An arithmetic progression has first term $a$ and common difference $d$.
The sum of the first 5 terms is equal to the sum of the first 8 terms.
Which one of the following expresses the relationship between $a$ and $d$ ?
A $a = - \frac { 38 } { 3 } d$
B $a = - 7 d$
C $a = - 6 d$
D $a = 6 d$
E $\quad a = 7 d$
F $\quad a = \frac { 38 } { 3 } d$

Sequence 1 is an arithmetic progression with first term 11 and common difference 3.
Sequence 2 is an arithmetic progression with first term 2 and common difference 5.
Some numbers that appear in Sequence 1 also appear in Sequence 2. Let $N$ be the 20th such number.
What is the remainder when $N$ is divided by 7 ?

In this question, $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ is an arithmetic progression, all of whose terms are integers.
Let $n$ be a positive integer. If the median of the first $n$ terms of the sequence is an integer, which of the following three statements must be true?
I The median of the first $n + 2$ terms is an integer.
II The median of the first $2 n$ terms is an integer.
III The median of $x _ { 2 } , x _ { 4 } , x _ { 6 } , \ldots , x _ { 2 n }$ is an integer.

An arithmetic series has $n$ terms, all of which are integers.
The sum of the series is 20 .
Which of the following statements must be true?
I The first term of the series is even.
II $n$ is even.
III The common difference is even.

An arithmetic sequence $T$ has first term $a$ and common difference $d$, where $a$ and $d$ are non-zero integers.
Property P is:
For some positive integer $m$, the sum of the first $m$ terms of the sequence is equal to the sum of the first $2 m$ terms of the sequence.
For example, when $a = 11$ and $d = - 2$, the sequence $T$ has property P , because
$$11 + 9 + 7 + 5 = 11 + 9 + 7 + 5 + 3 + 1 + ( - 1 ) + ( - 3 )$$
i.e. the sum of the first 4 terms equals the sum of the first 8 terms. Which of the following statements is/are true? I For $T$ to have property P , it is sufficient that $a d < 0$. II For $T$ to have property P , it is necessary that $d$ is even.
A neither of them
B I only
C II only
D I and II

A selection, $S$, of $n$ terms is taken from the arithmetic sequence $1,4,7,10 , \ldots , 70$. Consider the following statement: (*) There are two distinct terms in $S$ whose sum is 74 .
What is the smallest value of $n$ for which (*) is necessarily true?
A 12
B 13
C 14
D 21
E 22 F 23

Evaluate
$$\sum _ { n = 1 } ^ { 100 } \log _ { 10 } \left( 3 ^ { 1 - n } \right)$$

Evaluate
$$\sum _ { n = 0 } ^ { \infty } \frac { \sin \left( n \pi + \frac { \pi } { 3 } \right) } { 2 ^ { n } }$$

The base 10 number 0.03841 has the value
$$0 \times 10 ^ { - 1 } + 3 \times 10 ^ { - 2 } + 8 \times 10 ^ { - 3 } + 4 \times 10 ^ { - 4 } + 1 \times 10 ^ { - 5 } = 0.03841$$
Similarly, the base 2 number 0.01101 has the value
$$0 \times 2 ^ { - 1 } + 1 \times 2 ^ { - 2 } + 1 \times 2 ^ { - 3 } + 0 \times 2 ^ { - 4 } + 1 \times 2 ^ { - 5 } = \frac { 13 } { 32 }$$
What is the value of the recurring base 2 number $0 . \dot { 0 } 01 \dot { 1 } = 0.001100110011 \ldots$ ? A $\frac { 1 } { 3 }$ B $\frac { 1 } { 5 }$ C $\frac { 1 } { 15 }$ D $\frac { 2 } { 15 }$ E $\frac { 4 } { 15 }$ F $\frac { 3 } { 16 }$ G $\frac { 5 } { 16 }$ H $\frac { 6 } { 31 }$

$15 ^ { 13 } + 6 \cdot 15 ^ { 13 } + 8 \cdot 15 ^ { 13 }$
What is the result of this operation?
A) $15 ^ { 15 }$
B) $15 ^ { 14 }$
C) $14 \cdot 15 ^ { 13 }$
D) $10 \cdot 16 ^ { 13 }$
E) $16 ^ { 13 }$

$$\sum_{n=0}^{100} 3^{n}$$
What is the remainder when this sum is divided by 5?
A) 0
B) 1
C) 2
D) 3
E) 4

The sequences $\{a_{n}\}$ and $\{b_{n}\}$ are defined as follows. $$a_{n} = \begin{cases} 0, & \text{if } n \equiv 0 \pmod{3} \\ n, & \text{if } n \equiv 1 \pmod{3} \\ -n, & \text{if } n \equiv 2 \pmod{3} \end{cases}, \quad b_{n} = \sum_{k=0}^{n} a_{k}$$ Accordingly, what is $b_{4}$?
A) $-2$
B) $-1$
C) $0$
D) $2$
E) $3$

The geometric mean of numbers a and b is 3, and their arithmetic mean is 6.
Accordingly, what is the arithmetic mean of $\mathrm { a } ^ { 2 }$ and $\mathrm { b } ^ { 2 }$?
A) 67
B) 65
C) 63
D) 61
E) 57

On day 1, Ismail puts one of each of the following coins into his piggy bank: 5 Kr, 10 Kr, 25 Kr, 50 Kr, and 1 TL. On day 2, he puts two of each, and continuing in this manner, on day n he puts n of each.
If Ismail has saved 104.5 TL in his piggy bank, what is n?
A) 10 B) 11 C) 12 D) 13 E) 14

For positive real numbers x and y,
$$\begin{aligned} x \cdot y & = 5 \\ x ^ { 2 } + y ^ { 2 } & = 15 \end{aligned}$$
Given this, what is the value of the expression $x ^ { 3 } + y ^ { 3 }$?
A) 40
B) 45
C) 50
D) 60
E) 75

The sum of all two-digit natural numbers with digit A in the units place is 504. What is A?
A) 5
B) 6
C) 7
D) 8
E) 9

$$( n + 2 ) ! - ( n + 1 ) ! - n ! = 2 ^ { 3 } \cdot 3 \cdot 5 ^ { 2 } \cdot 7$$
Given this, what is $n$?
A) 5
B) 6
C) 7
D) 8
E) 9

$$A = 13 + 26 + 39 + \cdots + 169$$
Given this, what is the sum of the prime numbers that divide A?
A) 16
B) 18
C) 20
D) 22
E) 24

The geometric mean of real numbers a and b is 4, and the geometric mean of $a - 1$ and $b + 1$ is 6.
Accordingly, what is the difference $\mathbf { a - b }$?
A) 20
B) 21
C) 22
D) 23
E) 24

The following information is known about the athletes participating in a race.

• The jersey numbers of male athletes are consecutive odd numbers starting from 1.
• The jersey numbers of female athletes are consecutive even numbers starting from 2.
• The number of male athletes is 3 times the number of female athletes.
• The largest jersey number given to male athletes is 83.

Given this, what is the largest jersey number given to female athletes?
A) 28
B) 30
C) 32
D) 34
E) 36

The following information is known about the monthly salaries of Ahmet and Beyza, who started work on the same day at a workplace.

• Ahmet's initial salary is 2500 TL.
• Ahmet's salary increases by 50 TL every 4 months.
• Beyza's salary increases by 100 TL every 6 months.

Given that their salaries are equal 6 years after they receive their first salaries, what is Beyza's initial salary in TL?
A) 2000
B) 2100
C) 2200
D) 2300
E) 2400

Let a, b and c be three consecutive even integers arranged from smallest to largest such that the geometric mean of b and c is $\sqrt { 2 }$ times the geometric mean of a and b.
Accordingly, what is the sum $\mathrm { a } + \mathrm { b } + \mathrm { c }$?
A) 12
B) 18
C) 24
D) 30
E) 36

What is the sum of all natural numbers such that when divided by 6, the quotient and remainder are equal to each other?
A) 84
B) 91
C) 96
D) 105
E) 112

The function f on the set of real numbers is defined for every real number x as
$$f ( x ) = \left\{ \begin{array} { c c } x + 2 , & x < 0 \\ x , & x \geq 0 \end{array} \right.$$
Accordingly, what is the value of the sum $\sum _ { k = - 3 } ^ { 4 } f ( k )$?
A) 8
B) 10
C) 12
D) 14
E) 16

The smaller of two numbers is 3 less than the arithmetic mean of these two numbers, and the larger is 4 more than the geometric mean of these two numbers.
Accordingly, what is the sum of these two numbers?
A) 7
B) 9
C) 10
D) 12
E) 14