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	$\cdots$	$\cdots$	$\cdots$	$\cdots$

   
   

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

In a computer game, a cube is initially positioned as indicated in the figure.
Each displacement made by this cube always occurs in one of the directions defined by the three coordinate axes. When moving from the initial position, this cube moved 3 units closer to the $yz$ plane, moved 5 units away from the $xz$ plane, and moved 4 units closer to the $xy$ plane.
The figure that presents the orthogonal projections of this cube onto the three coordinate planes, after performing the described movements, is
(A), (B), (C), (D), or (E) as indicated in the figures.

In $\triangle \mathrm { ABC } , \mathrm { BE }$ is a median, and O the mid-point of BE. The line joining A and O meets BC at D. Find the ratio $\overline { \mathrm { AO } } : \overline { \mathrm { OD } }$ (Hint: Draw a line through E parallel to AD.)

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, on a plane $\alpha$ there is an equilateral triangle ABC with side length 3, and a sphere $S$ with radius 2 is tangent to the plane $\alpha$ at point A. For a point D on the sphere $S$ such that the segment AD passes through the center O of the sphere $S$, find the value of $| \overrightarrow { \mathrm { AB } } + \overrightarrow { \mathrm { DC } } | ^ { 2 }$. [4 points]

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

In coordinate space, the distance between point $\mathrm { P } ( 0,3,0 )$ and point $\mathrm { A } ( - 1,1 , a )$ is 2 times the distance between point P and point $\mathrm { B } ( 1,2 , - 1 )$. What is the value of the positive number $a$? [2 points]
(1) $\sqrt { 7 }$
(2) $\sqrt { 6 }$
(3) $\sqrt { 5 }$
(4) 2
(5) $\sqrt { 3 }$

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

In coordinate space, for two points $\mathrm { A } ( 1 , a , - 6 ) , \mathrm { B } ( - 3,2 , b )$, when the point that externally divides the line segment AB in the ratio $3 : 2$ lies on the $x$-axis, what is the value of $a + b$? [3 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

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]

2. Vector ${ } ^ { \mathbf { 1 } } = ( 2,4 )$ is collinear with vector ${ } ^ { \mathbf { 1 } } = ( x , 6 )$. Then the real number $x =$
(A) 2
(B) 3
(C) 4
(D) 6

4. Given $\vec { a } = ( 0 , - 1 ) , \vec { b } = ( - 1,2 )$, then $( 2 \vec { a } + \vec { b } ) \cdot \vec { a } =$
A. $- 1$
B. $0$
C. $1$
D. $2$