Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $ (a,0)$ to $ (3,3)$, divides the entire region into two regions of equal area. What is $ a$? [asy]size(200); defaultpen(linewidth(.8pt)+fontsize(8pt)); fill((2/3,0)--(3,3)--(3,1)--(2,1)--(2,0)--cycle,gray); xaxis("$x$",-0.5,4,EndArrow(HookHead,4)); yaxis("$y$",-0.5,4,EndArrow(4)); draw((0,1)--(3,1)--(3,3)--(2,3)--(2,0)); draw((1,0)--(1,2)--(3,2)); draw((2/3,0)--(3,3)); label("$(a,0)$",(2/3,0),S); label("$(3,3)$",(3,3),NE);[/asy]$ \textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac35\qquad \textbf{(C)}\ \frac23\qquad \textbf{(D)}\ \frac34\qquad \textbf{(E)}\ \frac45$

Given $S$ be the finite lattice (with integer coordinate) set in the $xy$-plane. $A$ is the subset of $S$ with most elements such that the line connecting any two points in $A$ is not parallel to $x$-axis or $y$-axis. $B$ is the subset of integer with least elements such that for any $(x,y)\in S$, $x \in B$ or $y \in B$ holds. Prove that $|A| \geq |B|$.

In the right parallelopiped $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$, with $AB=12\sqrt{3}$ cm and $AA^{\prime}=18$ cm, we consider the points $P\in AA^{\prime}$ and $N\in A^{\prime}B^{\prime}$ such that $A^{\prime}N=3B^{\prime}N$. Determine the length of the line segment $AP$ such that for any position of the point $M\in BC$, the triangle $MNP$ is right angled at $N$.

Let $P$ be a point inside a triangle $A_1A_2A_3$. For $i = 1, 2, 3$, line $PA_i$ intersects the side opposite to $A_i$ at $B_i$. Let $C_i$ and $D_i$ be the midpoints of $A_iB_i$ and $PB_i$, respectively. Prove that the areas of the triangles $C_1C_2C_3$ and $D_1D_2D_3$ are equal.

Given that $x^2 + y^2 = 14x + 6y + 6$, what is the largest possible value that $3x + 4y$ can have? $\text{(A)}\ 72 \qquad \text{(B)}\ 73 \qquad \text{(C)}\ 74 \qquad \text{(D)}\ 75\qquad \text{(E)}\ 76$

Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ? [i]Proposed by Hungary.[/i] [hide="Remark"]I'm not sure I'm posting this in a right Forum.[/hide]

In the Cartesian plane $Oxy$ we consider the points ${A_1}\left( {40,1} \right), {A_2}\left( {40,2} \right), \ldots , {A_{40}}\left( {40,40} \right)$ as well as the segments $O{A_1},O{A_2},\ldots,O{A_{40}}$. A point of the Cartesian plane $Oxy$ is called "good", if its coordinates are integers and it is internal of one segment $O{A_i}, i=1,2,3,\ldots,40$. Additionally, one of the segments $O{A_1},O{A_2},\ldots,O{A_{40}}$ is called "good" if it contains a "good" point. Find the number of "good" segments and "good" points.

Select numbers $ a$ and $ b$ between $ 0$ and $ 1$ independently and at random, and let $ c$ be their sum. Let $ A, B$ and $ C$ be the results when $ a, b$ and $ c$, respectively, are rounded to the nearest integer. What is the probability that $ A \plus{} B \equal{} C$? $ \textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac12 \qquad \textbf{(D)}\ \frac23 \qquad \textbf{(E)}\ \frac34$

A circle with center at the origin and radius $5$ intersects the abscissa in points $A$ and $B$. Let $P$ a point lying on the line $x = 11$, and the point $Q$ is the intersection point of $AP$ with this circle. We know what is the $Q$ point is the midpoint of the $AP$. Find the coordinates of the point $P$.

A sphere is centered at a point with integer coordinates and passes through the three points $(2,0,0), (0,4,0), (0,0,6),$ but not the origin $(0,0,0)$. If $r$ is the smallest possible radius of the sphere, compute $r^2$.

In the cartesian plane, consider the curves $x^2+y^2=r^2$ and $(xy)^2=1$. Call $F_r$ the convex polygon with vertices the points of intersection of these 2 curves. (if they exist) (a) Find the area of the polygon as a function of $r$. (b) For which values of $r$ do we have a regular polygon?

Four frogs are positioned at four points on a straight line such that the distance between any two neighbouring points is 1 unit length. Suppose the every frog can jump to its corresponding point of reflection, by taking any one of the other 3 frogs as the reference point. Prove that, there is no such case that the distance between any two neighbouring points, where the frogs stay, are all equal to 2008 unit length.

Each of $n$ black squares and $n$ white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to $n$ squares. [i](V. Dolnikov)[/i]

Given is a rectangular plane coordinate system. (a) Prove that it is impossible to find an equilateral triangle whose vertices have integer coordinates. (b) In the plane the vertices $A, B$ and $C$ lie with integer coordinates in such a way that $AB = AC$. Prove that $\frac{d(A,BC)}{BC}$ is rational.

There are lamps in every field of $n\times n$ table. At start all the lamps are off. A move consists of chosing $m$ consecutive fields in a row or a column and changing the status of that $m$ lamps. Prove that you can reach a state in which all the lamps are on only if $m$ divides $n.$

Let $ Q$ be the centre of the inscribed circle of a triangle $ ABC.$ Prove that for any point $ P,$ \[ a(PA)^2 \plus{} b(PB)^2 \plus{} c(PC)^2 \equal{} a(QA)^2 \plus{} b(QB)^2 \plus{} c(QC)^2 \plus{} (a \plus{} b \plus{} c)(QP)^2, \] where $ a \equal{} BC, b \equal{} CA$ and $ c \equal{} AB.$

Each face of a tetrahedron can be placed in a circle of radius $ 1$. Show that the tetrahedron can be placed in a sphere of radius $ \frac{3}{2\sqrt2}$.

A semicircle of diameter $ 1$ sits at the top of a semicircle of diameter $ 2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune. [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); filldraw(Circle((0,.866),.5),grey,black); label("1",(0,.866),S); filldraw(Circle((0,0),1),white,black); draw((-.5,.866)--(.5,.866),linetype("4 4")); clip((-1,0)--(1,0)--(1,2)--(-1,2)--cycle); draw((-1,0)--(1,0)); label("2",(0,0),S);[/asy]$ \textbf{(A)}\ \frac {1}{6}\pi \minus{} \frac {\sqrt {3}}{4} \qquad \textbf{(B)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{12}\pi \qquad \textbf{(C)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{24}\pi\qquad\textbf{(D)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{24}\pi$ $ \textbf{(E)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{12}\pi$

A circle centred at $(a, b)$ contains the origin $(0,0)$. Denote by $S^+$ the total area of the parts of the circle in the first and third quadrants, and by $S^-$ the total area of the parts of the circle in the second and the fourth quadrants. Compute $S^+ -S^-$. (G Galperin)

We are given a lattice and two pebbles $A$ and $B$ that are placed at two lattice points. At each step we are allowed to relocate one of the pebbles to another lattice point with the condition that the distance between pebbles is preserved. Is it possible after finite number of steps to switch positions of the pebbles?

Let $n>1$ be an integer. Given a simple graph $G$ on $n$ vertices $v_1, v_2, \dots, v_n$ we let $k(G)$ be the minimal value of $k$ for which there exist $n$ $k$-dimensional rectangular boxes $R_1, R_2, \dots, R_n$ in a $k$-dimensional coordinate system with edges parallel to the axes, so that for each $1\leq i<j\leq n$, $R_i$ and $R_j$ intersect if and only if there is an edge between $v_i$ and $v_j$ in $G$. Define $M$ to be the maximal value of $k(G)$ over all graphs on $n$ vertices. Calculate $M$ as a function of $n$.

In the coordinate space, define a square $S$, defined by the inequality $|x|\leq 1,\ |y|\leq 1$ on the $xy$-plane, with four vertices $A(-1,\ 1,\ 0),\ B(1,\ 1,\ 0),\ C(1,-1,\ 0), D(-1,-1,\ 0)$. Let $V_1$ be the solid by a rotation of the square $S$ about the line $BD$ as the axis of rotation, and let $V_2$ be the solid by a rotation of the square $S$ about the line $AC$ as the axis of rotation. (1) For a real number $t$ such that $0\leq t<1$, find the area of cross section of $V_1$ cut by the plane $x=t$. (2) Find the volume of the common part of $V_1$ and $V_2$.

Let $A_1$, $A_2$, $A_3$ be three points in the plane, and for convenience, let $A_4= A_1$, $A_5 = A_2$. For $n = 1$, $2$, and $3$, suppose that $B_n$ is the midpoint of $A_n A_{n+1}$, and suppose that $C_n$ is the midpoint of $A_n B_n$. Suppose that $A_n C_{n+1}$ and $B_n A_{n+2}$ meet at $D_n$, and that $A_n B_{n+1}$ and $C_n A_{n+2}$ meet at $E_n$. Calculate the ratio of the area of triangle $D_1 D_2 D_3$ to the area of triangle $E_1 E_2 E_3$.

Let $n$ be a positive integer and let $V$ be a $(2n-1)$-dimensional vector space over the two-element field. Prove that for arbitrary vectors $v_1,\dots,v_{4n-1} \in V,$ there exists a sequence $1\leq i_1<\dots<i_{2n}\leq 4n-1$ of indices such that $v_{i_1}+\dots+v_{i_{2n}}=0.$

Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence. [list] [*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence. [*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list] [i]Mitchell Lee and Benjamin Gunby.[/i]