Let $f , g , h$ be functions from $\mathbb { R }$ to $\mathbb { R }$ such that $$h ( f ( x ) + g ( y ) ) = x y$$ for all $x , y \in \mathbb { R }$. Show the following:
(A) $(2$ marks$)$ $h$ is surjective.
(B) $(3$ marks$)$ If $f$ is continuous then $f$ is strictly monotone.
(C) $(5$ marks$)$ There do not exist continuous functions $f , g , h$ satisfying $(*)$.

Let $f : [ 0,1 ] \longrightarrow \mathbb { R }$ and $g : \mathbb { R } \longrightarrow \mathbb { R }$ be continuous functions. Assume that $g$ is periodic with period 1. Show that $$\lim _ { n \mapsto \infty } \int _ { 0 } ^ { 1 } f ( x ) g ( n x ) d x = \left( \int _ { 0 } ^ { 1 } f ( x ) d x \right) \left( \int _ { 0 } ^ { 1 } g ( x ) d x \right)$$

Prove or disprove each of the statements below.
(A) (4 marks) Let $f : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R }$ be a continuous function that takes both positive and negative values. Then $f$ has infinitely many zeros.
(B) (6 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$ be a continuous function. Then $f$ is not open.

It is known that there exist surjective continuous maps $I \longrightarrow I ^ { 2 }$ where $I = [ 0,1 ]$ is the unit interval.
(A) (4 marks) Using the above result or otherwise, show that there exists a surjective continuous map $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$.
(B) (6 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$ be a surjective continuous map. Let $\Gamma = \{ ( x , f ( x ) ) \mid x \in \mathbb { R } \} \subset \mathbb { R } ^ { 3 }$. Show that $\mathbb { R } ^ { 3 } \setminus \Gamma$ is path connected.

For a natural number $k$, when $n = 5 ^ { k }$, $f ( n )$ satisfies $$f ( 5 n ) = f ( n ) + 3 , \quad f ( 5 ) = 4$$ Find the value of $\sum _ { k = 1 } ^ { 10 } f \left( 5 ^ { k } \right)$. [4 points]

For a natural number $p \geqq 2$, a sequence $\left\{ a _ { n } \right\}$ satisfies the following three conditions. Which of the following in are correct? [4 points] Conditions (가) $a _ { 1 } = 0$ (나) $a _ { k + 1 } = a _ { k } + 1 \quad ( 1 \leqq k \leqq p - 1 )$ (다) $a _ { k + p } = a _ { k } \quad ( k = 1,2,3 , \cdots )$ 〈Remarks〉 ㄱ. $a _ { 2 k } = 2 a _ { k }$ ㄴ. $a _ { 1 } + a _ { 2 } + \cdots + a _ { p } = \frac { p ( p - 1 ) } { 2 }$ ㄷ. $a _ { p } + a _ { 2 p } + \cdots + a _ { k p } = k ( p - 1 )$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

What is the value of $\lim _ { n \rightarrow \infty } \frac { n } { \sqrt { 4 n ^ { 2 } + 1 } + \sqrt { n ^ { 2 } + 2 } }$? [2 points]
(1) 1
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 1 } { 5 }$

For a natural number $n \geq 2$, let $C _ { n }$ be the circle obtained by translating the circle $C$ with center at the origin and radius 1 by $\frac { 2 } { n }$ in the $x$-direction. Let $l _ { n }$ be the length of the common chord of circles $C$ and $C _ { n }$. When $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { \left( n l _ { n } \right) ^ { 2 } } = \frac { q } { p }$, find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

Let $a _ { n }$ be the sum of all natural numbers such that when divided by a natural number $n$ ($n \geqq 2$), the quotient and remainder are equal. For example, when divided by 4, the natural numbers with equal quotient and remainder are $5, 10, 15$, so $a _ { 4 } = 5 + 10 + 15 = 30$. Find the minimum value of the natural number $n$ satisfying $a _ { n } > 500$. [4 points]

As shown in the figure, a square A with side length 2 and a square B with side length 1 have sides parallel to each other, and the intersection point of the two diagonals of A coincides with the intersection point of the two diagonals of B. Let R be the region of A and its interior excluding the interior of B.
For a natural number $n \geqq 2$, small squares with side length $\frac { 1 } { n }$ are drawn in R according to the following rule. (가) One side of each small square is parallel to a side of A. (나) The interiors of the small squares do not overlap with each other.
According to such rules, let $a _ { n }$ be the maximum number of small squares with side length $\frac { 1 } { n }$ that can be drawn in R. For example, $a _ { 2 } = 12$ and $a _ { 3 } = 20$. When $\lim _ { n \rightarrow \infty } \frac { a _ { 2 n + 1 } - a _ { 2 n } } { a _ { 2 n } - a _ { 2 n - 1 } } = c$, find the value of $100 c$. [4 points]

For a natural number $n \geq 2$, consider the set $$\left\{ 3 ^ { 2 k - 1 } \mid k \text{ is a natural number, } 1 \leqq k \leqq n \right\}$$ Let $S$ be the set containing only all possible values obtained by multiplying two distinct elements of this set, and let $f ( n )$ be the number of elements in $S$. For example, $f ( 4 ) = 5$. Find the value of $\sum _ { n = 2 } ^ { 11 } f ( n )$. [4 points]

For a natural number $n \geq 2$, consider the set $$\left\{ 3 ^ { 2 k - 1 } \mid k \text { is a natural number, } 1 \leqq k \leqq n \right\}$$ Let $S$ be the set containing only all possible values obtained by multiplying two distinct elements of this set, and let $f ( n )$ be the number of elements in $S$. For example, $f ( 4 ) = 5$. Find the value of $\sum _ { n = 2 } ^ { 11 } f ( n )$.

For a natural number $m$, blocks in the shape of identical cubes are stacked with 1 block in column 1, 2 blocks in column 2, 3 blocks in column 3, $\cdots$, and $m$ blocks in column $m$. The following trial is repeated until there are no columns with an even number of blocks remaining.
For each column with an even number of blocks, remove from that column a number of blocks equal to $\frac { 1 } { 2 }$ of the number of blocks in that column.
Let $f ( m )$ be the sum of the number of blocks remaining in columns 1 through $m$ after all block removal trials are completed. For example, $f ( 2 ) = 2 , f ( 3 ) = 5 , f ( 4 ) = 6$.
$$\lim _ { n \rightarrow \infty } \frac { f \left( 2 ^ { n + 1 } \right) - f \left( 2 ^ { n } \right) } { f \left( 2 ^ { n + 2 } \right) } = \frac { q } { p }$$
Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

For a natural number $m$, there are blocks in the shape of identical cubes stacked with 1 block in column 1, 2 blocks in column 2, 3 blocks in column 3, $\cdots$, and $m$ blocks in column $m$. The following procedure is repeated until there are no columns with an even number of blocks remaining.
For each column with an even number of blocks, remove $\frac { 1 } { 2 }$ of the blocks in that column from the column.
After completing all block removal procedures, let $f ( m )$ be the sum of the number of blocks remaining in columns 1 through $m$. For example, $f ( 2 ) = 2 , f ( 3 ) = 5 , f ( 4 ) = 6$. $$\lim _ { n \rightarrow \infty } \frac { f \left( 2 ^ { n + 1 } \right) - f \left( 2 ^ { n } \right) } { f \left( 2 ^ { n + 2 } \right) } = \frac { q } { p }$$ Find the value of $p + q$. (where $p$ and $q$ are coprime natural numbers.) [4 points]

The sequence $\left\{ a_n \right\}$ satisfies $a_1 = 4$ and $$a_{n+1} = n \cdot 2^n + \sum_{k=1}^{n} \frac{a_k}{k} \quad (n \geq 1)$$ The following is the process of finding the general term $a_n$.
From the given equation, $$a_n = (n-1) \cdot 2^{n-1} + \sum_{k=1}^{n-1} \frac{a_k}{k} \quad (n \geq 2)$$ Therefore, for natural numbers $n \geq 2$, $$a_{n+1} - a_n = \text{(가)} + \frac{a_n}{n}$$ so $$a_{n+1} = \frac{(n+1)a_n}{n} + \text{(가)}$$ If $b_n = \frac{a_n}{n}$, then $$b_{n+1} = b_n + \frac{(\text{가})}{n+1} \quad (n \geq 2)$$ and since $b_2 = 3$, $$b_n = \text{(나)} \quad (n \geq 2)$$ Therefore, $$a_n = \begin{cases} 4 & (n = 1) \\ n \times (\text{나}) & (n \geq 2) \end{cases}$$ If the expressions for (가) and (나) are $f(n)$ and $g(n)$, respectively, what is the value of $f(4) + g(7)$? [4 points]
(1) 90
(2) 95
(3) 100
(4) 105
(5) 110

The sequence $\left\{ a _ { n } \right\}$ has $a _ { 1 } = 4$ and satisfies
$$a _ { n + 1 } = n \cdot 2 ^ { n } + \sum _ { k = 1 } ^ { n } \frac { a _ { k } } { k } \quad ( n \geq 1 )$$
The following is the process of finding the general term $a _ { n }$.
From the given equation,
$$a _ { n } = ( n - 1 ) \cdot 2 ^ { n - 1 } + \sum _ { k = 1 } ^ { n - 1 } \frac { a _ { k } } { k } \quad ( n \geq 2 )$$
Therefore, for natural numbers $n \geq 2$,
$$a _ { n + 1 } - a _ { n } = \text { (a) } + \frac { a _ { n } } { n }$$
so
$$a _ { n + 1 } = \frac { ( n + 1 ) a _ { n } } { n } + \text { (a) }$$
If $b _ { n } = \frac { a _ { n } } { n }$, then
$$b _ { n + 1 } = b _ { n } + \frac { ( \text { a } ) } { n + 1 } ( n \geq 2 )$$
and since $b _ { 2 } = 3$,
$$b _ { n } = \text { (b) } \quad ( n \geq 2 )$$
Therefore,
$$a _ { n } = \left\{ \begin{array} { c c } 4 & ( n = 1 ) \\ n \times ( \boxed { ( \text{b} ) } ) & ( n \geq 2 ) \end{array} \right.$$
Let $f ( n )$ and $g ( n )$ be the expressions that fit (a) and (b), respectively. What is the value of $f ( 4 ) + g ( 7 )$? [4 points]
(1) 90
(2) 95
(3) 100
(4) 105
(5) 110

For the sequence $\left\{ a _ { n } \right\}$, $$\sum _ { k = 1 } ^ { 10 } \left( a _ { k } + 1 \right) ^ { 2 } = 28 , \sum _ { k = 1 } ^ { 10 } a _ { k } \left( a _ { k } + 1 \right) = 16$$ Find the value of $\sum _ { k = 1 } ^ { 10 } \left( a _ { k } \right) ^ { 2 }$. [4 points]

A sequence $\left\{ a _ { n } \right\}$ has $a _ { 1 } = 2$ and satisfies for all natural numbers $n$: $$a _ { n + 1 } = \left\{ \begin{array} { c c } \frac { a _ { n } } { 2 - 3 a _ { n } } & ( \text{when } n \text{ is odd} ) \\ 1 + a _ { n } & ( \text{when } n \text{ is even} ) \end{array} \right.$$ What is the value of $\sum _ { n = 1 } ^ { 40 } a _ { n }$? [3 points]
(1) 30
(2) 35
(3) 40
(4) 45
(5) 50

What is the value of $\lim _ { n \rightarrow \infty } \frac { 1 } { \sqrt { 4 n ^ { 2 } + 2 n + 1 } - 2 n }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$, $$\sum _ { k = 1 } ^ { 5 } a _ { k } = 8 , \quad \sum _ { k = 1 } ^ { 5 } b _ { k } = 9$$ What is the value of $\sum _ { k = 1 } ^ { 5 } \left( 2 a _ { k } - b _ { k } + 4 \right)$? [3 points]
(1) 19
(2) 21
(3) 23
(4) 25
(5) 27

A sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1$ and for all natural numbers $n$, $$\sum _ { k = 1 } ^ { n } \left( a _ { k } - a _ { k + 1 } \right) = - n ^ { 2 } + n$$ What is the value of $a _ { 11 }$? [3 points]
(1) 88
(2) 91
(3) 94
(4) 97
(5) 100

For a sequence $\left\{ a _ { n } \right\}$, $$\sum _ { k = 1 } ^ { 10 } a _ { k } - \sum _ { k = 1 } ^ { 7 } \frac { a _ { k } } { 2 } = 56 , \quad \sum _ { k = 1 } ^ { 10 } 2 a _ { k } - \sum _ { k = 1 } ^ { 8 } a _ { k } = 100$$ find the value of $a _ { 8 }$. [3 points]

All terms of a sequence $\left\{ a_{n} \right\}$ are integers and satisfy the following conditions. What is the sum of the values of $\left| a_{1} \right|$? [4 points] (가) For all natural numbers $n$, $$a_{n+1} = \begin{cases} a_{n} - 3 & \left(\left| a_{n} \right| \text{ is odd}\right) \\ \frac{1}{2}a_{n} & \left(a_{n} = 0 \text{ or } \left| a_{n} \right| \text{ is even}\right) \end{cases}$$ (나) The minimum value of the natural number $m$ such that $\left| a_{m} \right| = \left| a_{m+2} \right|$ is 3.

The sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions.

• $a _ { 1 } = 7$
• For natural numbers $n \geq 2$,

$$\sum _ { k = 1 } ^ { n } a _ { k } = \frac { 2 } { 3 } a _ { n } + \frac { 1 } { 6 } n ^ { 2 } - \frac { 1 } { 6 } n + 10$$
The following is the process of finding the value of $\sum _ { k = 1 } ^ { 12 } a _ { k } + \sum _ { k = 1 } ^ { 5 } a _ { 2 k + 1 }$.
For natural numbers $n \geq 2$, since $a _ { n + 1 } = \sum _ { k = 1 } ^ { n + 1 } a _ { k } - \sum _ { k = 1 } ^ { n } a _ { k }$, $$a _ { n + 1 } = \frac { 2 } { 3 } \left( a _ { n + 1 } - a _ { n } \right) + \text { (가) }$$ and simplifying this equation gives $$2 a _ { n } + a _ { n + 1 } = 3 \times \text { (가) } \quad \cdots \cdots \text { (ㄱ) }$$ From $$\sum _ { k = 1 } ^ { n } a _ { k } = \frac { 2 } { 3 } a _ { n } + \frac { 1 } { 6 } n ^ { 2 } - \frac { 1 } { 6 } n + 10 \quad ( n \geq 2 )$$ substituting $n = 2$ gives $$a _ { 2 } = \text { (나) }$$ By (ㄱ) and (ㄴ), $$\begin{aligned} \sum _ { k = 1 } ^ { 12 } a _ { k } + \sum _ { k = 1 } ^ { 5 } a _ { 2 k + 1 } & = a _ { 1 } + a _ { 2 } + \sum _ { k = 1 } ^ { 5 } \left( 2 a _ { 2 k + 1 } + a _ { 2 k + 2 } \right) \\ & = \text { (다) } \end{aligned}$$ Let $f ( n )$ be the expression that fits in (가), and let $p$ and $q$ be the numbers that fit in (나) and (다), respectively. Find the value of $\frac { p \times q } { f ( 12 ) }$. [4 points]

21. (12 points) Let $f _ { n } ( x )$ be the sum of the terms of the geometric sequence $1 , x , x ^ { 2 } , \cdots , x ^ { n }$, where $x > 0$, $n \in \mathrm {~N} , ~ n \geq 2$. (I) Prove that the function $\mathrm { F } _ { n } ( x ) = f _ { n } ( x ) - 2$ has exactly one zero in $\left( \frac { 1 } { 2 } , 1 \right)$ (denoted as $x _ { n }$), and $x _ { n } = \frac { 1 } { 2 } + \frac { 1 } { 2 } x _ { n } ^ { n + 1 }$; (II) Consider an arithmetic sequence with the same first term, last term, and number of terms as the above geometric sequence, with sum $g _ { n } ( x )$. Compare the sizes of $f _ { n } ( x )$ and $g _ { n } ( x )$, and provide a proof.

Choose one of questions 22, 23, or 24 to answer. If you do more than one, only the first one will be graded. Mark the box number of your chosen question with a 2B pencil on the answer sheet.