7. The point $(4/5, 2)$ is a vertex of a rectangle whose two sides lie on the lines $4x + y = 3$ and $4x - y = 5$. What is the maximum distance from the midpoint of the diagonal? (1) $\dfrac{\sqrt{17}}{2}$ (2) $\dfrac{\sqrt{17}}{4}$ (3) $2\sqrt{17}$ (4) $\sqrt{17}$
8. For the function $f(x) = \sqrt{x - 2\sqrt{mx-1}}$ on its natural domain, the line $y = 12 - x$ intersects the graph at 10 points on the $x$-axis. What is the value of $f(m+4)$? (1) $\dfrac{1}{2}$ (2) $\dfrac{1}{4}$ (3) $2$ (4) $1$
9. A quantity of an element exists. If each hour $\dfrac{1}{9}$ of the mass is lost, after how many minutes will $\dfrac{1}{6}$ of the element's mass remain? $\left(\log_2^5 = 1/4 \text{ and } \log_3^5 = 2/4\right)$ (1) $380$ (2) $360$ (3) $440$ (4) $420$
10. In the figure below, what is the value of $\cot\alpha$? (1) $1$ (2) $2$ (3) $\dfrac{1}{2}$ (4) $\dfrac{1}{3}$ [Figure: A right triangle with a vertical side of length 2, and the base divided into two segments each of length 2, with angle $\alpha$ at the top left.]
11. Triangle $ABC$ has sides $\sqrt{3}$, $6$, and $\alpha$ (with the angle between them being $\alpha$) and is inscribable in a circle. If the area of this triangle is at most $4.5$, what is the minimum value of $\alpha$? (1) $2$ (2) $3$ (3) $4$ (4) $5$ \hrule Workspace %% Page 5
12. If the figure below shows part of the graph of the function $f(x) = a + b\sin(cx - \dfrac{3\pi}{4})\cos(cx - \dfrac{3\pi}{4})$, what is the difference of the zeros of $f$ in the interval $[0, \pi]$? [Figure: A sinusoidal curve with maximum value 3 and minimum value $-1$, with a visible point at $x = \pi$ on the x-axis]
13. In the trigonometric equation $m(\cos x - \sin x) - 3\sqrt{6}\sin(2x) = \sqrt{6}$, if $\cos(x + \dfrac{\pi}{4}) = \dfrac{1}{\sqrt{3}}$, what is the value of $m$? \[
\text{(1) } -6 \qquad \text{(2) } -3 \qquad \text{(3) } 6 \qquad \text{(4) } 3
\]
14. Function $f$ is strictly decreasing and its range is the set of all negative values. If $f(m^2 - m - 5) < f(-3 + 2m - m^2)$, how many correct integer values does $m$ have? \[
\text{(1) } 1 \qquad \text{(2) } 2 \qquad \text{(3) } 3 \qquad \text{(4) zero}
\]
16. For a specific value of $k$, the function $$f(x) = \begin{cases} |x - [-x]| & x \in [x] \text{ even} \\ x - [x] + k & x \in [x] \text{ odd} \end{cases}$$ is continuous at $x = n$ and $x = -n$. Which case is correct regarding $n$ specifically? $(k, n \in \mathbb{N})$
18 -- Line $d$ is tangent to the parabola $y = x^2 + 1$, cuts the $x$-axis at two points, and the tangent lines drawn at those two points are perpendicular to each other. What are the coordinates of the $x$-intercept of line $d$? (1) $1/25$ (2) $3/25$ (3) $\circ/75$ (4) $2/75$
19 -- For how many positive and negative integer values of $k$, does the inflection point of $y = kx^3 + (k+1)x^2$ lie in the second quadrant? (1) $1$ (2) $2$ (3) more than $2$ (4) zero
20 -- What is the minimum distance from points on the curve $y = \sqrt{x - [x^2]}$ to the line $x - y + 2 = 0$? (1) $\dfrac{\sqrt{5}}{5}$ (2) $\dfrac{3\sqrt{5}}{8}$ (3) $\dfrac{\sqrt{5}}{10}$ (4) $\dfrac{3\sqrt{5}}{10}$
21 -- In how many ways can 4 ministers, each with one assistant, sit in two rows of 8 seats facing each other so that each minister sits exactly opposite their own assistant? (1) $24$ (2) $32$ (3) $48$ (4) $64$
22 -- In a group of 150 students, 40 bought only a ticket for film ``A'' and 75 bought only a ticket for film ``B''. If $P(A)$ and $P(B)$ are respectively the probabilities of buying a ticket for films ``A'' and ``B'', what is the maximum value of $\dfrac{P(A)}{P(B)}$? (1) $\dfrac{15}{29}$ (2) $\dfrac{38}{45}$ (3) $\dfrac{8}{15}$ (4) $\dfrac{15}{22}$
23 -- The integers from 9 to 19 are chosen at random. Two numbers are removed from these numbers and replaced by their difference. This process continues until all numbers are even, non-repeating, and the mean is as large as possible. What is the standard deviation of the new data? (1) $\sqrt{10}$ (2) $\sqrt{11}$ (3) $\sqrt{21}$ (4) $\sqrt{28}$
24 -- A device is designed so that it randomly receives one of two letters A or B as input and passes through three stages. At each stage, the input letter is printed with probability $\frac{1}{r}$ without change, or it moves to the next stage changed to the other letter. If the probability of selecting letter A as input is 2 times that of letter B, with what probability is ``A'' printed by the device equal to the probability of the input letter being A? (1) $\dfrac{14}{23}$ (2) $\dfrac{21}{22}$ (3) $\dfrac{9}{41}$ (4) $\dfrac{17}{41}$ %% Page 7
25. In a rhombus, the geometric mean of the two diagonals is the length of the rhombus's side. The smaller angle of the triangle formed by drawing the diagonals of this rhombus is how many degrees? (1) $10$ (2) $15$ (3) $30$ (4) $45$
26. In the figure below, $\hat{ABF} = C\hat{A}E = B\hat{C}D$, $DF = 2.5$, and $EF = 3$. What is the length of $AB$? [Figure: Triangle ABC with point D on side AB, point E on side BC, and point F inside the triangle, with lines drawn from vertices through F]
27. In a rectangle, lines from two opposite vertices are drawn perpendicular to a diagonal, and that diagonal is divided into three parts such that the middle part is twice each of the two side parts. The area of this rectangle is how many times the area of the smallest triangle formed inside the rectangle? (1) $24$ (2) $16$ (3) $12$ (4) $8$
28. In triangle $ABC$, the medians drawn from vertices $B$ and $C$ are perpendicular to each other. If the length of the median drawn from vertex $C$ is $4.5$ and the area of this triangle is $18$, what is the ratio of the lengths of the medians drawn from vertices $B$ and $C$? (1) $\dfrac{17}{9}$ (2) $\dfrac{19}{9}$ (3) $\dfrac{5}{3}$ (4) $\dfrac{4}{3}$
30. In the figure below, two tangent lines are drawn from point $A$. What is the radius of the circle? [Figure: Two tangent lines drawn from external point $A$ to a circle, with segments labeled 9, 8, 7 and point $B$, $C$ marked]
32. In the figure below, if $\widehat{DAC} = 3\widehat{BAD}$, what is the length of side $AC$? [Figure: Triangle with vertices $A$, $B$, $C$, point $D$ on $BC$ with $BD = 4$, $AB = 8$, $AD = 6$]
35. Suppose $\vec{a}$ and $\vec{b}$ are non-zero vectors whose dot product is $-\dfrac{3}{5}$ times the product of their magnitudes. What is the area of the triangle formed by the vectors $\left(\dfrac{3\vec{a}}{|\vec{a}|}+\dfrac{2\vec{b}}{|\vec{b}|}\right)$ and $\left(\dfrac{\vec{a}}{|\vec{a}|}-\dfrac{2\vec{b}}{|\vec{b}|}\right)$? (1) $6/4$ (2) $4/8$ (3) $3/2$ (4) $1/6$
36. Line $d$ has equation $y - x = 0$. A circle with center at the origin has a radius twice that of another circle. If line $d$ is tangent to the smaller circle with equation $x^2 + y^2 + 6x - 2y = r$, what is the product of the lengths of the chord(s) of intersection of the two circles? (1) $\dfrac{5}{2}$ (2) $\dfrac{5}{4}$ (3) $\dfrac{65}{32}$ (4) $\dfrac{65}{64}$