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

$$\left( \sum _ { k = 1 } ^ { 9 } k \right) \cdot \left( \sum _ { n = 1 } ^ { 8 } \frac { 1 } { n ( n + 1 ) } \right)$$
What is the result of this operation?
A) 27
B) 30
C) 32
D) 36
E) 40

Let $\left( a _ { n } \right)$ be an arithmetic sequence such that $$\begin{aligned}& a _ { 10 } + a _ { 7 } = 6 \\& a _ { 9 } - a _ { 6 } = 1\end{aligned}$$ the following equalities are given.\ Accordingly, what is $a _ { 1 }$?\ A) $\frac { 7 } { 3 }$\ B) $\frac { 5 } { 2 }$\ C) $\frac { 4 } { 3 }$\ D) $\frac { 5 } { 6 }$\ E) $\frac { 1 } { 2 }$

For an arithmetic sequence $(a_n)$ with distinct terms and common difference $r$,
$$\begin{aligned} & a _ { 1 } = 3 \cdot r \\ & a _ { 6 } = a _ { 2 } \cdot a _ { 4 } \end{aligned}$$
the equalities are given.
Accordingly, what is $\mathbf { a } _ { \mathbf { 1 0 } }$?
A) 10 B) 8 C) 6 D) 4 E) 2

Filiz creates cup towers by placing identical cardboard cups inside each other. The distance between the bases of every two consecutive cups is equal in all the cup towers she creates. Then, she places these towers on a table and measures their heights.
Filiz observes that the sum of the heights of two towers with 6 and 9 cups equals the height of the tower with 18 cups.
Accordingly, to what height of a cup tower is the sum of the heights of two towers with 8 and 12 cups equal?
A) 23
B) 24
C) 26
D) 27
E) 29

For an arithmetic sequence $(a_n)$:
$$\begin{gathered} a _ { 2 } = 2 a _ { 1 } + 1 \\ a _ { 6 } + a _ { 22 } = 34 \end{gathered}$$
equalities are given.
Accordingly, what is $a _ { 7 }$?
A) $61^3$
B) 7
C) 8
D) 9
E) 10

In a census conducted on January 1, 2015, a city with a population of 810,000 had population counts on January 1 each year from 2016 to 2023. In each of the first four years after 2015, the population increased by a ratio of $\frac{1}{10}$ compared to the previous year, and in each of the following four years, the population increased by a ratio of $\frac{1}{11}$ compared to the previous year.
Accordingly, what was the population of this city in the census conducted on January 1, 2023?
A) $2^{20}$ B) $3^{13}$ C) $5^{9}$ D) $6^{8}$ E) $10^{6}$

A painter displayed all of his paintings at his first exhibition and sold some of them. In all subsequent exhibitions, this painter displayed the paintings that were not sold at the previous exhibition together with new paintings he created.
The painter sold $\frac{3}{5}$ of the paintings he displayed at each exhibition. Also, for each exhibition after the first, he created as many new paintings as the number of paintings remaining from the previous exhibition.
Given that the painter sold 96 paintings at his 3rd exhibition, how many paintings did he display at his first exhibition?
A) 100 B) 150 C) 200 D) 250 E) 300

A geometric sequence $(b_n)$ with first two terms $b_{1} = \frac{4}{3}$ and $b_{2} = 2$ and an arithmetic sequence $(a_n)$ whose common difference equals the common ratio of this geometric sequence are given.
If $b_{7} = a_{11}$, what is $a_{1}$?
A) $\frac{1}{4}$ B) $\frac{1}{8}$ C) $\frac{3}{8}$ D) $\frac{3}{16}$ E) $\frac{5}{16}$

Selma designs a toy with identical clips in yellow and blue colors. In step 1, she places one clip on the ground. In each subsequent step, she attaches one clip to each blue part of all the clips she placed in the previous step as shown in the figure, and moves to the next step. Selma completes the first 3 steps of this toy using 7 clips.
Accordingly, after Selma completes step 12, how many more clips has she used in total compared to after completing step 10?
A) $3 \cdot 2^{10}$
B) $3 \cdot 2^{11}$
C) $7 \cdot 2^{9}$
D) $7 \cdot 2^{10}$
E) $2^{11}$

For an arithmetic sequence $(a_{n})$ with common difference $r$
$$\begin{aligned} & a_{30} - a_{25} < 53 \\ & a_{25} - a_{8} > 53 \end{aligned}$$
the inequalities are given.
What is the sum of the integer values that $r$ can take?
A) 49 B) 56 C) 64 D) 72 E) 84

Merve plans a 30-day walking program. On the first day, she walks for a certain duration, and plans to walk 1 minute longer each subsequent day than the previous day. Following this plan for the first 15 days, Merve began to struggle and reduced her walking time by 1 minute each day for the remaining days compared to the previous day.
After 30 days, Merve calculated her total walking time to be 1395 minutes.
Accordingly, how many minutes did Merve's first day walk last?
A) 30
B) 35
C) 40
D) 45
E) 50

For an arithmetic sequence $(a_{n})$,
$$\begin{aligned} & a_{1} \cdot a_{2} \cdot a_{3} = 2 \\ & a_{2} \cdot a_{3} \cdot a_{4} = 14 \end{aligned}$$
are given. Accordingly, what is the product $a_{3} \cdot a_{4} \cdot a_{5}$?
A) 28 B) 35 C) 42 D) 49 E) 56