The manufacturer of padlocks of the brand ``K'' wishes to print a logo for his company. This logo has the shape of a stylized capital letter K, inscribed in a square ABCD, with side length one unit of length, and respecting the following conditions C1 and C2:

• Condition C1: the letter K must consist of three lines:
• one of the lines is the segment $[AD]$;
• a second line has endpoints at point A and a point E on segment $[DC]$;
• the third line has endpoint at point B and a point G located on the second line.
• Condition C2: the area of each of the three surfaces delimited by the three lines drawn in the square must be between 0.3 and 0.4, with the unit of area being that of the square. These areas are denoted $r$, $s$, $t$.

We place ourselves in the orthonormal coordinate system $(A; \overrightarrow{AB}, \overrightarrow{AD})$.
Part A: study of Proposal A
In this proposal, the three lines are segments and the three areas are equal: $r = s = t = \frac{1}{3}$. Determine the coordinates of points E and G.

Question 156
Um estudante realizou um experimento e obteve os seguintes dados:

$x$	$y$
1	3
2	5
3	7
4	9

A função que melhor representa a relação entre $x$ e $y$ é
(A) $y = x + 2$ (B) $y = 2x + 1$ (C) $y = 3x$ (D) $y = x^2 + 2$ (E) $y = 2x^2 - 1$

Question 169
A figura mostra dois triângulos semelhantes $ABC$ e $DEF$.
[Figure]
Se $AB = 6$ cm, $BC = 8$ cm, $AC = 10$ cm e $DE = 9$ cm, o perímetro do triângulo $DEF$, em cm, é
(A) 24 (B) 30 (C) 36 (D) 40 (E) 45

A distância entre os pontos $A = (1, 2)$ e $B = (4, 6)$ no plano cartesiano é
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

A equação da reta que passa pelos pontos $(0, 3)$ e $(2, 7)$ é
(A) $y = x + 3$ (B) $y = 2x + 3$ (C) $y = 3x + 1$ (D) $y = 2x - 3$ (E) $y = x + 7$

O coeficiente angular da reta $3x - 2y + 6 = 0$ é
(A) $-3$ (B) $-\dfrac{3}{2}$ (C) $\dfrac{3}{2}$ (D) $2$ (E) $3$

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 139
A rectangular piece of land has dimensions 30 m by 20 m. The owner wants to build a fence around the entire perimeter of the land. If the fence costs R\$ 15.00 per meter, the total cost of the fence will be
(A) R\$ 900.00
(B) R\$ 1,200.00
(C) R\$ 1,500.00
(D) R\$ 1,800.00
(E) R\$ 2,100.00

QUESTION 144
The sum of the interior angles of a hexagon is
(A) $540^\circ$
(B) $600^\circ$
(C) $660^\circ$
(D) $720^\circ$
(E) $780^\circ$

QUESTION 150
In a right triangle, one leg measures 6 cm and the hypotenuse measures 10 cm. The other leg measures
(A) 4 cm
(B) 6 cm
(C) 7 cm
(D) 8 cm
(E) 9 cm

QUESTION 154
The equation of a line passing through the points $(1, 2)$ and $(3, 6)$ is
(A) $y = x + 1$
(B) $y = 2x$
(C) $y = 2x + 1$
(D) $y = 3x - 1$
(E) $y = x + 3$

QUESTION 158
The distance between the points $A(1, 2)$ and $B(4, 6)$ is
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

QUESTION 162
The lateral surface area of a cone with base radius 3 cm and slant height 5 cm is
(A) $10\pi$ cm$^2$
(B) $12\pi$ cm$^2$
(C) $15\pi$ cm$^2$
(D) $18\pi$ cm$^2$
(E) $20\pi$ cm$^2$

QUESTION 179
The midpoint of the segment with endpoints $A(2, 4)$ and $B(6, 8)$ is
(A) $(3, 5)$
(B) $(4, 6)$
(C) $(5, 7)$
(D) $(6, 8)$
(E) $(8, 12)$

The equation of the line passing through the points $(0, 3)$ and $(2, 7)$ is:
(A) $y = x + 3$
(B) $y = 2x + 3$
(C) $y = 3x + 1$
(D) $y = 2x + 1$
(E) $y = x + 5$

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) $y = 3x - 16$

In an isoceles $\triangle \mathrm { ABC }$ with A at the apex the height and the base are both equal to 1 cm. Points $\mathrm { D } , \mathrm { E }$ and F are chosen one from each side such that BDEF is a rhombus. Find the length of the side of this rhombus.

Let $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$ and $b _ { 1 } , b _ { 2 } , \ldots , b _ { n }$ be two arithmetic progressions. Prove that the points $\left( a _ { 1 } , b _ { 1 } \right) , \left( a _ { 2 } , b _ { 2 } \right) , \ldots , \left( a _ { n } , b _ { n } \right)$ are collinear.

In a rectangle ABCD , the length BC is twice the width AB . Pick a point P on side BC such that the lengths of AP and BC are equal. The measure of angle CPD is
(A) $75 ^ { \circ }$
(B) $60 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

a) Let $\mathrm { E } , \mathrm { F } , \mathrm { G }$ and H respectively be the midpoints of the sides $\mathrm { AB } , \mathrm { BC } , \mathrm { CD }$ and DA of a convex quadrilateral ABCD. Show that EFGH is a parallelogram whose area is half that of ABCD. b) Let $\mathrm { E } = ( 0,0 ) , \mathrm { F } = ( 0 , - 1 ) , \mathrm { G } = ( 1 , - 1 ) , \mathrm { H } = ( 1,0 )$. Find all points $\mathrm { A } = ( p , q )$ in the first quadrant such that $\mathrm { E } , \mathrm { F } , \mathrm { G }$ and H respectively are the midpoints of the sides $\mathrm { AB } , \mathrm { BC } , \mathrm { CD }$ and DA of a convex quadrilateral ABCD.

Let $S$ be a circle with center $O$. Suppose $A , B$ are points on the circumference of $S$ with $\angle A O B = 120 ^ { \circ }$. For triangle $A O B$, let $C$ be its circumcenter and $D$ its orthocenter (i.e., the point of intersection of the three lines containing the altitudes). For each statement below, write whether it is TRUE or FALSE. a) The triangle $A O C$ is equilateral.
Answer: $\_\_\_\_$ b) The triangle $A B D$ is equilateral.
Answer: $\_\_\_\_$ c) The point $C$ lies on the circle $S$.
Answer: $\_\_\_\_$ d) The point $D$ lies on the circle $S$.
Answer: $\_\_\_\_$

You are given a triangle ABC, a point D on segment AC, a point E on segment AB and a point F on segment BC. Let BD and CE intersect in point P. Join P with F. Suppose that $\angle\mathrm{EPB} = \angle\mathrm{BPF} = \angle\mathrm{FPC} = \angle\mathrm{CPD}$ and $\mathrm{PD} = \mathrm{PE} = \mathrm{PF}$.
For each statement below, state if it is true or false.
(i) AP must bisect $\angle\mathrm{BAC}$.
(ii) $\triangle\mathrm{ABC}$ must be isosceles.
(iii) $\mathrm{A}$, $\mathrm{P}$, $\mathrm{F}$ must be collinear.
(iv) $\angle\mathrm{BAC}$ must be $60^{\circ}$.

You are given a regular hexagon. We say that a square is inscribed in the hexagon if it can be drawn in the interior such that all the four vertices lie on the perimeter of the hexagon.
(a) A line segment has its endpoints on opposite edges of the hexagon. Show that it passes through the center of the hexagon if and only if it divides the two edges in the same ratio.
(b) Suppose a square $ABCD$ is inscribed in the hexagon such that $A$ and $C$ are on the opposite sides of the hexagon. Prove that center of the square is same as that of the hexagon.
(c) Suppose the side of the hexagon is of length 1. Then find the length of the side of the inscribed square whose one pair of opposite sides is parallel to a pair of opposite sides of the hexagon.
(d) Show that, up to rotation, there is a unique way of inscribing a square in a regular hexagon.

Suppose a rectangle $EBFD$ is given and a rhombus $ABCD$ is inscribed in it so that the point $A$ is on side $ED$ of the rectangle. The diagonals of $ABCD$ intersect at point $G$.
Statements
(5) Triangles $CGD$ and $DFB$ must be similar. (6) It must be true that $\frac { AC } { BD } = \frac { EB } { ED }$. (7) Triangle $CGD$ cannot be similar to triangle $AEB$. (8) For any given rectangle $EBFD$, a rhombus $ABCD$ as described above can be constructed.

On the coordinate plane, a line passes through the point $( 4,1 )$ and is perpendicular to the vector $\vec { n } = ( 1,2 )$. Let the coordinates of the points where this line meets the $x$-axis and $y$-axis be $( a , 0 ) , ( 0 , b )$ respectively. Find the value of $a + b$. [3 points]