Let $a$ and $b$ be real numbers, where $a > 0$. Consider the two quadratic functions $$f ( x ) = 2 x ^ { 2 } - 4 x + 5 , \quad g ( x ) = x ^ { 2 } + a x + b .$$ We are to find the values of $a$ and $b$ when the function $g ( x )$ satisfies the following two conditions. (i) The minimum value of $g ( x )$ is 8 less than the minimum value of $f ( x )$. (ii) There exists only one $x$ which satisfies $f ( x ) = g ( x )$. Since the minimum value of $f ( x )$ is $\mathbf { A }$, from condition (i), we derive the equality $$b = \frac { a ^ { 2 } } { \mathbf { B } } - \mathbf { C } \text {. }$$ Hence the equation from which we can find the $x$ satisfying $f ( x ) = g ( x )$ is $$x ^ { 2 } - ( a + \mathbf { D } ) x - \frac { a ^ { 2 } } { \mathbf { E } } + \mathbf { F G } = 0 .$$ Thus, since $a > 0$, from condition (ii) we obtain $$a = \mathbf { H } , \quad b = \mathbf { I J } .$$ In this case, the $x$ satisfying $f ( x ) = g ( x )$ is $\square \mathbf{ K }$.
Consider the sets $A = \{ 4 m \mid m$ is a natural number $\}$ and $B = \{ 6 m \mid m$ is a natural number $\}$. (1) For each of the following $\mathbf { L } \sim \mathbf{O}$, choose the correct answer from among (0) $\sim$ (3) below. Let $n$ be a natural number. (i) $n \in A$ is $\mathbf { L }$ for $n$ to be divisible by 2 . (ii) $n \in B$ is $\mathbf { M }$ for $n$ to be divisible by 24 . (iii) $n \in A \cup B$ is $\mathbf { N }$ for $n$ to be divisible by 3 . (iv) $n \in A \cap B$ is $\square\mathbf{O}$ for $n$ to be divisible by 12 . (0) a necessary and sufficient condition (1) a necessary condition but not a sufficient condition (2) a sufficient condition but not a necessary condition (3) neither a necessary condition nor a sufficient condition (2) Let $C = \{ m \mid m$ is a natural number satisfying $1 \leqq m \leqq 100 \}$. The number of elements which belong to $( \bar { A } \cup \bar { B } ) \cap C$ is $\mathbf { P Q }$, and the number of elements which belong to $\bar { A } \cap \bar { B } \cap C$ is $\mathbf { R S }$. Note that $\bar { A }$ and $\bar { B }$ denote the complements of $A$ and $B$, where the universal set is the set of all natural numbers.
Consider the permutations of the eight letters of the word ``POSITION''. (1) The number of permutations in which the two I's are adjacent and the two O's also are adjacent is $\mathbf { A B C }$. (2) The number of permutations such that the permutations both begin and end with the letter I and furthermore the two O's are adjacent is $\mathbf{DEF}$. (3) The number of permutations that both begin and end with the letter I is $\mathbf{GHI}$. (4) The number of permutations of the 4 letters I, I, O, O is $\mathbf { J }$. Also, the number of permutations of the 4 letters $\mathrm { N } , \mathrm { P } , \mathrm { S } , \mathrm { T }$ is $\mathbf { K L }$. Hence the number of permutations of POSITION which begin or end with either I or O, and furthermore in which none of letters $\mathrm { N } , \mathrm { P } , \mathrm { S } , \mathrm { T }$ are adjacent to each other is $\mathbf{MNO}$.
Suppose that an integer $x$ and a real number $y$ satisfy both the equation $$2 ( y + 1 ) = x ( 8 - x ) \tag{1}$$ and the inequality $$5 x - 4 y + 1 \leqq 0 . \tag{2}$$ We are to find $M$, the maximum value of $y$, and $m$, the minimum value of $y$. First of all, let us transform (1) into $$y = - \frac { 1 } { \mathbf { P } } ( x - \mathbf { Q } ) ^ { 2 } + \mathbf{R} .$$ Also, from (1) and (2) we obtain the inequality in $x$ $$2 x ^ { 2 } - \mathbf { S T } x + \mathbf { U V } \leqq 0 . \tag{3}$$ Thus when $x$ is an integer satisfying (3) if we consider the range of values which $y$ can take, we see that $y$ is maximized at $x = \square \mathbf{ V }$ and is minimized at $x = \square \mathbf{ W }$, and hence that $$M = \mathbf { X } , \quad m = \frac { \mathbf { Y } } { \mathbf{Z} } .$$
For each of $\mathbf{A} \sim \mathbf{D}$ in the following questions, choose the correct answer from among (0) $\sim$ (5) below each question. Consider the three quadratic inequalities $$\begin{aligned}
x ^ { 2 } + 3 x - 18 & < 0 \tag{1}\\
x ^ { 2 } - 2 x - 8 & > 0 \tag{2}\\
x ^ { 2 } + a x + b & < 0 . \tag{3}
\end{aligned}$$ (1) The range of $x$ which satisfies both of the inequalities (1) and (2) is $\mathbf { A }$. Also, the range of $x$ which satisfies neither inequality (1) nor (2) is $\mathbf{B}$. (0) $3 \leqq x \leqq 4$ (1) $- 6 \leqq x \leqq - 2$ (2) $3 < x < 4$ (3) $2 < x < 6$ (4) $- 6 < x < - 2$ (5) $- 4 \leqq x \leqq - 3$ (2) The range of $x$ that satisfies at least one of the inequalities (1) and (3) will be $- 6 < x < 7$, if and only if $a$ and $b$ satisfy the equation $\square \mathbf{C}$, and $a$ satisfies the inequality $\square \mathbf{D}$. (0) $b = 6 a - 36$ (1) $b = 7 a - 49$ (2) $b = - 7 a - 49$ (3) $- 10 < a \leqq - 3$ (4) $- 10 < a \leqq - 1$ (5) $- 1 \leqq a < 3$
Suppose that a quadrangle ABCD which is inscribed in a circle has the side lengths $$\mathrm { AB } = \sqrt { 2 } , \quad \mathrm { BC } = \mathrm { CD } = 2 , \quad \mathrm { DA } = \sqrt { 6 } .$$ (1) Let us set $\theta = \angle \mathrm { BAD }$. We have the two equalities $$\begin{aligned}
& \mathrm { BD } ^ { 2 } = \mathbf { A } - \mathbf { B } \sqrt { \mathbf { C } } \cos \theta , \\
& \mathrm { BD } ^ { 2 } = \mathbf { D } + \mathbf { E } \cos \theta .
\end{aligned}$$ Hence, $$\theta = \mathbf { F G } { } ^ { \circ } , \quad \mathrm { BD } = \mathbf { H } \text {. } \mathbf { I } \text {. }$$ (2) Furthermore, we have $$\angle \mathrm { BAC } = \mathbf { J K } ^ { \circ } , \quad \angle \mathrm { BCA } = \mathbf { L M } ^ { \circ } \text { and } \mathrm { AC } = \mathbf { N } + \sqrt { \mathbf { O } } \text {. }$$ We also have $$\sin \angle \mathrm { ADC } = \frac { \sqrt { \mathbf { P } } } { \mathbf{Q} } ( \sqrt { \mathbf { R } } + \mathbf { S } )$$ (3) Let us denote the point of intersection of the straight line AD and the straight line BC by E. We have $\mathrm { EB } = \mathbf { T } + \mathbf { U } \sqrt { \mathbf { V } }$.
Let $a$ and $b$ be real numbers, where $a > 0$. Consider the two quadratic functions $$f ( x ) = 2 x ^ { 2 } - 4 x + 5 , \quad g ( x ) = x ^ { 2 } + a x + b .$$ We are to find the values of $a$ and $b$ when the function $g ( x )$ satisfies the following two conditions. (i) The minimum value of $g ( x )$ is 8 less than the minimum value of $f ( x )$. (ii) There exists only one $x$ which satisfies $f ( x ) = g ( x )$. Since the minimum value of $f ( x )$ is $\mathbf{A}$, from condition (i), we derive the equality $$b = \frac { a ^ { 2 } } { \mathbf { B } } - \mathbf { C } \text {. }$$ Hence the equation from which we can find the $x$ satisfying $f ( x ) = g ( x )$ is $$x ^ { 2 } - ( a + \mathbf { D } ) x - \frac { a ^ { 2 } } { \mathbf { E } } + \mathbf { F G } = 0 .$$ Thus, since $a > 0$, from condition (ii) we obtain $$a = \mathbf { H } , \quad b = \mathbf { I J } .$$ In this case, the $x$ satisfying $f ( x ) = g ( x )$ is $\mathbf { K }$.
Consider the sets $A = \{ 4 m \mid m$ is a natural number $\}$ and $B = \{ 6 m \mid m$ is a natural number $\}$. (1) For each of the following $\mathbf { L } \sim \square \mathbf { O }$, choose the correct answer from among (0) $\sim$ (3) below. Let $n$ be a natural number. (i) $n \in A$ is $\mathbf { L }$ for $n$ to be divisible by 2 . (ii) $n \in B$ is $\mathbf { M }$ for $n$ to be divisible by 24 . (iii) $n \in A \cup B$ is $\mathbf { N }$ for $n$ to be divisible by 3 . (iv) $n \in A \cap B$ is $\mathbf{O}$ for $n$ to be divisible by 12 . (0) a necessary and sufficient condition (1) a necessary condition but not a sufficient condition (2) a sufficient condition but not a necessary condition (3) neither a necessary condition nor a sufficient condition (2) Let $C = \{ m \mid m$ is a natural number satisfying $1 \leqq m \leqq 100 \}$. The number of elements which belong to $( \bar { A } \cup \bar { B } ) \cap C$ is $\mathbf { P Q }$, and the number of elements which belong to $\bar { A } \cap \bar { B } \cap C$ is $\mathbf { R S }$. Note that $\bar { A }$ and $\bar { B }$ denote the complements of $A$ and $B$, where the universal set is the set of all natural numbers.
Let $S$ be a circle with its center at point O and a radius of 1. Let $\triangle \mathrm { ABC }$ be a triangle such that all its vertices are on $S$ and $\mathrm { AB } : \mathrm { AC } = 3 : 2$. As shown in the figure, let D be a point on the extension of side BC and $k$ be the number where $$\mathrm { BC } : \mathrm { CD } = 2 : k .$$ Moreover, set $$\overrightarrow { \mathrm { OA } } = \vec { a } , \quad \overrightarrow { \mathrm { OB } } = \vec { b } , \quad \overrightarrow { \mathrm { OC } } = \vec { c }$$ Answer the following questions. (1) When we express $\overrightarrow { \mathrm { OD } }$ in terms of $\vec { b } , \vec { c }$ and $k$, we have $$\overrightarrow { \mathrm { OD } } = \left( \frac { k } { \mathbf { A } } + \mathbf { B } \right) \vec { c } - \frac { k } { \mathbf { C } } \vec { b }$$ (2) Since the equality $$| \vec { b } - \vec { a } | = \frac { \mathbf { D } } { \mathbf { E } } | \vec { c } - \vec { a } |$$ holds, by expressing the inner product $\vec { a } \cdot \vec { b }$ in terms of the inner product $\vec { a } \cdot \vec { c }$, we have $$\vec { a } \cdot \vec { b } = \frac { \mathbf { F } } { \mathbf { G } } \vec { a } \cdot \vec { c } - \frac { \mathbf { H } } { \mathbf { I } }$$ (3) It follows that when the tangent to $S$ at the point A passes through the point D, $$k = \frac { \mathbf { J } } { \mathbf { K } } .$$
Let $p > 1$ and $q > 1$. Consider an equation in $x$ $$e ^ { 2 x } - a e ^ { x } + b = 0 \tag{1}$$ such that the equation in $t$ obtained by setting $t = e ^ { x }$ in (1) $$t ^ { 2 } - a t + b = 0 \tag{2}$$ has the solutions $\log _ { q ^ { 2 } } p$ and $\log _ { p ^ { 3 } } q$. We are to find the minimum value of $a$ and the solution of equation (1) at this minimum. (1) First of all, we see that $$b = \frac { \mathbf { A } } { \mathbf { A B } }$$ and $$a = \frac { \mathbf { C } } { \mathbf{D} } \log _ { q } p + \frac { \mathbf { E } } { \mathbf { F } } \log _ { p } q .$$ (2) As long as $p > 1$ and $q > 1$, it always follows that $\log _ { p } q > \mathbf { G }$. Hence, $a$ takes the minimum value $\frac { \sqrt { \mathbf { H } } } { \mathbf { I } }$ when $\log _ { p } q = \frac { \sqrt { \mathbf { J } } } { \mathbf { K } }$. In this case, the solution of (1) is $$x = - \frac { \mathbf { L } } { \mathbf { M } } \log _ { e } \mathbf { N } .$$
Let $a$ and $t$ be positive real numbers. Let $\ell$ be the tangent to the graph $C$ of $y = a x ^ { 3 }$ at a point $\mathrm { P } \left( t , a t ^ { 3 } \right)$, and let Q be the point at which $\ell$ intersects the curve $C$ again. Further, let $p$ be the line passing through the point P parallel to the $x$-axis; let $q$ be the line passing through the point Q parallel to the $y$-axis; and let R be the point of intersection of $p$ and $q$. Also, let us denote by $S _ { 1 }$ the area of the region bounded by the curve $C$, the straight line $p$ and the straight line $q$, and denote by $S _ { 2 }$ the area of the region bounded by the curve $C$ and the tangent $\ell$. We are to find the value of $\frac { S _ { 1 } } { S _ { 2 } }$. First, since the equation of the tangent $\ell$ is $$y = \mathbf { A } a t ^ { \mathbf{B} } x - \mathbf { C } a t ^ { \mathbf{D} } \text {, }$$ the $x$-coordinate of Q is $- \mathbf { E } t$. Hence, $S _ { 1 }$ is $$S _ { 1 } = \frac { \mathbf { F G } } { \mathbf { H } } a t ^ { \mathbf { I } } .$$ Also, since $S _ { 2 }$ is obtained by subtracting $S _ { 1 }$ from the area of the triangle PQR, we have $$S _ { 2 } = \frac { \mathbf { J K } } { \mathbf { L } } a t ^ { \mathbf { M } } .$$ Hence, the value of $\frac { S _ { 1 } } { S _ { 2 } }$ is always $$\frac { S _ { 1 } } { S _ { 2 } } = \mathbf { N } ,$$ independent of the values of $a$ and $t$.
Given the function in $x$ $$f _ { n } ( x ) = \sin ^ { n } x \quad ( n = 1,2,3 , \cdots ) ,$$ answer the following questions. (1) Consider the cases in which the equality $$\lim _ { x \rightarrow 0 } \frac { a - x ^ { 2 } - \left( b - x ^ { 2 } \right) ^ { 2 } } { f _ { n } ( x ) } = c$$ holds for three real numbers $a , b$ and $c$. (i) We have $a = b$. (ii) When $n = 2$, if $c = 6$, then $b = \frac { \mathbf { P } } { \mathbf { Q } }$. (iii) When $n = 4$, then $b = \frac { \mathbf { R } } { \mathbf { S } }$ and $c = - \mathbf { T }$. (2) For this $f _ { n } ( x )$, consider the definite integral $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } f _ { n } ( x ) \sin 2 x \, d x \quad ( n = 1,2,3 , \cdots )$$ When the integral is calculated, we have $$I _ { n } = \frac { \mathbf { U } } { n + \mathbf { V } } .$$ Hence we obtain $$\lim _ { n \rightarrow \infty } \left( I _ { n - 1 } + I _ { n } + I _ { n + 1 } + \cdots + I _ { 2 n - 2 } \right) = \int _ { 0 } ^ { \mathbf { W } } \frac { \mathbf { X } } { \mathbf { Y } + x } \, dx = \log \mathbf { Z }$$