Suppose that positive real numbers $a$ and $b$ satisfy
$$a ^ { 2 } = 3 + \sqrt { 5 } , \quad b ^ { 2 } = 3 - \sqrt { 5 } .$$
Let $c$ be the fractional portion of $a + b$. We are to find the value of $\frac { 1 } { c } - c$.
(1) We see that $( a b ) ^ { 2 } = \mathbf { N }$ and $( a + b ) ^ { 2 } = \mathbf { O P }$.
(2) Since $\mathbf { Q }$ $< a + b < \mathbf { Q } + 1$, the value of $c$ is $\sqrt { \mathbf { R S } } - \mathbf { T }$.
Thus we obtain $\frac { 1 } { c } - c = \mathbf { U }$.

Suppose that positive real numbers $a$ and $b$ satisfy
$$a ^ { 2 } = 3 + \sqrt { 5 } , \quad b ^ { 2 } = 3 - \sqrt { 5 } .$$
Let $c$ be the fractional portion of $a + b$. We are to find the value of $\frac { 1 } { c } - c$.
(1) We see that $( a b ) ^ { 2 } = \mathbf { N }$ and $( a + b ) ^ { 2 } = \mathbf { O P }$.
(2) Since $\mathbf { Q }$ $< a + b < \mathbf { Q } + 1$, the value of $c$ is $\sqrt { \mathbf { R S } } - \mathbf { T }$.
Thus we obtain $\frac { 1 } { c } - c = \mathbf { U }$.

For the real numbers $a$ and $b$ satisfying
$$a ^ { 3 } = \frac { 1 } { \sqrt { 5 } - 2 } , \quad b ^ { 3 } = 2 - \sqrt { 5 }$$
we are to find the value of $a + b$.
When we set $x = a + b$, we have
$$x ^ { 3 } = ( a + b ) ^ { 3 } = a ^ { 3 } + b ^ { 3 } + \mathbf { A } a b ( a + b ) .$$
Since $a b = \mathbf { B C }$, we know that this $x$ satisfies
$$x ^ { 3 } + \mathbf { D } x - \mathbf { E } = 0 .$$
The left side of this equation can be factorized as follows:
$$\begin{aligned} x ^ { 3 } + \mathbf { D } x - \mathbf { E } & = \left( x ^ { 3 } - \mathbf { F } \right) + \mathbf { D } \left( x - \frac { \mathbf { F } } { \mathbf { F } } \right) \\ & = ( x - \mathbf { F } ) \left( x ^ { 2 } + x + \mathbf { G } \right) . \end{aligned}$$
Since
$$x ^ { 2 } + x + \mathbf { G } = \left( x + \frac { \mathbf { H } } { \mathbf { I } } \right) ^ { 2 } + \frac { \mathbf { J K } } { \mathbf { L } } > 0 ,$$
we obtain $x = a + b = \mathbf { M }$.

For the real numbers $a$ and $b$ satisfying
$$a ^ { 3 } = \frac { 1 } { \sqrt { 5 } - 2 } , \quad b ^ { 3 } = 2 - \sqrt { 5 }$$
we are to find the value of $a + b$.
When we set $x = a + b$, we have
$$x ^ { 3 } = ( a + b ) ^ { 3 } = a ^ { 3 } + b ^ { 3 } + \mathbf { A } a b ( a + b ) .$$
Since $a b = \mathbf { B C }$, we know that this $x$ satisfies
$$x ^ { 3 } + \mathbf { D } x - \mathbf { E } = 0 .$$
The left side of this equation can be factorized as follows:
$$\begin{aligned} x ^ { 3 } + \mathbf { D } x - \mathbf { E } & = \left( x ^ { 3 } - \mathbf { F } \right) + \mathbf { D } \left( x - \frac { \mathbf { F } } { \mathbf { E } } \right) \\ & = ( x - \mathbf { F } ) \left( x ^ { 2 } + x + \mathbf { G } \right) . \end{aligned}$$
Since
$$x ^ { 2 } + x + \frac { \mathbf { G } } { \mathbf { H } } = \left( x + \frac { \mathbf { H } } { \mathbf { I } } \right) ^ { 2 } + \frac { \mathbf { J K } } { \mathbf { L } } > 0 ,$$
we obtain $x = a + b = \mathbf { M }$.

Let $x = \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }$ and $y = \frac { \sqrt { 6 } - \sqrt { 2 } } { \sqrt { 6 } + \sqrt { 2 } }$.
(1) We have $x = \mathbf { A } + \sqrt { \mathbf { B } }$ and $y = \mathbf { C } - \sqrt { \mathbf { C } }$. Hence we have
$$x + y = \mathbf { E } , \quad x y = \mathbf { F } , \quad \frac { 1 } { x ^ { 2 } } + \frac { 1 } { y ^ { 2 } } = \mathbf { G H } .$$
Also we have
$$5 \left( x ^ { 2 } - 4 x \right) + 3 \left( y ^ { 2 } - 4 y + 1 \right) = \square \mathbf { I J } .$$
(2) The values of the integers $m$ and $n$ such that $\frac { m } { x } + \frac { n } { y } = 4 + 4 \sqrt { 3 }$ are
$$m = \mathbf { K L } , \quad n = \mathbf { M } .$$

Q1 Let $a = \sqrt { 5 } + \sqrt { 3 }$ and $b = \sqrt { 5 } - \sqrt { 3 }$. We are to find the integers $x$ satisfying
$$2 \left| x - \frac { a } { b } \right| + x < 10$$
(1) We see that $\frac { a } { b } = \mathbf { A } + \sqrt { \mathbf { BC } }$. Hence the largest integer less than $\frac { a } { b }$ is $\mathbf{D}$.
(2) For $\mathbf { F }$ and $\mathbf { H }$ in the following sentence, choose the correct answer from among choices (0) $\sim$ (7) below, and for $\mathbf { E }$ and $\mathbf { G }$, enter the correct numbers.
When $x$ is an integer, the left side of the inequality can be expressed without using the absolute value symbol as follows:
$$\left\{ \begin{array} { l } \text { if } x \leqq \mathbf { E } , \text { then } 2 \left| x - \frac { a } { b } \right| + x = \mathbf { F } , \\ \text { if } x \geqq \mathbf { G } , \text { then } 2 \left| x - \frac { a } { b } \right| + x = \mathbf { H } . \end{array} \right.$$
(0) $x - 6 - 2 \sqrt { 10 }$
(1) $x + 8 + 2 \sqrt { 15 }$
(2) $- x + 8 + 2 \sqrt { 15 }$
(3) $- x + 6 + 2 \sqrt { 10 }$
(4) $3 x - 6 - 2 \sqrt { 10 }$
(5) $3 x - 8 - 2 \sqrt { 15 }$ (6) $- 3 x + 8 + 2 \sqrt { 15 }$ (7) $- 3 x + 6 + 2 \sqrt { 10 }$
(3) Thus, the integers $x$ satisfying inequality $2 \left| x - \frac { a } { b } \right| + x < 10$ are those greater than or equal to $\mathbf { I }$ and less than or equal to $\mathbf { J }$.

For positive real numbers x and y,
$$\begin{aligned} x \cdot y & = 5 \\ x ^ { 2 } + y ^ { 2 } & = 15 \end{aligned}$$
Given this, what is the value of the expression $x ^ { 3 } + y ^ { 3 }$?
A) 40
B) 45
C) 50
D) 60
E) 75