Points are marked on the coordinate plane in the following [Steps]. [Step 1] Mark a point at $( 0,1 )$. [Step 2] Mark 3 points at $( 0,3 ) , ( 1,3 ) , ( 2,3 )$ in this order. $\vdots$ [Step $k$ ] Mark $( 2 k - 1 )$ points at $( 0,2 k - 1 ) , ( 1,2 k - 1 ) , ( 2,2 k - 1 ) , \cdots$, $( 2 k - 2,2 k - 1 )$ in this order. (Here, $k$ is a natural number.) $\vdots$ When points are marked in this manner starting from [Step 1], the coordinates of the 100th marked point are $( p , q )$. What is the value of $p + q$? [4 points]
(1) 46
(2) 43
(3) 40
(4) 37
(5) 34

Find the number of ways to partition the natural number 11 into natural numbers between 3 and 7 (inclusive). [3 points]
(1) 2
(2) 4
(3) 6
(4) 8
(5) 10

(17 points) Let $m$ be a positive integer. The sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is an arithmetic sequence with nonzero common difference. If after removing two terms $a _ { i }$ and $a _ { j } ( i < j )$ , the remaining $4 m$ terms can be evenly divided into $m$ groups, and the 4 numbers in each group form an arithmetic sequence, then the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is called an $( i , j )$ -divisible sequence.
(1) Write out all pairs $( i , j )$ with $1 \leqslant i < j \leqslant 6$ such that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 6 }$ is an $( i , j )$ -divisible sequence;
(2) When $m \geqslant 3$ , prove that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is a $( 2,13 )$ -divisible sequence;
(3) From $1,2 , \cdots , 4 m + 2$ , randomly select two numbers $i$ and $j$ ( $i < j$ ) at once. Let $P _ { m }$ denote the probability that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is an $( i , j )$ -divisible sequence. Prove that $P _ { m } > \frac { 1 } { 8 }$ .

In how many ways can we choose $a _ { 1 } < a _ { 2 } < a _ { 3 } < a _ { 4 }$ from the set $\{ 1,2 , \ldots , 30 \}$ such that $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ are in arithmetic progression?
(A) 135
(B) 145
(C) 155
(D) 165

A pack contains $n$ cards numbered from 1 to $n$. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is $k$, then $k - 20 =$

The probability that $x_1, x_2, x_3$ are in an arithmetic progression, is
(A) $\frac{9}{105}$
(B) $\frac{10}{105}$
(C) $\frac{11}{105}$
(D) $\frac{7}{105}$

Let $X$ be the set consisting of the first 2018 terms of the arithmetic progression $1,6,11 , \ldots$, and $Y$ be the set consisting of the first 2018 terms of the arithmetic progression $9,16,23 , \ldots$. Then, the number of elements in the set $X \cup Y$ is $\_\_\_\_$.

Let $A P ( a ; d )$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d > 0$. If $$A P ( 1 ; 3 ) \cap A P ( 2 ; 5 ) \cap A P ( 3 ; 7 ) = A P ( a ; d )$$ then $a + d$ equals $\_\_\_\_$

A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after
(1) 19 months
(2) 20 months
(3) 21 months
(4) 18 months

The number of terms in an $A.P$. is even, the sum of the odd terms in it is 24 and that the even terms is 30 . If the last term exceeds the first term by $10 \frac { 1 } { 2 }$, then the number of terms in the $A.P$. is
(1) 4
(2) 8
(3) 16
(4) 12

The number of terms in an A.P. is even; the sum of the odd terms in it is 24 and that the even terms is 30. If the last term exceeds the first term by $10\frac{1}{2}$, then the number of terms in the A.P. is:
(1) 4
(2) 8
(3) 12
(4) 16

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is
(1) 262
(2) 190
(3) 225
(4) 157

If $\left\{ a _ { i } \right\} _ { i = 1 } ^ { \mathrm { n } }$, where $n$ is an even integer, is an arithmetic progression with common difference 1 , and $\sum _ { i = 1 } ^ { n } a _ { i } = 192 , \sum _ { i = 1 } ^ { \frac { n } { 2 } } a _ { 2 i } = 120$, then $n$ is equal to
(1) 18
(2) 36
(3) 96
(4) 48

The $8^{\text{th}}$ common term of the series $$\begin{aligned} & S_{1} = 3 + 7 + 11 + 15 + 19 + \ldots \\ & S_{2} = 1 + 6 + 11 + 16 + 21 + \ldots \end{aligned}$$ is

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total number of persons who participated in the tournament is $\_\_\_\_$.

Let the digits $a , b , c$ be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

The number of common terms in the progressions $4,9,14,19 , \ldots$. up to $25 ^ { \text {th} }$ term and $3,6,9,12 , \ldots$. up to $37 ^ { \text {th} }$ term is:
(1) 9
(2) 5
(3) 7
(4) 8

Let the positive integers be written in the form : If the $k ^ { \text {th} }$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is $\_\_\_\_$

Suppose that the number of terms in an A.P. is $2k , k \in N$. If the sum of all odd terms of the A.P. is 40 , the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27 , then k is equal to :
(1) 6
(2) 5
(3) 8
(4) 4

The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6 ^ { \circ }$. If the largest interior angle of the polygon is $219 ^ { \circ }$, then n is equal to

An electronic billboard continuously alternates between playing advertisements A and B ($A$, $B$, $A$, $B \ldots$), with each advertisement playing for $T$ minutes (where $T$ is an integer). A person passes by just as advertisement A starts playing. 30 minutes later, the person returns to the location and sees advertisement B just starting to play. Select the options that could be the value of $T$.
(1) $15$ (2) $10$ (3) $8$ (4) $6$ (5) $5$

From the 20 integers from 1 to 20, select three distinct numbers $a$, $b$, $c$ that form an arithmetic sequence with $a < b < c$. The number of ways to choose $(a, b, c)$ is $\square\square$.

For integers $\mathrm { x } , \mathrm { y }$ and z
$$2 x = 3 y = 5 z$$
Given this, which of the possible values of the sum $x + y + z$ is closest to 100?
A) 93
B) 96
C) 98
D) 103
E) 105

A school principal sends an electronic mail on Monday to some students of the school containing the note, ``Every student who receives this message should send it to two students the next day.'' The students who receive the message follow what is written in that note.
By the end of Friday of the same week, this message reaches all students in the school and each student receives this message only once.
Given that the number of students in the school is 930, how many students was this message initially sent to?
A) 6
B) 10
C) 15
D) 21
E) 30

On a circular playground, five players named Ali, Büşra, Cem, Deniz, and Ekin are playing with a ball in positions shown in the figure. In each turn of this game; the player holding the ball passes it to the third player after them in the direction of the arrow.
Initially, the ball is in Ali's hands and the game starts when Ali passes the ball to Deniz. Deniz received the ball on the 1st turn, Büşra on the 2nd turn, and the game continued in this way.
Accordingly, who received the ball on the 99th turn?
A) Ali B) Büşra C) Cem D) Deniz E) Ekin