Let $G$ be a finite group of odd order and $1 < d < | G |$ be a divisor of $| G |$. Assume that $G$ has exactly three subgroups $H _ { 1 } , H _ { 2 }$ and $H _ { 3 }$ of order $d$. Suppose that $H _ { 1 }$ is not normal in $G$. For each $i = 1,2,3$, let $N _ { i }$ denote the normalizer of $H _ { i }$. Let $S : = \left\{ H _ { 1 } , H _ { 2 } , H _ { 3 } \right\}$.
(A) (4 marks) For each $g \in G$ let $s _ { g }$ denote the cardinality of the set $\left\{ H \in S \mid g H g ^ { - 1 } = H \right\}$. Show that $\sum _ { g \in G } s _ { g } = \left| N _ { 1 } \right| + \left| N _ { 2 } \right| + \left| N _ { 3 } \right|$.
(B) (6 marks) Show that $G \neq N _ { 1 } \cup N _ { 2 } \cup N _ { 3 }$.

(A) (5 marks) Let $X$ be a non-empty finite set and let $R$ be the ring of $\mathbb { Z }$-valued functions on $X$, with pointwise addition and multiplication. Let $S$ be an additive subgroup of $R$ such that the multiplicative identity $1 _ { R } \notin S$ and such that for all $s , s ^ { \prime } \in S , s s ^ { \prime } \in S$. Show that there exists $x \in X$ and a prime number $p$ such that $f ( x )$ is divisible by $p$ for all $f \in S$ (Hint: Consider the sets $\{ f ( x ) : f \in S \}$ for all $x \in X$.)
(B) (5 marks) Let $K$ be a subfield of $\mathbb { C }$ with $[ K : \mathbb { Q } ] = 2$. Let $P \in \mathbb { Q } [ x ]$ be irreducible over $\mathbb { Q }$. Show that $P$ is either irreducible in $K [ x ]$ or splits as product of two irreducible polynomials in $K [ x ]$.

Let $A ( X ) , B ( X )$ be non-zero polynomials in $\mathbb { C } [ X ]$ such that $0 \leq \operatorname { deg } A \leq \operatorname { deg } B - 2$ and $A ( X )$ and $B ( X )$ do not share any roots. Let $\alpha _ { 1 } , \alpha _ { 2 } , \ldots , \alpha _ { k }$ be the roots of $B ( X )$. Suppose that each of them is a simple root.
Show that
$$\sum _ { j = 1 } ^ { k } \frac { A \left( \alpha _ { j } \right) } { B ^ { \prime } \left( \alpha _ { j } \right) } = 0$$

(A) (3 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R }$ be a continuous function such that $f ( r ) = f \left( r + \frac { 1 } { n } \right)$ for each $r \in \mathbb { Q }$ and each positive integer $n$. Prove or disprove the following statement: $f$ is a constant function.
(B) (7 marks) Let $a _ { n } , n \geq 1$ be a sequence of non-negative real numbers such that $a _ { m + n } \leq a _ { m } + a _ { n }$ for all $m , n$. Show that
$$\lim _ { n \longrightarrow \infty } \frac { a _ { n } } { n } = \inf \left\{ \left. \frac { a _ { n } } { n } \right\rvert\, n \geq 1 \right\}$$

Let $p$ be a prime number. Let $n \geq 2$ be an integer. Let $V$ be an $n$-dimensional $\mathbb { F } _ { p }$-vector space. Determine, with a proof, the number of two-dimensional $\mathbb { F } _ { p }$-subspaces of $V$.

Suppose $f$ is a function whose domain is $X$ and codomain is $Y$. It is given that $|X|>1$ and $|Y|>1$. No other information is known about $X$, $Y$ and $f$. Instruction: Write the number of a single correct option for the given statement S.
$\mathrm{S} =$ "For each $x$ in $X$, there exists $y$ in $Y$ such that $f(x) = y$." [1 point]
Options:

1. S is always true.
2. S is always false.
3. S is true if and only if $f$ is one-to-one.
4. If S is true then $f$ is one-to-one but the converse is false.
5. If $f$ is one-to-one then S is true but the converse is false.
6. S is true if and only if $f$ is onto.
7. If S is true then $f$ is onto but the converse is false.
8. If $f$ is onto then S is true but the converse is false.
9. S is true if and only if $f$ is a constant function.
10. If S is true then $f$ is a constant function but the converse is false.
11. If $f$ is a constant function then S is true but the converse is false.
12. None of the above.

Let $R$ be the ring of all the real-valued functions on $\mathbb { N } \times \mathbb { N }$. Show that $R$ contains a subring isomorphic to the polynomial ring $\mathbb { R } [ X , Y ]$.

Suppose $f$ is a function whose domain is $X$ and codomain is $Y$. It is given that $|X|>1$ and $|Y|>1$. No other information is known about $X$, $Y$ and $f$. Instruction: Write the number of a single correct option for the given statement S.
$\mathrm{S} =$ "For each $y$ in $Y$, there exists $x$ in $X$ such that $f(x) = y$." [1 point]
Options:

1. S is always true.
2. S is always false.
3. S is true if and only if $f$ is one-to-one.
4. If S is true then $f$ is one-to-one but the converse is false.
5. If $f$ is one-to-one then S is true but the converse is false.
6. S is true if and only if $f$ is onto.
7. If S is true then $f$ is onto but the converse is false.
8. If $f$ is onto then S is true but the converse is false.
9. S is true if and only if $f$ is a constant function.
10. If S is true then $f$ is a constant function but the converse is false.
11. If $f$ is a constant function then S is true but the converse is false.
12. None of the above.

Let $M _ { n } ( \mathbb { R } )$ be the space of $n \times n$ matrices with real entries, identified with the euclidean space $\mathbb { R } ^ { n ^ { 2 } }$. Let $X$ be a compact subset of $M _ { n } ( \mathbb { R } )$, and $S \subset \mathbb { C }$ be the set of all eigenvalues of the matrices in $X$. Show that $S$ is a compact subset of $\mathbb { C }$.

Suppose $f$ is a function whose domain is $X$ and codomain is $Y$. It is given that $|X|>1$ and $|Y|>1$. No other information is known about $X$, $Y$ and $f$. Instruction: Write the number of a single correct option for the given statement S.
$\mathrm{S} =$ "There exists a unique $x$ in $X$ such that for each $y$ in $Y$ it is true that $f(x) = y$." [1 point]
Options:

1. S is always true.
2. S is always false.
3. S is true if and only if $f$ is one-to-one.
4. If S is true then $f$ is one-to-one but the converse is false.
5. If $f$ is one-to-one then S is true but the converse is false.
6. S is true if and only if $f$ is onto.
7. If S is true then $f$ is onto but the converse is false.
8. If $f$ is onto then S is true but the converse is false.
9. S is true if and only if $f$ is a constant function.
10. If S is true then $f$ is a constant function but the converse is false.
11. If $f$ is a constant function then S is true but the converse is false.
12. None of the above.

Fix $0 < \lambda < 1$. Choose $\epsilon > 0$ such that $\epsilon + \frac { \lambda ^ { 2 } } { 4 } \leq \frac { \lambda } { 2 }$. Consider the metric space
$$X : = \left\{ \psi \in \mathcal { C } ^ { 1 } ( [ - \epsilon , \epsilon ] ) : | y + \psi ( y ) | \leq \frac { \lambda } { 2 } \text { for all } y \in [ - \epsilon , \epsilon ] \right\}$$
with the induced supremum metric from $\mathcal { C } ^ { 1 } ( [ - \epsilon , \epsilon ] )$, which we denote by $d$. (Recall: $\mathcal { C } ^ { 1 } ( [ - \epsilon , \epsilon ] )$ is the set of real-valued differentiable functions on $[ - \epsilon , \epsilon ]$ whose derivative is continuous.)
(A) (1 mark) Show that there is a function $A : X \longrightarrow X$ given by
$$( A \psi ) ( y ) = - ( y + \psi ( y ) ) ^ { 2 } .$$
(B) (2 marks) Show that if $\psi \in X$ is such that $A \psi = \psi$, then the function $x = y + \psi ( y )$ is an inverse to the function $y = x + x ^ { 2 }$, locally near the origin. In the next few steps, we show that such a $\psi$ exists.
(C) (2 marks) Show that $d \left( A \psi _ { 1 } , A \psi _ { 2 } \right) \leq \lambda d \left( \psi _ { 1 } , \psi _ { 2 } \right)$.
(D) (4 marks) Let $\phi \in X$. Show that the sequence $A ^ { n } \phi , n \geq 1$ is a Cauchy sequence, and it has a limit. By $A ^ { n }$, we mean the $n$-fold composition $A \circ A \circ \cdots \circ A$. (You may use the fact that $X$ is complete with respect to d.)
(E) (1 mark) Show that there exists $\psi \in X$ such that $A \psi = \psi$.

Suppose $f$ is a function whose domain is $X$ and codomain is $Y$. It is given that $|X|>1$ and $|Y|>1$. No other information is known about $X$, $Y$ and $f$. Instruction: Write the number of a single correct option for the given statement S.
$\mathrm{S} =$ "There exists a unique $y$ in $Y$ such that for each $x$ in $X$ it is true that $f(x) = y$." [1 point]
Options:

1. S is always true.
2. S is always false.
3. S is true if and only if $f$ is one-to-one.
4. If S is true then $f$ is one-to-one but the converse is false.
5. If $f$ is one-to-one then S is true but the converse is false.
6. S is true if and only if $f$ is onto.
7. If S is true then $f$ is onto but the converse is false.
8. If $f$ is onto then S is true but the converse is false.
9. S is true if and only if $f$ is a constant function.
10. If S is true then $f$ is a constant function but the converse is false.
11. If $f$ is a constant function then S is true but the converse is false.
12. None of the above.

Suppose a differentiable function $f$ from $\mathbb{R}$ to $\mathbb{R}$ has a local maximum at $(a, f(a))$ (This means there are numbers $m$ and $M$ such that (i) $m < a < M$ and (ii) $f(a) \geq f(x)$ for any $x \in [m,M]$.) The proof of a standard result is sketched below. Complete it as instructed.
Proof: For sufficiently $\_\_\_\_$ 1 $h > 0$, it is given that $f(a+h)$ $\_\_\_\_$ 2 3. Therefore for such $h$ the quantity $\_\_\_\_$ 4 must be $\_\_\_\_$ 5. By taking the limit of this quantity as $h \rightarrow 0$ from the right, we get that $\_\_\_\_$ 7 must be $\_\_\_\_$ 8. A parallel argument for suitable negative values of $h$ gives that $\_\_\_\_$ 10 must be $\_\_\_\_$ 11. Combining both conclusions gives the desired result: $\_\_\_\_$ 13 $\_\_\_\_$ 14. Note that the mentioned limits exist because $\_\_\_\_$ 16.
Write a sequence of 9 letters indicating the correct options to fill in the numbered blanks 1 to 9. Do not use any spaces, full stop or other punctuation. E.g., ABACDIJKB is in the correct format. [3 points]
Options: A. small B. large C. $\geq$ D. $>$ E. $\leq$ F. $<$ G. $=$ H. $\neq$ I. 0 J. $f(a)$ K. $\frac{f(a+h)-f(a)}{h}$ L. $f'(a)$ M. $f$ is differentiable N. $f$ is continuous

Suppose a differentiable function $f$ from $\mathbb{R}$ to $\mathbb{R}$ has a local maximum at $(a, f(a))$ (This means there are numbers $m$ and $M$ such that (i) $m < a < M$ and (ii) $f(a) \geq f(x)$ for any $x \in [m,M]$.) The proof of a standard result is sketched below. Complete it as instructed.
Proof: For sufficiently $\_\_\_\_$ 1 $h > 0$, it is given that $f(a+h)$ $\_\_\_\_$ 2 3. Therefore for such $h$ the quantity $\_\_\_\_$ 4 must be $\_\_\_\_$ 5. By taking the limit of this quantity as $h \rightarrow 0$ from the right, we get that $\_\_\_\_$ 7 must be $\_\_\_\_$ 8. A parallel argument for suitable negative values of $h$ gives that $\_\_\_\_$ 10 must be $\_\_\_\_$ 11. Combining both conclusions gives the desired result: $\_\_\_\_$ 13 $\_\_\_\_$ 14. Note that the mentioned limits exist because $\_\_\_\_$ 16.
Write a sequence of 7 letters indicating the correct options to fill in the numbered blanks 10 to 16. [2 points]
Options: A. small B. large C. $\geq$ D. $>$ E. $\leq$ F. $<$ G. $=$ H. $\neq$ I. 0 J. $f(a)$ K. $\frac{f(a+h)-f(a)}{h}$ L. $f'(a)$ M. $f$ is differentiable N. $f$ is continuous

Let $K$ be the splitting field of $X ^ { n } - 1$ over $\mathbb { F } _ { p }$, where $n$ is a positive integer. Pick the correct statement(s) from below.
(A) $K$ has $p ^ { n }$ elements.
(B) If $p \nmid n$, then the group of field automorphisms of $K$ is isomorphic to the multiplicative group $( \mathbb { Z } / n \mathbb { Z } ) ^ { \times }$.
(C) $K$ is a separable extension of $\mathbb { F } _ { p }$.
(D) There exists $m > n$ such that $K$ is the splitting field of $X ^ { m } - 1$ over $\mathbb { F } _ { p }$.

Consider the map $f : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R } ^ { 2 } , ( x , y ) \mapsto \left( - x - y ^ { 2 } , y + x ^ { 2 } \right)$. Pick the correct statement(s) from below.
(A) There exist infinitely many $( a , b ) \in \mathbb { R } ^ { 2 }$ such that there is an open neighbourhood $U$ of $( a , b )$ such that $\left. f \right| _ { U }$ is a homeomorphism from $U$ to $f ( U )$.
(B) The derivative $D f$ maps some non-zero tangent vector to $\mathbb { R } ^ { 2 }$ at $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$, to the zero tangent vector at $\left( - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right)$.
(C) There exist infinitely many $( a , b ) \in \mathbb { R } ^ { 2 }$ such that for every open neighbourhood $U$ of $( a , b ) , \left. f \right| _ { U }$ is not a homeomorphism from $U$ to $f ( U )$.
(D) For every differentiable curve $\gamma$ through $( 0,0 ) , f \circ \gamma$ is differentiable curve.

How many additional edges must be drawn to make a regular 10-gon graph into a complete graph? [3 points]
(1) 25
(2) 35
(3) 45
(4) 55
(5) 65

How many different spanning trees does the graph on the right have? [3 points]
(1) 10
(2) 12
(3) 15
(4) 18
(5) 21

The following table shows the tasks, task times, and prerequisite tasks required to repair the interior of a moving house.

Task	Task Time (minutes)	Prerequisite Task
Wallpaper (A)	210	None
Light Bulb Replacement (B)	20	A
Bathroom Repair (C)	60	B
Kitchen Repair (D)	100	B
Curtain Replacement (E)	50	A
Flooring Replacement (F)	150	C, D

What is the minimum time required to complete all the tasks? [4 points]
(1) 260 minutes
(2) 370 minutes
(3) 440 minutes
(4) 480 minutes
(5) 530 minutes

Graph $G$ has vertices $1, 2, 3, 4, 5, 6, 7, 8$, and all edges connect two distinct vertices that have a divisor or multiple relationship. How many vertices in graph $G$ have degree 3? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Let $A$ be the adjacency matrix of a graph $G$ with 5 vertices. The following represents $A^2$. $$\left( \begin{array} { l l l l l } 4 & 3 & 3 & 2 & 2 \\ 3 & 4 & 3 & 2 & 2 \\ 3 & 3 & 4 & 2 & 2 \\ 2 & 2 & 2 & 3 & 3 \\ 2 & 2 & 2 & 3 & 3 \end{array} \right)$$ Choose all correct statements about graph $G$ from the given options. [4 points]
Options ㄱ. There are 2 vertices with degree 3. ㄴ. It has a Hamiltonian circuit. ㄷ. There are at least 2 paths consisting of 2 edges connecting any two distinct vertices.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

The following shows a tree with 1023 vertices, where consecutive natural numbers from 1 to 1023 are assigned to each vertex according to a rule.
Let $M(a, b)$ be the maximum natural number corresponding to a vertex that is commonly included in both the path from the vertex corresponding to 1 to the vertex corresponding to $a$ and the path from the vertex corresponding to 1 to the vertex corresponding to $b$.
For example, $M(4, 11) = 2$ and $M(7, 12) = 3$.
If $M(33, 79) = k$, find the value of $10k$. [4 points]

A newly constructed building has 6 offices A, B, C, D, E, F. The cost required to establish a computer network between offices is shown in the table. What is the minimum cost required to establish a computer network so that all 6 offices are connected through the network? [3 points] (Unit: 1 million won)

Solve for $x$:
$$\sin ( 2 x ) + \cos ^ { 2 } ( x ) = \frac { \ln ( x ) } { e ^ { 2 x } + 1 }$$

Calculate the surface area of a torus with a major radius $R = 5$ units and a minor radius $r = 2$ units.