Let be a line $ d: 3x+4y-5=0 $ on a Cartesian plane. We mark with $ \mathcal{L} $ de locus of the planar points $ P $ such that the distance from $ P $ to $ d $ is double the distance from $ P $ to the origin. Let be $ B_{\lambda } ,C_{\lambda }\in\mathcal{L} $ such that $ C_{\lambda } -B_{\lambda } +\lambda =0. $ Find the locus of the middlepoints of the segments $ B_{\lambda }C_{\lambda }, $ if $ \lambda\in\mathbb{R} $ is variable.

Let $ABC$ be a triangle such that $CA \neq CB$, the points $A_{1}$ and $B_{1}$ are tangency points for the ex-circles relative to sides $CB$ and $CA$, respectively, and $I$ the incircle. The line $CI$ intersects the cincumcircle of the triangle $ABC$ in the point $P$. The line that trough $P$ that is perpendicular to $CP$, intersects the line $AB$ in $Q$. Prove that the lines $QI$ and $A_{1}B_{1}$ are parallels.

A line with slope $ 3$ intersects a line with slope $ 5$ at the point $ (10, 15)$. What is the distance between the $ x$-intercepts of these two lines? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 20$

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of $ABCD$, $K$ the intersection of the diagonals, $ L\neq O $ the intersection of the circles circumscribed to $OAC$ and $OBD$, and $G$ the intersection of the diagonals of the quadrilateral whose vertices are the midpoints of the sides of $ABCD$. Prove that $O, K, L$ and $G$ are collinear

Let $C$ be the curve determined by the equation $y = x^3$ in the rectangular coordinate system. Let $t$ be the tangent to $C$ at a point $P$ of $C$; t intersects $C$ at another point $Q$. Find the equation of the set $L$ of the midpoints $M$ of $PQ$ as $P$ describes $C$. Is the correspondence associating $P$ and $M$ a bijection of $C$ on $L$ ? Find a similarity that transforms $C$ into $L.$

A regular triangular prism has the altitude $h,$ and the two bases of the prism are equilateral triangles with side length $a.$ Dream-holes are made in the centers of both bases, and the three lateral faces are mirrors. Assume that a ray of light, entering the prism through the dream-hole in the upper base, then being reflected once by any of the three mirrors, quits the prism through the dream-hole in the lower base. Find the angle between the upper base and the light ray at the moment when the light ray entered the prism, and the length of the way of the light ray in the interior of the prism.

A six dimensional "cube" (a $6$-cube) has $64$ vertices at the points $(\pm 3,\pm 3,\pm 3,\pm 3,\pm 3,\pm 3).$ This $6$-cube has $192\text{ 1-D}$ edges and $240\text{ 2-D}$ edges. This $6$-cube gets cut into $6^6=46656$ smaller congruent "unit" $6$-cubes that are kept together in the tightly packaged form of the original $6$-cube so that the $46656$ smaller $6$-cubes share 2-D square faces with neighbors ($\textit{one}$ 2-D square face shared by $\textit{several}$ unit $6$-cube neighbors). How many 2-D squares are faces of one or more of the unit $6$-cubes?

Take a point $P\left(\frac 12,\ \frac 14\right)$ on the coordinate plane. Let two points $Q(\alpha ,\ \alpha ^ 2),\ R(\beta ,\ \beta ^2)$ move in such a way that 3 points $P,\ Q,\ R$ form an isosceles triangle with the base $QR$, find the locus of the barycenter $G(X,\ Y)$ of $\triangle{PQR}$. [i]2011 Tokyo University entrance exam[/i]

Let $\Lambda$ be the set of points in the plane whose coordinates are integers and let $F$ be the collection of all functions from $\Lambda$ to $\{1,-1\}.$ We call a function $f$ in $F$ [i]perfect[/i] if every function $g$ in $F$ that differs from $f$ at finitely many points satisfies the condition \[ \sum_{0<d(P,Q)<2010} \frac{f(P)f(Q)-g(P)g(Q)}{d(P,Q)} \geq 0 \] where $d(P,Q)$ denotes the distance between $P$ and $Q.$ Show that there exist infinitely many [i]perfect[/i] functions that are not translates of each other.

A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.

You plot weight $(y)$ against height $(x)$ for three of your friends and obtain the points $(x_{1},y_{1})$, $(x_{2},y_{2})$, $(x_{3},y_{3})$. If \[x_{1} < x_{2} < x_{3}\quad\text{ and }\quad x_{3} - x_{2} = x_{2} - x_{1},\] which of the following is necessarily the slope of the line which best fits the data? "Best fits" means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line. $ \textbf{(A)}\ \frac{y_{3} - y_{1}}{x_{3} - x_{1}}\qquad\textbf{(B)}\ \frac{(y_{2} - y_{1}) - (y_{3} - y_{2})}{x_{3} - x_{1}}\qquad\textbf{(C)}\ \frac{2y_{3} - y_{1} - y_{2}}{2x_{3} - x_{1} - x_{2}}\qquad\textbf{(D)}\ \frac{y_{2} - y_{1}}{x_{2} - x_{1}} + \frac{y_{3} - y_{2}}{x_{3} - x_{2}}\qquad\textbf{(E)}\ \text{none of these} $

In Cartesian coordinates, two areas $M,N$ are defined below: $M:y\geq0,y\leq x,y\leq 2-x$; $N:t\leq x\leq t+1$. $t$ is a real number that $t\in[0,1]$. Then the area of $M\cap N$ is $\text{(A)}-t^2+t+\frac{1}{2}\qquad\text{(B)}-2t^2+2t\qquad\text{(C)}1-2t^2\qquad\text{(D)}\frac{1}{2}(t-2)^2$

A triangle is formed by joining three points whose coordinates are integers. If the $ x$-coordinate and the $ y$-coordinate each have a value of $ 1$, then the area of the triangle, in square units: $ \textbf{(A)}\ \text{must be an integer}\qquad \textbf{(B)}\ \text{may be irrational}\qquad \textbf{(C)}\ \text{must be irrational}\qquad \textbf{(D)}\ \text{must be rational}\qquad \\ \textbf{(E)}\ \text{will be an integer only if the triangle is equilateral.}$

The diagram below shows a $ 4\times4$ rectangular array of points, each of which is $ 1$ unit away from its nearest neighbors. [asy]unitsize(0.25inch); defaultpen(linewidth(0.7)); int i, j; for(i = 0; i < 4; ++i) for(j = 0; j < 4; ++j) dot(((real)i, (real)j));[/asy]Define a [i]growing path[/i] to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let $ m$ be the maximum possible number of points in a growing path, and let $ r$ be the number of growing paths consisting of exactly $ m$ points. Find $ mr$.

A circle intersects the $y$-axis at two points $(0, a)$ and $(0, b)$ and is tangent to the line $x+100y = 100$ at $(100, 0)$. Compute the sum of all possible values of $ab - a - b$.

Let $ f(x)\equal{}x^2\plus{}6x\plus{}1$, and let $ R$ denote the set of points $ (x,y)$ in the coordinate plane such that \[ f(x)\plus{}f(y)\le0\text{ and }f(x)\minus{}f(y)\le0 \]The area of $ R$ is closest to $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 22 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 25$

Complex numbers $a$, $b$ and $c$ are the zeros of a polynomial $P(z) = z^3+qz+r$, and $|a|^2+|b|^2+|c|^2=250$. The points corresponding to $a$, $b$, and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h$. Find $h^2$.

A person moves in the $x-y$ plane moving along points with integer co-ordinates $x$ and $y$ only. When she is at a point $(x,y)$, she takes a step based on the following rules: (a) if $x+y$ is even she moves to either $(x+1,y)$ or $(x+1,y+1)$; (b) if $x+y$ is odd she moves to either $(x,y+1)$ or $(x+1,y+1)$. How many distinct paths can she take to go from $(0,0)$ to $(8,8)$ given that she took exactly three steps to the right $((x,y)$ to $(x+1,y))$?

Consider a convex pentagon circumscribed about a circle. We name the lines that connect vertices of the pentagon with the opposite points of tangency with the circle [i]gergonnians[/i]. (a) Prove that if four gergonnians are conncurrent, the all five of them are concurrent. (b) Prove that if there is a triple of gergonnians that are concurrent, then there is another triple of gergonnians that are concurrent.

Pyramid $EARLY$ is placed in $(x,y,z)$ coordinates so that $E=(10,10,0),A=(10,-10,0)$, $R=(-10,-10,0)$, $L=(-10,10,0)$, and $Y=(0,0,10)$. Tunnels are drilled through the pyramid in such a way that one can move from $(x,y,z)$ to any of the $9$ points $(x,y,z-1)$, $(x\pm 1,y,z-1)$, $(x,y\pm 1, z-1)$, $(x\pm 1, y\pm 1, z-1)$. Sean starts at $Y$ and moves randomly down to the base of the pyramid, choosing each of the possible paths with probability $\dfrac{1}{9}$. What is the probability that he ends up at the point $(8,9,0)$?

A regular triangular prism has the altitude $h,$ and the two bases of the prism are equilateral triangles with side length $a.$ Dream-holes are made in the centers of both bases, and the three lateral faces are mirrors. Assume that a ray of light, entering the prism through the dream-hole in the upper base, then being reflected once by any of the three mirrors, quits the prism through the dream-hole in the lower base. Find the angle between the upper base and the light ray at the moment when the light ray entered the prism, and the length of the way of the light ray in the interior of the prism.

Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\tfrac{12}5\sqrt2$. What is the volume of the tetrahedron? $\textbf{(A) }3\sqrt2\qquad\textbf{(B) }2\sqrt5\qquad\textbf{(C) }\dfrac{24}5\qquad\textbf{(D) }3\sqrt3\qquad\textbf{(E) }\dfrac{24}5\sqrt2$

If $m > 0$ and the points $(m,3)$ and $(1,m)$ lie on a line with slope $m$, then $m = $ $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \sqrt{2}\qquad\textbf{(C)}\ \sqrt{3}\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \sqrt{5} $

Where is AMC 10a No.19? Thanks

We match sets $ M$ of points in the coordinate plane to sets $ M*$ according to the rule that $ (x*,y*) \in M*$ if and only if $ x \cdot x* \plus{} y \cdot y* \leq 1$ whenever $ (x,y) \in M.$ Find all triangles $ Q$ such that $ Q*$ is the reflection of $ Q$ in the origin.