An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$.

In this diagram a scheme is indicated for associating all the points of segment $ \overline{AB}$ with those of segment $ \overline{A'B'}$, and reciprocally. To described this association scheme analytically, let $ x$ be the distance from a point $ P$ on $ \overline{AB}$ to $ D$ and let $ y$ be the distance from the associated point $ P'$ of $ \overline{A'B'}$ to $ D'$. Then for any pair of associated points, if $ x \equal{} a,\, x \plus{} y$ equals: [asy]defaultpen(linewidth(.8pt)); unitsize(.8cm); pair D= (0,9); pair E = origin; pair A = (3,9); pair P = (3.6,9); pair B = (4,9); pair F = (1,0); pair G = (2.6,0); pair H = (5,0); dot((0,0));dot((1,0));dot((2,0));dot((3,0));dot((4,0));dot((5,0)); dot((0,9));dot((1,9));dot((2,9));dot((3,9));dot((4,9));dot((5,9)); draw((D+(0,0.5))--(0,-0.5)); draw(A--H); draw(P--G); draw(B--F); draw(F--H); draw(A--B); label("$D$",D,NW); label("$D'$",E,NW); label("0",(0,0),SE); label("1",(1,0),SE); label("2",(2,0),SE); label("3",(3,0),SE); label("4",(4,0),SE); label("5",(5,0),SE); label("0",(0,9),SE); label("1",(1,9),SE); label("2",(2,9),SE); label("3",(3,9),SW); label("4",(4,9),SE); label("5",(5,9),SE); label("$B'$",F,NW); label("$P'$",G,S); label("$A'$",H,NE); label("$A$",A,NW); label("$P$",P,N); label("$B$",B,NE);[/asy] $ \textbf{(A)}\ 13a\qquad \textbf{(B)}\ 17a \minus{} 51\qquad \textbf{(C)}\ 17 \minus{} 3a\qquad \textbf{(D)}\ \frac {17 \minus{} 3a}{4}\qquad \textbf{(E)}\ 12a \minus{} 34$

Let $A,B,C$ be fixed points in the plane , and $D$ be a variable point on the circle $ABC$, distinct from $A,B,C$ . Let $I_{A},I_{B},I_{C},I_{D}$ be the Simson lines of $A,B,C,D$ with respect to triangles $BCD,ACD,ABD,ABC$ respectively. Find the locus of the intersection points of the four lines $I_{A},I_{B},I_{C},I_{D}$ when point $D$ varies.

The circular base of a hemisphere of radius $2$ rests on the base of a square pyramid of height $6$. The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid? $ \textbf{(A)}\ 3\sqrt{2} \qquad \textbf{(B)}\ \frac{13}{3} \qquad \textbf{(C)}\ 4\sqrt{2} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ \frac{13}{2} $

Triangle $ABC$ has $AB=21$, $AC=22$, and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Find the angle $\angle A$ of a triangle $ABC$, when we know it's altitudes $BD$ and $CE$ intersect in an interior point $H$ of the triangle and $BH=2HD$ and $CH=HE$.

In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.

An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 4, what is the area of the hexagon? $\textbf{(A)}\hspace{.05in}4 \qquad \textbf{(B)}\hspace{.05in}5 \qquad \textbf{(C)}\hspace{.05in}6 \qquad \textbf{(D)}\hspace{.05in}4\sqrt3 \qquad \textbf{(E)}\hspace{.05in}6\sqrt3 $

From a police station situated on a straight road innite in both directions, a thief has stolen a police car. Its maximal speed equals $90$% of the maximal speed of a police cruiser. When the theft is discovered some time later, a policeman starts to pursue the thief on a cruiser. However, he does not know in which direction along the road the thief has gone, nor does he know how long ago the car has been stolen. Is it possible for the policeman to catch the thief?

Let $\vartriangle ABC$ be an acute triangle, and let $A'$ and $B'$ be the feet of altitudes from $A$ to $BC$ and from $B$ to $CA$, respectively; the altitudes intersect at $H$. If $BH$ is equal to the circumradius of $\vartriangle ABC$, find $\frac{A'B}{AB}$ .

An amusement park has a collection of scale models, with ratio $1 : 20$, of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica to the nearest whole number? $\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad\textbf{(E) }20$

Let $ABCD$ be a quadrilateral with an inscribed circle, centre $O$. Let \[AO = 5, BO =6, CO = 7, DO = 8.\] If $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$, determine $\frac{OM}{ON}$ .

Given triangle $ABC$. Point $M$ is the midpoint of side $BC$, and point $P$ is the projection of $B$ to the perpendicular bisector of segment $AC$. Line $PM$ meets $AB$ in point $Q$. Prove that triangle $QPB$ is isosceles.

A painting $18''\ \text{X}\ 24''$ is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is: $\textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 1:2 \qquad\textbf{(C)}\ 2:3 \qquad\textbf{(D)}\ 3:4 \qquad\textbf{(E)}\ 1:1$

[color=darkblue]Let $ ABCD$ be a trapezoid with $ AB\parallel CD$. Exterior equilateral triangles $ ABE$ and $ CDF$ are constructed. Prove that lines $ AC$, $ BD$ and $ EF$ are concurrent.[/color]