Functions $f$ and $g$ are defined on the set of real numbers as $$\begin{aligned}& f ( x ) = \frac { x \cdot ( x - 2 ) } { 2 } \\& g ( x ) = \frac { x \cdot ( x - 1 ) \cdot ( x - 2 ) } { 3 }\end{aligned}$$ The sum of the $\mathbf{x}$ values satisfying the equality $$f ( 2 x ) = g ( x + 1 )$$ is what?\ A) 1\ B) 3\ C) 4\ D) 6\ E) 8

Let $m$ and $n$ be two non-zero and distinct real numbers,
$$x ^ { 2 } + ( m + 1 ) x + n - m = 0$$
one of the roots of the equation is the number $m - n$.
Accordingly, what is the ratio $\frac { \mathbf { n } } { \mathbf { m } }$?
A) 2 B) 3 C) 4 D) 5 E) 6

Where $a$ and $b$ are positive real numbers,
$$2ax^2 - 5bx + 8b = a$$
the roots of the equation are $a$ and $b$. Accordingly, what is the sum $a + b$?
A) 5
B) 6
C) 10
D) 12
E) 15

A group of students, each 7 years old, visited a botanical garden in 2015; another group of students, each 10 years old, visited in 2020. The official who guided the groups through the garden said about the same historical tree in the garden to both groups: ``The age of this tree is equal to the sum of all of your ages.''
From these two groups, if the number of students in the first group is 10 more than the number of students in the second group, how old is this tree in 2020?
A) 220
B) 230
C) 240
D) 250
E) 260

Uncle Ahmet has a rectangular field with side lengths $x + 20$ and $2x + 30$ meters. He grows sunflowers in a square-shaped part with side length $x$ meters as shown in the figure.
If the area of the remaining part of the field is 1400 square meters, what is the perimeter of the entire field in meters?
A) 148 B) 154 C) 160 D) 166 E) 172

Let $a$ and $b$ be positive real numbers. For each of the equations
$$\begin{aligned} & ax^{2} - 2x + b = 0 \\ & bx^{2} - 3bx + a = 0 \end{aligned}$$
the sum of roots is 1 more than the product of roots.
Which of the following could be the quadratic equation whose roots are $a$ and $b$?
A) $9x^{2} + 8x + 18 = 0$ B) $9x^{2} - 14x + 8 = 0$ C) $9x^{2} - 18x + 14 = 0$ D) $9x^{2} - 8x + 14 = 0$ E) $9x^{2} - 18x + 8 = 0$