Prove that on the coordinate plane it is impossible to draw a closed broken line such that [i](i)[/i] the coordinates of each vertex are rational; [i](ii)[/i] the length each of its edges is 1; [i](iii)[/i] the line has an odd number of vertices.

Consider parabolas $C_a: y=-2x^2+4ax-2a^2+a+1$ and $C: y=x^2-2x$ in the coordinate plane. When $C_a$ and $C$ have two intersection points, find the maximum area enclosed by these parabolas.

Suppose $\triangle ABC$ has circumcircle $\Gamma$, circumcentre $O$ and orthocentre $H$. Parallel lines $\alpha, \beta, \gamma$ are drawn through the vertices $A, B, C$, respectively. Let $\alpha ', \beta ', \gamma '$ be the reflections of $\alpha, \beta, \gamma$ in the sides $BC, CA, AB$, respectively. $(a)$ Show that $\alpha ', \beta ', \gamma '$ are concurrent if and only if $\alpha, \beta, \gamma$ are parallel to the Euler line $OH$. $(b)$ Suppose that $\alpha ', \beta ' , \gamma '$ are concurrent at the point $P$ . Show that $\Gamma$ bisects $OP$ .

I couldn't solve this problem and the only solution I was able to find was very unnatural (it was an official solution, I think) and I couldn't be satisfied with it, so I ask you if you can find some different solutions. The problem is really great one! If $M$ is the centroid of a triangle $ABC$, prove that the following inequality holds: \[\sin\angle CAM+\sin\angle CBM\leq\frac{2}{\sqrt3}.\] The equality occurs in a very strange case, I don't remember it.

Let $n$ be a positive integer and $m = \frac{(n+1)(n+2)}{2}$. In coordinate plane, there are $n$ distinct lines $L_1, L_2, \ldots, L_n$ and $m$ distinct points $A_1, A_2, \ldots, A_m$, satisfying the following conditions: [b][i]i)[/i][/b] Any two lines are non-parallel. [b][i]ii)[/i][/b] Any three lines are non-concurrent. [b][i]iii)[/i][/b] Only $A_1$ does not lies on any line $L_k$, and there are exactly $k + 1$ points $A_j$'s that lie on line $L_k$ $(k = 1, 2, \ldots, n).$ Prove that there exist a unique polynomial $p(x, y)$ with degree $n$ satisfying $p(A_1) = 1$ and $p(A_j) = 0$ for $j = 2, 3, \ldots, m.$

For $135^\circ < x < 180^\circ$, points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$? $ \textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3} $

In $\triangle ABC$ we have $BC>CA>AB$. The nine point circle is tangent to the incircle, $A$-excircle, $B$-excircle and $C$-excircle at the points $T,T_A,T_B,T_C$ respectively. Prove that the segments $TT_B$ and lines $T_AT_C$ intersect each other.

Given a point $(a,b)$ with $0<b<a$, determine the minimum perimeter of a triangle with one vertex at $(a,b)$, one on the $x$-axis, and one on the line $y=x$. You may assume that a triangle of minimum perimeter exists.

The vertices of triangle $ PQR$ have coordinates as follows: $ P(0,a),\,Q(b,0),\,R(c,d),$ where $ a,\,b,\,c$ and $ d$ are positive. The origin and point $ R$ lie on opposite sides of $ PQ$. The area of triangle $ PQR$ may be found from the expression: $ \textbf{(A)}\ \frac{ab \plus{} ac \plus{} bc \plus{} cd}{2} \qquad \textbf{(B)}\ \frac{ac \plus{} bd \minus{} ab}{2}\qquad \textbf{(C)}\ \frac{ab \minus{} ac \minus{} bd}{2}\qquad \textbf{(D)}\ \frac{ac \plus{} bd \plus{} ab}{2}\qquad \textbf{(E)}\ \frac{ac \plus{} bd \minus{} ab \minus{} cd}{2}$

Assume there are $ n\ge3$ points in the plane, Prove that there exist three points $ A,B,C$ satisfying $ 1\le\frac{AB}{AC}\le\frac{n\plus{}1}{n\minus{}1}.$

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A [i]tromino[/i] is an $L$-shape formed by three connected unit squares. $(a)$ For which values of $n$ is it possible to cover all the black squares with non-overlapping trominoes lying entirely on the chessboard? $(b)$ When it is possible, find the minimum number of trominoes needed.

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?

Consider a parameterized curve $C: x=e^{-t}\cos t,\ y=e^{-t}\sin t\left (0\leq t\leq \frac{\pi}{2}\right).$ (1) Find the length $L$ of $C$. (2) Find the area $S$ of the region enclosed by the $x,\ y$ axis and $C$. Please solve the problem without using the formula of area for polar coordinate for Japanese High School Students who don't study it in High School. [i]1997 Kyoto University entrance exam/Science[/i]

Let $M$ be a point which has coordinates $(p\times 1994,7p\times 1994)$ in the Cartesian plane ($p$ is a prime). Find the number of right-triangles satisfying the following conditions: (1) all vertexes of the triangle are lattice points, moreover $M$ is on the right-angled corner of the triangle; (2) the origin ($0,0$) is the incenter of the triangle.

Let $ \mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $ |8 \minus{} x| \plus{} y \le 10$ and $ 3y \minus{} x \ge 15$. When $ \mathcal{R}$ is revolved around the line whose equation is $ 3y \minus{} x \equal{} 15$, the volume of the resulting solid is $ \frac {m\pi}{n\sqrt {p}}$, where $ m$, $ n$, and $ p$ are positive integers, $ m$ and $ n$ are relatively prime, and $ p$ is not divisible by the square of any prime. Find $ m \plus{} n \plus{} p$.

Let $C_1$ and $C_2$ be two concentric circles in the plane with radii $R$ and $3R$ respectively. Show that the orthocenter of any triangle inscribed in circle $C_1$ lies in the interior of circle $C_2$. Conversely, show that every point in the interior of $C_2$ is the orthocenter of some triangle inscribed in $C_1$.

If ellipse $x^2+4(y-a)^2=4$ and parabola $x^2=2y$ have intersections, then the range value of $a$ is________.

A triangle $KLM$ is given in the plane together with a point $A$ lying on the half-line opposite to $KL$. Construct a rectangle $ABCD$ whose vertices $B, C$ and $D$ lie on the lines $KM, KL$ and $LM$, respectively. (We allow the rectangle to be a square.)

[asy] draw(unitsquare);draw((0,0)--(.4,1)^^(0,.6)--(1,.2)); label("D",(0,1),NW);label("E",(.4,1),N);label("C",(1,1),NE); label("P",(0,.6),W);label("M",(.25,.55),E);label("Q",(1,.2),E); label("A",(0,0),SW);label("B",(1,0),SE); //Credit to Zimbalono for the diagram[/asy] Inside square $ABCD$ (See figure) with sides of length $12$ inches, segment $AE$ is drawn where $E$ is the point on $DC$ which is $5$ inches from $D$. The perpendicular bisector of $AE$ is drawn and intersects $AE$, $AD$, and $BC$ at points $M$, $P$, and $Q$ respectively. The ratio of segment $PM$ to $MQ$ is $\textbf{(A) }5:12\qquad\textbf{(B) }5:13\qquad\textbf{(C) }5:19\qquad\textbf{(D) }1:4\qquad \textbf{(E) }5:21$

Consider the parabola consisting of the points $(x, y)$ in the real plane satisfying $$(y + x) = (y - x)^2 + 3(y - x) + 3.$$ Find the minimum possible value of $y$.

Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by　distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$. (1) Find $u(t),\ v(t)$. (2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$. (3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$. [i]2011 Tokyo University entrance exam/Science, Problem 3[/i]

The points $ (2,\minus{}3)$, $ (4,3)$, and $ (5, k/2)$ are on the same straight line. The value(s) of $ k$ is (are): $ \textbf{(A)}\ 12\qquad \textbf{(B)}\ \minus{}12\qquad \textbf{(C)}\ \pm 12\qquad \textbf{(D)}\ {12}\text{ or }{6}\qquad \textbf{(E)}\ {6}\text{ or }{6\frac{2}{3}}$

For any two points $A (x_1 , y_1)$ and $B (x_2, y_2)$, the distance $r (A, B)$ between them is determined by the equality $r(A, B) = | x_1- x_2 | + | y_1 - y_2 |$. Prove that the triangle inequality $r(A, C) + r(C, B) \ge r(A, B)$. holds for the distance introduced in this way . Let $A$ and $B$ be two points of the plane (you can take $A(1, 3)$, $B(3, 7)$). Find the locus of points $C$ for which a) $r(A, C) + r(C, B) = r(A, B)$ b) $r(A, C) = r(C, B).$

Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy\equal{}1$ and both branches of the hyperbola $ xy\equal{}\minus{}1.$ (A set $ S$ in the plane is called [i]convex[/i] if for any two points in $ S$ the line segment connecting them is contained in $ S.$)

Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation.