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 $V_2$.

Given a cube with edges of length 1, $e$ the midpoint of $[bc]$, and $m$ midpoint of the face $cdc_1d_1$, as on the figure. Find the area of intersection of the cube with the plane through the points $a,m,e$. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=279[/img]

For a set $A$ of integers, define $f(A)=\{x^2+xy+y^2: x,y\in A\}$. Is there a constant $c$ such that for all positive integers $n$, there exists a set $A$ of size $n$ such that $|f(A)|\le cn$? [i]David Yang.[/i]

Let $ABC$ be a triangle inscribed in circle $\Gamma$, centered at $O$ with radius $333.$ Let $M$ be the midpoint of $AB$, $N$ be the midpoint of $AC$, and $D$ be the point where line $AO$ intersects $BC$. Given that lines $MN$ and $BO$ concur on $\Gamma$ and that $BC = 665$, find the length of segment $AD$. [i]Author: Alex Zhu[/i]

Find the area of the region enclosed by the graph of $|x-60|+|y|=|x/4|.$

A convex pentagon $ABCDE$ is given in the coordinate plane with all vertices in lattice points. Prove that there must be at least one lattice point in the pentagon determined by the diagonals $AC$, $BD$, $CE$, $DA$, $EB$ or on its boundary.

The Rational Unit Jumping Frog starts at $(0, 0)$ on the Cartesian plane, and each minute jumps a distance of exactly $1$ unit to a point with rational coordinates. (a) Show that it is possible for the frog to reach the point $\left(\frac15,\frac{1}{17}\right)$ in a finite amount of time. (b) Show that the frog can never reach the point $\left(0,\frac14\right)$.

Consider the parallelogram $ABCD$ with obtuse angle $A$. Let $H$ be the feet of perpendicular from $A$ to the side $BC$. The median from $C$ in triangle $ABC$ meets the circumcircle of triangle $ABC$ at the point $K$. Prove that points $K,H,C,D$ lie on the same circle.

Find the slopes of all lines passing through the origin and tangent to the curve $y^2=x^3+39x-35$.

On the Cartesian coordinate plane, the graph of the parabola $y = x^2$ is drawn. Three distinct points $A$, $B$, and $C$ are marked on the graph with $A$ lying between $B$ and $C$. Point $N$ is marked on $BC$ so that $AN$ is parallel to the y-axis. Let $K_1$ and $K_2$ are the areas of triangles $ABN$ and $ACN$, respectively. Express $AN$ in terms of $K_1$ and $K_2$.

Let $n$ be an integer greater than $1$. In the Cartesian coordinate system we consider all squares with integer vertices $(x,y)$ such that $1\le x,y\le n$. Denote by $p_k\ (k=0,1,2,\ldots )$ the number of pairs of points that are vertices of exactly $k$ such squares. Prove that $\sum_k(k-1)p_k=0$.

Let $\mathcal{H}$ be the region of points $(x, y)$, such that $(1, 0), (x, y), (-x, y)$, and $(-1,0)$ form an isosceles trapezoid whose legs are shorter than the base between $(x, y)$ and $(-x,y)$. Find the least possible positive slope that a line could have without intersecting $\mathcal{H}$.

Point $ P$ is inside equilateral $ \triangle ABC$. Points $ Q, R$ and $ S$ are the feet of the perpendiculars from $ P$ to $ \overline{AB}, \overline{BC}$, and $ \overline{CA}$, respectively. Given that $ PQ \equal{} 1, PR \equal{} 2$, and $ PS \equal{} 3$, what is $ AB$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 3 \sqrt {3}\qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 4 \sqrt {3}\qquad \textbf{(E)}\ 9$

For certain ordered pairs $(a,b)$ of real numbers, the system of equations \begin{eqnarray*} && ax+by =1\\ &&x^2+y^2=50\end{eqnarray*} has at least one solution, and each solution is an ordered pair $(x,y)$ of integers. How many such ordered pairs $(a,b)$ are there?

On Cartesian plane, given a line defined by $y=x+\frac{1}{\sqrt{2}}$. a) Prove that every circle has center $I\in d$ and radius is $\frac{1}{8}$ has no integral point inside. b) Find the greatest $k>0$ such that the distance of every integral points to $d$ is greater or equal than $k$.

The circle $x^2 + y^2 = 25$ intersects the abscissa in points $A$ and $B$. Let $P$ be a point that lies on the line $x = 11$, $C$ is the intersection point of this line with the $Ox$ axis, and the point $Q$ is the intersection point of $AP$ with the given circle. It turned out that the area of the triangle $AQB$ is four times smaller than the area of the triangle $APC$. Find the coordinates of $Q$.

Cube $ABCDEFGH$, labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. [asy] draw((0,0)--(10,0)--(10,10)--(0,10)--cycle); draw((0,10)--(4,13)--(14,13)--(10,10)); draw((10,0)--(14,3)--(14,13)); draw((0,0)--(4,3)--(4,13), dashed); draw((4,3)--(14,3), dashed); dot((0,0)); dot((0,10)); dot((10,10)); dot((10,0)); dot((4,3)); dot((14,3)); dot((14,13)); dot((4,13)); dot((14,8)); dot((5,0)); label("A", (0,0), SW); label("B", (10,0), S); label("C", (14,3), E); label("D", (4,3), NW); label("E", (0,10), W); label("F", (10,10), SE); label("G", (14,13), E); label("H", (4,13), NW); label("M", (5,0), S); label("N", (14,8), E); [/asy]

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 $\lim_{n\to\infty} S_n$.

Let $f(x)=x^2-6x+5$. On the plane $(x, y)$ draw a set of points $M(x, y)$ whose coordinates satisfy the inequalities $$\begin{cases} f(x)+f(y)\le 0 \\ f(x)-f(y)\ge 0 \end{cases}$$

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coefficients is it true that $|4x+3y|\leq 1000$? $\textbf{(A) }38\qquad \textbf{(B) }40\qquad \textbf{(C) }42\qquad \textbf{(D) }44\qquad \textbf{(E) }46\qquad$

In the Cartesian plane, let $C$ be a unit circle with center at origin $O$. For any point $Q$ in the plane distinct from $O$, define $Q'$ to be the intersection of the ray $OQ$ and the circle $C$. Prove that for any $P\in C$ and any $k\in\mathbb{N}$ there exists a lattice point $Q(x,y)$ with $|x|=k$ or $|y|=k$ such that $PQ'<\frac{1}{2k}$.

A coordinate system was drawn on the board and points $A (1,2)$ and $B (3,1)$ were marked. The coordinate system was erased. Restore it by the two marked points.

2 curves $ y \equal{} x^3 \minus{} x$ and $ y \equal{} x^2 \minus{} a$ pass through the point $ P$ and have a common tangent line at $ P$. Find the area of the region bounded by these curves.

Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)

Let $z$ be a complex number and k a positive integer such that $z^k$ is a positive real number other than $1$. Let $f(n)$ denote the real part of the complex number $z^n$. Assume the parabola $p(n) = an^2 +bn+c$ intersects $f(n)$ four times, at $n = 0, 1, 2, 3$. Assuming the smallest possible value of $k$, find the largest possible value of $a$.