There is a rectangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ with $\overline { \mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } } = 1 , \overline { \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } } = 2$. As shown in the figure, let $\mathrm { M } _ { 1 }$ be the midpoint of segment $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$, and on segment $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$, determine two points $\mathrm { B } _ { 2 } , \mathrm { C } _ { 2 }$ such that $\angle \mathrm { A } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 2 } = \angle \mathrm { C } _ { 2 } \mathrm { M } _ { 1 } \mathrm { D } _ { 1 } = 15 ^ { \circ } , \angle \mathrm { B } _ { 2 } \mathrm { M } _ { 1 } \mathrm { C } _ { 2 } = 60 ^ { \circ }$. Let $S _ { 1 }$ be the sum of the area of triangle $\mathrm { A } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 2 }$ and the area of triangle $\mathrm { C } _ { 2 } \mathrm { M } _ { 1 } \mathrm { D } _ { 1 }$.
Quadrilateral $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is a rectangle with $\overline { \mathrm { B } _ { 2 } \mathrm { C } _ { 2 } } = 2 \overline { \mathrm {~A} _ { 2 } \mathrm {~B} _ { 2 } }$, and determine two points $\mathrm { A } _ { 2 } , \mathrm { D } _ { 2 }$ as shown in the figure. Let $\mathrm { M } _ { 2 }$ be the midpoint of segment $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 }$, and on segment $\mathrm { A } _ { 2 } \mathrm { D } _ { 2 }$, determine two points $\mathrm { B } _ { 3 } , \mathrm { C } _ { 3 }$ such that $\angle \mathrm { A } _ { 2 } \mathrm { M } _ { 2 } \mathrm {~B} _ { 3 } = \angle \mathrm { C } _ { 3 } \mathrm { M } _ { 2 } \mathrm { D } _ { 2 } = 15 ^ { \circ }$, $\angle \mathrm { B } _ { 3 } \mathrm { M } _ { 2 } \mathrm { C } _ { 3 } = 60 ^ { \circ }$. Let $S _ { 2 }$ be the sum of the area of triangle $\mathrm { A } _ { 2 } \mathrm { M } _ { 2 } \mathrm {~B} _ { 3 }$ and the area of triangle $\mathrm { C } _ { 3 } \mathrm { M } _ { 2 } \mathrm { D } _ { 2 }$. Continuing this process to obtain $S _ { n }$, what is the value of $\sum _ { n = 1 } ^ { \infty } S _ { n }$? [4 points]
(1) $\frac { 2 + \sqrt { 3 } } { 6 }$
(2) $\frac { 3 - \sqrt { 3 } } { 2 }$
(3) $\frac { 4 + \sqrt { 3 } } { 9 }$
(4) $\frac { 5 - \sqrt { 3 } } { 5 }$
(5) $\frac { 7 - \sqrt { 3 } } { 8 }$

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with nonzero common difference, the three terms $a _ { 2 } , a _ { 4 } , a _ { 9 }$ form a geometric sequence with common ratio $r$ in this order. Find the value of $6r$. [4 points]

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]

The sequence $\left\{ a _ { n } \right\}$ satisfies $$2 a _ { n + 1 } = a _ { n } + a _ { n + 2 }$$ for all natural numbers $n$. When $a _ { 2 } = - 1 , a _ { 3 } = 2$, what is the sum of the first 10 terms of the sequence $\left\{ a _ { n } \right\}$? [3 points]
(1) 95
(2) 90
(3) 85
(4) 80
(5) 75

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term $- 5$ and common difference 2, what is the value of $\sum _ { k = 11 } ^ { 20 } a _ { k }$? [3 points]
(1) 260
(2) 255
(3) 250
(4) 245
(5) 240

Three numbers $a , a + b , 2 a - b$ form an arithmetic sequence in this order, and three numbers $1 , a - 1, 3 b + 1$ form a geometric sequence with positive common ratio in this order. Find the value of $a ^ { 2 } + b ^ { 2 }$. [3 points]

Three numbers $a , a + b , 2 a - b$ form an arithmetic sequence in this order, and three numbers $1 , a - 1, 3 b + 1$ form a geometric sequence with positive common ratio in this order. Find the value of $a ^ { 2 } + b ^ { 2 }$. [3 points]

For a natural number $n$, a line passing through the focus F of the parabola $y ^ { 2 } = \frac { x } { n }$ intersects the parabola at two points P and Q, respectively. If $\overline { \mathrm { PF } } = 1$ and $\overline { \mathrm { FQ } } = a _ { n }$, what is the value of $\sum _ { n = 1 } ^ { 10 } \frac { 1 } { a _ { n } }$? [4 points]
(1) 210
(2) 205
(3) 200
(4) 195
(5) 190

For an arithmetic sequence $\left\{ a_n \right\}$, $$a_2 = 16, \quad a_5 = 10$$ Find the value of $k$ that satisfies $a_k = 0$. [3 points]

For natural numbers $n$, the point $\mathrm{P}_n$ on the coordinate plane is determined according to the following rules. (가) The coordinates of the three points $\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_3$ are $(-1, 0)$, $(1, 0)$, and $(-1, 2)$, respectively. (나) The midpoint of segment $\mathrm{P}_n \mathrm{P}_{n+1}$ and the midpoint of segment $\mathrm{P}_{n+2} \mathrm{P}_{n+3}$ are the same. For example, the coordinates of point $\mathrm{P}_4$ are $(1, -2)$. If the coordinates of point $\mathrm{P}_{25}$ are $(a, b)$, find the value of $a + b$. [4 points]

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 2, when $a _ { 9 } = 3 a _ { 3 }$, what is the value of $a _ { 5 }$? [3 points]
(1) 10
(2) 11
(3) 12
(4) 13
(5) 14

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 6 and common difference $d$, let $S _ { n }$ denote the sum of the first $n$ terms. When
$$\frac { a _ { 8 } - a _ { 6 } } { S _ { 8 } - S _ { 6 } } = 2$$
holds, what is the value of $d$? [3 points]
(1) $-1$
(2) $-2$
(3) $-3$
(4) $-4$
(5) $-5$

A sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions. (가) $a _ { 1 } = a _ { 2 } + 3$ (나) $a _ { n + 1 } = - 2 a _ { n } ( n \geq 1 )$ Find the value of $a _ { 9 }$. [3 points]

For a sequence $\left\{ a _ { n } \right\}$, if the sum of the first $n$ terms $S _ { n } = \frac { n } { n + 1 }$, what is the value of $a _ { 4 }$? [3 points]
(1) $\frac { 1 } { 22 }$
(2) $\frac { 1 } { 20 }$
(3) $\frac { 1 } { 18 }$
(4) $\frac { 1 } { 16 }$
(5) $\frac { 1 } { 14 }$

For an arithmetic sequence $\left\{ a _ { n } \right\}$ satisfying $\sum _ { k = 1 } ^ { n } a _ { 2 k - 1 } = 3 n ^ { 2 } + n$, what is the value of $a _ { 8 }$? [4 points]
(1) 16
(2) 19
(3) 22
(4) 25
(5) 28

For an arithmetic sequence $\left\{ a _ { n } \right\}$, when $a _ { 8 } - a _ { 4 } = 28$, find the common difference of the sequence $\left\{ a _ { n } \right\}$. [3 points]

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 2, $$2 \left( a _ { 2 } + a _ { 3 } \right) = a _ { 9 }$$ Find the common difference of the sequence $\left\{ a _ { n } \right\}$. [3 points]

An arithmetic sequence $\left\{ a _ { n } \right\}$ with positive common difference satisfies the following conditions. What is the value of $a _ { 2 }$? [4 points] (가) $a _ { 6 } + a _ { 8 } = 0$ (나) $\left| a _ { 6 } \right| = \left| a _ { 7 } \right| + 3$
(1) - 15
(2) - 13
(3) - 11
(4) - 9
(5) - 7

For the function $f ( x ) = \frac { 1 } { 2 } x + 2$, find the value of $\sum _ { k = 1 } ^ { 15 } f ( 2 k )$. [3 points]

An arithmetic sequence $\left\{ a _ { n } \right\}$ satisfies $$a _ { 5 } + a _ { 13 } = 3 a _ { 9 } , \quad \sum _ { k = 1 } ^ { 18 } a _ { k } = \frac { 9 } { 2 }$$ Find the value of $a _ { 13 }$. [4 points]
(1) 2
(2) 1
(3) 0
(4) $-1$
(5) $-2$

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]

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 4, $$a _ { 10 } - a _ { 7 } = 6$$ What is the value of $a _ { 4 }$? [3 points]
(1) 10
(2) 11
(3) 12
(4) 13
(5) 14

An arithmetic sequence $\left\{ a _ { n } \right\}$ with first term a natural number and common difference a negative integer, and a geometric sequence $\left\{ b _ { n } \right\}$ with first term a natural number and common ratio a negative integer, satisfy the following conditions. Find the value of $a _ { 7 } + b _ { 7 }$. [4 points] (가) $\sum _ { n = 1 } ^ { 5 } \left( a _ { n } + b _ { n } \right) = 27$ (나) $\sum _ { n = 1 } ^ { 5 } \left( a _ { n } + \left| b _ { n } \right| \right) = 67$ (다) $\sum _ { n = 1 } ^ { 5 } \left( \left| a _ { n } \right| + \left| b _ { n } \right| \right) = 81$

For an arithmetic sequence with first term 50 and common difference $- 4$, let $S _ { n }$ denote the sum of the first $n$ terms. What is the value of the natural number $m$ that maximizes $\sum _ { k = m } ^ { m + 4 } S _ { k }$? [4 points]
(1) 8
(2) 9
(3) 10
(4) 11
(5) 12

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