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Q4 Number Theory GCD, LCM, and Coprimality View

Let $a, b, c$ and $d$ be positive integers. $$\begin{aligned} & M = 6^{a} \cdot 5^{b} \\ & N = 10^{c} \cdot 9^{d} \end{aligned}$$ For the numbers $M$ and $N$ $$\begin{aligned} & \gcd(M, N) = 2^{3} \cdot 3^{2} \cdot 5 \\ & \text{lcm}(M, N) = 2^{5} \cdot 3^{3} \cdot 5^{5} \end{aligned}$$ are given. Accordingly, what is the sum $a + b + c + d$?
A) 8 B) 9 C) 10 D) 11 E) 12

Q5 Number Theory Prime Number Properties and Identification View

The sum of five distinct prime numbers equals 100, and their product equals a six-digit natural number ABCABC.
Accordingly, what is the sum $A + B + C$?
A) 8 B) 11 C) 14 D) 17 E) 20

Q6 Inequalities Ordering and Sign Analysis from Inequality Constraints View

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) $-\frac{1}{x}$

Q7 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View

Let $a$ and $b$ be real numbers. For the functions $f$ and $g$ defined on the set of real numbers as
$$\begin{aligned} & f(x) = \frac{x}{2} + 1 \\ & g(x) = 2x - 3 \end{aligned}$$
the equalities
$$\begin{aligned} & (f + g)(a) = f(a) \\ & (f - g)(b) = g(b) \end{aligned}$$
are satisfied. Accordingly, what is the value of $(f \circ g)(a \cdot b)$?
A) $\frac{1}{2}$ B) $\frac{5}{2}$ C) $\frac{9}{2}$ D) $\frac{13}{2}$ E) $\frac{17}{2}$

Q8 Curve Sketching Limit Reading from Graph View

In the rectangular coordinate plane, the graphs of the functions $f + g$ and $f \cdot g$ defined on the closed interval $[0, 10]$ are shown below.
For the real numbers $a, b$ and $c$ in the closed interval $[0, 10]$,

$f(a), f(b)$ and $g(b)$ values are positive,
$g(a), f(c)$ and $g(c)$ values are negative.

Accordingly, which of the following is the correct ordering of $a, b$ and $c$?
A) $a < c < b$ B) $b < a < c$ C) $b < c < a$ D) $c < a < b$ E) $c < b < a$

Q9 Conditional Probability Direct Conditional Probability Computation from Definitions View

Regarding the subsets $A, B$ and $C$ of the set of natural numbers, the propositions
$$\begin{aligned} & p : 9 \in A \cup B \\ & q : 9 \in A \cap C \\ & r : 9 \notin C \end{aligned}$$
are given. Given that the proposition $(p \Rightarrow q)' \wedge r'$ is true, which of the following statements are true?
I. $9 \in A$ II. $9 \in B$ III. $9 \in C$
A) Only I B) Only III C) I and II D) II and III E) I, II and III

Q10 Permutations & Arrangements Distribution of Objects into Bins/Groups View

Regarding the sets $A, B, C, K$ and $L$, $$K = A \times B$$ $$L = B \times C$$ are given.
Given that $K \cup L = \{(1,2), (1,3), (2,2), (3,2), (3,3)\}$, which of the following is the set $K \cap L$?
A) $\{(1,2)\}$ B) $\{(1,3)\}$ C) $\{(2,2)\}$ D) $\{(3,2)\}$ E) $\{(3,3)\}$

Q11 Discriminant and conditions for roots Condition for repeated (equal/double) roots View

Let $a$ and $b$ be positive real numbers. The equations
$$\begin{aligned} & x^{2} + ax + b = 0 \\ & ax^{2} + (b + 3)x + a = 0 \end{aligned}$$
are given. Given that the solution set of each of these equations has exactly 1 element, what is the product of the different values that the sum $a + b$ can take?
A) 24 B) 32 C) 45 D) 72 E) 120

Q12 Factor & Remainder Theorem Polynomial Construction from Root/Value Conditions View

Let $P(x)$ and $Q(x)$ be polynomials with real coefficients such that $P(x) + Q(x)$ is a second-degree polynomial and
$$\begin{aligned} & P(x) \cdot Q(x) = -4 \cdot (x-1)^{4} \cdot (x-2)^{2} \\ & P(3) = -16 \end{aligned}$$
are satisfied. Accordingly, what is the value of $Q(4)$?
A) 12 B) 24 C) 36 D) 48 E) 54

Q13 Arithmetic Sequences and Series Find Specific Term from Given Conditions View

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

Q14 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View

Let $n$ be a positive integer. In the expansion of
$$\left(x^{2} + x\right)^{n}$$
both the coefficient of the term containing $x^{19-n}$ and the coefficient of the term containing $x^{16-n}$ equal a positive integer $k$. Accordingly, what is $k$?
A) 6 B) 12 C) 15 D) 18 E) 21

Q15 Permutations & Arrangements Linear Arrangement with Constraints View

Sena has chosen a series A consisting of three films and a series B consisting of three films to watch a different film each of six specific evenings. For each series, Sena will not watch the second film without watching the first film of that series, and will not watch the third film without watching the second film of that series.
Accordingly, in how many different ways can Sena determine the order in which she will watch these six films?
A) 20 B) 24 C) 27 D) 30 E) 32

Q16 Probability Definitions Conditional Probability and Bayes' Theorem View

In front of the two doors of a shopping mall, there are 2 parking lots named Blue and Red in front of the first door, and 3 parking lots named Yellow, Orange and Green in front of the second door. Kartal, who came to this shopping mall, randomly came in front of one of the doors and randomly parked his car in one of the parking lots in front of that door and entered the shopping mall. When leaving the shopping mall, since Kartal forgot which parking lot he parked his car in and which door he entered the shopping mall from, he exited from one of the doors randomly and searched for his car in one of the parking lots in front of that door randomly.
Accordingly, what is the probability that the parking lot where Kartal searched for his car is the parking lot where he parked his car?
A) $\frac{1}{5}$ B) $\frac{5}{24}$ C) $\frac{6}{25}$ D) $\frac{7}{36}$ E) $\frac{11}{48}$

Q17 Laws of Logarithms Solve a Logarithmic Equation View

Let $a$ and $b$ be non-consecutive positive integers. The equality
$$\ln(a!) = \ln(b!) + 3 \cdot \ln 2 + 2 \cdot \ln 3 + \ln 7$$
is satisfied. Accordingly, what is the sum $a + b$?
A) 10 B) 13 C) 15 D) 18 E) 20

Q18 Laws of Logarithms Solve a Logarithmic Equation View

Let $a, b, c$ and $d$ be distinct positive real numbers. The sets $A$ and $B$ are defined as
$$\begin{aligned} & A = \left\{ \log_{2} a, \log_{2} b, \log_{2} c, \log_{2} d \right\} \\ & B = \left\{ \log_{\frac{1}{2}} a, \log_{\frac{1}{2}} b, \log_{\frac{1}{2}} c, \log_{\frac{1}{2}} d \right\} \end{aligned}$$
$$\begin{aligned} & s(A \cap B) = 3 \\ & a \cdot b \cdot c \cdot d = \frac{7}{5} \\ & a + b + c + d = \frac{38}{5} \end{aligned}$$
Given that, what is the sum $a^{2} + b^{2} + c^{2} + d^{2}$?
A) 20 B) 22 C) 24 D) 26 E) 28

Q19 Composite & Inverse Functions Find or Apply an Inverse Function Formula View

Let $a$ be a non-zero real number, and $b$ and $c$ be real numbers. For the function $f(x) = ax + b$ defined on the set of real numbers and its inverse function $f^{-1}$,
$$\begin{aligned} & \lim_{x \rightarrow b} \frac{f(x)}{f^{-1}(x)} = c \\ & f(1) = 3 \end{aligned}$$
are given. Accordingly, what is the sum of the different values that $c$ can take?
A) 6 B) 7 C) 10 D) 11 E) 14

Q20 Completing the square and sketching Determining coefficients from given conditions on function values or geometry View

Let $a$ and $b$ be real numbers. The function $f$ defined on the set of real numbers as
$$f(x) = \begin{cases} x^{2} - ax + 6 & , x \leq a \\ 2x + a & , a < x \leq b \\ 11 - 2x + b & , x > b \end{cases}$$
is continuous on its domain.
Accordingly, what is the product $a \cdot b$?
A) 4 B) 6 C) 8 D) 10 E) 12

Q21 Chain Rule Finding Composition Parameters from Derivative Conditions View

Let $n$ be a positive integer and $a$ be a non-zero real number. For a polynomial function $f$ with degree $n$ and leading coefficient $a$,
$$\left((f(x))^{3}\right)' = \left(f'(x)\right)^{4}$$
is satisfied.
Accordingly, what is the product $a \cdot n$?
A) $\frac{1}{4}$ B) $\frac{1}{3}$ C) $\frac{1}{2}$ D) $1$ E) $2$

Q22 Chain Rule Chain Rule with Composition of Explicit Functions View

In the rectangular coordinate plane, for a function $y \geq f(x)$,

the tangent line at the point $(2, f(2))$ is $y = 3x - 1$
the tangent line at the point $(5, f(5))$ is $y = 2x + 4$

Accordingly, for the function $$g(x) = x^{2} \cdot (f \circ f)(x)$$
what is the value of $g'(2)$?
A) 64 B) 72 C) 80 D) 88 E) 96

Q23 Stationary points and optimisation Analyze function behavior from graph or table of derivative View

In the rectangular coordinate plane, the graph of the derivative $f'$ of a continuous function $f$ defined on the set of real numbers is shown in the figure.
$$f(5) = f(20) = 0$$
Given that, what is the local minimum value of the function $f$?
A) $-18$ B) $-15$ C) $-12$ D) $-9$ E) $-6$

Q25 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

For a continuous function $f$ defined on the set of real numbers and the function $g(x) = 2x + 2$ defined as,
$$\begin{aligned} & \int_{-1}^{1} f(g(x))\, dx = 18 \\ & \int_{2}^{4} g(f(x))\, dx = 18 \end{aligned}$$
are satisfied. Accordingly, what is the value of the integral $\int_{0}^{2} f(x)\, dx$?
A) 20 B) 23 C) 26 D) 29 E) 32

Q26 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

Let $m$ be a positive real number. In the rectangular coordinate plane, the region between the graph of a function $f$ defined on the closed interval $[-m, m]$ and the $x$-axis is divided into four regions and these regions are colored as shown in the figure. The areas of these regions, which are different from each other, are denoted by $A, B, C$ and $D$ as shown in the figure.
$$\int_{-m}^{m} |f(x)|\, dx = \int_{-m}^{m} f(x)\, dx + \int_{0}^{m} 2 \cdot f(x)\, dx$$
Given that, which of the following is the integral $\int_{-m}^{m} f(x)\, dx$ equal to?
A) $A + B$ B) $A + C$ C) $A + D$ D) $B + C$ E) $C + D$

Q27 Reciprocal Trig & Identities View

The simplified form of the expression
$$\frac{1 - \cos(4x)}{\sin(4x) + 2 \cdot \sin(2x)}$$
is which of the following?
A) $\sin x$ B) $\tan x$ C) $\cot x$ D) $\sec x$ E) $\operatorname{cosec} x$

Q28 Standard trigonometric equations Evaluate trigonometric expression given a constraint View

Let $x, y$ and $z$ be distinct elements of the set $\left\{\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}\right\}$ such that
$$\sin x < \tan y < \sec z$$
Which of the following is the correct ordering of $x$, $y$ and $z$?
A) $x < y < z$ B) $y < x < z$ C) $y < z < x$ D) $z < x < y$ E) $z < y < x$

Q29 Addition & Double Angle Formulae Geometric Configuration with Trigonometric Identities View

For a triangle $ABC$ with side lengths $|BC| = a$ units, $|AC| = b$ units and $|AB| = c$ units,
$$2a^{2} = 2b^{2} + 2c^{2} + 3bc$$
is satisfied. Let $m(\widehat{BAC}) = x$. What is the value of $\tan x$?
A) $-\frac{\sqrt{2}}{3}$ B) $-\frac{\sqrt{3}}{3}$ C) $-\frac{\sqrt{5}}{3}$ D) $-\frac{\sqrt{6}}{3}$ E) $-\frac{\sqrt{7}}{3}$

Q30 Reciprocal Trig & Identities View

Let $0 < x < \frac{\pi}{2}$. Given that
$$2 \cdot \cos^{2} x + 9 \cdot \sin^{2} x + 2 \cdot \sin(2x) = 9$$
what is the value of $\cot x$?
A) $\frac{4}{7}$ B) $\frac{7}{6}$ C) $\frac{3}{5}$ D) $\frac{2}{3}$ E) $\frac{5}{2}$

Q31 Addition & Double Angle Formulae Geometric Configuration with Trigonometric Identities View

Let $a$ and $b$ be positive real numbers. In the rectangular coordinate plane, the acute angles that the lines $d_{1}$ and $d_{2}$ shown make with the $x$-axis are $A$ and $B$ respectively, as shown in the figure.
Accordingly, which of the following is the expression for the ratio $\frac{a}{b}$ in terms of $A$ and $B$?
A) $\frac{\tan A}{\tan B}$ B) $\cot A \cdot \cot B$ C) $\cot A - \tan B$ D) $1 + \cot A \cdot \tan B$ E) $1 - \tan A \cdot \cot B$

Q32 Circles Area and Geometric Measurement Involving Circles View

An equilateral triangle with red-colored sides and an equilateral triangle with blue-colored sides are drawn such that one vertex of each is on a side of the other triangle, as shown in the figure.
In the resulting figure, the area of the yellow-colored triangle equals 4 times the area of the gray-colored triangle.
Accordingly, what is the ratio of the area of the triangle with red-colored sides to the area of the triangle with blue-colored sides?
A) $\frac{2}{3}$ B) $\frac{5}{6}$ C) $\frac{8}{9}$ D) $\frac{25}{27}$ E) $\frac{25}{36}$

Q33 Circles Circle Equation Derivation View

Two identical blue ropes have one end each tied to two nails on a wall at equal heights from the ground and 48 units apart. Then a circular plate is hung on these ropes such that the other ends of the ropes are attached to two points on the circumference of the plate and the ropes are perpendicular to the ground, as shown in Figure 1. Later, one of these ropes broke and the plate hung on the remaining rope, and when the rope is perpendicular to the ground, the view in Figure 2 is obtained, and the height of the plate from the ground decreased by 16 units compared to the initial situation.
Accordingly, what is the radius of this plate in units?
A) 25 B) 26 C) 29 D) 30 E) 32

Q34 Circles Area and Geometric Measurement Involving Circles View

In the figure, point $C$ is on the line segment $[AB]$, point $D$ is on the semicircle with diameter $[AB]$, and $m(\widehat{BAD}) = 18^{\circ}$.
In the figure, the area of the yellow-colored region equals 4 times the area of the blue-colored region. Accordingly, what is the ratio $\frac{|AC|}{|BC|}$?
A) $\frac{3}{2}$ B) $\frac{5}{3}$ C) $\frac{7}{4}$ D) $\frac{7}{5}$ E) $\frac{9}{5}$

Q35 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View

In the rectangular coordinate plane, a triangle $OAB$ with one vertex at the origin and the other two vertices on the axes, and the line segment $[PR]$ connecting the points $P(6, -3)$ and $R(-2, 9)$ are drawn. The line segment $[PR]$ passes through the midpoints of both $[OA]$ and $[OB]$.
According to this, what is the area of triangle $OAB$ in square units?
A) 36 B) 42 C) 48 D) 54 E) 60

Q36 Straight Lines & Coordinate Geometry Geometric Figure on Coordinate Plane View

Let $a$ and $b$ be positive real numbers. In the rectangular coordinate plane, the region between the lines $y = -\sqrt{3}x$ and $y = ax + b$ and the $x$-axis forms an equilateral triangle with area $9\sqrt{3}$ square units.
Accordingly, what is the product $a \cdot b$?
A) 18 B) 24 C) 27 D) 30 E) 36

Q37 Circles Distance from Center to Line View

A circle drawn in the rectangular coordinate plane

has one common point with the line $d_{1}: y - \frac{4x}{3} - 46 = 0$,
has two common points with the line $d_{2}: y - \frac{4x}{3} - 6 = 0$,
has no common points with the line $d_{3}: y - \frac{4x}{3} - 1 \geqslant 0$.

It is known that. Accordingly, which of the following could be the radius of this circle in units?
A) 11 B) 13 C) 15 D) 17 E) 19

Q38 Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View

In the rectangular coordinate plane, when point $A$ is translated 15 units in the negative direction along the $x$-axis, the resulting point lies on the line $d: 4x - 3y + 24 = 0$.
Accordingly, if point $A$ is translated how many units in the positive direction along the $y$-axis, the resulting point will lie on line $d$?
A) 9 B) 12 C) 16 D) 20 E) 25

Q39 Circles Chord Length and Chord Properties View

Let $m$ and $n$ be real numbers. In the rectangular coordinate plane, a circle passing through point $A(4, 1)$ is drawn with equation
$$x^{2} + y^{2} - 2x + 6y = n$$
The line $y = mx$ drawn in the plane intersects this circle at points $B$ and $C$. Given that $m(\widehat{BAC}) = 90^{\circ}$, what is the sum $m + n$?
A) 8 B) 9 C) 10 D) 11 E) 12

Q40 Circles Sphere and 3D Circle Problems View

For research laboratories planned to be established on the Luna planet, two completely closed buildings with the same radii and volumes are designed to be placed on the ground as shown in the figure: one in the shape of a half right circular cylinder and the other in the shape of a hemisphere.
Accordingly, what is the ratio of the surface area of the half right circular cylinder building (excluding the ground) to the surface area of the hemisphere building (excluding the ground)?
A) $\frac{1}{2}$ B) $\frac{3}{5}$ C) $1$ D) $\frac{7}{6}$ E) $\frac{4}{3}$

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