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 $BT$.

How many hours does it take a train traveling at an average rate of $ 40$ mph between stops to travel $ a$ miles it makes $ n$ stops of $ m$ minutes each? $ \textbf{(A)}\ \frac{3a\plus{}2mn}{120} \qquad \textbf{(B)}\ 3a\plus{}2mn \qquad \textbf{(C)}\ \frac{3a\plus{}2mn}{12} \qquad \textbf{(D)}\ \frac{a\plus{}mn}{40} \qquad \textbf{(E)}\ \frac{a\plus{}40mn}{40}$

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$.

The two lines with slopes $2$ and $1/2$ pass through an arbitrary point $T$ on the axis $Oy$ and intersect the hyperbola $y=1/x$ at two points. [b]a)[/b] Prove that these four points lie on a circle. [b]b)[/b] The point $T$ runs through the entire $y$-axis. Find the locus of centers of such circles. [i](I. Gorodnin)[/i]

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$

Line $\ell_1$ has equation $3x-2y=1$ and goes through $A=(-1,-2)$. Line $\ell_2$ has equation $y=1$ and meets line $\ell_1$ at point $B$. Line $\ell_3$ has positive slope, goes through point $A$, and meets $\ell_2$ at point $C$. The area of $\triangle ABC$ is $3$. What is the slope of $\ell_3$? $ \textbf{(A)}\ \frac{2}{3}\qquad\textbf{(B)}\ \frac{3}{4}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{4}{3}\qquad\textbf{(E)}\ \frac{3}{2} $

In the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origin, and a circle with radius $10$ with center on the positive $y$-axis is tangent to the $x$-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect? $\textbf{(A)}\ \dfrac{2}{7} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{2}{\sqrt{29}} \qquad\textbf{(D)}\ \dfrac{1}{\sqrt{29}} \qquad\textbf{(E)}\ \dfrac{2}{5}$

A cubic polynomial $f$ with a positive leading coefficient has three different positive zeros. Show that $f'(a)+ f'(b)+ f'(c) > 0$.

In triangle $ABC$, $AB=\frac{20}{11} AC$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AC$ and $BM$. The ratio of $CP$ to $PA$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

If $ A$ and $ B$ are fixed points on a given circle and $ XY$ is a variable diameter of the same circle, determine the locus of the point of intersection of lines $ AX$ and $ BY$. You may assume that $ AB$ is not a diameter.

A line passes through $ A(1,1)$ and $ B(100,1000)$. How many other points with integer coordinates are on the line and strictly between $ A$ and $ B$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square? $ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27 $

[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$. [b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.

$ 2n\plus{}1$ lines are drawn in the plane, in such a way that every 3 lines define a triangle with no right angles. What is the maximal possible number of acute triangles that can be made in this way?