Questions that define quantities using conjugate surd expressions and require computing sums, products, or symmetric functions using identities like (a+b)² or a³+b³.
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 $x = a + b = \mathbf { M }$.
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