Find the slope of a line passing through the point $(0,\ 1)$ with which the area of the part bounded by the line and the parabola $y=x^2$ is $\frac{5\sqrt{5}}{6}.$

Let $ n$ be a given positive integer. In the coordinate set, consider the set of points $ \{P_{1},P_{2},...,P_{4n\plus{}1}\}\equal{}\{(x,y)|x,y\in \mathbb{Z}, xy\equal{}0, |x|\le n, |y|\le n\}.$ Determine the minimum of $ (P_{1}P_{2})^{2} \plus{} (P_{2}P_{3})^{2} \plus{}...\plus{} (P_{4n}P_{4n\plus{}1})^{2} \plus{} (P_{4n\plus{}1}P_{1})^{2}.$

A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point? $ \textbf{(A)}\ \sqrt{13}\qquad\textbf{(B)}\ \sqrt{14}\qquad\textbf{(C)}\ \sqrt{15}\qquad\textbf{(D)}\ \sqrt{16}\qquad\textbf{(E)}\ \sqrt{17}$

On a plane with a fixed coordinate system, there is a convex polygon whose all vertices have integer coordinates. Prove that twice the area of this polygon is an integer.

All vertices of a convex $n$-gon ($n\ge3$) in the plane have integer coordinates. Show that its area is at least $\frac{n-2}2$.

A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0),$ $(2020, 0),$ $(2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth$?$ $\textbf{(A) } 0.3 \qquad \textbf{(B) } 0.4 \qquad \textbf{(C) } 0.5 \qquad \textbf{(D) } 0.6 \qquad \textbf{(E) } 0.7$

A triangle $ ABC$ is given in the plane. Let $ M$ be a point inside the triangle and $ A'$, $ B'$, $ C'$ be its projections on the sides $ BC$, $ CA$, $ AB$, respectively. Find the locus of $ M$ for which $ MA \cdot MA' \equal{} MB \cdot MB' \equal{} MC \cdot MC'$.

Define $f(x)$ to be the nearest integer to $x$, with the greater integer chosen if two integers are tied for being the nearest. For example, $f(2.3) = 2$, $f(2.5) = 3$, and $f(2.7) = 3$. Define $[A]$ to be the area of region $A$. Define region $R_n$, for each positive integer $n$, to be the region on the Cartesian plane which satisfies the inequality $f(|x|) + f(|y|) < n$. We pick an arbitrary point $O$ on the perimeter of $R_n$, and mark every two units around the perimeter with another point. Region $S_{nO}$ is defined by connecting these points in order. [b]a)[/b] Prove that the perimeter of $R_n$ is always congruent to $4 \pmod{8}$. [b]b)[/b] Prove that $[S_{nO}]$ is constant for any $O$. [b]c)[/b] Prove that $[R_n] + [S_{nO}] = (2n-1)^2$. [i]Proposed by Lewis Chen[/i]

A straight line passing through the point $(0,4)$ is perpendicular to the line $x-3y-7=0$. Its equation is: $\textbf{(A)}\ y+3x-4=0 \qquad \textbf{(B)}\ y+3x+4=0 \qquad \textbf{(C)}\ y-3x-4=0 \qquad\\ \textbf{(D)}\ 3y+x-12=0 \qquad \textbf{(E)}\ 3y-x-12=0 $

Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC=8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$, respectively, so that $EI=EJ=EK=2$. A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{AE}$, and containing the edges, $\overline{IJ}$, $\overline{JK}$, and $\overline{KI}$. The surface area of $S$, including the walls of the tunnel, is $m+n\sqrt{p}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p$.

It is known that the graph of a quadratic trinomial $y = x^2 + px + q$ touches the graph of a straight line $y = 2x + p$. Prove that all such quadratic trinomials have the same minimum value. Find this smallest value.

Let $a,\ b$ be positive real numbers. Consider the circle $C_1: (x-a)^2+y^2=a^2$ and the ellipse $C_2: x^2+\frac{y^2}{b^2}=1.$ (1) Find the condition for which $C_1$ is inscribed in $C_2$. (2) Suppose $b=\frac{1}{\sqrt{3}}$ and $C_1$ is inscribed in $C_2$. Find the coordinate $(p,\ q)$ of the point of tangency in the first quadrant for $C_1$ and $C_2$. (3) Under the condition in (1), find the area of the part enclosed by $C_1,\ C_2$ for $x\geq p$. 60 point

On the Cartesian plane consider the set $V$ of all vectors with integer coordinates. Determine all functions $f : V \rightarrow \mathbb{R}$ satisfying the conditions: (i) $f(v) = 1$ for each of the four vectors $v \in V$ of unit length. (ii) $f(v+w) = f(v)+f(w)$ for every two perpendicular vectors $v, w \in V$ (Zero vector is considered to be perpendicular to every vector).

Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a $ 14\times 16$ rectangle. What maximum number of knights can this rectangle contain?

Two points $P$ and $Q$ lying on side $BC$ of triangle $ABC$ and their distance from the midpoint of $BC$ are equal.The perpendiculars from $P$ and $Q$ to $BC$ intersect $AC$ and $AB$ at $E$ and $F$,respectively.$M$ is point of intersection $PF$ and $EQ$.If $H_1$ and $H_2$ be the orthocenters of triangles $BFP$ and $CEQ$, respectively, prove that $ AM\perp H_1H_2 $. Author:Mehdi E'tesami Fard , Iran

A set $ S$ of points in space satisfies the property that all pairwise distances between points in $ S$ are distinct. Given that all points in $ S$ have integer coordinates $ (x,y,z)$ where $ 1 \leq x,y, z \leq n,$ show that the number of points in $ S$ is less than $ \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).$ [i]Dan Schwarz, Romania[/i]

Four lighthouses are located at points $ A$, $ B$, $ C$, and $ D$. The lighthouse at $ A$ is $ 5$ kilometers from the lighthouse at $ B$, the lighthouse at $ B$ is $ 12$ kilometers from the lighthouse at $ C$, and the lighthouse at $ A$ is $ 13$ kilometers from the lighthouse at $ C$. To an observer at $ A$, the angle determined by the lights at $ B$ and $ D$ and the angle determined by the lights at $ C$ and $ D$ are equal. To an observer at $ C$, the angle determined by the lights at $ A$ and $ B$ and the angle determined by the lights at $ D$ and $ B$ are equal. The number of kilometers from $ A$ to $ D$ is given by $ \displaystyle\frac{p\sqrt{r}}{q}$, where $ p$, $ q$, and $ r$ are relatively prime positive integers, and $ r$ is not divisible by the square of any prime. Find $ p\plus{}q\plus{}r$,

There are four basketball players $A,B,C,D$. Initially the ball is with $A$. The ball is always passed from one person to a different person. In how many ways can the ball come back to $A$ after $\textbf{seven}$ moves? (for example $A\rightarrow C\rightarrow B\rightarrow D\rightarrow A\rightarrow B\rightarrow C\rightarrow A$, or $A\rightarrow D\rightarrow A\rightarrow D\rightarrow C\rightarrow A\rightarrow B\rightarrow A)$.

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$

On the coordinate plane, draw a set of points whose coordinates $(x, y)$ satisfy the inequality $$2 arc \cos x \ge arc \cos y$$

Let $S$ be a set of in $3-$ space such that each of the points in $S$ has integer coordinates $(x,y,z)$ with $1 \le x,y,z \le n $. Suppose the pairwise distances between these points are all distinct. Prove that $$|S| \le min \{(n+2)\sqrt{\frac{n}{3}},n\sqrt{6} \}$$

Rectangle $ ABCD$ and a semicircle with diameter $ AB$ are coplanar and have nonoverlapping interiors. Let $ \mathcal{R}$ denote the region enclosed by the semicircle and the rectangle. Line $ \ell$ meets the semicircle, segment $ AB$, and segment $ CD$ at distinct points $ N$, $ U$, and $ T$, respectively. Line $ \ell$ divides region $ \mathcal{R}$ into two regions with areas in the ratio $ 1: 2$. Suppose that $ AU \equal{} 84$, $ AN \equal{} 126$, and $ UB \equal{} 168$. Then $ DA$ can be represented as $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

Define a [i]growing spiral[/i] in the plane to be a sequence of points with integer coordinates $P_0=(0,0),P_1,\dots,P_n$ such that $n\ge 2$ and: • The directed line segments $P_0P_1,P_1P_2,\dots,P_{n-1}P_n$ are in successive coordinate directions east (for $P_0P_1$), north, west, south, east, etc. • The lengths of these line segments are positive and strictly increasing. \[\begin{picture}(200,180) \put(20,100){\line(1,0){160}} \put(100,10){\line(0,1){170}} \put(0,97){West} \put(180,97){East} \put(90,0){South} \put(90,180){North} \put(100,100){\circle{1}}\put(100,100){\circle{2}}\put(100,100){\circle{3}} \put(115,100){\circle{1}}\put(115,100){\circle{2}}\put(115,100){\circle{3}} \put(115,130){\circle{1}}\put(115,130){\circle{2}}\put(115,130){\circle{3}} \put(40,130){\circle{1}}\put(40,130){\circle{2}}\put(40,130){\circle{3}} \put(40,20){\circle{1}}\put(40,20){\circle{2}}\put(40,20){\circle{3}} \put(170,20){\circle{1}}\put(170,20){\circle{2}}\put(170,20){\circle{3}} \multiput(100,99.5)(0,.5){3}{\line(1,0){15}} \multiput(114.5,100)(.5,0){3}{\line(0,1){30}} \multiput(40,129.5)(0,.5){3}{\line(1,0){75}} \multiput(39.5,20)(.5,0){3}{\line(0,1){110}} \multiput(40,19.5)(0,.5){3}{\line(1,0){130}} \put(102,90){P0} \put(117,90){P1} \put(117,132){P2} \put(28,132){P3} \put(30,10){P4} \put(172,10){P5} \end{picture}\] How many of the points $(x,y)$ with integer coordinates $0\le x\le 2011,0\le y\le 2011$ [i]cannot[/i] be the last point, $P_n,$ of any growing spiral?

The function $f(x)$ is defined for all $x\ge 0$ and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope $m$, it intersects the graph $y=f(x)$ at precisely one point, which is $\frac{1}{\sqrt{m}}$ units from the origin. Let $a$ be the unique real number for which $f$ takes on its maximum value at $x=a$ (you may assume that such an $a$ exists). Find $\int_{0}^{a}f(x) \, dx$.

Let $ ABCD$ be a rectangle with center $ O$, $ AB\neq BC$. The perpendicular from $ O$ to $ BD$ cuts the lines $ AB$ and $ BC$ in $ E$ and $ F$ respectively. Let $ M,N$ be the midpoints of the segments $ CD,AD$ respectively. Prove that $ FM \perp EN$.