Consider the two quadratic functions
$$f ( x ) = - 2 x ^ { 2 } , \quad g ( x ) = x ^ { 2 } + a x + b$$
Function $g ( x )$ satisfies the following two conditions:
(i) the value of $g ( x )$ is minimized at $x = 3$;
(ii) $g ( 4 ) = f ( 4 )$.
(1) From condition (i) we see that $a = -$ A . Further, from condition (ii) we see that $b = - \mathbf { B C }$. Hence the minimum value of function $g ( x )$ is $- \mathbf { D E }$.
(2) Let us find the value of $x$ such that $f ( x ) = g ( x )$ and $x$ is not 4 . Since $x$ satisfies
$$x ^ { 2 } - \mathbf { F } x - \mathbf { G } \mathbf { G } = 0 \text {, }$$
we obtain $x = - \mathbf { H }$.
(3) The value of $f ( x ) - g ( x )$ on $- \mathrm { H } \leqq x \leqq 4$ is maximized at $x = \square$, and its maximum value is JK.

For a game, each of two people, A and B , has a bag containing three cards on which the numbers 1, 2, and 3 are written, each number on a different card. In the game, A and B each take out one card from their own bag and compare the numbers. If the numbers are the same, the game is a draw. If the numbers are different, the person with the greater number wins.
(1) For a single game the probability of a draw is $\frac { \mathbf { L } } { \mathbf { M } }$.
(2) If this game is successively played four times, replacing the cards after each game, let us find the probabilities for the following.
(i) The probability that A wins three times or more is $\frac { \mathbf { N } } { \mathbf{O} }$.
(ii) The probability that A wins once and loses once and two games are draws is $\frac { \mathrm { P } } { \mathrm { QR } }$.
(iii) The probability that the number of times that A wins and the number of times that B wins are the same is $\frac { \mathbf { S T } } { \mathbf { U V } }$. Hence, the probability that the number of times that A wins is greater than the number of times that B wins is $\frac { \mathbf { W } \mathbf { X } } { \mathbf{UV} }$.

Answer the following questions.
(1) The positive integers $m$ and $n$ which simultaneously satisfy the following two inequalities
$$\frac { m } { 3 } < \sqrt { 3 } < \frac { n } { 4 } , \quad \frac { n } { 3 } < \sqrt { 6 } < \frac { m } { 2 }$$
are
$$m = \mathbf { A } , \quad n = \mathbf { B } .$$
(2) Using the results of (1), let us compare the sizes of numbers (1) $\sim$ (5).
(1) $( \sqrt { ( - 3 ) ( - 4 ) } ) ^ { 3 }$
(2) $6 \sqrt { ( - 2 ) ^ { 3 } ( - 3 ) }$
(3) $\sqrt { \left\{ ( - 4 ) ( - 3 ) ^ { 2 } \right\} ^ { 2 } }$
(4) $( - 1 ) ^ { 3 } \sqrt { \left\{ ( - 2 ) ^ { 5 } \right\} ^ { 2 } }$
(5) $\left( \frac { 5 \sqrt { 3 } } { 1 - \sqrt { 6 } } \right) ^ { 2 }$
When the denominator of (5) is rationalized, we have
$$\left( \frac { 5 \sqrt { 3 } } { 1 - \sqrt { 6 } } \right) ^ { 2 } = \mathbf { C D } + \mathbf { E } \sqrt { \mathbf { F } }$$
Of the five numbers, there are $\mathbf { G }$ number(s) greater than 35 and $\mathbf { H }$ negative number(s).
When we arrange the five numbers in the ascending order of their size using the numbers (1) $\sim$ (5), we have
$$\mathbf { I } < \mathbf { J } < \mathbf { K } < \mathbf { L } < \mathbf { M } .$$

The function $f ( x ) = x ^ { 2 } + a x + b$ satisfies the following two conditions:
(i) $\quad f ( 3 ) = 1$;
(ii) $13 \leqq f ( - 1 ) \leqq 25$.
We are to express the minimum value $m$ of $f ( x )$ in terms of $a$. In addition, we are to find the maximum and minimum values of $m$.
From condition (i), $a$ and $b$ satisfy
$$\mathbf { N } a + b + \mathbf { O } = 0 \text {. }$$
From this, $f ( x )$ can be expressed in terms of $a$ as
$$f ( x ) = x ^ { 2 } + a x - \mathbf { P } a - \mathbf { Q } .$$
Hence from condition (ii), $a$ satisfies
$$- \mathbf { R } \leqq a \leqq - \mathbf { S } .$$
On the other hand, $m$ can be expressed in terms of $a$ as
$$m = - \frac { 1 } { \mathbf { T } } ( a + \mathbf { U } ) ^ { 2 } + \mathbf { V }$$
Thus $m$ is maximized at $a = - \mathbf { W }$, and its maximum value is $\mathbf { X }$; it is minimized at $a = - \mathbf { Y }$, and its minimum value is $\mathbf { Z }$.

Let $N$ be a positive integer. Both when it is written in base 5 and when it is written in base 9, it is a 3-digit number, but the order of the numerals is reversed. We are to represent $N$ in base 10 (decimal) and in base 4.
Let $N$ be $abc$ in base 5 and $cba$ in base 9. Then we have
$$\mathbf { A } \leq a \leq \mathbf { B } , \quad \mathbf { C } \leq b \leq \mathbf { D } , \quad \mathbf { E } \leqq c \leqq \mathbf { F } \text {. }$$
Since we also have
$$N = \mathbf { G H } a + \mathbf { I } \quad b + c = \mathbf { J K } c + \mathbf { L } \quad b + a ,$$
we obtain
$$b = \mathbf { M } a - \mathbf { N O } c .$$
The $a$, $b$ and $c$ satisfying (1) and (2) are
$$a = \mathbf { P } , \quad b = \mathbf { Q } , \quad c = \mathbf { R } .$$
Thus $N$ expressed in base 10 is $\mathbf { S T U }$, and $N$ expressed in base 4 is $\mathbf { V W X Y }$.

In a triangle ABC, let $\angle \mathrm { B } = 45 ^ { \circ }$ and $\angle \mathrm { C } = 75 ^ { \circ }$, and let D be the intersection of the bisector of $\angle \mathrm { A }$ and side BC.
(1) From the law of sines we have
$$\mathrm { AC } = \frac { \sqrt { \mathbf { A } } } { \sqrt { \mathbf { B } } } \mathrm { BC } , \quad \mathrm { AD } = \sqrt { \mathbf { C } } \mathrm { BD } .$$
In particular, from $\angle \mathrm { ADC } = \mathbf { D E } ^ { \circ }$ we see that
$$\mathrm { BD } : \mathrm { BC } = \mathbf { F } : \sqrt { \mathbf { G } }$$
and hence we have
$$\mathrm { AB } : \mathrm { AC } = \mathbf { H } : \left( \sqrt { \mathbf { I } } - \frac { \mathbf { J } }{\mathbf{J} } \right) .$$
(2) Let $\mathrm { O } _ { 1 }$ be the center of the circumscribed circle of triangle ABD, and let $\mathrm { O } _ { 2 }$ be the center of the circumscribed circle of triangle ADC. Let us find the ratio of the areas of triangle ABC and triangle $\mathrm { AO } _ { 1 } \mathrm { O } _ { 2 }$, $\triangle \mathrm { ABC } : \triangle \mathrm { AO } _ { 1 } \mathrm { O } _ { 2 }$.
Since $\angle \mathrm { AO } _ { 1 } \mathrm { D } = \mathbf { K } \mathbf { L } ^ { \circ }$ and $\angle \mathrm { AO } _ { 2 } \mathrm { O } _ { 1 } = \mathbf { M N } ^ { \circ }$, by the same reasoning as (1), we have
$$\mathrm { AC } = \sqrt { \mathbf { O } } \mathrm { AO } _ { 1 } , \quad \mathrm { AO } _ { 2 } = ( \sqrt { \mathbf { P } } - \mathbf { Q } ) \mathrm { AO } _ { 1 } .$$
Hence we obtain
$$\triangle \mathrm { ABC } : \triangle \mathrm { AO } _ { 1 } \mathrm { O } _ { 2 } = \mathbf { R } : ( \mathbf { S } - \sqrt { \mathbf { T } } ) .$$

Consider the two quadratic functions
$$f ( x ) = - 2 x ^ { 2 } , \quad g ( x ) = x ^ { 2 } + a x + b$$
Function $g ( x )$ satisfies the following two conditions:
(i) the value of $g ( x )$ is minimized at $x = 3$;
(ii) $\quad g ( 4 ) = f ( 4 )$.
(1) From condition (i) we see that $a = -$ A . Further, from condition (ii) we see that $b = - \mathbf { B C }$. Hence the minimum value of function $g ( x )$ is $- \mathbf { D E }$.
(2) Let us find the value of $x$ such that $f ( x ) = g ( x )$ and $x$ is not 4 . Since $x$ satisfies
$$x ^ { 2 } - \mathbf { F } x - \mathbf { G } \mathbf { G } = 0 \text {, }$$
we obtain $x = - \mathbf { H }$.
(3) The value of $f ( x ) - g ( x )$ on $- \mathrm { H } \leqq x \leqq 4$ is maximized at $x = \square$, and its maximum value is JK.

For a game, each of two people, A and B , has a bag containing three cards on which the numbers 1, 2, and 3 are written, each number on a different card. In the game, A and B each take out one card from their own bag and compare the numbers. If the numbers are the same, the game is a draw. If the numbers are different, the person with the greater number wins.
(1) For a single game the probability of a draw is $\frac { \mathbf { L } } { \mathbf { M } }$.
(2) If this game is successively played four times, replacing the cards after each game, let us find the probabilities for the following.
(i) The probability that A wins three times or more is $\frac { \mathbf { N } } { \mathbf{O} }$.
(ii) The probability that A wins once and loses once and two games are draws is $\frac { \mathrm { P } } { \mathrm { QR } }$.
(iii) The probability that the number of times that A wins and the number of times that B wins are the same is $\frac { \mathbf { S T } } { \mathbf { U V } }$. Hence, the probability that the number of times that A wins is greater than the number of times that B wins is $\frac { \mathbf { W } \mathbf { X } } { \mathbf{UV} }$.

For $\mathbf { C } , \mathbf { D } , \mathbf { E } , \mathbf { F } , \mathbf { G }$ in the following sentences, choose the correct answer from among choices (0) $\sim$ (9) below. For the other $\square$, enter the correct number.
Consider a regular tetrahedron OABC with sides of length 1. Let $x$ be a number satisfying $0 < x < 1$, and let P be the point that divides side AB by the ratio $x : ( 1 - x )$ and Q be the point that divides side BC by the ratio $x : ( 1 - x )$. Also, let $\overrightarrow { \mathrm { OA } } = \vec { a } , \overrightarrow { \mathrm { OB } } = \vec { b }$ and $\overrightarrow { \mathrm { OC } } = \vec { c }$. We are to find the range of values of $\cos \angle \mathrm { POQ }$.
The vectors $\vec { a } , \vec { b }$ and $\vec { c }$ satisfy
$$\vec { a } \cdot \vec { b } = \vec { b } \cdot \vec { c } = \vec { c } \cdot \vec { a } = \frac { \mathbf { A } } { \mathbf { B } }$$
Next, since we can express $\overrightarrow { \mathrm { OP } }$ and $\overrightarrow { \mathrm { OQ } }$ as $\overrightarrow { \mathrm { OP } } = \square \mathbf { C }$ and $\overrightarrow { \mathrm { OQ } } = \mathbf { D }$, we have
$$| \overrightarrow { \mathrm { OP } } | = | \overrightarrow { \mathrm { OQ } } | = \sqrt { \vec { E } } , \quad \overrightarrow { \mathrm { OP } } \cdot \overrightarrow { \mathrm { OQ } } = \square \mathbf { F } .$$
Hence we obtain
$$\cos \angle \mathrm { POQ } = \frac { 1 } { \mathbf { G } } - \frac { \mathbf { H } } { \mathbf { I } } .$$
From this we finally obtain
$$\frac { \square \mathbf { J } } { \mathbf { K } } < \cos \angle \mathrm { POQ } \leqq \frac { \mathbf { L } } { \mathbf { M } } .$$
(0) $( 1 - x ) \vec { a } + x \vec { b }$
(1) $x \vec { a } + ( 1 - x ) \vec { b }$
(2) $( 1 - x ) \vec { b } + x \vec { c }$
(3) $x \vec { b } + ( 1 - x ) \vec { c }$
(4) $x ^ { 2 } + x + 1$
(5) $x ^ { 2 } - x + 1$ (6) $x ^ { 2 } - x - 1$ (7) $\frac { 1 } { 2 } \left( - x ^ { 2 } + x + 1 \right)$ (8) $\frac { 1 } { 2 } \left( - x ^ { 2 } - x + 1 \right)$ (9) $\frac { 1 } { 2 } \left( - x ^ { 2 } + x - 1 \right)$

We have a triangle ABC on the complex plane whose vertices are the three points $\mathrm { A } ( \alpha )$, $\mathrm { B } ( \beta )$ and $\mathrm { C } ( \gamma )$ that satisfy
$$\frac { \gamma - \alpha } { \beta - \alpha } = 1 - i$$
(In the following, the range of an argument $\theta$ is $0 \leqq \theta < 2 \pi$.)
(1) When we express the complex number $\frac { \gamma - \alpha } { \beta - \alpha }$ in polar form, we have
$$\frac { \gamma - \alpha } { \beta - \alpha } = \sqrt { \mathbf { N } } \left( \cos \frac { \mathbf { O } } { \mathbf { P } } \pi + i \sin \frac { \mathbf { O } } { \mathbf { P } } \pi \right) .$$
Hence we see that point C is the point resulting from rotating point B by $\frac { \square \mathbf { Q } } { \mathbf{R} } \pi$ around point A and then changing its distance from point A to its distance multiplied by $\sqrt { \mathbf { S } }$. From this we also see that the absolute value and the argument of the complex number $w = \frac { \gamma - \beta } { \alpha - \beta }$ are
$$| w | = \mathbf { T } \quad \text { and } \quad \arg w = \frac { \mathbf { U } } { \mathbf { 4 } } \pi .$$
(2) If $\alpha + \beta + \gamma = 0$, then we have that
$$| \alpha | : | \beta | : | \gamma | = \sqrt { \mathbf { W } } : \sqrt { \mathbf { X } } : \sqrt { \mathbf { Y } } .$$

We are to find the minimum value of the function
$$f ( x ) = 8 ^ { x } + 8 ^ { - x } - 3 \left( 4 ^ { 1 + x } + 4 ^ { 1 - x } - 2 ^ { 4 + x } - 2 ^ { 4 - x } \right) - 24$$
and the value of $x$ at which the function takes this minimum value.
First, let us set $2 ^ { x } + 2 ^ { - x } = t$. Then, since
$$4 ^ { x } + 4 ^ { - x } = t ^ { 2 } - \mathbf { A } \quad \text { and } \quad 8 ^ { x } + 8 ^ { - x } = t ^ { 3 } - \mathbf { B } t ,$$
$f ( x )$ can be expressed as
$$f ( x ) = t ^ { 3 } - \mathbf { C D } t ^ { 2 } + \mathbf { E F } t$$
When we consider the right side as a function of $t$ and denote it by $g ( t )$, its derivative is
$$g ^ { \prime } ( t ) = \mathbf { G } ( t - \mathbf { H } ) ( t - \mathbf { I } ) \text {, }$$
where $\mathrm { H } < \mathrm { I }$. Here, since $2 ^ { x } + 2 ^ { - x } = t$, the range of the values which $t$ takes is
$$t \geqq \mathbf { J }$$
When $t = \mathbf { J }$, we see that $g ( \mathbf { J } ) = \mathbf { K L }$. When $t > \mathbf { J }$, $g ( t )$ is locally maximized at $t = \mathbf { M }$, and its local maximum is $\mathbf { N O }$, and furthermore, it is locally minimized at $t = \mathbf { P }$, and its local minimum is $\mathbf { Q R }$.
Thus, the minimum value of $f ( x )$ is $\mathbf { S T }$, which is taken at
$$x = \mathbf { U } \quad \text { and } \quad x = \log _ { 2 } ( \mathbf { V } \pm \sqrt { \mathbf { W X } } ) - \mathbf { Y } .$$

Let $k$ be a positive real number. Consider the two curves
$$C _ { 1 } : y = \sin ^ { 2 } x , \quad C _ { 2 } : y = k \cos 2 x \quad \left( 0 \leqq x \leqq \frac { \pi } { 2 } \right)$$
Let $S _ { 1 }$ be the area of the region bounded by the two curves $C _ { 1 } , C _ { 2 }$ and the $y$-axis, and let $S _ { 2 }$ be the area of the region bounded by the two curves $C _ { 1 } , C _ { 2 }$ and the straight line $x = \frac { \pi } { 2 }$. We are to show that the value of $S _ { 2 } - S _ { 1 }$ is a constant independent of the value of $k$.
When we denote the $x$ satisfying the equation $\sin ^ { 2 } x = k \cos 2 x$ by $\alpha$, we have
$$\sin \alpha = \sqrt { \frac { k } { \mathbf { A } k + \mathbf { B } } } , \quad \cos \alpha = \sqrt { \frac { k + \mathbf { C } } { \mathbf { D } k + \mathbf { E } } } .$$
Then we have
$$\begin{aligned} & S _ { 1 } = \frac { \mathbf { F } } { \mathbf { F G } } \int _ { 0 } ^ { \alpha } \{ ( \mathbf { H } k + \mathbf { I } ) \cos \mathbf { J } x - 1 \} d x \\ & = \frac { \mathbf { K } } { \mathbf { L } } \{ \sqrt { k ( k + \mathbf { M } ) } - \alpha \} , \\ & S _ { 2 } = \frac { \mathbf { N } } { \mathbf { O } } \{ \sqrt { k ( k + \mathbf{P} ) } - \alpha \} + \frac { \pi } { \mathbf { Q } } . \end{aligned}$$
Hence, we obtain
$$S _ { 2 } - S _ { 1 } = \frac { \pi } { \mathbf { R } } ,$$
which shows that the value of $S _ { 2 } - S _ { 1 }$ is a constant independent of the value of $k$.