In coordinate space, there is a tetrahedron ABCD with vertices at four points $\mathrm { A } ( 2,0,0 ) , \mathrm { B } ( 0,1,0 ) , \mathrm { C } ( - 3,0,0 )$, $\mathrm { D } ( 0,0,2 )$. For point P moving on edge BD, let the coordinates of point P that minimize $\overline { \mathrm { PA } } ^ { 2 } + \overline { \mathrm { PC } } ^ { 2 }$ be $( a , b , c )$. If $a + b + c = \frac { q } { p }$, find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

In coordinate space, for two points $\mathrm { A } ( a , 1,3 ) , \mathrm { B } ( a + 6,4,12 )$, the point that divides the line segment AB internally in the ratio $1 : 2$ has coordinates $( 5,2 , b )$. What is the value of $a + b$? [2 points]
(1) 7
(2) 8
(3) 9
(4) 10
(5) 11

In coordinate space, for two points $\mathrm { A } ( a , 5,2 ) , \mathrm { B } ( - 2,0,7 )$, the point that divides segment AB internally in the ratio $3 : 2$ has coordinates $( 0 , b , 5 )$. What is the value of $a + b$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

In coordinate space, for two points $\mathrm { A } ( 2 , a , - 2 ) , \mathrm { B } ( 5 , - 3 , b )$, when the point that divides segment AB internally in the ratio $2 : 1$ lies on the $x$-axis, what is the value of $a + b$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

In coordinate space, for three points $\mathrm { A } ( a , 0,5 ) , \mathrm { B } ( 1 , b , - 3 ) , \mathrm { C } ( 1,1,1 )$ that are vertices of a triangle, when the centroid of the triangle has coordinates $( 2,2,1 )$, what is the value of $a + b$? [2 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

For two points $\mathrm { A } ( 1,6,4 ) , \mathrm { B } ( a , 2 , - 4 )$ in coordinate space, the point that divides segment AB internally in the ratio $1 : 3$ has coordinates $( 2,5,2 )$. What is the value of $a$? [2 points]
(1) 1
(2) 3
(3) 5
(4) 7
(5) 9

For two points $\mathrm { A } ( 2 , a , - 2 ) , \mathrm { B } ( 5 , - 2,1 )$ in coordinate space, when the point that divides segment AB internally in the ratio $2 : 1$ lies on the $x$-axis, what is the value of $a$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two points $\mathrm { A } ( 2,0,1 ) , \mathrm { B } ( 3,2,0 )$ in coordinate space, if the coordinates of a point on the $y$-axis that is equidistant from both points is $( 0 , a , 0 )$, what is the value of $a$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

In coordinate space, let P be the point obtained by reflecting point $\mathrm { A } ( 2,1,3 )$ across the xy-plane, and let Q be the point obtained by reflecting point A across the yz-plane. What is the length of segment PQ? [2 points]
(1) $5 \sqrt { 2 }$
(2) $2 \sqrt { 13 }$
(3) $3 \sqrt { 6 }$
(4) $2 \sqrt { 14 }$
(5) $2 \sqrt { 15 }$

For two points $\mathrm{A}(a, -2, 6)$ and $\mathrm{B}(9, 2, b)$ in coordinate space, the midpoint of segment AB has coordinates $(4, 0, 7)$. What is the value of $a + b$? [2 points]
(1) 1
(2) 3
(3) 5
(4) 7
(5) 9

Let $O$ be the origin and let $P Q R$ be an arbitrary triangle. The point $S$ is such that
$$\overrightarrow { O P } \cdot \overrightarrow { O Q } + \overrightarrow { O R } \cdot \overrightarrow { O S } = \overrightarrow { O R } \cdot \overrightarrow { O P } + \overrightarrow { O Q } \cdot \overrightarrow { O S } = \overrightarrow { O Q } \cdot \overrightarrow { O R } + \overrightarrow { O P } \cdot \overrightarrow { O S }$$
Then the triangle $P Q R$ has $S$ as its
[A] centroid
[B] circumcentre
[C] incentre
[D] orthocenter

Let the position vectors of the points $P , Q , R$ and $S$ be $\vec { a } = \hat { i } + 2 \hat { j } - 5 \hat { k } , \vec { b } = 3 \hat { i } + 6 \hat { j } + 3 \hat { k }$, $\vec { c } = \frac { 17 } { 5 } \hat { i } + \frac { 16 } { 5 } \hat { j } + 7 \hat { k }$ and $\vec { d } = 2 \hat { i } + \hat { j } + \hat { k }$, respectively. Then which of the following statements is true?
(A) The points $P , Q , R$ and $S$ are NOT coplanar
(B) $\frac { \vec { b } + 2 \vec { d } } { 3 }$ is the position vector of a point which divides $P R$ internally in the ratio $5 : 4$
(C) $\frac { \vec { b } + 2 \vec { d } } { 3 }$ is the position vector of a point which divides $P R$ externally in the ratio $5 : 4$
(D) The square of the magnitude of the vector $\vec { b } \times \vec { d }$ is 95

$ABC$ is a triangle in a plane with vertices $A ( 2,3,5 ) , B ( - 1,3,2 )$ and $C ( \lambda , 5 , \mu )$. If the median through $A$ is equally inclined to the coordinate axes, then the value of $\left( \lambda ^ { 3 } + \mu ^ { 3 } + 5 \right)$ is
(1) 1130
(2) 1348
(3) 1077
(4) 676

Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors $A , B , C$ and $P$ are $\vec{a}, \vec{b}, \vec{c}$ and $\frac { \vec { a } + \vec { b } + \vec { c } } { 4 }$ respectively, then the position vector of the orthocentre of this triangle, is :
(1) $- \left( \frac { \vec { a } + \vec { b } + \vec { c } } { 2 } \right)$
(2) $\vec { a } + \vec { b } + \vec { c }$
(3) $\frac { ( \vec { a } + \vec { b } + \vec { c } ) } { 2 }$
(4) $\overrightarrow { 0 }$

If a point $R ( 4 , y , z )$ lies on the line segment joining the points $P ( 2 , - 3 , 4 )$ and $Q ( 8 , 0 , 10 )$, then the distance of $R$ from the origin is
(1) $2 \sqrt { 21 }$
(2) $\sqrt { 53 }$
(3) 6
(4) $2 \sqrt { 14 }$

If the points with position vectors $\alpha \hat { \mathrm { i } } + 10 \hat { \mathrm { j } } + 13 \hat { \mathrm { k } } , 6 \hat { \mathrm { i } } + 11 \hat { \mathrm { j } } + 11 \hat { \mathrm { k } } , \frac { 9 } { 2 } \hat { \mathrm { i } } + \beta \hat { \mathrm { j } } - 8 \hat { \mathrm { k } }$ are collinear, then $( 19 \alpha - 6 \beta ) ^ { 2 }$ is equal to
(1) 36
(2) 25
(3) 49
(4) 16

Let two vertices of a triangle $ABC$ be $(2, 4, 6)$ and $(0, -2, -5)$, and its centroid be $(2, 1, -1)$. If the image of the third vertex in the plane $x + 2y + 4z = 11$ is $(\alpha, \beta, \gamma)$, then $\alpha\beta + \beta\gamma + \gamma\alpha$ is equal to
(1) 70
(2) 76
(3) 74
(4) 72

Consider a $\triangle ABC$ where $A(1,3,2)$, $B(-2,8,0)$ and $C(3,6,7)$. If the angle bisector of $\angle BAC$ meets the line $BC$ at $D$, then the length of the projection of the vector $\overrightarrow{AD}$ on the vector $\overrightarrow{AC}$ is:
(1) $\frac{37}{2\sqrt{38}}$
(2) $\frac{\sqrt{38}}{2}$
(3) $\frac{39}{2\sqrt{38}}$
(4) $\sqrt{19}$

The position vectors of the vertices $A , B$ and $C$ of a triangle are $2 \hat { i } - 3 \hat { j } + 3 \hat { k } , \quad 2 \hat { i } + 2 \hat { j } + 3 \hat { k }$ and $- \hat { i } + \hat { j } + 3 \hat { k }$ respectively. Let $l$ denotes the length of the angle bisector AD of $\angle \mathrm { BAC }$ where D is on the line segment BC , then $2 l ^ { 2 }$ equals :
(1) 49
(2) 42
(3) 50
(4) 45

Let the position vectors of the vertices $A , B$ and $C$ of a tetrahedron $A B C D$ be $\hat { \mathbf { i } } + 2 \hat { \mathbf { j } } + \hat { \mathbf { k } } , \hat { \mathbf { i } } + 3 \hat { \mathbf { j } } - 2 \hat { k }$ and $2 \hat { i } + \hat { j } - \hat { k }$ respectively. The altitude from the vertex $D$ to the opposite face $A B C$ meets the median line segment through $A$ of the triangle $A B C$ at the point $E$. If the length of $A D$ is $\frac { \sqrt { } \overline { 110 } } { 3 }$ and the volume of the tetrahedron is $\frac { \sqrt { 805 } } { 6 \sqrt { 2 } }$, then the position vector of $E$ is
(1) $\frac { 1 } { 12 } ( 7 \hat { \mathbf { i } } + 4 \hat { \mathbf { j } } + 3 \hat { k } )$
(2) $\frac { 1 } { 2 } ( \hat { i } + 4 \hat { j } + 7 \hat { k } )$
(3) $\frac { 1 } { 6 } ( 12 \hat { i } + 12 \hat { j } + \hat { k } )$
(4) $\frac { 1 } { 6 } ( 7 \hat { \mathrm { i } } + 12 \hat { \mathrm { j } } + \hat { \mathrm { k } } )$

Let the position vectors of three vertices of a triangle be $4\vec{p} + \vec{q} - 3\vec{r}$, $-5\vec{p} + \vec{q} + 2\vec{r}$ and $2\vec{p} - \vec{q} + 2\vec{r}$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\frac{\vec{p} + \vec{q} + \vec{r}}{4}$ and $\alpha\vec{p} + \beta\vec{q} + \gamma\vec{r}$ respectively, then $\alpha + 2\beta + 5\gamma$ is equal to:
(1) 3
(2) 4
(3) 1
(4) 6

Let $\mathrm { A } ( x , y , z )$ be a point in $xy$-plane, which is equidistant from three points $( 0,3,2 ) , ( 2,0,3 )$ and $( 0,0,1 )$. Let $\mathrm { B } = ( 1,4 , - 1 )$ and $\mathrm { C } = ( 2,0 , - 2 )$. Then among the statements (S1) : $\triangle \mathrm { ABC }$ is an isosceles right angled triangle, and (S2) : the area of $\triangle \mathrm { ABC }$ is $\frac { 9 \sqrt { 2 } } { 2 }$,
(1) both are true
(2) only (S2) is true
(3) only (S1) is true
(4) both are false

Suppose that a triangle ABC which is inscribed in a circle O of radius 2 satisfies
$$3\overrightarrow{\mathrm{OA}} + 4\overrightarrow{\mathrm{OB}} + 2\overrightarrow{\mathrm{OC}} = \vec{0}.$$
Let D denote the point of intersection of the straight line AO and the segment BC. We are to find the lengths of the segments AD and BD.
(1) When we set $\overrightarrow{\mathrm{OD}} = k\overrightarrow{\mathrm{OA}}$ where $k$ is a real number, we have
$$\overrightarrow{\mathrm{OD}} = -\frac{\mathbf{A}}{\mathbf{B}}k\overrightarrow{\mathrm{OB}} - \frac{\mathbf{C}}{\mathbf{D}}k\overrightarrow{\mathrm{OC}}.$$
As the three points B, C and D are located on a straight line, we obtain $k = \frac{\mathbf{EF}}{\mathbf{GH}}$. From this we derive that $\mathrm{OD} = \mathbf{H}$ and finally obtain
$$\mathrm{AD} = 1.$$
(2) From (1) we see that $\mathrm{BD} = \frac{\mathbf{J}}{\mathbf{K}}\mathrm{BC}$. So in order to find the length of the segment BD, we should find the length of the segment BC.
First we note that
$$\mathrm{BC}^2 = \mathbf{L} - \mathbf{M}\overrightarrow{\mathrm{OB}} \cdot \overrightarrow{\mathrm{OC}}$$
where $\overrightarrow{\mathrm{OB}} \cdot \overrightarrow{\mathrm{OC}}$ represents the inner product of $\overrightarrow{\mathrm{OB}}$ and $\overrightarrow{\mathrm{OC}}$. Since we know from (1) that $|4\overrightarrow{\mathrm{OB}} + 2\overrightarrow{\mathrm{OC}}|^2 = \mathbf{NO}$, we have
$$\overrightarrow{\mathrm{OB}} \cdot \overrightarrow{\mathrm{OC}} = \frac{\mathbf{PQR}}{\mathbf{S}}.$$
Hence we obtain $\mathrm{BC} = \frac{\square\sqrt{\mathbf{U}}}{\square\mathbf{V}}$ and finally from that
$$\mathrm{BD} = \frac{\sqrt{\mathrm{W}}}{\mathrm{X}}.$$

In coordinate space, let $O$ be the origin, and let point $P$ be the intersection of three planes $x - 3y - 5z = 0$, $x - 3y + 2z = 0$, $x + y = t$, where $t > 0$. If $\overline{OP} = 10$, then $t =$ （9）（10）（11）. (Express as a simplified radical)

In $\triangle A B C$, $\overline { A B } = 6 , \overline { A C } = 5 , \overline { B C } = 4$. Let $D$ be the midpoint of $\overline { A B }$, and $P$ be the intersection of the angle bisector of $\angle A B C$ and $\overline { C D }$, as shown in the figure. Select the correct options.
(1) $\overline { C P } = \frac { 3 } { 7 } \overline { C D }$
(2) $\overrightarrow { A P } = \frac { 3 } { 7 } \overrightarrow { A B } + \frac { 2 } { 7 } \overrightarrow { A C }$
(3) $\cos \angle B A C = \frac { 3 } { 4 }$
(4) The area of $\triangle A C P$ is $\frac { 15 } { 14 } \sqrt { 7 }$
(5) (Dot product) $\overrightarrow { A P } \cdot \overrightarrow { A C } = \frac { 120 } { 7 }$