For real numbers $a, b$ and $c$,
$$a > a \cdot b > 2 \cdot a > a \cdot c$$
is known to hold.
Accordingly, which of the following could be the representation of the numbers $\mathbf{a, b}$ and $\mathbf{c}$ on the number line?
A) [number line A] B) [number line B] C) [number line C] D) [number line D] E) [number line E]

Let $a$ and $b$ be positive integers,
$$\begin{aligned} & (x - a)(x + 2a) < 0 \\ & (x - b)(x + 2b) > 0 \end{aligned}$$
Given this system of inequalities.
If $a + b = 8$ and the solution set of this system of inequalities contains 16 integers, what is the product $a \cdot b$?
A) 7 B) 10 C) 12 D) 15 E) 16

A coffee machine detects the length of the cup placed in its cup holder. For the coffee machine to operate, the distance between the coffee reservoir and the top of the cup must be at least 11.5 cm and at most 12.4 cm, as shown in the figure.
When a cup of length $14.5 \mathrm{~cm}$ is placed in the cup holder of this coffee machine, the machine operates; when a cup of length $15.2 \mathrm{~cm}$ is placed, the machine does not operate.
Accordingly, when cups of the following lengths are placed in the cup holder of this coffee machine:
I. $14.2 \mathrm{~cm}$ II. $14.4 \mathrm{~cm}$ III. $14.6 \mathrm{~cm}$
For which of these will the machine definitely operate?
A) Only I B) Only II C) Only III D) I and II

Let $a$ be an integer. There are exactly 4 integer values of $x$ satisfying
$$0 < \left| x^{2} - 2x + 2 \right| - x^{2} - x < a$$
What is the sum of the different integer values that $a$ can take?
A) 33 B) 36 C) 39 D) 42 E) 45

In a store, the cost prices of a dishwasher, a washing machine, and a refrigerator are 18 thousand, 22 thousand, and $b$ TL, respectively. These products are sold together as a triple white goods package. This package is sold at twice the cost price of a refrigerator plus 30 thousand TL, the store makes a profit; when sold at three times the cost price of a refrigerator minus 20 thousand TL, the store makes a loss.
Accordingly, which of the following inequalities expresses all possible values of the cost price of a refrigerator?
A) $|b - 20000| < 10000$
B) $|b - 15000| < 5000$
C) $|2b - 15000| < 12000$
D) $|2b - 24000| < 17000$
E) $|3b - 16000| < 8000$

Let $x$ be a real number different from $-1, 0$ and $1$.
$$\left\{ x^{3}, x^{2}, x, -x, -\frac{1}{x} \right\}$$
When the elements of the set are arranged from smallest to largest, which element never occupies the exact middle position?
A) $x^{3}$ B) $x^{2}$ C) $x$ D) $-x$ E) $-\frac{1}{x}$