Exercise 3 -- Common to all candidates

In a vast plain, a network of sensors makes it possible to detect lightning and to produce an image of storm phenomena. The radar screen is divided into forty sectors denoted by a letter and a number between 1 and 5. The lightning sensor is represented by the center of the screen; five concentric circles corresponding to the respective radii $20, 40, 60, 80$ and 100 kilometres delimit five zones numbered from 1 to 5, and eight segments starting from the sensor delimit eight portions of the same angular opening, named in the trigonometric direction from A to H.
We assimilate the radar screen to a part of the complex plane by defining an orthonormal coordinate system $(\mathrm{O}, \vec{u}, \vec{v})$:

• the origin O marks the position of the sensor;
• the abscissa axis is oriented from West to East;
• the ordinate axis is oriented from South to North;
• the chosen unit is the kilometre.

In the following, a point on the radar screen is associated with a point with affix $z$.

Part A

1. We denote $z_{P}$ the affix of the point P located in sector B3 on the graph. We call $r$ the modulus of $z_{P}$ and $\theta$ its argument in the interval $]-\pi ; \pi]$. Among the four following propositions, determine the only one that proposes a correct bound for $r$ and for $\theta$ (no justification is required):
   

   Proposition A	Proposition B	Proposition C	Proposition D
   \begin{tabular}{ c } $40 < r < 60$
   and
   $0 < \theta < \frac{\pi}{4}$

   &

   $20 < r < 40$

   and

   $\frac{\pi}{2} < \theta < \frac{3\pi}{4}$

   &

   $40 < r < 60$

   and

   $\frac{\pi}{4} < \theta < \frac{\pi}{2}$

   &

   $0 < r < 60$

   and

   $-\frac{\pi}{2} < \theta < -\frac{\pi}{4}$

   \hline \end{tabular}
   
2. A lightning impact is materialized on the screen at a point with affix $z$. In each of the two following cases, determine the sector to which this point belongs: a. $z = 70 \mathrm{e}^{-\mathrm{i}\frac{\pi}{3}}$; b. $z = -45\sqrt{3} + 45\mathrm{i}$.

Part B

We assume in this part that the sensor displays an impact at point P with affix $50\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$. When the sensor displays the impact point P with affix $50\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$, the affix $z$ of the actual lightning impact point admits:

• a modulus that can be modeled by a random variable $M$ following a normal distribution with mean $\mu = 50$ and standard deviation $\sigma = 5$;
• an argument that can be modeled by a random variable $T$ following a normal distribution with mean $\frac{\pi}{3}$ and standard deviation $\frac{\pi}{12}$.

We assume that the random variables $M$ and $T$ are independent. In the following, probabilities will be rounded to $10^{-3}$ near.

1. Calculate the probability $P(M < 0)$ and interpret the result obtained.
2. Calculate the probability $P(M \in ]40 ; 60[)$.
3. We admit that $P\left(T \in \left]\frac{\pi}{4} ; \frac{\pi}{2}\right[\right) = 0.819$. Deduce the probability that the lightning actually struck sector B3 according to this modeling.

Exercise 3

Common to all candidates

Two species of turtles endemic to a small island in the Pacific Ocean, green turtles and hawksbill turtles, meet during different breeding episodes on two of the island's beaches to lay eggs. This island, being the convergence point of many turtles, specialists decided to take advantage of this to collect various data on them. They first observed that the corridors used in the ocean by each of the two species to reach the island could be assimilated to rectilinear trajectories. In what follows, space is referred to an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ) with unit 100 meters. The plane ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ) represents the water level and we admit that a point $M ( x ; y ; z )$ with $z < 0$ is located in the ocean. The specialists' model establishes that:

• the trajectory used in the ocean by green turtles is supported by the line $\mathscr { D } _ { 1 }$ whose parametric representation is:

$$\left\{ \begin{aligned} x & = 3 + t \\ y & = 6 t \text { with } t \text { real; } \\ z & = - 3 t \end{aligned} \right.$$

• the trajectory used in the ocean by hawksbill turtles is supported by the line $\mathscr { D } _ { 2 }$ whose parametric representation is:

$$\left\{ \begin{aligned} x & = 10 k \\ y & = 2 + 6 k \text { with } k \text { real; } \\ z & = - 4 k \end{aligned} \right.$$

1. Prove that the two species are never likely to cross before arriving on the island.
2. The objective of this question is to estimate the minimum distance separating these two trajectories. a. Verify that the vector $\vec { n } \left( \begin{array} { c } 3 \\ 13 \\ 27 \end{array} \right)$ is normal to the lines $\mathscr { D } _ { 1 }$ and $\mathscr { D } _ { 2 }$. b. It is admitted that the minimum distance between the lines $\mathscr { D } _ { 1 }$ and $\mathscr { D } _ { 2 }$ is the distance $\mathrm { HH } ^ { \prime }$ where $\overrightarrow { \mathrm { HH } ^ { \prime } }$ is a vector collinear to $\vec { n }$ with H belonging to the line $\mathscr { D } _ { 1 }$ and $\mathrm { H } ^ { \prime }$ belonging to the line $\mathscr { D } _ { 2 }$. Determine an approximate value in meters of this minimum distance. One may use the results below provided by a computer algebra system

\multicolumn{2}{|l|}{$\triangleright$ Computer algebra}
1	\begin{tabular}{ l } Solve $( \{ 10 * k - 3 - t = 3 * l , 2 + 6 * k - 6 * t = 13 * l , - 4 * k + 3 * t = 27 * l \} , \{ k , l , t \} )$
$\rightarrow \left\{ \left\{ k = \frac { 675 } { 1814 } , \ell = \frac { 17 } { 907 } , t = \frac { 603 } { 907 } \right\} \right\}$

\hline \end{tabular}

1. The scientists decide to install a beacon at sea.

It is located at point B with coordinates ( $2 ; 4 ; 0$ ). a. Let $M$ be a point on the line $\mathscr { D } _ { 1 }$.
Determine the coordinates of the point $M$ such that the distance $\mathrm { B} M$ is minimal. b. Deduce the minimum distance, rounded to the nearest meter, between the beacon and the green turtles.

As shown below, there is a right isosceles triangle with the two legs forming the right angle each having length 1. Let $R _ { 1 }$ be the figure obtained by coloring a square that has 2 vertices on the hypotenuse and the remaining 2 vertices on the two legs forming the right angle.
Let $R _ { 2 }$ be the figure obtained by coloring 2 congruent squares in the figure $R _ { 1 }$, where each square has 2 vertices on the hypotenuse of each of 2 congruent right isosceles triangles and the remaining 2 vertices on the two legs forming the right angle.
Let $R _ { 3 }$ be the figure obtained by coloring 4 congruent squares in the figure $R _ { 2 }$, where each square has 2 vertices on the hypotenuse of each of 4 congruent right isosceles triangles and the remaining 2 vertices on the two legs forming the right angle.
Continuing this process, let $S _ { n }$ be the sum of the areas of all colored squares in the figure $R _ { n }$ obtained at the $n$-th step. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 3 \sqrt { 2 } } { 20 }$
(2) $\frac { \sqrt { 2 } } { 5 }$
(3) $\frac { 3 } { 10 }$
(4) $\frac { \sqrt { 3 } } { 5 }$
(5) $\frac { 2 } { 5 }$

In the coordinate plane, let $f ( n )$ denote the number of triangles OAB satisfying the following conditions for a natural number $n$. Find the value of $f ( 1 ) + f ( 2 ) + f ( 3 )$. (Here, O is the origin.) [4 points] (가) The coordinates of point A are $\left( - 2,3 ^ { n } \right)$. (나) If the coordinates of point B are $( a , b )$, then $a$ and $b$ are natural numbers and satisfy $b \leq \log _ { 2 } a$. (다) The area of triangle OAB is at most 50.

7. Two numbers $x$ and $y$ are randomly chosen from the interval $[0,1]$. Let $p_1$ be the probability of the event ``$x + y \geq \frac{1}{2}$'', $p_2$ be the probability of the event ``$|x - y| \leq \frac{1}{2}$'', and $p_3$ be the probability of the event ``$xy \leq \frac{1}{2}$''. Then
A. $p_1 < p_2 < p_3$
B. $p_2 < p_3 < p_1$
C. $p_3 < p_1 < p_2$
D. $p_3 < p_2 < p_1$

7. As shown in the figure, the angle between the oblique line segment AB and plane $\alpha$ is $60 ^ { \circ }$ , B is the foot of the oblique line, and the moving point P on plane $\alpha$ satisfies $\angle \mathrm { PAB } = 30 ^ { \circ }$ . Then the locus of point P is
A. a line
B. a parabola
C. an ellipse
D. one branch of a hyperbola

8. Two numbers $\mathrm { x } , \mathrm { y }$ are randomly selected from the interval $[ 0,1 ]$. Let $p _ { 1 }$ be the probability of the event ``$x + y \leq \frac { 1 } { 2 }$'', and $P _ { 2 }$ be the probability of the event ``$x y \leq \frac { 1 } { 2 }$'', then
A. $p _ { 1 } < p _ { 2 } < \frac { 1 } { 2 }$
B. $p _ { 2 } < \frac { 1 } { 2 } < p _ { 1 }$
C. $\frac { 1 } { 2 } < p _ { 2 } < p _ { 1 }$
D. $p _ { 1 } < \frac { 1 } { 2 } < p _ { 2 }$

11. Let the complex number $z = ( x - 1 ) + y i ( x , y \in R )$. If $| z | \leq 1$, the probability that $y \geq x$ is
A. $\frac { 3 } { 4 } + \frac { 1 } { 2 \pi }$
B. $\frac { 1 } { 4 } - \frac { 1 } { 2 \pi }$
C. $\frac { 1 } { 2 } - \frac { 1 } { \pi }$
D. $\frac { 1 } { 2 } + \frac { 1 } { \pi }$

13. In the figure, the coordinates of point $A$ are $( 1,0 )$, the coordinates of point $C$ are $( 2,4 )$, and the function $f ( x ) = x ^ { 2 }$. If a point is randomly selected inside rectangle $A B C D$, then the probability that this point is in the shaded region equals $\_\_\_\_$.

A company's shuttle bus departs at 7:30, 8:00, and 8:30. Xiaoming arrives at the bus station between 7:50 and 8:30 to board the shuttle, and his arrival time is random. The probability that his waiting time does not exceed 10 minutes is
(A) $\frac { 1 } { 3 }$
(B) $\frac { 1 } { 2 }$
(C) $\frac { 2 } { 3 }$
(D) $\frac { 3 } { 4 }$

As shown in the figure, the pattern inside square $ABCD$ comes from the ancient Chinese Tai Chi diagram with central symmetry about the center. If a point is randomly selected inside the square, the probability that this point is in the shaded region is
A. $\frac { 1 } { 4 }$
B. $\frac { \pi } { 8 }$
C. $\frac { 1 } { 2 }$
D. $\frac { \pi } { 4 }$

As shown in the figure, the pattern inside square $ABCD$ comes from the ancient Chinese Tai Chi diagram. The black and white parts in the inscribed circle of the square are symmetric with respect to the center of the square. If a point is randomly selected from the square, the probability that it falls in the black part is
A. $\frac{1}{4}$
B. $\frac{\pi}{8}$
C. $\frac{1}{2}$
D. $\frac{\pi}{4}$

Let $O$ be the origin of the coordinate system. A point is randomly selected in the region $\left\{ ( x , y ) \mid 1 \leqslant x ^ { 2 } + y ^ { 2 } \leqslant 4 \right\}$. The probability that the inclination angle of line $OA$ does not exceed $\frac { \pi } { 4 }$ is
A. $\frac { 1 } { 8 }$
B. $\frac { 1 } { 6 }$
C. $\frac { 1 } { 4 }$
D. $\frac { 1 } { 2 }$

145- If $A_i = \left[-i,\, \dfrac{9-i}{2}\right]$, $i \in \{1,2,3,\ldots,9\}$, then the set $(A_1 \cap A_2) - (A_2 \cap A_5)$ is which of the following?
(1) $(-2,-1) \cup (1,2]$ (2) $[-2,-1] \cup [1,2]$ (3) $[-1,1]$ (4) $\phi$

147. Inside a regular hexagon with side $2\sqrt{3}$ units, a point is chosen at random. What is the probability that the distance from this point to each side of the hexagon is greater than one unit?
(1) $\dfrac{4}{9}$ (2) $\dfrac{2}{3}$ (3) $\dfrac{5}{9}$ (4) $\dfrac{3}{4}$

148- A point is chosen randomly inside an equilateral triangle with side $\sqrt{2\pi\sqrt{3}}$. What is the probability that the distance from this point to each vertex of the triangle is more than 1 unit?

(1) $\dfrac{1}{3}$

(2) $\dfrac{1}{2}$

(3) $\dfrac{2}{3}$

(4) $\dfrac{3}{4}$

147. Inside an equilateral triangle with side length 8 units, we randomly select a point. What is the probability that this point is more than $\sqrt{3}$ units away from every side of the triangle?
(1) $\dfrac{1}{16}$ (2) $\dfrac{1}{9}$ (3) $\dfrac{1}{8}$ (4) $\dfrac{3}{16}$

148- Two numbers are randomly chosen between 2 and 5. What is the probability that both numbers are between $0.3$ and $0.5$?
(1) $0.2$ (2) $0.25$ (3) $0.3$ (4) $0.35$

Let $P_0 = (0,0)$, $P_1 = (0,4)$, $P_2 = (4,0)$, $P_3 = (-4,-4)$, $P_4 = (2,4)$, $P_5 = (4,6)$ (or similar points). Find the region of all points closer to $P_0$ than to any of $P_1, P_2, P_3, P_4, P_5$, and compute its perimeter.

Let $R = \{(x, y) : x^2 + y^2 \leq 100,\ \sin(x+y) > 0\}$. Find the area of the region $R$.
(A) $25\pi$ (B) $50\pi$ (C) $100\pi$ (D) $75\pi$

Let $X = \left\{ ( x , y ) \in \mathbb { Z } \times \mathbb { Z } : \frac { x ^ { 2 } } { 8 } + \frac { y ^ { 2 } } { 20 } < 1 \right.$ and $\left. y ^ { 2 } < 5 x \right\}$. Three distinct points $P , Q$ and $R$ are randomly chosen from $X$. Then the probability that $P , Q$ and $R$ form a triangle whose area is a positive integer, is
(A) $\frac { 71 } { 220 }$
(B) $\frac { 73 } { 220 }$
(C) $\frac { 79 } { 220 }$
(D) $\frac { 83 } { 220 }$

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $( 0,0 ) , ( 0,41 )$ and $( 41,0 )$ is
(1) 780
(2) 901
(3) 861
(4) 820

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0, 0)$, $(0, 41)$ and $(41, 0)$, is:
(1) 820
(2) 780
(3) 901
(4) 861

If a point $A ( x , y )$ lies in the region bounded by the $y$-axis, straight lines $2 y + x = 6$ and $5 x - 6 y = 30$, then the probability that $y < 1$ is
(1) $\frac { 1 } { 6 }$
(2) $\frac { 5 } { 6 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 6 } { 7 }$

On the coordinate plane, there is an annular region formed by the intersection of the exterior of the circle $x ^ { 2 } + y ^ { 2 } = 3$ and the interior of the circle $x ^ { 2 } + y ^ { 2 } = 4$ . A person wants to use a straight scanning rod of length 1 to scan a certain region $R$ above the $x$-axis of this annular region. He designs the scanning rod with black and white ends moving respectively on the semicircles $C _ { 1 } : x ^ { 2 } + y ^ { 2 } = 3 ( y \geq 0 )$ and $C _ { 2 } : x ^ { 2 } + y ^ { 2 } = 4 ( y \geq 0 )$ . Initially, the black end of the scanning rod is at point $A ( \sqrt { 3 } , 0 )$ and the white end is at point $B$ on $C _ { 2 }$ . Then the black and white ends move counterclockwise along $C _ { 1 }$ and $C _ { 2 }$ respectively until the white end reaches point $B ^ { \prime } ( - 2,0 )$ on $C _ { 2 }$ , at which point scanning stops.
What are the coordinates of point $B$? (Single-choice question, 3 points)
(1) $( 0,2 )$
(2) $( 1 , \sqrt { 3 } )$
(3) $( \sqrt { 2 } , \sqrt { 2 } )$
(4) $( \sqrt { 3 } , 1 )$
(5) $( 2,0 )$