Given the four equations: $\textbf{(1)}\ 3y-2x=12 \qquad \textbf{(2)}\ -2x-3y=10 \qquad \textbf{(3)}\ 3y+2x=12 \qquad \textbf{(4)}\ 2y+3x=10$ The pair representing the perpendicular lines is: $\textbf{(A)}\ \text{(1) and (4)} \qquad \textbf{(B)}\ \text{(1) and (3)} \qquad \textbf{(C)}\ \text{(1) and (2)} \qquad \textbf{(D)}\ \text{(2) and (4)} \qquad \textbf{(E)}\ \text{(2) and (3)}$

Prove no three lattice points in the plane form an equilateral triangle.

Prove that the lines joining the middle-points of non-adjacent sides of an convex quadrilateral and the line joining the middle-points of diagonals, are concurrent. Prove that the intersection point is the middle point of the three given segments.

Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.

For any integers $1\leq n\leq m\leq5$, how many hyperbolas does the equation $\rho=\frac{1}{1-\text{C}_m^n \cos\theta}$ represent? Note: $\text{C}_m^n=\frac{m!}{n!(m-n)!}$. $\text{(A)}15\qquad\text{(B)}10\qquad\text{(C)}7\qquad\text{(D)}6$

We call a coloring $f$ of the elements in the set $M = \{(x, y) | x = 0, 1, \dots , kn - 1; y = 0, 1, \dots , ln - 1\}$ with $n$ colors allowable if every color appears exactly $k$ and $ l$ times in each row and column and there are no rectangles with sides parallel to the coordinate axes such that all the vertices in $M$ have the same color. Prove that every allowable coloring $f$ satisfies $kl \leq n(n + 1).$

Let $n$ be a positive integer, $n \ge 2$, and put $\theta=\frac{2\pi}{n}$. Define points $P_k=(k,0)$ in the [i]xy[/i]-plane, for $k=1,2,\dots,n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $P_k$. Let $R$ denote the map obtained by applying in order, $R_1$, then $R_2$, ..., then $R_n$. For an arbitrary point $(x,y)$, find and simplify the coordinates of $R(x,y)$.

Draw on the coordinate plane a set of points $M(a, b)$ such that the equation $x^4+ax+b=0$ has a unique root satisfying the condition $0 \le x \le 1$.

Square $\mathcal S$ has vertices $(1,0)$, $(0,1)$, $(-1,0)$ and $(0,-1)$. Points $P$ and $Q$ are independently selected, uniformly at random, from the perimeter of $\mathcal S$. Determine, with proof, the probability that the slope of line $PQ$ is positive. [i]Proposed by Isabella Grabski[/i]

Consider a $6 \times 6$ grid. Define a [i]diagonal[/i] to be the six squares whose coordinates $(i,j)$ ($1 \le i,j \le 6)$ satisfy $i-j \equiv k \pmod 6$ for some $k=0,1,\dots,5$. Hence there are six diagonals. Determine if it is possible to fill it with the numbers $1,2,\dots,36$ (each exactly once) such that each row, each column, and each of the six diagonals has the same sum.

Let $\mathcal{C}$ be the family of circumferences in $\mathbb{R}^2$ that satisfy the following properties: (i) if $C_n$ is the circumference with center $(n,1/2)$ and radius $1/2$, then $C_n\in \mathcal{C}$, for all $n\in \mathbb{Z}$. (ii) if $C$ and $C'$, both in $\mathcal{C}$, are externally tangent, then the circunference externally tangent to $C$ and $C'$ and tanget to $x$-axis also belongs to $\mathcal{C}$. (iii) $\mathcal{C}$ is the least family which these properties. Determine the set of the real numbers which are obtained as the first coordinate of the points of intersection between the elements of $\mathcal{C}$ and the $x$-axis.

Given a right triangle $T$, where the coordinates of its vertices are integers, let $E$ be the number of points of integer coordinates that belong to the edge of the triangle $T$, $I$ the number of points of integer coordinates that belong to the interior of the triangle $T$. Show that the area $A(T)$ of triangle $T$ is given by: $A(T) = \frac{E}{2}+I -1$.

Let $ ABC$ a triangle and $ X$, $ Y$ and $ Z$ points at the segments $ BC$, $ AC$ and $ AB$, respectively.Let $ A'$, $ B'$ and $ C'$ the circuncenters of triangles $ AZY$,$ BXZ$,$ CYX$, respectively.Prove that $ 4(A'B'C')\geq(ABC)$ with equality if and only if $ AA'$, $ BB'$ and $ CC'$ are concurrents. Note: $ (XYZ)$ denotes the area of $ XYZ$

A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\sqrt{x},\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k$ be the area of the region enclosed by the image of quadrilateral $ABCD.$ Find the greatest integer that does not exceed $k.$

A square $ABCD$ lies in the coordinate plane with its vertices $A$ and $C$ lying on different coordinate axes. Prove that one of the vertices $B$ or $D$ lies on the line $y = x$ and the other one on $y = -x$.

On the infinite chessboard several rectangular pieces are placed whose sides run along the grid lines. Each two have no squares in common, and each consists of an odd number of squares. Prove that these pieces can be painted in four colours such that two pieces painted in the same colour do not share any boundary points.

Find the area of a figure consisting of points whose coordinates satisfy the inequality $$(y^3 - arcsin x)(x^3 + arcsin y) \ge 0.$$

(1) Let $x>0,\ y$ be real numbers. For variable $t$, find the difference of Maximum and minimum value of the quadratic function $f(t)=xt^2+yt$ in $0\leq t\leq 1$. (2) Let $S$ be the domain of the points $(x,\ y)$ in the coordinate plane forming the following condition: For $x>0$ and all real numbers $t$ with $0\leq t\leq 1$ , there exists real number $z$ for which $0\leq xt^2+yt+z\leq 1$ . Sketch the outline of $S$. (3) Let $V$ be the domain of the points $(x,\ y,\ z) $ in the coordinate space forming the following condition: For $0\leq x\leq 1$ and for all real numbers $t$ with $0\leq t\leq 1$, $0\leq xt^2+yt+z\leq 1$ holds. Find the volume of $V$. [i]2011 Tokyo University entrance exam/Science, Problem 6[/i]

Let $T=\text{TNFTPP}$. Let $R$ be the region consisting of the points $(x,y)$ of the cartesian plane satisfying both $|x|-|y|\leq T-500$ and $|y|\leq T-500$. Find the area of region $R$.

A sled loaded with children starts from rest and slides down a snowy $25^\circ$ (with respect to the horizontal) incline traveling $85$ meters in $17$ seconds. Ignore air resistance. What is the coefficient of kinetic friction between the sled and the slope? $ \textbf {(A) } 0.36 \qquad \textbf {(B) } 0.40 \qquad \textbf {(C) } 0.43 \qquad \textbf {(D) } 1.00 \qquad \textbf {(E) } 2.01 $

[b]a)[/b] Let $ D_1,D_2,D_3 $ be pairwise skew lines. Through every point $ P_2\in D_2 $ there is an unique common secant of these three lines that intersect $ D_1 $ at $ P_1 $ and $ D_3 $ at $ P_3. $ Let coordinate systems be introduced on $ D_2 $ and $ D_3 $ having as origin $ O_2, $ respectively, $ O_3. $ Find a relation between the coordinates of $ P_2 $ and $ P_3. $ [b]b)[/b] Show that there exist four pairwise skew lines with exactly two common secants. Also find examples with exactly one and with no common secants. [b]c)[/b] Let $ F_1,F_2,F_3,F_4 $ be any four secants of $ D_1,D_2, D_3. $ Prove that $ F_1,F_2, F_3, F_4 $ have infinitely many common secants.

Let's consider the graph of a square trinomial having roots $1$ and $4$. Let's draw two tangents to it from point $O$ (the origin of coordinates), touching it at points $A$ and $B$. What values can the cosine of angle $\angle AOB$ take?

$A, B$ and $C$ are points in the plane with integer coordinates. The lengths of the sides of triangle $ABC$ are integer numbers. Prove that the perimeter of the triangle is an even number.

The graph of $y^2+2xy+40|x|=400$ partitions the plane into several regions. What is the area of the bounded region?

Let $P$ be a polygon (not necessarily convex) with $n$ vertices, such that all its sides and diagonals are less or equal with 1 in length. Prove that the area of the polygon is less than $\dfrac {\sqrt 3} 2$.