Find the length of the curve with equation
$$2 \log _ { 10 } ( x - y ) = \log _ { 10 } ( 2 - 2 x ) + \log _ { 10 } ( y + 5 )$$
A 5 B 10 C 15 D $3 \pi$ E $9 \pi$ F $12 \pi$

Find the complete set of values of $p$ for which the equation
$$x ^ { 2 } - 2 p x + y ^ { 2 } - 6 y - p ^ { 2 } + 8 p + 9 = 0$$
describes a circle in the $x y$-plane.

The point $P$ has coordinates $( p , q )$, and the equation of a circle is
$$x ^ { 2 } + 2 f x + y ^ { 2 } + 2 g y + h = 0$$
where $f , g , h , p$ and $q$ are all real constants. Let $L$ be the distance between the centre of the circle and the point $P$. Which one of the following is sufficient on its own to be able to calculate $L$ ?
A the values of $f , g$ and $h$
B the values of $f , g , p$ and $q$
C the values of $f , h , p$ and $q$
D the values of $g , h , p$ and $q$
E none of the options A-D is sufficient on its own

The diagram shows a kite $P Q R S$ whose diagonals meet at $O$.
$$\begin{aligned} & O P = x \\ & O Q = y \\ & O R = x \\ & O S = z \end{aligned}$$
Which of the following is necessary and sufficient for angle $S P Q$ to be a right angle?
A $x = y = z$
B $2 x = y + z$
C $\quad x ^ { 2 } = y z$
D $y = z$
E $y ^ { 2 } = x ^ { 2 } + z ^ { 2 }$

A circle has centre $O$ and radius 6 .
$P , Q$ and $R$ are points on the circumference with angle $P O Q \geq \frac { \pi } { 2 }$
The area of the triangle $P O Q$ is $9 \sqrt { 3 }$
What is the greatest possible area of triangle $P R Q$ ?

Point $P$ lies on the circle with equation $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 16$
Point $Q$ lies on the circle with equation $( x - 4 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 16$
What is the maximum possible length of $P Q$ ?

Problem 4
In a three-dimensional Cartesian coordinate system $x y z$, consider the positional relationship among three planes defined by Equations (1)-(3), and the positional relationship among the three planes and a sphere defined by Equation (4).
$$\begin{aligned} & a _ { 11 } x + a _ { 12 } y + a _ { 13 } z = b _ { 1 } , \\ & a _ { 21 } x + a _ { 22 } y + a _ { 23 } z = b _ { 2 } , \\ & a _ { 31 } x + a _ { 32 } y + a _ { 33 } z = b _ { 3 } , \\ & x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3 , \end{aligned}$$
where $a _ { i j }$ and $b _ { i } ( i , j = 1,2,3 )$ are constants.
For the three planes, let $\mathrm { A } = \left( \begin{array} { l l l } a _ { 11 } & a _ { 12 } & a _ { 13 } \\ a _ { 21 } & a _ { 22 } & a _ { 23 } \\ a _ { 31 } & a _ { 32 } & a _ { 33 } \end{array} \right)$ be the coefficient matrix and $\mathbf { B } = \left( \begin{array} { l l l l } a _ { 11 } & a _ { 12 } & a _ { 13 } & b _ { 1 } \\ a _ { 21 } & a _ { 22 } & a _ { 23 } & b _ { 2 } \\ a _ { 31 } & a _ { 32 } & a _ { 33 } & b _ { 3 } \end{array} \right)$ be the augmented coefficient matrix.
I. Let $\mathbf { A } = \left( \begin{array} { l l l } 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { c c c c } 1 & 1 & 1 & 3 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & - c \end{array} \right)$ where $c$ is a positive constant.

1. Find $\operatorname { rank } ( \mathrm { A } )$ and $\operatorname { rank } ( \mathrm { B } )$.
2. Among the three planes, the plane that is tangential to the sphere defined by Equation (4) at a point $\mathrm { P } ( 1,1,1 )$ is called Plane 1. Between the other two planes, the plane with the shorter distance to P is called Plane 2. Find the distance between P and Plane 2. Then, find the volume of the part of the sphere existing between Planes 1 and 2.

II. When the three planes intersect in a line, find $\operatorname { rank } ( \mathrm { A } )$ and $\operatorname { rank } ( \mathrm { B } )$.
III. Suppose that the three planes are tangential to the sphere at three different points. Illustrate all possible positional relationships among the three planes and the sphere. In addition, for each relationship, find $\operatorname { rank } ( \mathrm { A } )$ and $\operatorname { rank } ( \mathrm { B } )$.

Consider an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ in the $xy$-plane. Here, $a$ and $b$ are constants satisfying $a > b > 0$. Answer the following questions.

1. Find the equation of the tangent line at a point $(X, Y)$ on the ellipse in the first quadrant.
2. The tangent line obtained in Question I. 1 intersects the $x$- and $y$-axes. Find the coordinates $(X, Y)$ at the tangent point that minimizes the length of the segment connecting the two intersects and obtain the minimum length of the segment.
3. Consider a region bounded by the segment obtained in Question I. 2 and the $x$- and $y$-axes, and let $C_{1}$ be a cone formed by rotating the region around the $x$-axis. Next, let $C_{2}$ be a cone with the maximum volume while having the same surface area (including a base area) as the cone $C_{1}$. Find $\frac{S_{2}}{S_{1}}$, where $S_{1}$ and $S_{2}$ are the base areas of the cones $C_{1}$ and $C_{2}$, respectively.

3

Let $a$ be a real number, and let $C$ be the circumference of the circle with center $(0,\, a)$ and radius $1$ in the coordinate plane.

1. [(1)] Find the range of $a$ such that $C$ is entirely contained in the region represented by the inequality $y > x^2$.
   
2. [(2)] Suppose $a$ is in the range found in (1). Let $S$ be the part of $C$ satisfying $x \geq 0$ and $y < a$. For a point $\mathrm{P}$ on $S$, let $L_{\mathrm{P}}$ be the length of the chord cut off from the tangent line to $C$ at $\mathrm{P}$ by the parabola $y = x^2$. Find the range of $a$ such that there exist two distinct points $\mathrm{Q}$, $\mathrm{R}$ on $S$ satisfying $L_{\mathrm{Q}} = L_{\mathrm{R}}$.

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4

Let $f(x) = -\dfrac{\sqrt{2}}{4}x^2 + 4\sqrt{2}$. For a real number $t$ satisfying $0 < t < 4$, let $C_t$ be the circle that passes through the point $(t,\, f(t))$ on the coordinate plane, has a common tangent line with the parabola $y = f(x)$ at this point, and has its center on the $x$-axis.

1. [(1)] Let the center of circle $C_t$ be $(c(t),\, 0)$ and the radius be $r(t)$. Express $c(t)$ and $\{r(t)\}^2$ as polynomials in $t$.
   
2. [(2)] Suppose the real number $a$ satisfies $0 < a < f(3)$. How many real numbers $t$ in the range $0 < t < 4$ are there such that circle $C_t$ passes through the point $(3,\, a)$?

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Let $f(x) = -\dfrac{\sqrt{2}}{4}x^2 + 4\sqrt{2}$. For a real number $t$ satisfying $0 < t < 4$, let $C_t$ be the circle that passes through the point $(t, f(t))$ in the coordinate plane, has a common tangent line with the parabola $y = f(x)$ at this point, and has its center on the $x$-axis.

1. [(1)] Let the center of circle $C_t$ have coordinates $(c(t), 0)$ and radius $r(t)$. Express $c(t)$ and $\{r(t)\}^2$ as polynomials in $t$.
   
2. [(2)] Suppose the real number $a$ satisfies $0 < a < f(3)$. How many real numbers $t$ in the range $0 < t < 4$ are there such that the circle $C_t$ passes through the point $(3, a)$?

$$|\mathrm{OM}| = 2 \text{ units}$$
In the rectangular coordinate plane, a semicircle with center at point M and a quarter circle with center at the origin intersect at point A as shown in the figure.
Accordingly, what is the x-coordinate of point A?
A) $\frac{5}{3}$ B) $\sqrt{2}$ C) $\frac{\sqrt{3}}{2}$ D) $\frac{3}{2}$ E) $\sqrt{3}$

The figure below shows the construction used to obtain a square with an area equal to that of a given rectangle.
ABCD is a rectangle, HDFG is a square, semicircle with center O
$$A ( ABCD ) = A ( HDFG )$$
The F vertex of the square HDFG in the figure lies on the semicircle with center O.
Given that the perimeter of rectangle ABCD is 36 cm, what is the diameter of the circle in cm?
A) 12
B) 15
C) 18
D) 21
E) 24

Below, two concentric circles with center $O$ and a circle with center $M$ tangent to both circles are given.
The radius of the small circle with center $O$ is 4 units less than the radius of the large circle with center $O$, and 1 unit more than the radius of the circle with center $M$.
Accordingly, what is the area of the shaded region in square units?
A) $28 \pi$ B) $32 \pi$ C) $36 \pi$ D) $39 \pi$ E) $45 \pi$

ABC and BDE are equilateral triangles\ $[ \mathrm { BD } ] \perp [ \mathrm { AC } ]$\ $[ \mathrm { BF } ] \perp [ \mathrm { DE } ]$\ $[ \mathrm { FH } ] \perp [ \mathrm { BE } ]$\ $| \mathrm { AB } | = 16$ units
Accordingly, what is the area of triangle BFH in square units?\ A) $12 \sqrt { 3 }$\ B) $15 \sqrt { 3 }$\ C) $18 \sqrt { 3 }$\ D) $20 \sqrt { 3 }$\ E) $24 \sqrt { 3 }$

$ABC$ right triangle\ $[ \mathrm { AC } ] \perp [ \mathrm { BC } ]$\ $[AB]$ // $[DE]$\ $[BC]$ // $[FH]$\ $| \mathrm { AD } | = | \mathrm { DH } | = | \mathrm { HC } |$\ $| \mathrm { GE } | = 4$ units\ $| \mathrm { GF } | = 2$ units
Accordingly, what is the area of triangle ABC in square units?\ A) $9 \sqrt { 3 }$\ B) $12 \sqrt { 3 }$\ C) $15 \sqrt { 3 }$\ D) $18 \sqrt { 3 }$\ E) $20 \sqrt { 3 }$

Below are given squares $\mathrm { ABCD }$, $\mathrm { BLPR }$, and KLMN with side lengths of 3, 2, and 1 units respectively.
In the figure, points $\mathrm { A }$, $\mathrm { B }$, $\mathrm { K }$, and L are collinear.\ Accordingly, what is the area of triangle DNP in square units?\ A) 3\ B) 4\ C) 5\ D) 6\ E) 8

ABCD right trapezoid, ABD equilateral triangle\ $[AB]$ // $[DC]$\ $| \mathrm { BF } | = 4 | \mathrm { DF } |$\ $| \mathrm { AB } | = 8$ units
Accordingly, what is the area of right trapezoid ABCE in square units?\ A) $10 \sqrt { 3 }$\ B) $12 \sqrt { 3 }$\ C) $16 \sqrt { 3 }$\ D) $18 \sqrt { 3 }$\ E) $20 \sqrt { 3 }$

ABCD kite\ $[ \mathrm { AC } ] \perp [ \mathrm { BD } ]$\ $| \mathrm { AB } | = | \mathrm { BC } |$\ $| \mathrm { AD } | = | \mathrm { DC } |$\ $| \mathrm { BE } | = 4 | \mathrm { ED } |$\ $| \mathrm { AC } | = 16$ units
The area of kite ABCD in the figure is 160 square units.\ Accordingly, what is the perimeter of kite ABCD in units?\ A) $20 \sqrt { 5 }$\ B) $24 \sqrt { 5 }$\ C) $28 \sqrt { 5 }$

In the Cartesian coordinate plane, two circles with one centered at $(12,0)$ and the other centered at $(0,9)$ intersect only at point $(4,6)$.
What is the distance between the points on these circles that are closest to the origin?
A) $\sqrt { 5 }$ B) $\sqrt { 10 }$ C) $\sqrt { 13 }$ D) $2 \sqrt { 5 }$ E) $2 \sqrt { 10 }$

In the rectangular coordinate plane, a circle divided into two equal parts by the line $x + y = 4$ intersects the x-axis at a single point and the y-axis at two different points. Given that the distance between the points where the circle intersects the y-axis is 4 units, what is the circumference of the circle in units?
A) $4 \pi$
B) $5 \pi$
C) $6 \pi$
D) $7 \pi$
E) $8 \pi$

In a plane, three circles with radius r are constructed with the vertices of a right triangle $ABC$ as centers, and these circles do not intersect each other. The lengths of the parts on the sides of the triangle that are not inside these circles are given as 2 units, 3 units, and 5 units. Accordingly, what is the total area of the regions inside the circles but outside the triangle in square units?
A) $6 \pi$
B) $8 \pi$
C) $9 \pi$
D) $\frac { 9 \pi } { 2 }$
E) $\frac { 15 \pi } { 2 }$

A quarter-circle slice is cut out from a circular piece of paper with radius 8 units. The remaining part is joined together as shown in the figure with the red lines coinciding to form a right circular cone.
Accordingly, what is the height of the formed cone in units?
A) $2 \sqrt { 3 }$
B) $2 \sqrt { 5 }$
C) $2 \sqrt { 7 }$
D) $3 \sqrt { 2 }$
E) $3 \sqrt { 3 }$

In the rectangular coordinate plane, the line $y = m x$
$$x ^ { 2 } - 26 x + y ^ { 2 } + 144 = 0$$
intersects the circle at two different points.
Accordingly, which of the following is the interval showing all possible values of $m$?
A) $\left( - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right)$
B) $\left( - \frac { 3 } { 8 } , \frac { 3 } { 8 } \right)$
C) $\left( - \frac { 4 } { 9 } , \frac { 4 } { 9 } \right)$
D) $\left( - \frac { 5 } { 12 } , \frac { 5 } { 12 } \right)$
E) $\left( - \frac { 7 } { 24 } , \frac { 7 } { 24 } \right)$

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