=A subset $ S$ of $ \mathbb R^2$ is called an algebraic set if and only if there is a polynomial $ p(x,y)\in\mathbb R[x,y]$ such that \[ S \equal{} \{(x,y)\in\mathbb R^2|p(x,y) \equal{} 0\} \] Are the following subsets of plane an algebraic sets? 1. A square [img]http://i36.tinypic.com/28uiaep.png[/img] 2. A closed half-circle [img]http://i37.tinypic.com/155m155.png[/img]

Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac 12,\frac 13,\frac 16\right).$ The area of $\mathcal{S}$ divided by the area of $\mathcal{T}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal. [i]David Yang.[/i]

Consider two parabolas $y = x^2$ and $y = x^2 - 1$. Let $U$ be the set of points between the parabolas (including the points on the parabolas themselves). Does $U$ contain a line segment of length greater than $10^6$ ? Alexey Tolpygo

In a three-dimensional Euclidean space, by $\overrightarrow{u_1}$ , $\overrightarrow{u_2}$ , $\overrightarrow{u_3}$ are denoted the three orthogonal unit vectors on the $x, y$, and $z$ axes, respectively. a) Prove that the point $P(t) = (1-t)\overrightarrow{u_1} +(2-3t)\overrightarrow{u_2} +(2t-1)\overrightarrow{u_3}$ , where $t$ takes all real values, describes a straight line (which we will denote by $L$). b) What describes the point $Q(t) = (1-t^2)\overrightarrow{u_1} +(2-3t^2)\overrightarrow{u_2} +(2t^2 -1)\overrightarrow{u_3}$ if $t$ takes all the real values? c) Find a vector parallel to $L$. d) For what values of $t$ is the point $P(t)$ on the plane $2x+ 3y + 2z +1 = 0$? e) Find the Cartesian equation of the plane parallel to the previous one and containing the point $Q(3)$. f) Find the Cartesian equation of the plane perpendicular to $L$ that contains the point $Q(2)$.

All the points with integer coordinates in the $ xy$-Plane are coloured using three colours, red, blue and green, each colour being used at least once. It is known that the point $ (0,0)$ is red and the point $ (0,1)$ is blue. Prove that there exist three points with integer coordinates of distinct colours which form the vertices of a right-angled triangle.

Let $w_{1}$ and $w_{2}$ denote the circles $x^{2}+y^{2}+10x-24y-87=0$ and $x^{2}+y^{2}-10x-24y+153=0$, respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_{2}$ and internally tangent to $w_{1}$. Given that $m^{2}=p/q$, where $p$ and $q$ are relatively prime integers, find $p+q$.

Let $A,B,C$ be $3$ points on the plane with integral coordinates. Prove that there exists a point $P$ with integral coordinates distinct from $A,B$ and $C$ such that the interiors of the segments $PA,PB$ and $PC$ do not contain points with integral coordinates.

In a Cartesian coordinate system (with the same scale on the x and y axes)there is a graph of the exponential function $y=3^x$. Then the y-axis and all marks on the x-axis erased. Only the graph of the function and the x-axis remained without a scale and a mark of $0$. How can you restore the y-axis using a compass and ruler?

Does there exist a sphere passing through only one rational point? (A rational point is a point whose Cartesian coordinates are all rational numbers.) (A Rubin)

Let $a$ and $b$ be natural numbers. We consider the set $M$ of the points of the plane with an integer $x$-coordinate from $1$ to $a$ and integer $y$-coordinate from $1$ to $b$. For two points $P = (x, y)$ and $Q = (\tilde x, \tilde y)$ in M we write $P\le Q$ if $x\le \tilde x$ and $y \le \tilde y$, we say $P$ is [i]less [/i] than $Q$ when $P\le Q$ and $P \ne Q$. A subset $S$ of $M$ is now called [i]cute [/i] if for every point $P \in S$ it also contains all smaller points. From an arbitrary subset $S$ of $M$ we can now create new subsets in four ways to construct: (a) the complement $K (S) = \overline{S}$, (b) the subset $\min (S)$ of its minima, i.e. those points for which there is no smaller in $S$ occurs, (c) the cute set $P (S)$ of all those points in M that are less than or equal to some point are from $S$, (d) you do all these things one after the other and get a set $Z (S) = P (\min (K (S)))$. Let $S$ be cute. Prove that $$\underset{a+b\,\, times\,\, Z}{Z(Z(...(Z(S))...))=S}$$

We call a set of points [i]free[/i] if there is no equilateral triangle with the vertices among the points of the set. Prove that every set of $n$ points in the plane contains a [i]free[/i] subset with at least $\sqrt{n}$ elements.

Triangle $ABC$ has area $1$. Points $E$, $F$, and $G$ lie, respectively, on sides $BC$, $CA$, and $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$.

In the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origin, and a circle with radius $10$ with center on the positive $y$-axis is tangent to the $x$-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect? $\textbf{(A)}\ \dfrac{2}{7} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{2}{\sqrt{29}} \qquad\textbf{(D)}\ \dfrac{1}{\sqrt{29}} \qquad\textbf{(E)}\ \dfrac{2}{5}$

A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are (3,17) and (48,281)? (Include both endpoints of the segment in your count.) $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 46$

On an infinite chessboard (whose squares are labeled by $ (x, y)$, where $ x$ and $ y$ range over all integers), a king is placed at $ (0, 0)$. On each turn, it has probability of $ 0.1$ of moving to each of the four edge-neighboring squares, and a probability of $ 0.05$ of moving to each of the four diagonally-neighboring squares, and a probability of $ 0.4$ of not moving. After $ 2008$ turns, determine the probability that the king is on a square with both coordinates even. An exact answer is required.

The number of points with positive rational coordinates selected from the set of points in the xy-plane such that $x+y\leq 5$, is: $\text{(A)} \ 9 \qquad \text{(B)} \ 10 \qquad \text{(C)} \ 14 \qquad \text{(D)} \ 15 \qquad \text{(E)} \ \text{infinite}$

Arpon chooses a positive real number $k$. For each positive integer $n$, he places a marker at the point $(n,nk)$ in the $(x,y)$ plane. Suppose that two markers whose $x$-coordinates differ by $4$ have distance $31$. What is the distance between the markers $(7,7k)$ and $(19,19k)$?

Let $ P,Q $ be the midpoints of the diagonals $ BD, $ respectively, $ AC, $ of the quadrilateral $ ABCD, $ and points $ M,N,R,S $ on the segments $ BC,CD,PQ, $ respectively $ AC, $ except their extremities, such that $$ \frac{BM}{MC}=\frac{DN}{NC}=\frac{PR}{RQ}=\frac{AS}{SC} . $$ Show that the center of mass of the triangle $ AMN $ is situated on the segment $ RS. $

Let $ r$ and $ s$ two orthogonal lines that does not lay on the same plane. Let $ AB$ be their common perpendicular, where $ A\in{}r$ and $ B\in{}s$(*).Consider the sphere of diameter $ AB$. The points $ M\in{r}$ and $ N\in{s}$ varies with the condition that $ MN$ is tangent to the sphere on the point $ T$. Find the locus of $ T$. Note: The plane that contains $ B$ and $ r$ is perpendicular to $ s$.

Let points $ A = (0,0) , \ B = (1,2), \ C = (3,3), $ and $ D = (4,0) $. Quadrilateral $ ABCD $ is cut into equal area pieces by a line passing through $ A $. This line intersects $ \overline{CD} $ at point $ \left (\frac{p}{q}, \frac{r}{s} \right ) $, where these fractions are in lowest terms. What is $ p + q + r + s $? $ \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D)} \ 70 \qquad \textbf{(E)} \ 75 $

Each point of the plane with two integer coordinates is the center of a disk with radius $ \frac {1} {1000}$. Prove that there exists an equilateral triangle whose vertices belong to distinct disks. Prove that such a triangle has side-length greater than 96.

Prove that among any five distinct rays $ Ox$, $ Oy$, $ Oz$, $ Ot$, $ Or$ in space there exist two which form an angle less than or equal to $ 90^{\circ}$.

Assume $m\times n$ matrix which is filled with just 0, 1 and any two row differ in at least $n/2$ members, show that $m \leq 2n$. ( for example the diffrence of this two row is only in one index 110 100) [i]Edited by Myth[/i]

The line $x-2y-1=0$ insects the parabola $y^2=4x$ at two different points $A, B$. Let $C$ be a point on the parabola such that $\angle ACB=\frac{\pi}{2}$. Find the coordinate of point $C$.