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

The graph of the line $ y \equal{} mx \plus{} b$ is shown. Which of the following is true? [asy]import math; unitsize(8mm); defaultpen(linewidth(1pt)+fontsize(6pt)); dashed=linetype("4 4")+linewidth(.8pt); draw((-2,-2.5)--(-2,2.5)--(2.5,2.5)--(2.5,-2.5)--cycle,white); label("$-1$",(-1,0),SW); label("$1$",(1,0),SW); label("$2$",(2,0),SW); label("$1$",(0,1),NE); label("$2$",(0,2),NE); label("$-1$",(0,-1),SW); label("$-2$",(0,-2),SW); drawline((0,0),(1,0)); drawline((0,0),(0,1)); drawline((0,0.8),(1.8,0)); drawline((1,0),(1,1),dashed); drawline((2,0),(2,1),dashed); drawline((-1,0),(-1,1),dashed); drawline((0,1),(1,1),dashed); drawline((0,2),(1,2),dashed); drawline((0,-1),(1,-1),dashed); drawline((0,-2),(1,-2),dashed);[/asy] $ \textbf{(A)}\ mb < \minus{} 1 \qquad \textbf{(B)}\ \minus{} 1 < mb < 0 \qquad \textbf{(C)}\ mb \equal{} 0$ $ \textbf{(D)}\ 0 < mb < 1\qquad \textbf{(E)}\ mb > 1$

Let $ABCD$ be a square, and let $E$ and $F$ be points in $AB$ and $BC$ respectively such that $BE=BF$. In the triangle $EBC$, let N be the foot of the altitude relative to $EC$. Let $G$ be the intersection between $AD$ and the extension of the previously mentioned altitude. $FG$ and $EC$ intersect at point $P$, and the lines $NF$ and $DC$ intersect at point $T$. Prove that the line $DP$ is perpendicular to the line $BT$.

A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Consider the $ 12$-sided polygon $ ABCDEFGHIJKL$, as shown. Each of its sides has length $ 4$, and each two consecutive sides form a right angle. Suppose that $ \overline{AG}$ and $ \overline{CH}$ meet at $ M$. What is the area of quadrilateral $ ABCM$? [asy]unitsize(13mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(1,3), B=(2,3), C=(2,2), D=(3,2), Ep=(3,1), F=(2,1), G=(2,0), H=(1,0), I=(1,1), J=(0,1), K=(0,2), L=(1,2); pair M=intersectionpoints(A--G,H--C)[0]; draw(A--B--C--D--Ep--F--G--H--I--J--K--L--cycle); draw(A--G); draw(H--C); dot(M); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,NE); label("$D$",D,NE); label("$E$",Ep,SE); label("$F$",F,SE); label("$G$",G,SE); label("$H$",H,SW); label("$I$",I,SW); label("$J$",J,SW); label("$K$",K,NW); label("$L$",L,NW); label("$M$",M,W);[/asy]$ \textbf{(A)}\ \frac {44}{3}\qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ \frac {88}{5}\qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ \frac {62}{3}$

In the usual coordinate system $ xOy$ a line $ d$ intersect the circles $ C_1:$ $ (x\plus{}1)^2\plus{}y^2\equal{}1$ and $ C_2:$ $ (x\minus{}2)^2\plus{}y^2\equal{}4$ in the points $ A,B,C$ and $ D$ (in this order). It is known that $ A\left(\minus{}\frac32,\frac{\sqrt3}2\right)$ and $ \angle{BOC}\equal{}60^{\circ}$. All the $ Oy$ coordinates of these $ 4$ points are positive. Find the slope of $ d$.

[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 set $\mathbb Q^2$ of points in $\mathbb R^2$, both of whose coordinates are rational. [b](a)[/b] Prove that the union of segments with vertices from $\mathbb Q^2$ is the entire set $\mathbb R^2$. [b](b)[/b] Is the convex hull of $\mathbb Q^2$ (i.e., the smallest convex set in $\mathbb R^2$ that contains $\mathbb Q^2$) equal to $\mathbb R^2$ ?

For which real numbers $c$ is there a straight line that intersects the curve \[ y = x^4 + 9x^3 + cx^2 + 9x + 4\] in four distinct points?

Find the number of integers $n$ such that \[1+\left\lfloor\dfrac{100n}{101}\right\rfloor=\left\lceil\dfrac{99n}{100}\right\rceil.\]

A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figures of the third kind is even. [img]http://up.iranblog.com/Files7/dda310bab8b6455f90ce.jpg[/img]

$56$ lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly $594$ points, what is the maximum number of them that could have the same slope?

Draw a tangent with positive slope to a parabola $y=x^2+1$. Find the $x$-coordinate such that the area of the figure bounded by the parabola, the tangent and the coordinate axisis is $\frac{11}{3}.$

Find polynomials $f(x)$, $g(x)$, and $h(x)$, if they exist, such that for all $x$, \[|f(x)|-|g(x)|+h(x)=\begin{cases}-1 & \text{if }x<-1\\3x+2 &\text{if }-1\leq x\leq 0\\-2x+2 & \text{if }x>0.\end{cases}\]

Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$. [i]Proposed by Akshaj[/i]

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 $ n$ be a positive integer. Consider a rectangle $ (90n\plus{}1)\times(90n\plus{}5)$ consisting of unit squares. Let $ S$ be the set of the vertices of these squares. Prove that the number of distinct lines passing through at least two points of $ S$ is divisible by $ 4$.

In square $ABCD$ (labeled clockwise), let $P$ be any point on $BC$ and construct square $APRS$ (labeled clockwise). Prove that line $CR$ is tangent to the circumcircle of triangle $ABC$.

Let $m$ and $n$ be positive integers. A sequence of points $(A_0,A_1,\ldots,A_n)$ on the Cartesian plane is called [i]interesting[/i] if $A_i$ are all lattice points, the slopes of $OA_0,OA_1,\cdots,OA_n$ are strictly increasing ($O$ is the origin) and the area of triangle $OA_iA_{i+1}$ is equal to $\frac{1}{2}$ for $i=0,1,\ldots,n-1$. Let $(B_0,B_1,\cdots,B_n)$ be a sequence of points. We may insert a point $B$ between $B_i$ and $B_{i+1}$ if $\overrightarrow{OB}=\overrightarrow{OB_i}+\overrightarrow{OB_{i+1}}$, and the resulting sequence $(B_0,B_1,\ldots,B_i,B,B_{i+1},\ldots,B_n)$ is called an [i]extension[/i] of the original sequence. Given two [i]interesting[/i] sequences $(C_0,C_1,\ldots,C_n)$ and $(D_0,D_1,\ldots,D_m)$, prove that if $C_0=D_0$ and $C_n=D_m$, then we may perform finitely many [i]extensions[/i] on each sequence until the resulting two sequences become identical.

If $x$ and $y$ are real numbers with $(x+y)^4=x-y$, what is the maximum possible value of $y$?

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

In triangle $ABC$, $D$, $E$ and $F$ are the points of tangency of incircle with the center of $I$ to $BC$, $CA$ and $AB$ respectively. Let $M$ be the foot of the perpendicular from $D$ to $EF$. $P$ is on $DM$ such that $DP = MP$. If $H$ is the orthocenter of $BIC$, prove that $PH$ bisects $ EF$.

Let $ABC$ be a triangle with sides $AB = 6$, $BC = 10$, and $CA = 8$. Let $M$ and $N$ be the midpoints of $BA$ and $BC$, respectively. Choose the point $Y$ on ray $CM$ so that the circumcircle of triangle $AMY$ is tangent to $AN$. Find the area of triangle $NAY$.

Consider the parallelogram with vertices $(10,45),$ $(10,114),$ $(28,153),$ and $(28,84).$ A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

A block of mass $m$ on a frictionless inclined plane of angle $\theta$ is connected by a cord over a small frictionless, massless pulley to a second block of mass $M$ hanging vertically, as shown. If $M=1.5m$, and the acceleration of the system is $\frac{g}{3}$, where $g$ is the acceleration of gravity, what is $\theta$, in degrees, rounded to the nearest integer? [asy]size(12cm); pen p=linewidth(1), dark_grey=gray(0.25), ll_grey=gray(0.90), light_grey=gray(0.75); pair B = (-1,-1); pair C = (-1,-7); pair A = (-13,-7); path inclined_plane = A--B--C--cycle; draw(inclined_plane, p); real r = 1; // for marking angles draw(arc(A, r, 0, degrees(B-A))); // mark angle label("$\theta$", A + r/1.337*(dir(C-A)+dir(B-A)), (0,0), fontsize(16pt)); // label angle as theta draw((C+(-r/2,0))--(C+(-r/2,r/2))--(C+(0,r/2))); // draw right angle real h = 1.2; // height of box real w = 1.9; // width of box path box = (0,0)--(0,h)--(w,h)--(w,0)--cycle; // the box // box on slope with label picture box_on_slope; filldraw(box_on_slope, box, light_grey, black); label(box_on_slope, "$m$", (w/2,h/2)); pair V = A + rotate(90) * (h/2 * dir(B-A)); // point with distance l/2 from AB pair T1 = dir(125); // point of tangency with pulley pair X1 = intersectionpoint(T1--(T1 - rotate(-90)*(2013*dir(T1))), V--(V+B-A)); // construct midpoint of right side of box draw(T1--X1); // string add(shift(X1-(w,h/2))*rotate(degrees(B-A), (w,h/2)) * box_on_slope); // picture for the hanging box picture hanging_box; filldraw(hanging_box, box, light_grey, black); label(hanging_box, "$M$", (w/2,h/2)); pair T2 = (1,0); pair X2 = (1,-3); draw(T2--X2); // string add(shift(X2-(w/2,h)) * hanging_box); // Draws the actual pulley filldraw(unitcircle, grey, p); // outer boundary of pulley wheel filldraw(scale(0.4)*unitcircle, light_grey, p); // inner boundary of pulley wheel path pulley_body=arc((0,0),0.3,-40,130)--arc((-1,-1),0.5,130,320)--cycle; // defines "arm" of pulley filldraw(pulley_body, ll_grey, dark_grey+p); // draws the arm filldraw(scale(0.18)*unitcircle, ll_grey, dark_grey+p); // inner circle of pulley[/asy][i](Proposed by Ahaan Rungta)[/i]