The complex plane is equipped with a direct orthonormal coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). We denote by $\mathscr{C}$ the set of points $M$ in the plane with affix $z$ such that $|z - 2| = 1$.

1. Justify that $\mathscr{C}$ is a circle, and specify its center and radius.
2. Let $a$ be a real number. We call $\mathscr{D}$ the line with equation $y = ax$. Determine the number of intersection points between $\mathscr{C}$ and $\mathscr{D}$ as a function of the values of the real number $a$.

A circunferência de equação $x^2 + y^2 - 4x + 6y - 3 = 0$ tem centro e raio iguais a
(A) centro $(2, -3)$ e raio $4$ (B) centro $(-2, 3)$ e raio $4$ (C) centro $(2, -3)$ e raio $16$ (D) centro $(-2, 3)$ e raio $16$ (E) centro $(4, -6)$ e raio $3$

QUESTION 155
The area of a circle with radius 5 cm is
(A) $10\pi$ cm$^2$
(B) $15\pi$ cm$^2$
(C) $20\pi$ cm$^2$
(D) $25\pi$ cm$^2$
(E) $30\pi$ cm$^2$

Bowls is a sport played on courts, which are flat and level grounds, limited by perimeter wooden boards. The objective of this sport is to throw bowls, which are balls made of synthetic material, in such a way as to place them as close as possible to the jack, which is a smaller ball made, preferably, of steel, previously thrown. Suppose that a player threw a bowl with radius 5 cm that ended up touching the jack with radius 2 cm, as illustrated in Figure 2.
Consider point $C$ as the center of the bowl, and point $O$ as the center of the jack. It is known that $A$ and $B$ are the points where the bowl and the jack, respectively, touch the ground of the court, and that the distance between $A$ and $B$ is equal to $d$. Under these conditions, what is the ratio between $d$ and the radius of the jack?
(A) 1
(B) $\frac{2\sqrt{10}}{5}$
(C) $\frac{\sqrt{10}}{2}$
(D) 2
(E) $\sqrt{10}$

A waiter needs to choose a tray with a rectangular base to serve four glasses of sparkling wine that need to be arranged in a single row, parallel to the longer side of the tray, and with their bases completely supported on the tray. The base and upper edge of the glasses are circles with radius 4 cm and 5 cm, respectively.
The tray to be chosen should have a minimum area, in square centimeters, equal to
(A) 192.
(B) 300.
(C) 304.
(D) 320.
(E) 400.

A circle has equation $x^2 + y^2 - 4x + 6y - 3 = 0$. What is its radius?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Let $C _ { 1 } , C _ { 2 }$ be two circles of equal radii $R$. If $C _ { 1 }$ passes through the centre of $C _ { 2 }$ prove that the area of the region common to them is $\frac { R ^ { 2 } } { 6 } ( 4 \pi - \sqrt { 27 } )$.

You are given the following: a circle, one of its diameters $AB$ and a point $X$.
(a) Using only a straight-edge, show in the given figure how to draw a line perpendicular to $AB$ passing through $X$. No credit will be given without full justification. (Recall that a straight-edge is a ruler without any markings. Given two points, a straight-edge can be used to draw the line passing through the given points.)
(b) Do NOT draw any of your work for this part in the given figure. Reconsider your procedure to see if it can be made to work if the point $X$ is in some other position, e.g., when it is inside the circle or to the ``left/right'' of the circle. Clearly specify all positions of the point $X$ for which your procedure in part (a), or a small extension/variation of it, can be used to obtain the perpendicular to $AB$ through $X$. Justify your answer.

Fill in the blanks. Let $C_1$ be the circle with center $(-8, 0)$ and radius 6. Let $C_2$ be the circle with center $(8, 0)$ and radius 2. Given a point $P$ outside both circles, let $\ell_i(P)$ be the length of a tangent segment from $P$ to circle $C_i$. The locus of all points $P$ such that $\ell_1(P) = 3\ell_2(P)$ is a circle with radius \_\_\_ and center at (\_\_\_, \_\_\_).

[7 points] Suppose $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ are points on a circle such that AC and BD are diameters of that circle. Suppose $\mathrm{AB} = 12$ and $\mathrm{BC} = 5$. Let P be a point on the arc of the circle from A to B (the arc that does not contain points C and D). Let the distances of P from $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and D be $a, b, c$ and $d$ respectively. Find the values of $\frac{a+b}{c+d}$ and $\frac{a-b}{d-c}$. You may assume $d \neq c$ so the second ratio makes sense.

Solve the following two independent problems.
(i) Let $f$ be a function from domain $S$ to codomain $T$. Let $g$ be another function from domain $T$ to codomain $U$. For each of the blanks below choose a single letter corresponding to one of the four options listed underneath. (It is not necessary that each choice is used exactly once.) Write your answers as a sequence of four letters in correct order. Do NOT explain your answers.
If $g \circ f$ is one-to-one then $f$ $\_\_\_\_$ and $g$ $\_\_\_\_$ . If $g \circ f$ is onto then $f$ $\_\_\_\_$ and $g$ $\_\_\_\_$ .
Option A: must be one-to-one and must be onto. Option B: must be one-to-one but need not be onto. Option C: need not be one-to-one but must be onto. Option D: need not be one-to-one and need not be onto. Recall: $g \circ f$ is the function defined by $g \circ f ( a ) = g ( f ( a ) )$. The function $f$ is said to be one-to-one if, for any $a _ { 1 }$ and any $a _ { 2 }$ in $S , f \left( a _ { 1 } \right) = f \left( a _ { 2 } \right)$ implies $a _ { 1 } = a _ { 2 }$. The function $f$ is said to be onto if, for any $b$ in $T$, there is an $a$ in $S$ such that $f ( a ) = b$.
(ii) In the given figure $ABCD$ is a square. Points $X$ and $Y$, respectively on sides $BC$ and $CD$, are such that $X$ lies on the circle with diameter $AY$. What is the area of the square $ABCD$ if $AX = 4$ and $AY = 5$? (Figure is schematic and not to scale.)

In coordinate space, the $xy$-plane, $yz$-plane, and $zx$-plane divide the space into 8 regions. How many of these 8 regions does the sphere
$$( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = 24$$
pass through? [4 points]
(1) 8
(2) 7
(3) 6
(4) 5
(5) 4

For two points $\mathrm { A } ( 1 , \sqrt { 3 } ) , \mathrm { B } ( 1 , - \sqrt { 3 } )$ on the coordinate plane, what is the total length of the figure represented by point $\mathrm { P } ( x , y )$ satisfying the following two conditions? [4 points] (가) $x ^ { 2 } + y ^ { 2 } = 4$ (나) For any point $( 1 , a )$ on segment AB, the matrix $\left( \begin{array} { c c } x & y \\ 1 & a \end{array} \right)$ has an inverse matrix.
(1) $\frac { 1 } { 3 } \pi$
(2) $\frac { 1 } { 2 } \pi$
(3) $\pi$
(4) $\frac { 4 } { 3 } \pi$
(5) $\frac { 3 } { 2 } \pi$

On the coordinate plane, for two points $\mathrm { A } ( 1 , \sqrt { 3 } ) , \mathrm { B } ( 1 , - \sqrt { 3 } )$, what is the total length of the figure represented by points $\mathrm { P } ( x , y )$ satisfying the following two conditions? [4 points]
(가) $x ^ { 2 } + y ^ { 2 } = 4$
(나) For any point $( 1 , a )$ on the line segment AB, the matrix $\left( \begin{array} { l l } x & y \\ 1 & a \end{array} \right)$ has an inverse matrix.
(1) $\frac { 1 } { 3 } \pi$
(2) $\frac { 1 } { 2 } \pi$
(3) $\pi$
(4) $\frac { 4 } { 3 } \pi$
(5) $\frac { 3 } { 2 } \pi$

As shown in the figure, draw a circle $\mathrm { O } _ { 1 }$ centered at the origin with radius 3, and let the four points where circle $\mathrm { O } _ { 1 }$ meets the coordinate axes be $\mathrm { A } _ { 1 } ( 0,3 )$, $\mathrm { B } _ { 1 } ( - 3,0 ) , \mathrm { C } _ { 1 } ( 0 , - 3 ) , \mathrm { D } _ { 1 } ( 3,0 )$ respectively. Two circles passing through both points $\mathrm { B } _ { 1 }$ and $\mathrm { D } _ { 1 }$ and centered at points $\mathrm { A } _ { 1 }$ and $\mathrm { C } _ { 1 }$ respectively meet the $y$-axis inside circle $\mathrm { O } _ { 1 }$ at points $\mathrm { C } _ { 2 }$ and $\mathrm { A } _ { 2 }$ respectively. Let $S _ { 1 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 1 } \mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$ and arc $\mathrm { B } _ { 1 } \mathrm { A } _ { 2 } \mathrm { D } _ { 1 }$, and let $T _ { 1 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ and arc $\mathrm { B } _ { 1 } \mathrm { C } _ { 2 } \mathrm { D } _ { 1 }$. Draw circle $\mathrm { O } _ { 2 }$ with diameter $\mathrm { A } _ { 2 } \mathrm { C } _ { 2 }$, and let the two points where circle $\mathrm { O } _ { 2 }$ meets the $x$-axis be $\mathrm { B } _ { 2 }$ and $\mathrm { D } _ { 2 }$ respectively. Two circles passing through both points $\mathrm { B } _ { 2 }$ and $\mathrm { D } _ { 2 }$ and centered at points $\mathrm { A } _ { 2 }$ and $\mathrm { C } _ { 2 }$ respectively meet the $y$-axis inside circle $\mathrm { O } _ { 2 }$ at points $\mathrm { C } _ { 3 }$ and $\mathrm { A } _ { 3 }$ respectively. Let $S _ { 2 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 2 } \mathrm { A } _ { 2 } \mathrm { D } _ { 2 }$ and arc $\mathrm { B } _ { 2 } \mathrm { A } _ { 3 } \mathrm { D } _ { 2 }$, and let $T _ { 2 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ and arc $\mathrm { B } _ { 2 } \mathrm { C } _ { 3 } \mathrm { D } _ { 2 }$. Continuing this process, let $S _ { n }$ be the area of the region enclosed by arc $\mathrm { B } _ { n } \mathrm { A } _ { n } \mathrm { D } _ { n }$ and arc $\mathrm { B } _ { n } \mathrm { A } _ { n + 1 } \mathrm { D } _ { n }$, and let $T _ { n }$ be the area of the region enclosed by arc $\mathrm { B } _ { n } \mathrm { C } _ { n } \mathrm { D } _ { n }$ and arc $\mathrm { B } _ { n } \mathrm { C } _ { n + 1 } \mathrm { D } _ { n }$. What is the value of $\sum _ { n = 1 } ^ { \infty } \left( S _ { n } + T _ { n } \right)$? [4 points]
(1) $6 ( \sqrt { 2 } + 1 )$
(2) $6 ( \sqrt { 3 } + 1 )$
(3) $6 ( \sqrt { 5 } + 1 )$
(4) $9 ( \sqrt { 2 } + 1 )$
(5) $9 ( \sqrt { 3 } + 1 )$

As shown in the figure, draw a circle $\mathrm { O } _ { 1 }$ centered at the origin with radius 3, and let the four points where circle $\mathrm { O } _ { 1 }$ meets the coordinate axes be $\mathrm { A } _ { 1 } ( 0,3 )$, $\mathrm { B } _ { 1 } ( - 3,0 ) , \mathrm { C } _ { 1 } ( 0 , - 3 ) , \mathrm { D } _ { 1 } ( 3,0 )$ respectively. Two circles passing through both points $\mathrm { B } _ { 1 } , \mathrm { D } _ { 1 }$ and centered at points $\mathrm { A } _ { 1 } , \mathrm { C } _ { 1 }$ respectively meet the $y$-axis inside circle $\mathrm { O } _ { 1 }$ at points $\mathrm { C } _ { 2 } , \mathrm {~A} _ { 2 }$ respectively.
Let $S _ { 1 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 1 } \mathrm {~A} _ { 1 } \mathrm { D } _ { 1 }$ and arc $\mathrm { B } _ { 1 } \mathrm {~A} _ { 2 } \mathrm { D } _ { 1 }$, and let $T _ { 1 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ and arc $\mathrm { B } _ { 1 } \mathrm { C } _ { 2 } \mathrm { D } _ { 1 }$.
Draw circle $\mathrm { O } _ { 2 }$ with segment $\mathrm { A } _ { 2 } \mathrm { C } _ { 2 }$ as diameter, and let the two points where circle $\mathrm { O } _ { 2 }$ meets the $x$-axis be $\mathrm { B } _ { 2 } , \mathrm { D } _ { 2 }$ respectively. Two circles passing through both points $\mathrm { B } _ { 2 } , \mathrm { D } _ { 2 }$ and centered at points $\mathrm { A } _ { 2 } , \mathrm { C } _ { 2 }$ respectively meet the $y$-axis inside circle $\mathrm { O } _ { 2 }$ at points $\mathrm { C } _ { 3 } , \mathrm {~A} _ { 3 }$ respectively.
Let $S _ { 2 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 2 } \mathrm {~A} _ { 2 } \mathrm { D } _ { 2 }$ and arc $\mathrm { B } _ { 2 } \mathrm {~A} _ { 3 } \mathrm { D } _ { 2 }$, and let $T _ { 2 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ and arc $\mathrm { B } _ { 2 } \mathrm { C } _ { 3 } \mathrm { D } _ { 2 }$.
Continuing this process, let $S _ { n }$ be the area of the region enclosed by arc $\mathrm { B } _ { n } \mathrm {~A} _ { n } \mathrm { D } _ { n }$ and arc $\mathrm { B } _ { n } \mathrm {~A} _ { n + 1 } \mathrm { D } _ { n }$ obtained in the $n$-th step, and let $T _ { n }$ be the area of the region enclosed by arc $\mathrm { B } _ { n } \mathrm { C } _ { n } \mathrm { D } _ { n }$ and arc $\mathrm { B } _ { n } \mathrm { C } _ { n + 1 } \mathrm { D } _ { n }$. What is the value of $\sum _ { n = 1 } ^ { \infty } \left( S _ { n } + T _ { n } \right)$? [4 points]
(1) $6 ( \sqrt { 2 } + 1 )$
(2) $6 ( \sqrt { 3 } + 1 )$
(3) $6 ( \sqrt { 5 } + 1 )$
(4) $9 ( \sqrt { 2 } + 1 )$
(5) $9 ( \sqrt { 3 } + 1 )$

As shown in the figure, there are two circular disks with distance between centers $\sqrt { 3 }$ and radius 1, and a plane $\alpha$. The line $l$ passing through the centers of each disk is perpendicular to the planes of the two disks and makes an angle of $60 ^ { \circ }$ with plane $\alpha$. When sunlight shines perpendicular to plane $\alpha$ as shown in the figure, what is the area of the shadow cast by the two disks on plane $\alpha$? (Note: the thickness of the disks is negligible.) [4 points]
(1) $\frac { \sqrt { 3 } } { 3 } \pi + \frac { 3 } { 8 }$
(2) $\frac { 2 } { 3 } \pi + \frac { \sqrt { 3 } } { 4 }$
(3) $\frac { 2 \sqrt { 3 } } { 3 } \pi + \frac { 1 } { 8 }$
(4) $\frac { 4 } { 3 } \pi + \frac { \sqrt { 3 } } { 16 }$
(5) $\frac { 2 \sqrt { 3 } } { 3 } \pi + \frac { 3 } { 4 }$

As shown in the figure, in the coordinate plane, for two points $\mathrm { A } , \mathrm { B }$ on the $x$-axis, the parabola $p _ { 1 }$ with vertex at A and the parabola $p _ { 2 }$ with vertex at B satisfy the following conditions. What is the area of triangle ABC? [4 points] (가) The focus of $p _ { 1 }$ is B, and the focus of $p _ { 2 }$ is the origin O. (나) $p _ { 1 }$ and $p _ { 2 }$ meet at two points $\mathrm { C } , \mathrm { D }$ on the $y$-axis. (다) $\overline { \mathrm { AB } } = 2$
(1) $4 ( \sqrt { 2 } - 1 )$
(2) $3 ( \sqrt { 3 } - 1 )$
(3) $2 ( \sqrt { 5 } - 1 )$
(4) $\sqrt { 3 } + 1$
(5) $\sqrt { 5 } + 1$

In the coordinate plane, two lines $l _ { 1 } , l _ { 2 }$ tangent to the parabola $y ^ { 2 } = 8 x$ have slopes $m _ { 1 } , m _ { 2 }$ respectively. When $m _ { 1 } , m _ { 2 }$ are the two distinct roots of the equation $2 x ^ { 2 } - 3 x + 1 = 0$, what is the $x$-coordinate of the intersection point of $l _ { 1 }$ and $l _ { 2 }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

In coordinate space, a sphere $S$ with center coordinates all positive has center at $(x, y, z)$ where $x > 0, y > 0, z > 0$, is tangent to the $x$-axis and $y$-axis respectively, and intersects the $z$-axis at two distinct points. The area of the circle formed by the intersection of sphere $S$ and the $xy$-plane is $64 \pi$, and the distance between the two intersection points with the $z$-axis is 8. What is the radius of sphere $S$? [4 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15

As shown in the figure, there is a point $\mathrm { A } ( 0 , a )$ on the $y$-axis and a point P moving on the ellipse $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$ with foci $\mathrm { F } , \mathrm { F } ^ { \prime }$. When the minimum value of $\overline { \mathrm { AP } } - \overline { \mathrm { FP } }$ is 1, find the value of $a ^ { 2 }$. [4 points]

As shown in the figure, a line passes through the focus F of the parabola $y ^ { 2 } = 12 x$ and meets the parabola at two points $\mathrm { A } , \mathrm { B }$. Let C and D be the feet of the perpendiculars from A and B to the directrix $l$ respectively. When $\overline { \mathrm { AC } } = 4$, what is the length of segment BD? [3 points]
(1) 12
(2) $\frac { 25 } { 2 }$
(3) 13
(4) $\frac { 27 } { 2 }$
(5) 14

For the parabola $y ^ { 2 } = 4 x$, let $l$ be the tangent line at point $\mathrm { A } ( 4,4 )$. Let B be the intersection of line $l$ and the directrix of the parabola, C be the intersection of line $l$ and the $x$-axis, and D be the intersection of the directrix and the $x$-axis. What is the area of triangle BCD? [3 points]
(1) $\frac { 7 } { 4 }$
(2) 2
(3) $\frac { 9 } { 4 }$
(4) $\frac { 5 } { 2 }$
(5) $\frac { 11 } { 4 }$

In coordinate space, find the sum of all real numbers $k$ such that the plane $x + 8 y - 4 z + k = 0$ is tangent to the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 2 y - 3 = 0$. [3 points]

For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{6} = 1$, what is the slope of the tangent line at the point $(\sqrt{3}, -2)$ on the ellipse? (where $a$ is a positive number) [3 points]
(1) $\sqrt{3}$
(2) $\frac{\sqrt{3}}{2}$
(3) $\frac{\sqrt{3}}{3}$
(4) $\frac{\sqrt{3}}{4}$
(5) $\frac{\sqrt{3}}{5}$