The figure on the right shows 6 semicircles with center at $( 1,1 )$ and radii of lengths $\frac { 1 } { 3 } , \frac { 2 } { 3 } , 1 , \frac { 4 } { 3 } , \frac { 5 } { 3 } , 2$ respectively. Three functions $$\begin{aligned} & y = \log _ { \frac { 1 } { 4 } } x \\ & y = \left( \frac { 2 } { 3 } \right) ^ { x } \\ & y = 3 ^ { x } \end{aligned}$$ Let $a , b , c$ be the number of intersection points where the graphs of these functions meet the semicircles, respectively. Which of the following correctly represents the relationship between $a , b , c$? (Here, $x \geqq 1$ and the semicircles include the endpoints of the diameter.) [4 points]
(1) $a < b < c$
(2) $a < c < b$
(3) $b < c < a$
(4) $c < a < b$
(5) $c < b < a$

In the coordinate plane, for a natural number $n$, let $A _ { n }$ be a square with vertices at the four points $$\left( n ^ { 2 } , n ^ { 2 } \right) , \left( 4 n ^ { 2 } , n ^ { 2 } \right) , \left( 4 n ^ { 2 } , 4 n ^ { 2 } \right) , \left( n ^ { 2 } , 4 n ^ { 2 } \right)$$ Let $a _ { n }$ be the number of natural numbers $k$ such that the square $A _ { n }$ and the graph of the function $y = k \sqrt { x }$ intersect. Which of the following statements in the given options are correct? [4 points] Given Options ㄱ. $a _ { 5 } = 15$ ㄴ. $a _ { n + 2 } - a _ { n } = 7$ ㄷ. $\sum _ { k = 1 } ^ { 10 } a _ { k } = 200$
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

In the coordinate plane, let $f ( n )$ denote the number of triangles OAB satisfying the following conditions for a natural number $n$. Find the value of $f ( 1 ) + f ( 2 ) + f ( 3 )$. (Here, O is the origin.) [4 points] (가) The coordinates of point A are $\left( - 2,3 ^ { n } \right)$. (나) If the coordinates of point B are $( a , b )$, then $a$ and $b$ are natural numbers and satisfy $b \leq \log _ { 2 } a$. (다) The area of triangle OAB is at most 50.

On the coordinate plane, the function $$f ( x ) = \begin{cases} - x + 10 & ( x < 10 ) \\ ( x - 10 ) ^ { 2 } & ( x \geq 10 ) \end{cases}$$ and for a natural number $n$, there is a circle $O _ { n }$ centered at $( n , f ( n ) )$ with radius 3. Let $A _ { n }$ be the number of all points with integer coordinates that are inside circle $O _ { n }$ and below the graph of the function $y = f ( x )$, and let $B _ { n }$ be the number of all points with integer coordinates that are inside circle $O _ { n }$ and above the graph of the function $y = f ( x )$. What is the value of $\sum _ { n = 1 } ^ { 20 } \left( A _ { n } - B _ { n } \right)$? [4 points]
(1) 19
(2) 21
(3) 23
(4) 25
(5) 27

In the coordinate plane, find the number of points with both natural number coordinates that are contained in the interior of the region enclosed by the curve $y = \frac { 1 } { 2 x - 8 } + 3$ and the $x$-axis and $y$-axis. [3 points]
(1) 3
(2) 4
(3) 5
(4) 6
(5) 7

There are many beautifully shaped and meaningful curves in mathematics. The curve $C : x ^ { 2 } + y ^ { 2 } = 1 + | x | y$ is one of them (as shown in the figure). Three conclusions are given: (1) The curve $C$ passes through exactly 6 lattice points (points with both integer coordinates); (2) The distance from any point on curve $C$ to the origin does not exceed $\sqrt { 2 }$; (3) The area of the ``heart-shaped'' region enclosed by curve $C$ is less than 3. The sequence numbers of all correct conclusions are (A) (1) (B) (2) (C) (1)(2) (D) (1)(2)(3)

A person uses single-point perspective with a point on the horizon as the vanishing point to draw six vertical pillars $A , B , C , D , E , F$ on a coordinate plane. The coordinates of the top and base of each pillar are shown in the table below, with point $V ( 4,9 )$ representing the vanishing point, as shown in the figure. Since the base line and top line of pillars $A$ and $F$ in the figure are both parallel to the horizon, the actual heights of pillars $A$ and $F$ are equal. Based on the above, select the pillar with the maximum actual height.

Pillar	$A$	$B$	$C$	$D$	$E$	$F$
Top coordinate	$( 0,8 )$	$( 2,3 )$	$( 4,6 )$	$( 6,8 )$	$( 8,5 )$	$( 10,8 )$
Base coordinate	$( 0,6 )$	$( 2,0 )$	$( 4,3 )$	$( 6,5 )$	$( 8,1 )$	$( 10,6 )$

(1) $A$
(2) $B$
(3) $C$
(4) $D$
(5) $E$