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
(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
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
1. If each term in the sequence $a_{1}, a_{2}, \ldots, a_{k}, \ldots, a_{10}$ is either 1 or $-1$, how many possible values can $a_{1} + a_{2} + \cdots + a_{k} + \cdots + a_{10}$ take? (1) 10 (2) 11 (3) $P_{2}^{10}$ (4) $C_{2}^{10}$ (5) $2^{10}$
A robot cat starts from the origin on a number line and moves in the positive direction. Its movement method is as follows: With an 8-second cycle, it moves at a constant speed of 4 units per second for 6 seconds, then rests for 2 seconds. Continuing this way, the robot cat will reach the position with coordinate 116 on the number line after $\underbrace{(14)(15)}$ seconds from the start of movement.
A light show display uses color-changing flashing lights. After each activation, the flashing color changes periodically according to the following sequence: Blue–White–Red–White–Blue–White–Red–White–Blue–White–Red–White…, with one cycle every four flashes. Blue light lasts 5 seconds each time, white light lasts 2 seconds each time, and red light lasts 6 seconds each time. Assuming the time to change lights is negligible, select the light color(s) between the 99th and 101st seconds after activation. (1) All blue lights (2) All white lights (3) All red lights (4) Blue light first, then white light (5) White light first, then red light
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 $\square\square$.