In the orthonormal frame $(A ; \overrightarrow{AB} ; \overrightarrow{AD} ; \overrightarrow{AE})$, $ABCDEFGH$ denotes a cube with side length 1.
a. The dot product $\overrightarrow{AF} \cdot \overrightarrow{BG}$ is equal to 0. b. The dot product $\overrightarrow{AF} \cdot \overrightarrow{BG}$ is equal to $(-1)$. c. The dot product $\overrightarrow{AF} \cdot \overrightarrow{BG}$ is equal to 1. d. The dot product $\overrightarrow{AF} \cdot \overrightarrow{BG}$ is equal to 2.

Exercise 4 (5 points)

Candidates who have not followed the specialization course

A radio-controlled scooter moves in a straight line at the constant speed of $1\,\mathrm{m.s}^{-1}$. It is pursued by a dog that moves at the same speed. We represent the situation from above in an orthonormal coordinate system of the plane with unit 1 meter. The origin of this coordinate system is the initial position of the dog. The scooter is represented by a point belonging to the line with equation $x = 5$. It moves on this line in the direction of increasing ordinates.

Part A - Modeling using a sequence

At the initial instant, the scooter is represented by the point $S_0$. The dog pursuing it is represented by the point $M_0$. We consider that at each second, the dog instantly orients itself in the direction of the scooter and moves in a straight line over a distance of 1 meter. We then model the trajectories of the dog and the scooter by two sequences of points denoted $(M_n)$ and $(S_n)$. After $n$ seconds, the coordinates of point $S_n$ are $(5; n)$. We denote $(x_n; y_n)$ the coordinates of point $M_n$.

1. Construct on graph $\mathrm{n}^\circ 1$ given in the appendix the points $M_2$ and $M_3$.
2. We denote $d_n$ the distance between the dog and the scooter $n$ seconds after the start of the pursuit, $d_n = M_nS_n$. Calculate $d_0$ and $d_1$.
3. Justify that the point $M_2$ has coordinates $\left(1 + \frac{4}{\sqrt{17}}; \frac{1}{\sqrt{17}}\right)$.
4. We admit that, for every natural integer $n$: $$\left\{\begin{array}{l} x_{n+1} = x_n + \dfrac{5 - x_n}{d_n} \\[6pt] y_{n+1} = y_n + \dfrac{n - y_n}{d_n} \end{array}\right.$$ a. The table below, obtained using a spreadsheet, gives the coordinates of points $M_n$ and $S_n$ as well as the distance $d_n$ as a function of $n$. What formulas should be written in cells C5 and F5 and copied downward to fill columns C and F?
   

   	A	B	C	D	E	F
   1	$n$	\multicolumn{2}{|c|}{$M_n$}	\multicolumn{2}{|c|}{$S_n$}	$d_n$
   2		$x_n$	$y_n$	5	n
   3	0	0	0	5	0	5
   4	1	1	0	5	1	4.12310563
   5	2	1.9701425	$\cdots$	$\cdots$	$\cdots$	$\cdots$

   
   

We consider a cube $ABCDEFGH$. The point M is the midpoint of $[\mathrm{BF}]$, I is the midpoint of [BC], the point N is defined by the relation $\overrightarrow{\mathrm{CN}} = \frac{1}{2}\overrightarrow{\mathrm{GC}}$ and the point P is the center of the face ADHE.

Part A:

1. Justify that the line (MN) intersects the segment [BC] at its midpoint I.
2. Construct, on the figure provided in the appendix, the cross-section of the cube by the plane (MNP).

Part B:

We equip space with the orthonormal coordinate system ($A; \overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE}$).

1. Justify that the vector $\vec{n}\left(\begin{array}{l}1\\2\\2\end{array}\right)$ is a normal vector to the plane (MNP).
   Deduce a Cartesian equation of the plane (MNP).
2. Determine a system of parametric equations of the line (d) passing through G and orthogonal to the plane (MNP).
3. Show that the line (d) intersects the plane (MNP) at the point K with coordinates $\left(\frac{2}{3}; \frac{1}{3}; \frac{1}{3}\right)$. Deduce the distance GK.
4. We admit that the four points M, E, D and I are coplanar and that the area of the quadrilateral MEDI is $\frac{9}{8}$ square units. Calculate the volume of the pyramid GMEDI.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

PART A

ABCDEFGH is a cube with edge length 1. The points I, J, K, L and M are the midpoints respectively of the edges [AB], [BF], [AE], [CD] and [DH].
Statement 1: $\ll \overrightarrow { \mathrm { JH } } = 2 \overrightarrow { \mathrm { BI } } + \overrightarrow { \mathrm { DM } } - \overrightarrow { \mathrm { CB } } \gg$ Statement 2: ``The triplet of vectors ( $\overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AH } } , \overrightarrow { \mathrm { AG } }$ ) is a basis of space.'' Statement 3: ``$\overrightarrow { \mathrm { IB } } \cdot \overrightarrow { \mathrm { LM } } = - \frac { 1 } { 4 }$.''

PART B

In space equipped with an orthonormal coordinate system, we consider:

• the plane $\mathscr { P }$ with Cartesian equation $2 x - y + 3 z + 6 = 0$
• the points $\mathrm { A } ( 2 ; 0 ; - 1 )$ and $\mathrm { B } ( 5 ; - 3 ; 7 )$

Statement 4: ``The plane $\mathscr { P }$ and the line ( AB ) are parallel.'' Statement 5: ``The plane $\mathscr { P } ^ { \prime }$ parallel to $\mathscr { P }$ passing through B has Cartesian equation $- 2 x + y - 3 z + 34 = 0$'' Statement 6: ``The distance from point A to plane $\mathscr { P }$ is equal to $\frac { \sqrt { 14 } } { 2 }$.'' We denote by (d) the line with parametric representation
$$\left\{ \begin{array} { r l } x & = - 12 + 2 k \\ y & = 6 \\ z & = 3 - 5 k \end{array} , \text { where } k \in \mathbb { R } \right.$$
Statement 7: ``The lines (AB) and (d) are not coplanar.''

A group of junior scouts, in an activity at the city park where they live, set up a tent as shown in the photo in Figure 1. Figure 2 shows the diagram of this tent's structure, in the form of a right prism, in which metal rods were used.
After assembling the rods, one of the scouts observed an insect moving on them, starting from vertex $A$ toward vertex $B$, from there toward vertex $E$ and, finally, made the journey from vertex $E$ to $C$. Consider that all these movements were made by the shortest distance path between the points.
The projection of the insect's displacement on the plane containing the base $ABCD$ is given by (see answer options with figures).

If $\vec{u} = (1, 2)$ and $\vec{v} = (3, 4)$, what is the dot product $\vec{u} \cdot \vec{v}$?
(A) 7
(B) 9
(C) 11
(D) 13
(E) 15

On the coordinate plane, there are two arbitrary distinct vectors $\overrightarrow { \mathrm { OP } } , \overrightarrow { \mathrm { OQ } }$ with initial point at the origin O. When the endpoints $\mathrm { P } , \mathrm { Q }$ of the two vectors are translated 3 units in the $x$-direction and 1 unit in the $y$-direction to points $\mathrm { P } ^ { \prime } , \mathrm { Q } ^ { \prime }$ respectively, which of the following statements in are always true? [3 points]

ㄱ. $\left| \overrightarrow { \mathrm { OP } } - \overrightarrow { \mathrm { OP } ^ { \prime } } \right| = \sqrt { 10 }$ ㄴ. $| \overrightarrow { \mathrm { OP } } - \overrightarrow { \mathrm { OQ } } | = \left| \overrightarrow { \mathrm { OP } ^ { \prime } } - \overrightarrow { \mathrm { OQ } ^ { \prime } } \right|$ ㄷ. $\overrightarrow { \mathrm { OP } } \cdot \overrightarrow { \mathrm { OQ } } = \overrightarrow { \mathrm { OP } ^ { \prime } } \cdot \overrightarrow { \mathrm { OQ } ^ { \prime } }$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

As shown in the figure, in a rectangular parallelepiped $\mathrm { ABCD } - \mathrm { EFGH }$ with $\overline { \mathrm { AB } } = \overline { \mathrm { AD } } = 4$ and $\overline { \mathrm { AE } } = 8$, let P be the point that divides the edge AE in the ratio $1 : 3$, and let Q, R, S be the midpoints of edges $\mathrm { AB }$, $\mathrm { AD }$, and $\mathrm { FG }$, respectively. Let T be the midpoint of segment QR. Find the value of the dot product $\overrightarrow { \mathrm { TP } } \cdot \overrightarrow { \mathrm { QS } }$ of vectors $\overrightarrow { \mathrm { TP } }$ and $\overrightarrow { \mathrm { QS } }$. [3 points]

As shown in the figure, three cylinders with radius $\sqrt{3}$ and different heights are mutually externally tangent and placed on a plane $\alpha$. Let $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$ be the centers of the bases of the three cylinders that do not meet plane $\alpha$. Triangle $\mathrm{QPR}$ is an isosceles triangle, and the angle between plane $\mathrm{QPR}$ and plane $\alpha$ is $60°$. If the heights of the three cylinders are $8$, $a$, and $b$ respectively, find the value of $a + b$. (Given: $8 < a < b$) [4 points]

In the plane, the pentagon ABCDE satisfies $$\overline { \mathrm { AB } } = \overline { \mathrm { BC } } , \overline { \mathrm { AE } } = \overline { \mathrm { ED } } , \angle \mathrm {~B} = \angle \mathrm { E } = 90 ^ { \circ }$$ Which of the following statements in are correct? [4 points]
Remarks ㄱ. For the midpoint M of segment BE, $\overrightarrow { \mathrm { AB } } + \overrightarrow { \mathrm { AE } }$ and $\overrightarrow { \mathrm { AM } }$ are parallel to each other. ㄴ. $\overrightarrow { \mathrm { AB } } \cdot \overrightarrow { \mathrm { AE } } = - \overrightarrow { \mathrm { BC } } \cdot \overrightarrow { \mathrm { ED } }$ ㄷ. $| \overrightarrow { \mathrm { BC } } + \overrightarrow { \mathrm { ED } } | = | \overrightarrow { \mathrm { BE } } |$
(1) ᄀ
(2) ᄃ
(3) ᄀ, ᄂ
(4) ㄴ,ㄷ
(5) ᄀ, ᄂ, ᄃ

As shown in the figure, there is an equilateral triangle ABC and a circle O with diameter AC on a plane. Point D on segment BC is determined such that $\angle \mathrm { DAB } = \frac { \pi } { 15 }$. When point X moves on circle O, let P be the point where the dot product $\overrightarrow { \mathrm { AD } } \cdot \overrightarrow { \mathrm { CX } }$ of the two vectors $\overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { CX } }$ is minimized. If $\angle \mathrm { ACP } = \frac { q } { p } \pi$, find the value of $p + q$. (Note: $p$ and $q$ are coprime natural numbers.) [4 points]

In triangle ABC,
$$\overline { \mathrm { AB } } = 2 , \quad \angle \mathrm {~B} = 90 ^ { \circ } , \quad \angle \mathrm { C } = 30 ^ { \circ }$$
When point P satisfies $\overrightarrow { \mathrm { PB } } + \overrightarrow { \mathrm { PC } } = \overrightarrow { 0 }$, what is the value of $| \overrightarrow { \mathrm { PA } } | ^ { 2 }$? [3 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9

In an equilateral triangle ABC with side length 2, let H be the foot of the perpendicular from vertex A to side BC. When point P moves on line segment AH, find the maximum value of $| \overrightarrow { \mathrm { PA } } \cdot \overrightarrow { \mathrm { PB } } |$, which is $\frac { q } { p }$. Find the value of $p + q$. (Given that $p$ and $q$ are coprime natural numbers.)

For two vectors $\vec { a } = ( 1,3 ) , \vec { b } = ( 5 , - 6 )$, what is the sum of all components of the vector $\vec { a } - \vec { b }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two vectors $\vec { a } = ( 3 , - 1 ) , \vec { b } = ( 1,2 )$, what is the sum of all components of vector $\vec { a } + \vec { b }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two vectors $\vec { a } = ( 1 , - 2 ) , \vec { b } = ( - 1,4 )$, what is the sum of all components of the vector $\vec { a } + 2 \vec { b }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

In the coordinate plane, for a triangle ABC with area 9, let P, Q, R be points moving on the three sides AB, BC, CA respectively. When $$\overrightarrow { \mathrm { AX } } = \frac { 1 } { 4 } ( \overrightarrow { \mathrm { AP } } + \overrightarrow { \mathrm { AR } } ) + \frac { 1 } { 2 } \overrightarrow { \mathrm { AQ } }$$ is satisfied, the area of the region represented by point X is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

For two vectors $\vec { a } = ( 3,1 ) , \vec { b } = ( - 2,4 )$, what is the sum of all components of the vector $\vec { a } + \frac { 1 } { 2 } \vec { b }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Four distinct points $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ on a circle satisfy the following conditions. What is the value of $| \overrightarrow { \mathrm { AD } } | ^ { 2 }$? [4 points] (가) $| \overrightarrow { \mathrm { AB } } | = 8 , \overrightarrow { \mathrm { AC } } \cdot \overrightarrow { \mathrm { BC } } = 0$ (나) $\overrightarrow { \mathrm { AD } } = \frac { 1 } { 2 } \overrightarrow { \mathrm { AB } } - 2 \overrightarrow { \mathrm { BC } }$
(1) 32
(2) 34
(3) 36
(4) 38
(5) 40

As shown in the figure, there is a rectangle $\mathrm { AB } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ with $\overline { \mathrm { AB } _ { 1 } } = 2$ and $\overline { \mathrm { AD } _ { 1 } } = 4$. Let $\mathrm { E } _ { 1 }$ be the point that divides segment $\mathrm { AD } _ { 1 }$ internally in the ratio $3 : 1$, and let $\mathrm { F } _ { 1 }$ be a point inside rectangle $\mathrm { AB } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ such that $\overline { \mathrm { F } _ { 1 } \mathrm { E } _ { 1 } } = \overline { \mathrm { F } _ { 1 } \mathrm { C } _ { 1 } }$ and $\angle \mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 } = \frac { \pi } { 2 }$. Triangle $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 }$ is drawn. The figure obtained by shading quadrilateral $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ is called $R _ { 1 }$. In figure $R _ { 1 }$, a rectangle $\mathrm { AB } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is drawn with vertices at point $\mathrm { B } _ { 2 }$ on segment $\mathrm { AB } _ { 1 }$, point $\mathrm { C } _ { 2 }$ on segment $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 }$, point $\mathrm { D } _ { 2 }$ on segment $\mathrm { AE } _ { 1 }$, and point A, such that $\overline { \mathrm { AB } _ { 2 } } : \overline { \mathrm { AD } _ { 2 } } = 1 : 2$. Using the same method as for obtaining figure $R _ { 1 }$, triangle $\mathrm { E } _ { 2 } \mathrm {~F} _ { 2 } \mathrm { C } _ { 2 }$ is drawn in rectangle $\mathrm { AB } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ and quadrilateral $\mathrm { E } _ { 2 } \mathrm {~F} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is shaded to obtain figure $R _ { 2 }$. Continuing this process, let $S _ { n }$ be the area of the shaded region in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 441 } { 103 }$
(2) $\frac { 441 } { 109 }$
(3) $\frac { 441 } { 115 }$
(4) $\frac { 441 } { 121 }$
(5) $\frac { 441 } { 127 }$

In the coordinate plane, for a parallelogram OACB with $\overline { \mathrm { OA } } = \sqrt { 2 } , \overline { \mathrm { OB } } = 2 \sqrt { 2 }$ and $\cos ( \angle \mathrm { AOB } ) = \frac { 1 } { 4 }$, point P satisfies the following conditions. (가) $\overrightarrow { \mathrm { OP } } = s \overrightarrow { \mathrm { OA } } + t \overrightarrow { \mathrm { OB } } ( 0 \leq s \leq 1, 0 \leq t \leq 1 )$ (나) $\overrightarrow { \mathrm { OP } } \cdot \overrightarrow { \mathrm { OB } } + \overrightarrow { \mathrm { BP } } \cdot \overrightarrow { \mathrm { BC } } = 2$
For a point X moving on a circle centered at O and passing through point A, let $M$ and $m$ be the maximum and minimum values of $| 3 \overrightarrow { \mathrm { OP } } - \overrightarrow { \mathrm { OX } } |$ respectively. When $M \times m = a \sqrt { 6 } + b$, find the value of $a ^ { 2 } + b ^ { 2 }$. (Given that $a$ and $b$ are rational numbers.) [4 points]

For two vectors $\vec{a}$ and $\vec{b}$, $$|\vec{a}| = \sqrt{11}, \quad |\vec{b}| = 3, \quad |2\vec{a} - \vec{b}| = \sqrt{17}$$ What is the value of $|\vec{a} - \vec{b}|$? [3 points]
(1) $\frac{\sqrt{2}}{2}$
(2) $\sqrt{2}$
(3) $\frac{3\sqrt{2}}{2}$
(4) $2\sqrt{2}$
(5) $\frac{5\sqrt{2}}{2}$

In the coordinate plane, there is an equilateral triangle ABC with side length 4. Let D be the point that divides segment AB in the ratio $1:3$, E be the point that divides segment BC in the ratio $1:3$, and F be the point that divides segment CA in the ratio $1:3$. Four points $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$, and $\mathrm{X}$ satisfy the following conditions. (가) $|\overrightarrow{\mathrm{DP}}| = |\overrightarrow{\mathrm{EQ}}| = |\overrightarrow{\mathrm{FR}}| = 1$ (나) $\overrightarrow{\mathrm{AX}} = \overrightarrow{\mathrm{PB}} + \overrightarrow{\mathrm{QC}} + \overrightarrow{\mathrm{RA}}$ When $|\overrightarrow{\mathrm{AX}}|$ is maximized, let the area of triangle PQR be $S$. Find the value of $16S^2$. [4 points]

In the coordinate plane, there is a square ABCD with side length 4. $$|\overrightarrow{\mathrm{XB}} + \overrightarrow{\mathrm{XC}}| = |\overrightarrow{\mathrm{XB}} - \overrightarrow{\mathrm{XC}}|$$ Let $S$ be the figure formed by points X satisfying this condition. For a point P on figure $S$, $$4\overrightarrow{\mathrm{PQ}} = \overrightarrow{\mathrm{PB}} + 2\overrightarrow{\mathrm{PD}}$$ Let Q be the point satisfying this condition. If the maximum and minimum values of $\overrightarrow{\mathrm{AC}} \cdot \overrightarrow{\mathrm{AQ}}$ are $M$ and $m$ respectively, find the value of $M \times m$. [4 points]

6. Given point $A ( 1 , - 2 )$, if vector $\overrightarrow { AB }$ is in the same direction as $\vec { a } = \{ 2,3 \}$, [Figure] and $| \overrightarrow { AB } | = 2 \sqrt { 13 }$, then the coordinates of point B are $\_\_\_\_$.