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 $a$.

Exercise 3 — Part C
We are given the point $\mathrm{D}(9; 1; 1)$ which is one of the two solution points from question 4.c. of Part B. The four vertices of tetrahedron ABCD are located on a sphere.
Using the results from the questions in Parts A and B above, determine the coordinates of the center of this sphere and calculate its radius.

Question 147
Um triângulo retângulo tem catetos medindo 6 cm e 8 cm. A área do círculo circunscrito a esse triângulo, em cm², é
(A) $16\pi$ (B) $25\pi$ (C) $36\pi$ (D) $49\pi$ (E) $100\pi$

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 $3$

In recent years, television has undergone a true revolution in terms of image quality, sound and interactivity with viewers. This transformation is due to the conversion of the analog signal to the digital signal. However, many cities still do not have this new technology. Seeking to bring these benefits to three cities, a television station intends to build a new transmission tower that sends signal to antennas A, B and C, already existing in these cities. The locations of the antennas are represented in the Cartesian plane.
The tower must be located in a place equidistant from the three antennas.
The appropriate location for the construction of this tower corresponds to the point with coordinates
(A) $(65; 35)$. (B) $(53; 30)$. (C) $(45; 35)$. (D) $(50; 20)$. (E) $(50; 30)$.

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

In a digital game, there are three characters: one hero and two villains. The programming is done in such a way that the hero will always be attacked by the villain closest to him. One way to ``confuse'' the villains is to move the hero along trajectories that keep him equidistant from the villains, creating uncertainty between them, and thus preventing him from being attacked.
For the programming of one of the stages of this game, the programmer considered, in the Cartesian plane, the square STUV as the region of movement of the characters, where V and $T$ represent the fixed positions of the villains, and $S$, the initial position of the hero, as shown in the figure.
What is the equation of the trajectory along which the hero can move without being attacked?
(A) $y = -3x + 20$
(B) $y = -3x + 16$
(C) $y = -3x - 20$
(D) $y = 3x + 16$
(E) $y = 3x - 16$

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 $\frac { R ^ { 2 } } { 6 } ( 4 \pi - \sqrt { 27 } )$.

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 (\_\_\_, \_\_\_).

By definition the region inside the parabola $y = x^{2}$ is the set of points $(a,b)$ such that $b \geq a^{2}$. We are interested in those circles all of whose points are in this region. A bubble at a point $P$ on the graph of $y = x^{2}$ is the largest such circle that contains $P$. (You may assume the fact that there is a unique such circle at any given point on the parabola.)
(a) A bubble at some point on the parabola has radius 1. Find the center of this bubble.
(b) Find the radius of the smallest possible bubble at some point on the parabola. Justify.

[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.

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

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) $\frac { 3 } { 2 } \pi$

Two spheres $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 81 , x ^ { 2 } + ( y - 5 ) ^ { 2 } + z ^ { 2 } = 56$ are denoted by $S _ { 1 } , S _ { 2 }$ respectively. Let P be a point on the circle formed by the intersection of the two spheres $S _ { 1 } , S _ { 2 }$, and let $\mathrm { P } ^ { \prime }$ be the orthogonal projection of point P onto the $xy$-plane. Let Q and R be the points where the sphere $S _ { 1 }$ intersects the $y$-axis. Find the maximum volume of the tetrahedron $\mathrm { PQP } ^ { \prime } \mathrm { R }$. [4 points]

Let C be the circle formed by the intersection of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ and the plane $z = - 1$. When a plane $\alpha$ containing the $x$-axis intersects the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ to form a circle that meets C at exactly one point, and a normal vector to plane $\alpha$ is $\vec { n } = ( a , 3 , b )$, find the value of $a ^ { 2 } + b ^ { 2 }$. [4 points]

On a parabola $y ^ { 2 } = x$ with focus F, there is a point P such that $\overline { \mathrm { FP } } = 4$. As shown in the figure, point Q is taken on the extension of segment FP such that $\overline { \mathrm { FP } } = \overline { \mathrm { PQ } }$. What is the $x$-coordinate of point Q? [3 points]
(1) $\frac { 29 } { 4 }$
(2) 7
(3) $\frac { 27 } { 4 }$
(4) $\frac { 13 } { 2 }$
(5) $\frac { 25 } { 4 }$

Inside a rectangle with width 6 and height 8, a circle with center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the rectangle is drawn to obtain figure $R _ { 1 }$. From figure $R _ { 1 }$, four rectangles are drawn with each segment from a vertex of the rectangle to the intersection of the diagonal and circle as the diagonal. Inside each of the newly drawn rectangles, a circle with center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the newly drawn rectangle is drawn to obtain figure $R _ { 2 }$.
In figure $R _ { 2 }$, for each of the four congruent rectangles, four rectangles are drawn with each segment from a vertex to the intersection of the diagonal and circle as the diagonal. Inside each of the newly drawn rectangles, a circle with center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the newly drawn rectangle is drawn to obtain figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the sum of the areas of all circles in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? (Here, the widths and heights of all rectangles are parallel to each other, respectively.) [4 points]
(1) $\frac { 37 } { 9 } \pi$
(2) $\frac { 34 } { 9 } \pi$
(3) $\frac { 31 } { 9 } \pi$
(4) $\frac { 28 } { 9 } \pi$
(5) $\frac { 25 } { 9 } \pi$

On the coordinate plane, there is a circle $C _ { 1 } : ( x - 4 ) ^ { 2 } + y ^ { 2 } = 1$. As shown in the figure, a tangent line $l$ with positive slope is drawn from the origin to the circle $C _ { 1 }$, and the point of tangency is called $\mathrm { P } _ { 1 }$.
A circle $C _ { 2 }$ has its center on the line $l$, passes through the point $\mathrm { P } _ { 1 }$, and is tangent to the $x$-axis. Let $\mathrm { P } _ { 2 }$ be the point of tangency between this circle and the $x$-axis.
A circle $C _ { 3 }$ has its center on the $x$-axis, passes through the point $\mathrm { P } _ { 2 }$, and is tangent to the line $l$. Let $\mathrm { P } _ { 3 }$ be the point of tangency between this circle and the line $l$.
A circle $C _ { 4 }$ has its center on the line $l$, passes through the point $\mathrm { P } _ { 3 }$, and is tangent to the $x$-axis. Let $\mathrm { P } _ { 4 }$ be the point of tangency between this circle and the $x$-axis. Continuing this process, let $S _ { n }$ be the area of circle $C _ { n }$. What is the value of $\sum _ { n = 1 } ^ { \infty } S _ { n }$? (Note: The radius of circle $C _ { n + 1 }$ is smaller than the radius of circle $C _ { n }$.) [4 points]
(1) $\frac { 3 } { 2 } \pi$
(2) $2 \pi$
(3) $\frac { 5 } { 2 } \pi$
(4) $3 \pi$
(5) $\frac { 7 } { 2 } \pi$

On the coordinate plane, there is a circle $C _ { 1 } : ( x - 4 ) ^ { 2 } + y ^ { 2 } = 1$. As shown in the figure, when a tangent line $l$ with positive slope is drawn from the origin to the circle $C _ { 1 }$, let the point of tangency be $\mathrm { P } _ { 1 }$. Let $C _ { 2 }$ be a circle with center on the line $l$, passing through the point $\mathrm { P } _ { 1 }$, and tangent to the $x$-axis, and let $\mathrm { P } _ { 2 }$ be the point of tangency with the $x$-axis. Let $C _ { 3 }$ be a circle with center on the $x$-axis, passing through the point $\mathrm { P } _ { 2 }$, and tangent to the line $l$, and let $\mathrm { P } _ { 3 }$ be the point of tangency with the line $l$. Let $C _ { 4 }$ be a circle with center on the line $l$, passing through the point $\mathrm { P } _ { 3 }$, and tangent to the $x$-axis, and let $\mathrm { P } _ { 4 }$ be the point of tangency with the $x$-axis. Continuing this process, let $S _ { n }$ be the area of the circle $C _ { n }$. What is the value of $\sum _ { n = 1 } ^ { \infty } S _ { n }$? (Note: the radius of circle $C _ { n + 1 }$ is smaller than the radius of circle $C _ { n }$.) [4 points]
(1) $\frac { 3 } { 2 } \pi$
(2) $2 \pi$
(3) $\frac { 5 } { 2 } \pi$
(4) $3 \pi$
(5) $\frac { 7 } { 2 } \pi$

Let Q be the point where the tangent line at point $\mathrm { P } ( a , b )$ on the parabola $y ^ { 2 } = 4 x$ meets the $x$-axis. When $\overline { \mathrm { PQ } } = 4 \sqrt { 5 }$, what is the value of $a ^ { 2 } + b ^ { 2 }$? [3 points]
(1) 21
(2) 32
(3) 45
(4) 60
(5) 77

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) $9 ( \sqrt { 3 } + 1 )$