) and (2) the quadratic equation in $m$
$$\mathbf { O } m ^ { 2 } + m - \mathbf { P } t ^ { 2 } - t - \mathbf { Q } = 0 .$$
When we solve this for $m$, noting that $m > 0$, we have
$$m = - \frac { 1 } { \mathbf { R } } + \sqrt { \left( t + \frac { 1 } { \mathbf { S } } \right) ^ { 2 } + \mathbf{T} } .$$
This shows that $m$ is minimized when $t = - \dfrac { 1 } { \mathbf{U} }$. In this case, $p q = - \dfrac { 1 } { \mathbf{U} }$ and $p ^ { 2 } + q ^ { 2 } = \dfrac { \mathbf { V } } { \mathbf { V } }$, and so we have $p + q = \pm \mathbf { W }$.
Thus the coordinates of the M nearest to the $x$-axis are $\left( \pm \dfrac { 1 } { \mathbf { X } } , \dfrac { \mathbf { Y } } { \mathbf { Z } } \right)$.

(1) Answer the following questions.
(i) Consider an integer $a$. When $a$ is divided by 5, the remainder is 4. So, $a$ can be represented as
$$a = \mathbf { A } \, k + \mathbf { B } \quad ( k \text{ : an integer}).$$
Hence, when $a ^ { 2 }$ is divided by 5, the remainder is $\mathbf { C }$.
(ii) The number written as the three-digit number $120_{(3)}$ in the base-3 system is $\mathbf{DE}$ in the decimal system.
The greatest natural number that can be expressed in three digits using the base-3 system is $\mathbf { F G }$ in the decimal system, and the smallest is $\mathbf { H }$ in the decimal system.
(2) For each of $\mathbf { I }$, $\mathbf { J }$ in the following statements, choose the correct answer from among (0) $\sim$ (3) below.
In the following, let $a$ be an integer and $b$ be a natural number.
(i) "When $a$ is divided by 5, the remainder is 4" is $\mathbf { I }$ for "when $a ^ { 2 }$ is divided by 5, the remainder is $\mathbf { C }$".
(ii) "$b$ satisfies $6 \leqq b \leqq 30$" is $\mathbf { J }$ for "$b$ is a three-digit number in the base-3 system".
(0) a necessary condition but not a sufficient condition
(1) a sufficient condition but not a necessary condition
(2) a necessary and sufficient condition
(3) neither a necessary condition nor a sufficient condition

Consider a triangle ABC where $\angle \mathrm{ BAC } = 60 ^ { \circ }$.
Let D be the point of intersection of the bisector of $\angle \mathrm{ BAC }$ and the side BC. Let DE and DF be the line segments perpendicular to sides AB and AC, respectively. Let us set
$$x = \frac { \mathrm{ AB } } { \mathrm{ AC } } , \quad k = \frac { \triangle \mathrm{ DEF } } { \triangle \mathrm{ ABC } } .$$
Note that $\triangle \mathrm{ ABC }$ denotes the area of the triangle ABC, and similarly for other triangles.
(1) We are to represent $k$ in terms of $x$. Since $\triangle \mathrm{ ABD } + \triangle \mathrm{ ACD } = \triangle \mathrm{ ABC }$, when we set $b = \mathrm{ AB }$, $c = \mathrm{ AC }$ and $d = \mathrm{ AD }$, we have
$$d = \frac { \sqrt { \mathbf { A } } \, b c } { b + c } .$$
Next, since $\mathrm{ DE } = \mathrm{ DF } = \dfrac { \mathbf { B } } { \mathbf { C } } d$, we have
$$\triangle \mathrm{ DEF } = \frac { \sqrt { \mathbf { D } } } { \mathbf { EF } } d ^ { 2 } .$$
From (1) and (2), we see that
$$k = \frac { d ^ { 2 } } { \mathbf { G } \, b c } = \frac { \mathbf { H } \, b c } { \mathbf { I } ( b + c ) ^ { 2 } } .$$
Since $x = \dfrac { b } { c }$, we have
$$k = \frac { \mathbf { J } \, x } { \mathbf { K } ( x + \mathbf { L } ) ^ { 2 } } .$$
(2) If $\mathrm{ BD } = 8$ and $\mathrm{ BC } = 10$, then $x = \mathbf { M }$ and $k = \dfrac { \mathbf { N } } { \mathbf { O P } }$.

Consider the quadratic function in $x$
$$y = a x ^ { 2 } + b x + c .$$
The function (1) takes its maximum value 16 at $x = 1$, its graph intersects the $x$-axis at two points, and the length of the segment connecting those two points is 8. We are to find the values of $a$, $b$ and $c$.
From the conditions, (1) can be represented as
$$y = a ( x - \mathbf { A } ) ^ { 2 } + \mathbf { B } \mathbf { C } ,$$
and the coordinates of the two points at which the graph of (1) and the $x$-axis intersect are
$$( - \mathbf { D } , 0 ) , \quad ( \mathbf { E } , 0 ) .$$
Thus we obtain $a = \mathbf { F G }$. Hence we have
$$b = \mathbf { H } , \quad c = \mathbf { I J } .$$

In a box there are ten cards on which the numbers from 0 to 9 have been written successively. We take three cards out of the box using two methods and consider the probabilities.
(1) We take out three cards simultaneously.
(i) The probability that each number on the three cards is 2 or more and 6 or less is $\dfrac{\mathbf{KL}}{\mathbf{MN}}$.
(ii) The probability that the smallest number is 2 or less and the greatest number is 8 or more is $\dfrac { \mathbf { N O } } { \mathbf { P Q } }$.
(2) Three times we take out one card from the box, check its number, and then return it to the box. The probability that the smallest number is 2 or more and the greatest number is 6 or less is $\dfrac { \mathbf { R } } { \mathbf { S } }$.

Consider a sequence of positive numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , \cdots$ which satisfies
$$\begin{aligned} a _ { 1 } & = 1 , \quad a _ { 2 } = 10 , \\ \left( a _ { n } \right) ^ { 2 } a _ { n - 2 } & = \left( a _ { n - 1 } \right) ^ { 3 } \quad ( n = 3,4 , \cdots ) . \end{aligned}$$
We are to find $\lim _ { n \rightarrow \infty } a _ { n }$.
By finding the common logarithm of both sides of (1), we obtain
$$\mathbf { A } \log _ { 10 } a _ { n } + \log _ { 10 } a _ { n - 2 } = \mathbf { B } \log _ { 10 } a _ { n - 1 } .$$
When we set $b _ { n } = \log _ { 10 } a _ { n }$ $(n = 1,2 , \cdots)$, this equality is expressed as
$$\mathbf { A } b _ { n } + b _ { n - 2 } = \mathbf { B } b _ { n - 1 } .$$
By transforming (2), we have
$$b _ { n } - b _ { n - 1 } = \frac { 1 } { \mathbf { C } } \left( b _ { n - 1 } - b _ { n - 2 } \right) \quad ( n = 3,4 , \cdots ) ,$$
which gives
$$b _ { n } - b _ { n - 1 } = \left( \frac { 1 } { \mathbf { C } } \right) ^ { n - \mathbf { D } } \left( b _ { 2 } - b _ { 1 } \right) \quad ( n = 2,3 , \cdots ) .$$
Since $b _ { 1 } = \mathbf { E }$ and $b _ { 2 } = \mathbf { F }$, from (3) we get
$$b _ { n } = \sum _ { k = 2 } ^ { n } \left( \frac { 1 } { \mathbf { C } } \right) ^ { k - \mathbf { G } }$$
and hence
$$b _ { n } = \mathbf { H } - \left( \frac { 1 } { \mathbf { C } } \right) ^ { n - \mathbf{I} }$$
Finally, we obtain
$$\lim _ { n \rightarrow \infty } a _ { n } = \mathbf { J K L } .$$

Let $\alpha$ and $\beta$ be the two solutions of the quadratic equation $x ^ { 2 } + \sqrt { 3 } x + 1 = 0$, where $0 < \arg \alpha < \arg \beta < 2 \pi$. Consider the complex numbers $z$ satisfying the following three conditions:
$$\begin{cases} \arg \dfrac { \alpha - z } { \beta - z } = \dfrac { \pi } { 2 } & \ldots\ldots\ldots (1)\\ ( 1 + i ) z + ( 1 - i ) \bar { z } + k = 0 & \ldots\ldots\ldots (2)\\ \dfrac { \pi } { 2 } < \arg z < \pi , & \ldots\ldots\ldots (3) \end{cases}$$
where $k$ is a real number.
Let us denote the points on the complex number plane which express $\alpha$, $\beta$ and $z$ by $\mathrm{ A }$, $\mathrm{ B }$ and P.
(1) The arguments of $\alpha$ and $\beta$ are
$$\arg \alpha = \frac { \mathbf { A } } { \mathbf { B } } \pi \quad \text{ and } \quad \arg \beta = \frac { \mathbf { C } } { \mathbf { D } } \pi .$$
(2) For each of $\mathbf { E }$ $\sim$ $\mathbf { Q }$ in the following sentences, choose the correct answer from among (0) $\sim$ (9) below.
Since $\mathbf { E } = \dfrac { \pi } { 2 }$ from (1), the point P is located on the circumference of the circle with the center $-\dfrac{ \sqrt{\mathbf{F}} }{ \mathbf{G} }$ and the radius $\dfrac { \mathbf { H } } { \mathbf { I } }$.
On the other hand, from (2), the point P is on the straight line which has the slope $\mathbf{J}$ and passes through a certain point.
From these, we see that when $n$ is the number of complex numbers $z$ which simultaneously satisfy (1), (2) and (3), the maximum value of $n$ is $\mathbf { M }$, and in this case the range of values of $k$ is
$$\mathbf { N } + \sqrt { \mathbf { O } } < k < \sqrt { \mathbf { P } } + \sqrt { \mathbf { Q } }$$
where $\mathbf { P } < \mathbf { Q }$.
(0) 0
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5 (6) 6 (7) $\angle \mathrm{ PAB }$ (8) $\angle \mathrm{ PBA }$ (9) $\angle \mathrm{ APB }$

Let $x$ satisfy the inequality
$$2 \left( \log _ { \frac { 1 } { 3 } } x \right) ^ { 2 } + 9 \log _ { \frac { 1 } { 3 } } x + 9 \leqq 0 .$$
We are to find the maximum value of the function
$$f ( x ) = \left( \log _ { 3 } x \right) \left( \log _ { 3 } \frac { x } { 3 } \right) \left( \log _ { 3 } \frac { x } { 9 } \right) .$$
The range of values of $x$ satisfying (1) is
$$\mathbf { A } \sqrt { \mathbf { B } } \leqq x \leqq \mathbf { C D } .$$
When we set $t = \log _ { 3 } x$, the range of values of $t$ is
$$\frac { \mathbf { E } } { \mathbf { F } } \leqq t \leqq \mathbf { G } .$$
When we express the right side of (2) in terms of $t$ and consider it as a function $g ( t )$, its derivative is
$$g ^ { \prime } ( t ) = \mathbf { H } t ^ { 2 } - \mathbf { I } t + \mathbf { J } .$$
Hence $f ( x )$ is maximized at $x = \mathbf { K L }$, and its maximum value is $\mathbf { M }$.

Let $a > 0$. Consider the region of a plane bounded by the curve $y = \sqrt { x } e ^ { - x }$, the $x$-axis, and the straight line $x = a$ which passes through the point $\mathrm{ A }( a , 0 )$, and let $V$ be the volume of the solid obtained by rotating this region once about the $x$-axis.
(1) $V$ is expressed as a function in $a$ by
$$V = - \frac { \pi } { 4 } \left\{ ( \mathbf { N } a + \mathbf { O } ) e ^ { - \mathbf { P } a } - \mathbf { Q R } \right\} .$$
(2) Suppose that the point A starts at the origin and moves along the $x$-axis in the positive direction and that its speed at $t$ seconds is $4t$. Then the rate of change of $V$ at $t$ seconds is
$$\frac { d V } { d t } = \mathbf { R } \pi t ^ { \mathbf { S } } e ^ { - \mathbf { T } t ^ { \mathbf { U } } } .$$
This rate of change is maximized at
$$t = \frac { \sqrt { \mathbf { V } } } { 4 } ,$$
and the value of $V$ at this time is
$$V = - \frac { \pi } { 8 } \left( \mathbf { W } e ^ { - \frac { \mathbf { X } } { \mathbf { Y } } } - \mathbf { Z } \right) .$$