For positive real numbers $a, x$ and $y$ $$\begin{aligned}
& -2x^{2} + y^{2} = 2a \\
& 3x^{2} - 2y^{2} = -6a
\end{aligned}$$ what is the ratio $\dfrac{y}{x}$? A) 1 B) $\sqrt{2}$ C) $\sqrt{3}$ D) 2 E) 3
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 the rectangular coordinate plane, the graphs of functions $f$ and $g$ defined on the closed interval $[0,1]$ and the line $y = x$ are given below. For real numbers $a, b$ and $c$ in the open interval $(0,1)$ $$\begin{aligned}
& a < f(a) < g(a) \\
& g(b) < b < f(b) \\
& c < g(c) < f(c)
\end{aligned}$$ If these inequalities are satisfied, which of the following orderings is correct? A) $a < b < c$ B) $a < c < b$ C) $b < a < c$ D) $c < a < b$ E) $c < b < a$
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) $9x^{2} - 18x + 8 = 0$
Let $a$ and $b$ be real numbers. The function $f$ defined on the set of real numbers as $$f(x) = x^{3} + 9x^{2} + ax + b$$ takes positive values on positive real numbers and negative values on negative real numbers. What is the smallest integer value that $a$ can take? A) 9 B) 13 C) 17 D) 21 E) 25
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
Let $a$, $x$ and $y$ be positive real numbers. The numbers $$\log_{a} x, \quad \log_{a} y, \quad \log_{a}(x+y)$$ arranged from smallest to largest are consecutive integers. What is the value of $\log_{a}(2a+1)$? A) $-2$ B) $-1$ C) $2$ D) $3$ E) $4$
Melisa will stack 10 circular cardboards with radii $1\text{ cm}, 2\text{ cm}, 3\text{ cm}, \cdots, 10\text{ cm}$ on a table with their centers coinciding, where each has a different natural number radius. In how many different ways can Melisa arrange them so that when viewed from above, exactly one of the circles is completely hidden? A) 36 B) 40 C) 45 D) 48 E) 54
A certain bus arrives at the bus stop near Duru's house with probability $\dfrac{7}{10}$ at exactly 09:02 and with probability $\dfrac{3}{10}$ at exactly 09:03. Duru leaves home at exactly 09:00 to catch this bus. The time it takes for Duru to reach the stop is 100 seconds with probability $\dfrac{1}{2}$, 150 seconds with probability $\dfrac{3}{10}$, and 250 seconds with probability $\dfrac{1}{5}$. What is the probability that Duru is at the stop when the bus arrives? A) $\dfrac{55}{100}$ B) $\dfrac{59}{100}$ C) $\dfrac{63}{100}$ D) $\dfrac{67}{100}$ E) $\dfrac{71}{100}$
Let $m$ and $n$ be natural numbers. If the constant term in the expansion of $$\left(x + \frac{5}{x^{m}}\right)^{n}$$ is 60, what is $m + n$? A) 36 B) 35 C) 31 D) 27 E) 23
$$\lim_{x \rightarrow 1} \frac{(1 - \sqrt{x}) \cdot (\sqrt[3]{x} - 2)}{-x^{2} + 9x - 8}$$ What is the value of this limit? A) 1 B) $\dfrac{1}{2}$ C) $\dfrac{1}{7}$ D) $\dfrac{1}{14}$ E) $\dfrac{1}{18}$
The graph of a function $f$ in the rectangular coordinate plane is given below. A function $g$ defined on the set of real numbers has a limit at all points where it is defined, and $\lim_{x \rightarrow 3} g(x) = 14$ is calculated. If the function $f \cdot g$ is continuous on the set of real numbers, what is the value of $g(3)$? A) 4 B) 6 C) 8 D) 10
Let $a$ and $b$ be real numbers. In the rectangular coordinate plane, the parabola $y = x^{2} + ax + b$ is tangent to the $x$-axis and to the line $y = x$. What is the product $a \cdot b$? A) $\dfrac{1}{2}$ B) $\dfrac{1}{4}$ C) $\dfrac{1}{8}$ D) $\dfrac{1}{16}$ E) $\dfrac{1}{32}$
Let $a$ and $b$ be real numbers. The function $f$ defined as $$f(x) = ax^{3} + bx^{2} + x + 7$$ is always increasing. If $f(-1) = 0$, what is the sum of the different integer values that $b$ can take? A) 11 B) 13 C) 15 D) 17 E) 19
Let $k$ and $m$ be real numbers. The functions $f$ and $g$ defined on the set of real numbers are $$\begin{aligned}
& f(x) = 2x^{3} - 9x^{2} - mx - k \\
& g(x) = x^{3} \cdot f(x)
\end{aligned}$$ The functions $f$ and $g$ have local extrema at $x = -1$. What is the sum $k + m$? A) 31 B) 33 C) 35 D) 37 E) 39
In the rectangular coordinate plane, the shaded region between the lines $y = 2 + 3x$ and $y = 12 - x$ and the positive $x$ and $y$-axes is given below. Rectangles are constructed in this region with one side on the $x$-axis and one vertex each on the lines $y = 2 + 3x$ and $y = 12 - x$. What is the length of the side on the $x$-axis of the rectangle with the largest area? A) 6 B) $\dfrac{19}{3}$ C) $\dfrac{20}{3}$ D) 7 E) $\dfrac{22}{3}$
$$\int_{1}^{2} (x+2) \cdot \sqrt[3]{x^{2} + 4x - 4}\, dx$$ What is the value of this integral? A) $\dfrac{45}{8}$ B) $\dfrac{47}{8}$ C) $\dfrac{49}{8}$ D) $\dfrac{45}{4}$ E) $\dfrac{47}{4}$
For the function $f$ whose graph is given above in the rectangular coordinate plane $$\begin{aligned}
& \int_{a}^{c} |f(x)|\, dx = 20 \\
& \int_{a}^{c} f(x)\, dx = 8
\end{aligned}$$ the equalities are satisfied. What is the value of $$\int_{a/2}^{b/2} f(2x)\, dx$$ ? A) $-3$ B) $-4$ C) $-5$ D) $-6$ E) $-7$
Let $a$ be a real number. In the rectangular coordinate plane, the graphs of the functions $y = a\sqrt{x}$ and $y = \sqrt{x}$ are given below. Let the area of the blue shaded region be $A_{1}$ and the area of the yellow shaded region be $A_{2}$. If $$A_{1} \cdot A_{2} = 96$$ what is $a$? A) 2 B) 3 C) 4 D) 5 E) 6
The functions $f$ and $g$ are defined and differentiable on the set of real numbers and satisfy $$\begin{aligned}
& \int_{1}^{2} f^{\prime}(3x)\, dx = 4 \\
& \int f(2x)\, dx = g(x) + C, \quad (C \text{ constant})
\end{aligned}$$ If $f(3) = 5$, what is the value of the derivative $g^{\prime}(3)$? A) 1 B) 5 C) 9 D) 13 E) 17
Let $a = \sin(40^{\circ})$ $$\begin{aligned}
& b = \sec(40^{\circ}) \\
& c = \tan(40^{\circ})
\end{aligned}$$ Which of the following is the correct ordering of the numbers $a$, $b$ and $c$? A) $a < b < c$ B) $a < c < b$ C) $b < a < c$ D) $b < c < a$ E) $c < a < b$
$$\frac{\cos(2x+y) + \sin(2x-y)}{\cos(2x) + \sin(2x)}$$ Which of the following is the simplified form of this expression? A) $\cos y - \sin y$ B) $\cos y + \sin y$ C) $\cos x - \sin y$ D) $\sin x - \cos y$ E) $\sin x - \cos x$
Let $0 < a < \dfrac{\pi}{2}$ and $$\cos^{2} a - \cos(2a) = \sin(2a)$$ For the value of $a$ satisfying this equality, which of the following is correct? A) $\tan a = \dfrac{1}{5}$ B) $\cot a = \dfrac{2}{\sqrt{5}}$ C) $\cos a = \dfrac{1}{\sqrt{5}}$ D) $\operatorname{cosec} a = \sqrt{5}$ E) $\sin(2a) = \dfrac{3}{5}$
Let $0 \leq x \leq \pi$ and $$\sqrt{2}\sin(4x) - \cos(8x) = 1$$ What is the sum of the $x$ values satisfying this equality? A) $\dfrac{\pi}{3}$ B) $\dfrac{3\pi}{4}$ C) $\pi$ D) $\dfrac{3\pi}{2}$ E) $2\pi$
O is the center of a semicircle, ABCD is a rectangle. A, O and B are collinear. $|AE| = |ED| = \dfrac{1}{2}$ unit, $m(\widehat{AOE}) = x$ Points C and D lie on the semicircle with center O. What is the length of a diagonal of rectangle ABCD in terms of $x$? A) $\tan x$ B) $\operatorname{cosec} x$ C) $\sec x$ D) $\sin x$ E) $\cos x$
In a triangle $ABC$, the length of side $AB$ is equal to half the length of side $BC$. If two of the altitudes of this triangle have lengths 4 units and 10 units, which of the following could be the length of the other altitude? I. 2 units II. 5 units III. 8 units A) Only I B) Only II C) Only III D) I and II E) II and III
In a rectangular coordinate plane, points $A(9,2)$, $B(10,1)$, $C$, $D(4,13)$, $E(3,6)$ and $F$ are given. Given that the centroid of triangle $ABC$ and the centroid of triangle $DEF$ are the same point, what is the distance between points $C$ and $F$ in units? A) 10 B) 13 C) 15 D) 17 E) 20
In the rectangular coordinate plane below, a red square with one side on the $y$-axis and a blue square with one side on the $x$-axis share a common vertex. One vertex of each of the red and blue squares lies on the line $\dfrac{x}{2} + \dfrac{y}{3} = 1$. According to this, what is the side length of the red square in units? A) $\dfrac{14}{15}$ B) $\dfrac{15}{16}$ C) $\dfrac{16}{17}$ D) $\dfrac{17}{18}$ E) $\dfrac{18}{19}$
In a rectangular coordinate plane, what is the area of the triangular region bounded by the lines $2x - y = 0$, $x + 2y = 0$ and $x - 8y + 30 = 0$ in square units? A) 9 B) 12 C) 15 D) 18 E) 21
In a rectangular coordinate plane, point $A(a, b)$; its reflection with respect to point $B(3, 0)$ is point $C$, and its reflection with respect to the $y$-axis is point $D$. Given that the equation of the line passing through points $C$ and $D$ is $y = -x - 1$, what is the sum $a + b$? A) 7 B) 13 C) 15 D) 19 E) 24
In a rectangular coordinate plane, point $A(11, 9)$ is located in the interior of a circle that is tangent to the line $y = x$ at point $B(7, 7)$. Accordingly, what is the smallest integer value that the radius of this circle can take in units? A) 6 B) 8 C) 10 D) 12 E) 14