How many natural numbers $x$ satisfy the inequality $\left( \frac { 1 } { 9 } \right) ^ { x } < 3 ^ { 21 - 4 x }$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

What is the value of $\left( 2 ^ { \sqrt { 3 } } \times 4 \right) ^ { \sqrt { 3 } - 2 }$? [2 points]
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) 1
(4) 2
(5) 4

What is the value of $\left( \frac { 4 } { 2 ^ { \sqrt { 2 } } } \right) ^ { 2 + \sqrt { 2 } }$? [2 points]
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) 1
(4) 2
(5) 4

Find the value of $\sqrt[3]{24} \times 3^{\frac{2}{3}}$. [2 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

What is the value of $\sqrt[3]{5} \times 25^{\frac{1}{3}}$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

What is the value of $9 ^ { \frac { 1 } { 4 } } \times 3 ^ { - \frac { 1 } { 2 } }$? [2 points]
(1) 1
(2) $\sqrt { 3 }$
(3) 3
(4) $3 \sqrt { 3 }$
(5) 9

Exercise I

I-A- $\quad \frac { ( 2 \sqrt { 3 } ) ^ { 2 } \times 12 ^ { 3 } \times 3 ^ { 2 } } { 3 ^ { - 4 } \times ( \sqrt { 2 } ) ^ { 4 } } = 3 ^ { 10 } \times 2 ^ { 8 }$. I-B- $\quad 2 \sqrt { 27 } - ( 2 \sqrt { 3 } - 1 ) ^ { 2 } = 10 \sqrt { 3 } - 13$. I-C- $\quad \ln \left( \frac { e } { 4 } \right) + \ln \left( \frac { 1 } { 9 e } \right) + \ln ( 36 e ) = 1$. I-D- $\quad e ^ { 2 \ln 3 + \ln 5 } + e ^ { - 2 \ln 5 } = 20$. I-E- For every real number $x$ different from $-2$ and from $2$, $\frac { 2 } { x + 2 } - \frac { 1 } { x - 2 } + \frac { 8 } { x ^ { 2 } - 4 } = \frac { 1 } { x - 2 }$. I-F- For every real number $x$, $\frac { e ^ { 2 x } + 2 e ^ { x } + 1 } { e ^ { x } + 1 } = e ^ { x } + 1$.
For each statement, indicate whether it is TRUE or FALSE.

101- What is the value of $2\sqrt[4]{2} \cdot \sqrt[4]{2\sqrt{2}} \cdot (\sqrt{2-\sqrt{3}}+\sqrt{2+\sqrt{3}})$?
(1) $\sqrt{3}$ (2) $2$ (3) $1+\sqrt{3}$ (4) $2\sqrt{3}$

102. If $A = \sqrt[6]{4\sqrt[3]{16}}\left(\dfrac{1}{2}\right)^{-\frac{1}{3}}$, then $(2A)^{-\frac{4}{3}}$ equals which of the following?
(1) $0.25$ (2) $0.5$ (3) $0.75$ (4) $1$

The last digit of $( 2004 ) ^ { 5 }$ is
(A) 4
(B) 8
(C) 6
(D) 2

The last digit of $( 2004 ) ^ { 5 }$ is
(A) 4
(B) 8
(C) 6
(D) 2

Let $a \geq b \geq c \geq 0$ be integers such that $2 ^ { a } + 2 ^ { b } - 2 ^ { c } = 144$. Then, $a + b - c$ equals:
(A) 7
(B) 8
(C) 9
(D) 10 .

Any positive real number $x$ can be expanded as $x = a _ { n } \cdot 2 ^ { n } + a _ { n - 1 } \cdot 2 ^ { n - 1 } + \cdots + a _ { 1 } \cdot 2 ^ { 1 } + a _ { 0 } \cdot 2 ^ { 0 } + a _ { - 1 } \cdot 2 ^ { - 1 } + a _ { - 2 } \cdot 2 ^ { - 2 } + \cdots$, for some $n \geq 0$, where each $a _ { i } \in \{ 0,1 \}$. In the above-described expansion of 21.1875, the smallest positive integer $k$ such that $a _ { - k } \neq 0$ is:
(A) 3
(B) 2
(C) 1
(D) 4

If $x = 1 + \sqrt[5]{2} + \sqrt[5]{4} + \sqrt[5]{8} + \sqrt[5]{16}$, then the value of $\left(1 + \frac{1}{x}\right)^{30}$ is
(A) 2
(B) 5
(C) 32
(D) 64

The set of all real numbers $x$ for which $3^{2^{1-x^2}}$ is an integer has
(A) 3 elements
(B) 15 elements
(C) 24 elements
(D) infinitely many elements

Let $S = \{ a + b \sqrt { 2 } : a , b \in \mathbb { Z } \} , T _ { 1 } = \left\{ ( - 1 + \sqrt { 2 } ) ^ { n } : n \in \mathbb { N } \right\}$, and $T _ { 2 } = \left\{ ( 1 + \sqrt { 2 } ) ^ { n } : n \in \mathbb { N } \right\}$.
Then which of the following statements is (are) TRUE?
(A) $\mathbb { Z } \bigcup T _ { 1 } \bigcup T _ { 2 } \subset S$
(B) $T _ { 1 } \cap \left( 0 , \frac { 1 } { 2024 } \right) = \phi$, where $\phi$ denotes the empty set.
(C) $T _ { 2 } \cap ( 2024 , \infty ) \neq \phi$
(D) For any given $a , b \in \mathbb { Z } , \cos ( \pi ( a + b \sqrt { 2 } ) ) + i \sin ( \pi ( a + b \sqrt { 2 } ) ) \in \mathbb { Z }$ if and only if $b = 0$, where $i = \sqrt { - 1 }$.

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 $\frac { 1 } { c } - c = \mathbf { U }$.

[1] Let $x$ be a positive real number satisfying $x ^ { 2 } + \frac { 4 } { x ^ { 2 } } = 9$. Then
$$\left( x + \frac { 2 } { x } \right) ^ { 2 } = \text { A B }$$
Therefore, $x + \frac { 2 } { x } = \sqrt { \text { A B } }$. Furthermore,
$$\begin{aligned} x ^ { 3 } + \frac { 8 } { x ^ { 3 } } & = \left( x + \frac { 2 } { x } \right) \left( x ^ { 2 } + \frac { 4 } { x ^ { 2 } } - \right. \\ & = \text { D } \sqrt { \text { E F } } \end{aligned}$$
□ C
Also,
$$x ^ { 4 } + \frac { 16 } { x ^ { 4 } } = \text { G H }$$
[2] Let $p$ and $q$ be two conditions regarding the real number $x$:
$$\begin{array} { l l } p : & x = 1 \\ q : & x ^ { 2 } = 1 \end{array}$$
Also, let $\bar { p }$ and $\bar { q }$ denote the negations of conditions $p$ and $q$, respectively.
(1) Choose one from the options (0)–(3) below for each of the blanks K, L, M, N. You may select the same option more than once. $q$ is a \text{____} condition for $p$ (K) $\bar { p }$ is a \text{____} condition for $q$ (L) ($p$ or $\bar { q }$) is a \text{____} condition for $q$ (M) ($\bar { p }$ and $q$) is a \text{____} condition for $q$ (N) (0) necessary but not sufficient
(1) sufficient but not necessary
(2) necessary and sufficient
(3) neither necessary nor sufficient
(2) Let $r$ be a condition regarding the real number $x$:
$$r : x > 0$$
Choose one from options ⓪–⑦ below for the blank O.
Consider the three propositions:
A: ``($p$ and $q$) $\Longrightarrow r$'' B: ``$q \Longrightarrow r$'' C: ``$\bar { q } \Longrightarrow \bar { p }$''
The correct statement about the truth values of these propositions is □ O. (0) A is true, B is true, C is true
(1) A is true, B is true, C is false
(2) A is true, B is false, C is true
(3) A is true, B is false, C is false
(4) A is false, B is true, C is true
(5) A is false, B is true, C is false (6) A is false, B is false, C is true (7) A is false, B is false, C is false
[3] Let $a$ be a constant, and let $g ( x ) = x ^ { 2 } - 2 \left( 3 a ^ { 2 } + 5 a \right) x + 18 a ^ { 4 } + 30 a ^ { 3 } + 49 a ^ { 2 } + 16$. The vertex of the parabola $y = g ( x )$ is
$$\text { ( P } a ^ { 2 } + \text { Q } a , \text { R } a ^ { 4 } + \text { S T } a ^ { 2 } + \text { U V } \text { ) }$$
When $a$ varies over all real numbers, the minimum value of the $x$-coordinate of the vertex is $- \frac { \text { W X } } { \text { Y Z } }$.
Next, let $t = a ^ { 2 }$. Then the $y$-coordinate of the vertex can be expressed as
$$\text { R } t ^ { 2 } + \text { S T } t + \text { U V }$$
Therefore, when $a$ varies over all real numbers, the minimum value of the $y$-coordinate of the vertex is AA.

We are to find the positive number $a$ satisfying $a ^ { 3 } = 9 + \sqrt { 80 }$.
Let us consider the positive number $b$ which satisfies $b ^ { 3 } = 9 - \sqrt { 80 }$. Then
$$\left\{ \begin{aligned} a ^ { 3 } + b ^ { 3 } & = \mathbf { A B } \\ a b & = \mathbf { C } \end{aligned} \right.$$
holds.
First, using (2), (1) can be transformed into
$$( a + b ) ^ { 3 } - \mathbf { D } ( a + b ) = \mathbf { A B } .$$
Then, setting $x = a + b$, we have
$$x ^ { 3 } - \mathrm { D } x = \mathrm { AB } .$$
Transforming this equation, we obtain
$$x ^ { 3 } - 27 = \mathbf { D } ( x - \mathbf { E } ) \text {, }$$
which gives
$$( x - \mathbf { F } ) \left( x ^ { 2 } + \mathbf { G } x + \mathbf{H} \right) = 0 .$$
From that we have $x = \mathbf{I}$ and hence
$$a + b = \mathbf{I} \text {. }$$
Thus, from (2), (3) and $a > b$, we have
$$a = \frac { \mathbf { J } + \sqrt { \mathbf { K } } } { \mathbf{L} } .$$

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
$$m = \mathbf { K L } , \quad n = \mathbf { M } .$$

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
$$\mathbf { I } < \mathbf { J } < \mathbf { K } < \mathbf { L } < \mathbf { M } .$$

Write $( \sqrt [ 3 ] { 49 } ) ^ { 100 }$ in scientific notation as $( \sqrt [ 3 ] { 49 } ) ^ { 100 } = a \times 10 ^ { n }$, where $1 \leq a < 10$ and $n$ is a positive integer. If the integer part of $a$ is $m$, then the ordered pair $( m , n ) = ($ (25) )(26).

If a positive integer $N$ is entered into a calculator, and then the ``$\sqrt{ }$'' key (taking the positive square root) is pressed 3 times consecutively, the display shows the answer as 2. Then $N$ equals which of the following options?
(1) $2^{3}$
(2) $2^{4}$
(3) $2^{6}$
(4) $2^{8}$
(5) $2^{12}$

Given $a = 6$, $b = \frac{20}{3}$, $c = 2\sqrt{10}$, and $d$, where $d$ is a rational number. These four numbers are marked on a number line as $A(a)$, $B(b)$, $C(c)$, and $D(d)$. Select the correct options.
(1) $a + b + c + d$ must be a rational number (2) $abcd$ must be an irrational number (3) Point $D$ could possibly be at a distance of $2\sqrt{10} + 6$ from point $C$ (4) The midpoint of points $A$ and $B$ is to the right of point $C$ (5) Among all points on the number line at a distance less than 8 from point $B$, there are 14 positive integers and 1 negative integer