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.

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 $60^{\circ}$.

Under what condition are the lines $\Delta(q, \vec{u})$ and $\Delta(r, \vec{v})$ identical?

Let $A , B , C$ be finite subsets of the plane such that $A \cap B , B \cap C$ and $C \cap A$ are all empty. Let $S = A \cup B \cup C$. Assume that no three points of $S$ are collinear and also assume that each of $A , B$ and $C$ has at least 3 points. Which of the following statements is always true?
(A) There exists a triangle having a vertex from each of $A , B , C$ that does not contain any point of $S$ in its interior.
(B) Any triangle having a vertex from each of $A , B , C$ must contain a point of $S$ in its interior.
(C) There exists a triangle having a vertex from each of $A , B , C$ that contains all the remaining points of $S$ in its interior.
(D) There exist 2 triangles, both having a vertex from each of $A , B , C$ such that the two triangles do not intersect.

Consider three points $P = ( - \sin ( \beta - \alpha ) , - \cos \beta ) , Q = ( \cos ( \beta - \alpha ) , \sin \beta )$ and $R = ( \cos ( \beta - \alpha + \theta ) , \sin ( \beta - \theta ) )$, where $0 < \alpha , \beta , \theta < \frac { \pi } { 4 }$. Then,
(A) $P$ lies on the line segment $R Q$
(B) $Q$ lies on the line segment $P R$
(C) $R$ lies on the line segment $Q P$
(D) $P , Q , R$ are non-collinear

Consider the lines given by
$$\begin{aligned} & L _ { 1 } : x + 3 y - 5 = 0 \\ & L _ { 2 } : 3 x - k y - 1 = 0 \\ & L _ { 3 } : 5 x + 2 y - 12 = 0 \end{aligned}$$
Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.
Column I
(A) $L _ { 1 } , L _ { 2 } , L _ { 3 }$ are concurrent, if
(B) One of $L _ { 1 } , L _ { 2 } , L _ { 3 }$ is parallel to at least one of the other two, if
(C) $L _ { 1 } , L _ { 2 } , L _ { 3 }$ form a triangle, if
(D) $L _ { 1 } , L _ { 2 } , L _ { 3 }$ do not form a triangle, if
Column II
(p) $k = - 9$
(q) $k = - \frac { 6 } { 5 }$
(r) $k = \frac { 5 } { 6 }$
(s) $k = 5$

The point of intersection of the lines $\left( a ^ { 3 } + 3 \right) x + a y + a - 3 = 0$ and $\left( a ^ { 5 } + 2 \right) x + ( a + 2 ) y + 2 a + 3 = 0$ (a real) lies on the $y$-axis for
(1) no value of $a$
(2) more than two values of $a$
(3) exactly one value of $a$
(4) exactly two values of $a$

The line parallel to $x$-axis and passing through the point of intersection of lines $a x + 2 b y + 3 b = 0$ and $b x - 2 a y - 3 a = 0$, where $( a , b ) \neq ( 0,0 )$ is
(1) above $x$-axis at a distance $2/3$ from it
(2) above $x$-axis at a distance $3/2$ from it
(3) below $x$-axis at a distance $3/2$ from it
(4) below $x$-axis at a distance $2/3$ from it

Let $a , b , c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4 a x + 2 a y + c = 0$ \& $5 b x + 2 b y + d = 0$ lies in the fourth quadrant and is equidistant from the two axes then
(1) $3 b c - 2 a d = 0$
(2) $3 b c + 2 a d = 0$
(3) $2 b c - 3 a d = 0$
(4) $2 b c + 3 a d = 0$

Let the area of the triangle with vertices $A ( 1 , \alpha ) , B ( \alpha , 0 )$ and $C ( 0 , \alpha )$ be 4 sq. units. If the points $( \alpha , - \alpha ) , ( - \alpha , \alpha )$ and $\left( \alpha ^ { 2 } , \beta \right)$ are collinear, then $\beta$ is equal to
(1) 64
(2) - 8
(3) - 64
(4) 512

If the sum of squares of all real values of $\alpha$, for which the lines $2 x - y + 3 = 0,6 x + 3 y + 1 = 0$ and $\alpha x + 2 y - 2 = 0$ do not form a triangle is $p$, then the greatest integer less than or equal to $p$ is $\_\_\_\_$ .

Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to
(1) 84
(2) 113
(3) 91
(4) 101

In an open space, there are three utility poles perpendicular to the ground with equal heights and equally spaced bases on a straight line. A person uses one-point perspective to draw these three utility poles on a canvas. A coordinate system is set up on the canvas so that the utility poles are parallel to the $y$-axis. The three base points are $A_{1}(0,0)$, $A_{2}$, $A_{3}$, all on the line $L: x + 3y = 0$; the three top points are $B_{1}(0,3)$, $B_{2}$, $B_{3}$, all on the line $M: 2x - 3y + 9 = 0$, as shown in the figure. It is known that $\overline{A_{3}B_{3}} = 2\overline{A_{1}B_{1}}$, and by one-point perspective, the intersection of lines $A_{1}B_{3}$ and $A_{3}B_{1}$ lies on line $A_{2}B_{2}$. Let $P$ be the intersection of $L$ and $M$ (this point is also called the "vanishing point").
Find the coordinates of points $P$ and $B_{3}$.

Let m be a real number. In the rectangular coordinate plane,

• the slope of a line passing through the point $( 0,1 )$ is $m$,
• the slope of a line passing through the point $( 0,0 )$ is $2 m$,
• the slope of a line passing through the point $( 1,0 )$ is $3 m$, and these three lines intersect at one point.

Accordingly, what is the value of $m$?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 3 } { 5 }$
E) $\frac { 4 } { 5 }$