101- If the sum and product of the real roots of the equation $x^4 - 7x^2 - 5 = 0$ are $S$ and $P$ respectively, what is the value of $2P^2 - 3SP + 2S$? (1) $59 - 7\sqrt{69}$ (2) $7 + \sqrt{69}$ (3) $50$ (4) $59 + 7\sqrt{69}$
104- Suppose the solution set of the inequality $\dfrac{((m^2-1)x^2 - 4mx + 4)(x - 3\sqrt{x} + 2)}{3x - 3} > 0$, for $x > \dfrac{3}{2}$, is $[2, 4]$. What is the value of $m$? (1) $-2$ (2) zero (3) $1$ (4) $2$
105- If $\tan\!\left(\dfrac{\alpha}{2}\right) = \dfrac{1}{4}$, what is the value of $\dfrac{\tan(\alpha) - \sin(\alpha)}{\sin(\alpha) - \cos(\alpha)}$? (1) $-\dfrac{91}{105}$ (2) $-\dfrac{19}{105}$ (3) $\dfrac{16}{105}$ (4) $\dfrac{91}{105}$
106- If $f(\alpha) = 4\sin(\alpha)\cos(2\alpha) + 2\sin(\alpha)$, what is the value of $f\!\left(\dfrac{41\pi}{9}\right)$? (1) $-\sqrt{3}$ (2) $\sqrt{3}$ (3) $1$ (4) $-1$
107- Suppose $A$ is the solution set of the trigonometric equation $\left(1+\cos(2\alpha)\right)\!\left(1+\cos(4\alpha)\right)\!\left(1+\cos(8\alpha)\right) = \dfrac{1}{8}$, in the interval $[0, \pi]$. What is the maximum element of $A$? (1) $\dfrac{5}{7}\pi$ (2) $\dfrac{6}{7}\pi$ (3) $\dfrac{7}{9}\pi$ (4) $\dfrac{8}{9}\pi$
108- $P(x)$ is a second-degree polynomial with natural coefficients. If $P(x)$ is divided by $P'(x)$ (the derivative of $P(x)$), the quotient and remainder are $\dfrac{1}{2}x + 1$ and $2-$, respectively. What is the minimum value of the sum of the coefficients of $P(x)$? (1) $4$ (2) $6$ (3) $7$ (4) $9$ %% Page 4 121-AMathematicsPage 3
109. Suppose the general term of the logistic recurrence sequence $a_{n+1} = \dfrac{1}{a_n} + 1$ with condition $a_1 = 1$ equals $\dfrac{k}{m}$. The ninth and eighth terms of the sequence, which one is correct? (1) $\dfrac{k-m}{2m-k}$ (2) $\dfrac{k-2m}{k-m}$ (3) $\dfrac{k-m}{k-2m}$ (4) $\dfrac{2m-k}{k-m}$
110. The sequence $a_n = \begin{cases} 2^k & ; \ n = 2k \\ -2k+4 & ; \ n = 2k+1 \\ \left[\dfrac{n}{k+2}\right]+a & ; \ n = 2k+2 \end{cases}$ is defined for integer values of $n$, and is assumed. If the sum of the first 10 terms of this sequence is 19, then the value of $a_2 + a_5 + a_4 + \cdots + a_{29}$ is: (1) $-2$ (2) zero (3) $2$ (4) $1$
111. Find the range of the function $f(x) = 2^{\frac{1}{3}\sqrt{9\cos^2(x)-1}} - 2^{\frac{1}{3}\sqrt{1-9\cos^2(x)}}$ in the form $[a, b]$. The value of $b - a$ is: (1) $\dfrac{9}{5}$ (2) $\dfrac{15}{4}$ (3) $\dfrac{9}{2}$ (4) $\dfrac{21}{4}$
113. The curve $y = \sqrt{4-x}$ is translated $k$ units in the horizontal direction and $k-2$ units in the vertical direction such that the new curve intersects its own inverse at a point with width 1. Then the resulting curve is shifted 1 unit in the downward direction. The length of the intersection point of the curve with the $x$-axis is: (1) $-4$ (2) $-3$ (3) $1$ (4) $2$
114. Suppose $f(x) = \begin{cases} -1 & x < -1 \\ x & -1 \leq x \leq 1 \\ 1 & x > 1 \end{cases}$ and $g(x) = 1 - x^2$. The number of elements of the set of points where $g \circ f$ and $f \circ g$ are not differentiable is: (1) $2$ (2) $3$ (3) $4$ (4) $5$
115. The graph of the function $f(x) = 9^{\log_3 x}$ is: [Figure: Four graph options labeled (1), (2), (3), (4) showing different curve shapes in coordinate planes] \rule{\textwidth}{0.4pt} Calculation Space %% Page 5
116. Suppose $a$ is known and $a + n$ is given. Find the value of $\displaystyle\lim_{x \to 0^+} \frac{\tan^2\!\left(\dfrac{1}{\sqrt{1-x^2}}-1\right)}{\left(1-\cos(\sqrt{7x})\right)^n} = a$. What is $a+n$? (1) $\dfrac{7}{4}$ (2) $\dfrac{9}{4}$ (3) $\dfrac{15}{4}$ (4) $\dfrac{17}{4}$
118. The function $f(x) = \dfrac{ax^3 - bx^2 + 2}{ax^3 - bx + 2}$ is discontinuous at exactly two points and has exactly two asymptotes parallel to the coordinate axes. What are the values of $a$ and $b$? (1) $a = 0,\ b = 2$ (2) $a = 8,\ b = 10$ (3) $a = -2,\ b = 0$ (4) $a = -8,\ b = -6$
119. If $\displaystyle\lim_{x \to -\infty} \frac{\sqrt[5\circ]{(a^2x^2-1)(a^4x^4-1)\cdots(a^{100}x^{100}-1)}}{a^{49}x^k - 1} = -1$, what are the values of $a$ and $k$? (1) $k = 51,\ a = -1$ (2) $k = 51,\ a = 1$ (3) $k = 49,\ a = -1$ (4) $k = 49,\ a = 1$
120. Suppose $f(x) = \cos^2(2x) + ax^2 + b$, $\displaystyle\lim_{x \to 0^+} \frac{f(x)}{x} = 0$, and $\displaystyle\lim_{x \to 0^-} \frac{f'(x)}{x} = 2$. What is $a + b$? (1) $8$ (2) $6$ (3) $4$ (4) $-8$ 121. The tangent lines to the curve $f(x) = |\sin(2x)| + 1$ at the point $x = 0$ with length 4 are drawn. If $A$ and $B$ are respectively the second and fourth intersection points of the tangent lines with the $x$-axis, what is the length of segment $AB$? (1) zero (2) $\dfrac{2\sqrt{7}}{3}$ (3) $\dfrac{4\sqrt{7}}{3}$ (4) $2\sqrt{2}$
122. For the function $f(x) = 2\sqrt{x} - \dfrac{3}{2\sqrt[3]{x^2}-1}$, which statement is correct? (1) $f$ is increasing on $(1,\infty) \cup (0,1)$. (2) $f$ is increasing on $(1,\infty)$ and $(0,1)$. (3) $f$ is increasing on $(1,\infty)$ and decreasing on $(0,1)$. (4) $f$ is decreasing on $(1,\infty)$ and increasing on $(0,1)$.
123. Consider the intervals on which $f(x) = \dfrac{x^4}{x^3 - 8}$ is strictly decreasing. What is the minimum total length of these intervals? (1) $2$ (2) $\sqrt[4]{4}-1$ (3) $2\sqrt[4]{4}$ (4) $2(\sqrt[4]{4}-1)$
124. Suppose $A$ and $B$ are the extreme points of $f(x) = 2x^3 - 3x^2 - 12x + 1$. How many points on the curve $f$ have a tangent line parallel to line $AB$? (1) zero (2) $1$ (3) $2$ (4) $3$ %% Page 6 121-AMathematicsPage 5
128- In the frequency table below, the median is $13.5$ and the first quartile minus the third quartile is $17$. If we add $4$ units to each data value in the table, what is the new variance?
130- We write the numbers 1 to 21 each on a card and place them in a bag. We then randomly draw two cards one after another without replacement and place them aside. The set of all possible outcomes forms set $A$. We select one number from set $A$. What is the probability that the selected number is divisible by 6? (1) $\dfrac{13}{84}$ (2) $\dfrac{65}{417}$ (3) $\dfrac{11}{70}$ (4) $\dfrac{67}{417}$
132- The number of positive multiples of positive integer $x = 3^m \times 5^n$ that are also multiples of positive integer $\dfrac{x}{40}$ is more than 12. What is the minimum value of $x$? (1) $640$ (2) $800$ (3) $1000$ (4) $1280$
133- What is the average of the largest and smallest three-digit numbers of the form $\overline{aba}$ that are multiples of 12? (1) $348$ (2) $450$ (3) $570$ (4) $574$
134- If the integer part of dividing natural number $a > 9$ by 11 is 3 more than its remainder, what is the probability that $9 - a$ is divisible by 24? (1) $\dfrac{13}{22}$ (2) $\dfrac{6}{11}$ (3) $\dfrac{1}{2}$ (4) $\dfrac{5}{11}$ \rule{\textwidth}{0.4pt} Workspace %% Page 7 121-AMathematicsPage 6
135. If $m$ is the largest natural number such that $36 \equiv (m)! \pmod{15}$, then $m^{123}$ divided by $15$, the remainder is which of the following? (1) $1$ (2) $2$ (3) $4$ (4) $6$
136. In the first jar there are 3 blue marbles and 6 red marbles, and in the second jar there are 4 blue marbles and 5 red marbles. Two coins are tossed. If the total number of heads is more than 9, one marble is taken from the first jar and added to the second jar. Otherwise, one marble is taken from the second jar and added to the first jar. Now one marble is selected from the jar with more marbles. What is the probability that this marble is red? (1) $\dfrac{157}{270}$ (2) $\dfrac{165}{270}$ (3) $\dfrac{173}{270}$ (4) $\dfrac{185}{270}$
137. The number of correct non-negative integer solutions of the equation $x_1 + x_2 + x_3 = \dfrac{10}{x_4}$ is which of the following? (1) $60$ (2) $72$ (3) $81$ (4) $96$
141. Vector $\vec{a} = (-1, \alpha, 1)$ makes a $45°$ angle with the $z$-axis in space. If $\vec{b} = \left(-\dfrac{4}{3}, \dfrac{2}{3}, 2\right)$ and the angle of vector $\vec{a} \times \vec{b}$ with the $z$-axis is $\theta$, then $\cos\theta$ is which of the following? (1) $-\dfrac{\sqrt{3}}{3}$ (2) $-\dfrac{\sqrt{3}}{4}$ (3) $\dfrac{\sqrt{3}}{4}$ (4) $\dfrac{\sqrt{3}}{2}$
144. Suppose $A = \begin{bmatrix} 1 & -1 \\ 2 & 1 \\ 3 & 1 \end{bmatrix}$. If $BA^T A = 52I$, what is the maximum value of the entries of matrix $B$? (1) $14$ (2) $18$ (3) $24$ (4) $28$
146. The parabola $6 = 6y - 12y - (x-1)^2$ has vertex $F$ and focus $F'$. An ellipse has foci $F$ and $F'$ and eccentricity $0.6$. What is the distance from the center of the ellipse to the origin? (1) $1$ (2) $\sqrt{2}$ (3) $\sqrt{3}$ (4) $2$
152. In the figure below, line segment $AC$ is tangent to the circle. If $\dfrac{AC}{BC} = \sqrt{3}$, then what is $\dfrac{DB}{BC}$? [Figure: Circle with points A, B, C, D, where AC is tangent to the circle at C]
153. According to the figure below, rectangle $ABCD$ is circumscribed about a circle with radius 3, and the circumference is $M\widehat{B}N = 120°$. What is the area of quadrilateral $OMNC$? [Figure: Rectangle ABCD with inscribed circle centered at O, points M on AB and N inside, shaded region OMNC]
154. Suppose lines $x + y = 1$ and $x - y = 3$ are the diameters of a circle, and the line $4x + 3y + 5 = 0$ is tangent to it. What is the distance of point $M(4, -2)$ from the circle?
155. Suppose the length of the radical axis of two circles with radii $1 - 6a$ and $6 - 2a^2$ equals 6 units. If the two circles have exactly one common tangent, what is the average of the possible values of $a$?