Aaron takes a square sheet of paper, with one corner labeled $A$. Point $P$ is chosen at random inside of the square and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$. Let $p$ be the probability that the remaining piece is a pentagon. Find the integer nearest to $100p$. [i]Proposed by Aaron Lin[/i]

In triangle $ABC$, angle $A$ is equal to $a$ and angle $B$ is equal to $2a$. A circle with center at point $C$ of radius $CA$ intersects the line containing the bisector of the exterior angle at vertex $B$, at points $M$ and $N$. Find the angles of triangle $MAN$.

[asy] size(150); pair A=(0,0),B=(1,0),C=(0,1),D=(-1,0),E=(0,.5),F=(sqrt(2)/2,.25); draw(circle(A,1)^^D--B); draw(circle(E,.5)^^circle( F ,.25)); label("$A$", D, W); label("$K$", A, S); label("$B$", B, dir(0)); label("$L$", E, N); label("$M$",shift(-.05,.05)*F); //Credit to Klaus-Anton for the diagram[/asy] In the adjoining figure, circle $\mathit{K}$ has diameter $\mathit{AB}$; cirlce $\mathit{L}$ is tangent to circle $\mathit{K}$ and to $\mathit{AB}$ at the center of circle $\mathit{K}$; and circle $\mathit{M}$ tangent to circle $\mathit{K}$, to circle $\mathit{L}$ and $\mathit{AB}$. The ratio of the area of circle $\mathit{K}$ to the area of circle $\mathit{M}$ is $\textbf{(A) }12\qquad\textbf{(B) }14\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad \textbf{(E) }\text{not an integer}$

Let $ABC$ be a triangle and let $D, E$ and $F$ be the midpoints of sides $BC, CA$ and $AB$, respectively. Let $X\ne A$ be the intersection of $AD$ with the circumcircle of $ABC$. Let $\Omega$ be the circle through $D$ and $X$, tangent to the circumcircle of $ABC$. Let $Y$ and $Z$ be the intersections of the tangent to $\Omega$ at $D$ with the perpendicular bisectors of segments $DE$ and $DF$, respectively. Let $P$ be the intersection of $YE$ and $ZF$ and let $G$ be the centroid of $ABC$. Show that the tangents at $B$ and $C$ to the circumcircle of $ABC$ and the line $PG$ are concurrent.

We divided a regular octagon into parallelograms. Prove that there are at least $2$ rectangles between the parallelograms.

Let $n \ge 3$ be an integer. What is the largest possible number of interior angles greater than $180^\circ$ in an $n$-gon in the plane, given that the $n$-gon does not intersect itself and all its sides have the same length?

Three given rectangles cover the sides of a triangle completely and each rectangle has a side parallel to a given line. Show that the rectangles also cover the interior of the triangle.

Given a circle $G$ with center $O$ and radius $r$. Let $AB$ be a fixed diameter of $G$. Let $K$ be a fixed point of segment $AO$. Denote by $t$ the line tangent to at $A$. For any chord $CD$ (other than $AB$) passing through $K$. Let $P$ and $Q$ be the points of intersection of lines $BC$ and $BD$ with $t$. Prove that the product $AP\cdot AQ$ remains costant as the chord $CD$ varies.

Let $\omega$ be the circumcircle of $\triangle{ABC}$. $P$ is an interior point of $\triangle{ABC}$. $A_{1}, B_{1}, C_{1}$ are the intersections of $AP, BP, CP$ respectively and $A_{2}, B_{2}, C_{2}$ are the symmetrical points of $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of side $BC, CA, AB$. Show that the circumcircle of $\triangle{A_{2}B_{2}C_{2}}$ passes through the orthocentre of $\triangle{ABC}$.

Prove that for a given convex polygon of area $A$ and perimeter $P$ there exists a circle of radius $\frac AP$ that is contained in the interior of the polygon.

The diagonals of some quadrilateral with sides $a,b,c,d$ are perpendicular. Prove that the diagonals of any other quadrilateral with sides $a,b,c,d $ also are perpendicular

In quadrilateral $ABCD$, $\angle B = \angle D = 90$ and $AC = BC + DC$. Point $P$ of ray $BD$ is such that $BP = AD$. Prove that line $CP$ is parallel to the bisector of angle $ABD$. [i](Proposed by A.Trigub)[/i]

In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$, let $M$ be the midpoint of $\overline{BC}$. Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$. Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$. Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$, and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$. Prove that $A$, $Y$, and $Z$ are collinear. [i]Tiger Zhang[/i]

Let $ABC$ be a triangle and let $D, E$ are points on its circumscribed circle, such that $D$ lies on arc $AB, E$ lies on arc $AC$ (smaller arcs) and $BD \parallel CE$ . Let the point F be the intersection of the lines $DA$ and $CE$, and the intersection of the lines $EA$ and $BD$ is $G$. Let $P$ be the second intersection of the circumscribed circles of $\vartriangle ABG$ and $\vartriangle ACF$. Prove that the line$ AP$ passes through the mid point of the side $BC$.

Convex quadrilateral $ABCD$ is cyclic in circle $(O)$, $P$ is the intersection of the diagonals $AC$ and $BD$. Circle $(O_{1})$ passes through $P$ and $B$, circle $(O_{2})$ passes through $P$ and $A$, Circles $(O_{1})$ and $(O_{2})$ intersect at $P$ and $Q$. $(O_{1})$, $(O_{2})$ intersect $(O)$ at another points $E$, $F$ (besides $B$, $A$), respectively. Prove that $PQ$, $CE$, $DF$ are concurrent or parallel.

Consider points $D,E$ and $F$ on sides $BC,AC$ and $AB$, respectively, of a triangle $ABC$, such that $AD, BE$ and $CF$ concurr at a point $G$. The parallel through $G$ to $BC$ cuts $DF$ and $DE$ at $H$ and $I$, respectively. Show that triangles $AHG$ and $AIG$ have the same areas.

Find the sides of a triangle if it is known that the inscribed circle meets one of its medians in two points and these points divide the median into three equal segments and the area of the triangle is equal to $6\sqrt{14}\text{ cm}^2$.

Let $\alpha,\beta,\gamma$ be angles of a triangle. Two of them can be expressed using an auxiliary angle $\varphi$ in a way that $$\alpha=\varphi+\frac\pi4,\quad\beta=\pi-3\varphi.$$ Show that $\alpha>\gamma.$

Let $ABCD$ be a rectangle with area $2012$. There exist points $E$ on $AB$ and $F$ on $CD$ such that $DE = EF = F B$. Diagonal $AC$ intersects $DE$ at $X$ and $EF$ at $Y$ . Compute the area of triangle $EXY$ .

Let $ABCD$ be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at $H$. Denote by $M$ and $N$ the midpoints of $\overline{BC}$ and $\overline{CD}$. Rays $MH$ and $NH$ meet $\overline{AD}$ and $\overline{AB}$ at $S$ and $T$, respectively. Prove that there exists a point $E$, lying outside quadrilateral $ABCD$, such that [list] [*] ray $EH$ bisects both angles $\angle BES$, $\angle TED$, and [*] $\angle BEN = \angle MED$. [/list] [i]Proposed by Evan Chen[/i]

Two parallel chords $c, d$ in a circle have lengths $10$ and $14$ respectively, and the distance between them is $6$. If the length of the chord that is equidistant from $c$ and $d$ and parallel to $c$ and $d$ is $x$, find $x^2$.

An acute triangle $ABC$ is given. Let us denote by $D$ and $E$ the orthogonal projections, respectively of points $ B$ and $C$ on the bisector of the external angle $BAC$. Let $F$ be the point of intersection of the lines $BE$ and $CD$. Show that the lines $AF$ and $DE$ are perpendicular.

Rhombus $ABCD$ has side length $2$ and $\angle B = 120 ^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$? $ \textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1+\frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2 $

Prove that an $m \times n$ rectangle can be constructed using copies of the following shape if and only if $mn$ is a multiple of $8$ where $m>1$ and $n>1$ [asy] draw ((0,0)--(0,1)); draw ((0,0)--(1.5,0)); draw ((0,1)--(.5,1)); draw ((.5,1)--(.5,0)); draw ((0,.5)--(1.5,.5)); draw ((1.5,.5)--(1.5,0)); draw ((1,.5)--(1,0)); [/asy]

Let $ \alpha,\beta,\gamma$ be the angles of a triangle. Prove that $csc^2\frac{\alpha}{2}+csc^2\frac{\beta}{2}+csc^2\frac{\gamma}{2} \ge 12$ and find the conditions for equality.