A triangle with integral sides has perimeter $8$. The area of the triangle is $\textbf{(A) }2\sqrt{2}\qquad \textbf{(B) }\dfrac{16}{9}\sqrt{3}\qquad \textbf{(C) }2\sqrt{3}\qquad \textbf{(D) }4\qquad \textbf{(E) }4\sqrt{2}$

The diagram below contains eight line segments, all the same length. Each of the angles formed by the intersections of two segments is either a right angle or a $45$ degree angle. If the outside square has area $1000$, find the largest integer less than or equal to the area of the inside square. [asy] size(130); real r = sqrt(2)/2; defaultpen(linewidth(0.8)); draw(unitsquare^^(r,0)--(0,r)^^(1-r,0)--(1,r)^^(r,1)--(0,1-r)^^(1-r,1)--(1,1-r)); [/asy]

let $ ABC $ an equilateral triangle with circum circle $ w $ let $ P $ a point on arc $ BC $ ( point $ A $ is on the other side ) pass a tangent line $ d $ through point $ P $ such that $ P \cap AB = F $ and $ AC \cap d = L $ let $ O $ the center of the circle $ w $ prove that $ \angle LOF > 90^{0} $

Let $\Omega $ be the circumcincle of the quadrilateral $ABCD $ , and $E $ the intersection point of the diagonals $AC $ and $BD $ . A line passing through $E $ intersects $AB $ and $BC$ in points $P $ and $Q $ . A circle ,that is passing through point $D $ , is tangent to the line $PQ $ in point $E $ and intersects $\Omega$ in point $R $ , different from $D $ . Prove that the points $B,P,Q,$ and $R $ are concyclic .

In a circle, consider two chords $[AB], [CD]$ that intersect at $E$, lines $AC$ and $BD$ meet at $F$. Let $G$ be the projection of $E$ onto $AC$. We denote by $M,N,K$ the midpoints of the segment lines $[EF] ,[EA]$ and $[AD]$, respectively. Prove that the points $M, N,K,G$ are concyclic.

Let $H$ be the orthocenter of $\triangle ABC$. The points $A_{1} \not= A$, $B_{1} \not= B$ and $C_{1} \not= C$ lie, respectively, on the circumcircles of $\triangle BCH$, $\triangle CAH$ and $\triangle ABH$ and satisfy $A_{1}H=B_{1}H=C_{1}H$. Denote by $H_{1}$, $H_{2}$ and $H_{3}$ the orthocenters of $\triangle A_{1}BC$, $\triangle B_{1}CA$ and $\triangle C_{1}AB$, respectively. Prove that $\triangle A_{1}B_{1}C_{1}$ and $\triangle H_{1}H_{2}H_{3}$ have the same orthocenter.

4-Point $ P$ is taken on the segment $ BC$ of the scalene triangle $ ABC$ such that $ AP \neq AB,AP \neq AC$.$ l_1,l_2$ are the incenters of triangles $ ABP,ACP$ respectively. circles $ W_1,W_2$ are drawn centered at $ l_1,l_2$ and with radius equal to $ l_1P,l_2P$,respectively. $ W_1,W_2$ intersects at $ P$ and $ Q$. $ W_1$ intersects $ AB$ and $ BC$ at $ Y_1( \mbox{the intersection closer to B})$ and $ X_1$,respectively. $ W_2$ intersects $ AC$ and $ BC$ at $ Y_2(\mbox{the intersection closer to C})$ and $ X_2$,respectively.PROVE THE CONCURRENCY OF $ PQ,X_1Y_1,X_2Y_2$.

Triangle $ABC$ is a right triangle with $AC=7,$ $BC=24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD=BD=15.$ Given that the area of triangle $CDM$ may be expressed as $\frac{m\sqrt{n}}{p},$ where $m,$ $n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p.$

In an aquarium there are nine small fish. The aquarium is cube shaped with a side length of two meters and is completely filled with water. Show that it is always possible to find two small fish with a distance of less than $\sqrt3$ meters.

Let $k$ be a circle with diameter $AB$. Let $C$ be a point on the straight line $AB$, so that $B$ between $A$ and $C$ lies. Let $T$ be a point on $k$ such that $CT$ is a tangent to $k$. Let $l$ be the parallel to $CT$ through $A$ and $D$ the intersection of $l$ and the perpendicular to $AB$ through $T$. Show that the line $DB$ bisects segment $CT$.

A line is called $good$ if it bisects perimeter and area of a figure at the same time.Prove that: [i]a)[/i] all of the good lines in a triangle concur. [i]b)[/i] all of the good lines in a regular polygon concur too.

Evaluate $\lim_{n\to \infty} cos\frac{x}{2}cos\frac{x}{2^2} cos\frac{x}{2^3}...cos\frac{x}{2^n}$

Let $ABC$ be a triangle. The incircle of $ABC$ touches $BC,AC,AB$ at $D,E,F$, respectively. Each pair of the incircles of triangles $AEF, BDF,CED$ has two pair of common external tangents, one of them being one of the sides of $ABC$. Show that the other three tangents divide triangle $DEF$ into three triangles and three parallelograms.

The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear. [i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]

A triangle has side lengths $a, b, c$. Prove that $$a(b -c)^2 + b(c - a)^2 + c(a - b)^2 + 4abc > a^3 + b^3 + c^3$$

In prism $JHOPKINS$, quadrilaterals $JHOP$ and $KINS$ are parallel and congruent bases that are kites, where $JH = JP = KI = KS$ and $OH = OP = NI = NS$; the longer two sides of each kite have length $\tfrac{4 + \sqrt{5}}{2}$, and the shorter two sides of each kite have length $\tfrac{5 + \sqrt{5}}{4}$. Assume that $\overline{JK}$, $\overline{HI}$, $\overline{ON}$, and $\overline{PS}$ are congruent edges of $JHOPKINS$ perpendicular to the planes containing $JHOP$ and $KINS$. Vertex $J$ is part of a regular pentagon $JAZZ'Y$ that can be inscribed in prism $JHOPKINS$ such that $A \in \overline{HI}$, $Z \in \overline{NI}$, $Z' \in \overline{NS}$, $Y \in \overline{PS}$, $AI = YS$, and $ZI = Z'S$. Compute the height of $JHOPKINS$ (that is, the distance between the bases).

Let $\omega$ be the circumcircle of the acute nonisosceles triangle $\Delta ABC$. Point $P$ lies on the altitude from $A$. Let $E$ and $F$ be the feet of the altitudes from P to $CA$, $BA$ respectively. Circumcircle of triangle $\Delta AEF$ intersects the circle $\omega$ in $G$, different from $A$. Prove that the lines $GP$, $BE$ and $CF$ are concurrent.

Two circles are tangent externaly at a point $B$. A line tangent to one of the circles at a point $A$ intersects the other circle at points $C$ and $D$. Show that $A$ is equidistant to the lines $BC$ and $BD$.

Points $ K$, $ L$, $ M$, and $ N$ lie in the plane of the square $ ABCD$ so that $ AKB$, $ BLC$, $ CMD$, and $ DNA$ are equilateral triangles. If $ ABCD$ has an area of $ 16$, find the area of $ KLMN$. [asy]unitsize(2cm); defaultpen(fontsize(8)+linewidth(0.8)); pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5); pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0); draw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$D$",D,NE); label("$K$",K,NNW); label("$L$",L,E); label("$M$",M,S); label("$N$",N,W);[/asy] $ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 16 \plus{} 16\sqrt {3} \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 32 \plus{} 16\sqrt {3} \qquad \textbf{(E)}\ 64$

On the bisector of angle $A$ of triangle $ABC$, points $D$ and $F$ are taken inside the triangle so that $\angle DBC = \angle FBA$. Prove that: a) $\angle DCB = \angle FCA$ b) a circle passing through $B$ and $F$ and tangent to the segment $BC$ is tangle to the circumscribed circle of the triangle $ABC$.

Six regular octagons each with sides of length 2 are placed in a two-by-three array and inscribed in a square as shown. The area of the square can be written in the form $m+n\sqrt{2}$ where $m$ and $n$ are positive integers. Find $m+n$. [asy] size(175); pair A=dir(-45/2); real h=A.x; path P=A; for(int k=1;k<8;++k) P=P--rotate(45*k)*A; P=P--cycle; for(real x=-2*h;x<3*h;x+=2*h) for(real y:new real[]{-h,h} ) draw (shift((x,y))*P); pair C=dir(45/2)+(2*h,h), X=(C.x+C.y,0), Y=(0,C.x+C.y); draw(X--Y--(-X)--(-Y)--cycle);[/asy]

A parallelogram is inscribed in a quadrilateral, two opposite vertices of which are the midpoints of the opposite sides of the quadrilateral. Determine the area of ​​such a parallelogram if the area of ​​the quadrilateral is equal to $S_o$.

A circle of radius $ 2$ is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded? [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; filldraw(Arc((0,0),4,0,180)--cycle,gray,black); filldraw(Circle((0,2),2),white,black); dot((0,2)); draw((0,2)--((0,2)+2*dir(60))); label("$2$",midpoint((0,2)--((0,2)+2*dir(60))),SE);[/asy]$ \textbf{(A)}\ \frac{1}{2}\qquad \textbf{(B)}\ \frac{\pi}{6}\qquad \textbf{(C)}\ \frac{2}{\pi}\qquad \textbf{(D)}\ \frac{2}{3}\qquad \textbf{(E)}\ \frac{3}{\pi}$

In isosceles triangle $ABC$, $AB=AC$. Extend segment $BC$ through $C$ to $P$. Points $X$ and $Y$ lie on lines $AB$ and $AC$, respectively, such that $PX \parallel AC$ and $PY \parallel AB$. Point $T$ lies on the circumcircle of triangle $ABC$ such that $PT \perp XY$. Prove that $\angle BAT = \angle CAT$.

In $\vartriangle ABC, PQ // BC$ where $P$ and $Q$ are points on $AB$ and $AC$ respectively. The lines $PC$ and $QB$ intersect at $G$. It is also given $EF//BC$, where $G \in EF, E \in AB$ and $F\in AC$ with $PQ = a$ and $EF = b$. Find value of $BC$.