Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n=\triangle ABC$ and $D,E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB,BC$ and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD,BE,$ and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$? $ \textbf{(A)}\ \frac{1509}{8} \qquad \textbf{(B)}\ \frac{1509}{32} \qquad \textbf{(C)}\ \frac{1509}{64} \qquad \textbf{(D)}\ \frac{1509}{128} \qquad \textbf{(E)}\ \frac{1509}{256} $

Suppose that $ ABCD$ is a convex quadrilateral. Let $ F \equal{} AB\cap CD$, $ E \equal{} AD\cap BC$ and $ T \equal{} AC\cap BD$. Suppose that $ A,B,T,E$ lie on a circle which intersects with $ EF$ at $ P$. Prove that if $ M$ is midpoint of $ AB$, then $ \angle APM \equal{} \angle BPT$.

A square with sides $1$ and four circles of radius $1$ considered each having a vertex of have the square as the center. Find area of the shaded part (see figure). [img]https://cdn.artofproblemsolving.com/attachments/b/6/6e28d94094d69bac13c2702853ac2c906a80a1.png[/img]

Let $ABCD$ be a convex quadrilateral with $\angle B = \angle D = 90^{\circ}$. Let $E$ be the point of intersection of $BC$ with $AD$ and let $M$ be the midpoint of $AE$. On the extension of $CD$, beyond the point $D$, we pick a point $Z$ such that $MZ = \frac{AE}{2}$. Let $U$ and $V$ be the projections of $A$ and $E$ respectively on $BZ$. The circumcircle of the triangle $DUV$ meets again $AE$ at the point $L$. If $I$ is the point of intersection of $BZ$ with $AE$, prove that the lines $BL$ and $CI$ intersect on the line $AZ$.

In a triangle $ABC$, $\angle C=45$. $AD$ is the altitude of the triangle. $X$ is on $AD$ such that $\angle XBC=90-\angle B$ ($X$ is in the triangle). $AD$ and $CX$ cut the circumcircle of $ABC$ in $M$ and $N$ respectively. if tangent to circumcircle of $ABC$ at $M$ cuts $AN$ at $P$, prove that $P$,$B$ and $O$ are collinear.(25 points) the exam time was 4 hours and 30 minutes.

Johann and Nicole are playing a game on the coordinate plane. First, Johann draws any polygon $\mathcal{S}$ and then Nicole can shift $\mathcal{S}$ to wherever she wants. Johann wins if there exists a point with coordinates $(x, y)$ in the interior of $\mathcal{S}$, where $x$ and $y$ are coprime integers. Otherwise, Nicole wins. Determine who has a winning strategy.

Does there exist a triangle $ABC$ whose sides are rational numbers and $BC$ equals to the altitude from $A$?

Inside a circle with radius $6$ lie four smaller circles with centres $A,B,C$ and $D$. The circles touch each other as shown. The point where the circles with centres $A$ and $C$ touch each other is the centre of the big circle. Calculate the area of quadrilateral $ABCD$. [img]https://1.bp.blogspot.com/-FFsiOOdcjao/XzT_oJYuQAI/AAAAAAAAMVk/PpyUNpDBeEIESMsiElbexKOFMoCXRVaZwCLcBGAsYHQ/s0/2012%2BMohr%2Bp1.png[/img]

Let $ P$ be a point in the plane of a triangle $ ABC$. Lines $ AP,BP,CP$ respectively meet lines $ BC,CA,AB$ at points $ A',B',C'$. Points $ A'',B'',C''$ are symmetric to $ A,B,C$ with respect to $ A',B',C',$ respectively. Show that: $ S_{A''B''C''}\equal{}3S_{ABC}\plus{}4S_{A'B'C'}$.

A circle $C$ with radius $3$ has an equilateral triangle inscribed in it. Let $D$ be a circle lying outside the equilateral triangle, tangent to $C$, and tangent to the equilateral triangle at the midpoint of one of its sides. The radius of $D$ can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Let $D$ be the foot of the altitude from $A$ in a triangle $ABC$. The angle bisector at $C$ intersects $AB$ at a point $E$. Given that $\angle CEA=\frac\pi4$, compute $\angle EDB$.

A cone with semivertical angle $30^{\circ}$ is half filled with water. What is the angle it must be tilted by so that water starts spilling?

Let $ABCD$ be a square. Point $P, Q, R, S$ are chosen on the sides $AB$, $BC$, $CD$, $DA$, respectively, such that $AP + CR \ge AB \ge BQ + DS$. Prove that $$area \,\, (PQRS) \le \frac12 \,\, area \,\, (ABCD)$$ and determine all cases when equality holds.

A triangle $ABC$ with $AB < AC$ is inscribed in a circle $\omega$. Circles $\gamma_1$ and $\gamma_2$ touch the lines $AB$ and $AC$, and their centres lie on the circumference of $\omega$. Prove that $C$ lies on a common external tangent to $\gamma_1$ and $\gamma_2$.

Given the parallelogram $ABCD$. The trisection points of side $AB$ are: $H_1, H_2$, ($AH_1 = H_1H_2 =H_2B$). The trisection points of the side $DC$ are $G_1, G_2$, ($DG_1 = G_1G_2 = G_2C$), and $AD = 1, AC = 2$. Prove that triangle $AH_2G_1$ is isosceles.

Given a rectangle $ABCD$ such that $AB = b > 2a = BC$, let $E$ be the midpoint of $AD$. On a line parallel to $AB$ through point $E$, a point $G$ is chosen such that the area of $GCE$ is $$(GCE)= \frac12 \left(\frac{a^3}{b}+ab\right)$$ Point $H$ is the foot of the perpendicular from $E$ to $GD$ and a point $I$ is taken on the diagonal $AC$ such that the triangles $ACE$ and $AEI$ are similar. The lines $BH$ and $IE$ intersect at $K$ and the lines $CA$ and $EH$ intersect at $J$. Prove that $KJ \perp AB$.

Let $AD$ be the angle bisector of triangle $ABC$, and let the line $\ell$ touch circumcircles of triangles $ADB$ and $ADC$ at points $M$ and $N$ accordingly. Prove that the circle passing through the midpoints of the segments $BD$, $DC$ and $MN$ is tangent to the line $\ell$.

$BB_1$ is the angle bisector of $\triangle ABC$, and $I$ is its incenter. The perpendicular bisector of segment $AC$ intersects the circumcircle of $\triangle AIC$ at $D$ and $E$. Point $F$ is on the segment $B_1C$ such that $AB_1=CF$.Prove that the four points $B, D, E$ and $F$ are concyclic.

Functions f and g are defined on the whole real line and are mutually inverse: g(f(x))=x, f(g(y))=y for all x, y. It is known that f can be written as a sum of periodic and linear functions: f(x)=kx+h(x) for some number k and a periodic function h(x). Show that g can also be written as a sum of periodic and linear functions. (A functions h(x) is called periodic if there exists a non-zero number d such that h(x+d)=h(x) for any x.)

p1. Bahri lives quite close to the clock gadang in the city of Bukit Tinggi West Sumatra. Bahri has an antique clock. On Monday $4$th March $2013$ at $10.00$ am, Bahri antique clock is two minutes late in comparison with Clock Tower. A day later, the antique clock was four minutes late compared to the Clock Tower. March $6$, $2013$ the clock is late six minutes compared to Jam Gadang. The following days Bahri observed that his antique clock exhibited the same pattern of delay. On what day and what date in $2014$ the antique Bahri clock (hand short and long hands) point to the same number as the Clock Tower? p2. In one season, the Indonesian Football League is participated by $20$ teams football. Each team competes with every other team twice. The result of each match is $3$ if you win, $ 1$ if you draw, and $0$ if you lose. Every week there are $10$ matches involving all teams. The winner of the competition is the team that gets the highest total score. At the end what week is the fastest possible, the winner of the competition on is the season certain? p3. Look at the following picture. The quadrilateral $ABCD$ is a cyclic. Given that $CF$ is perpendicular to $AF$, $CE$ is perpendicular to $BD$, and $CG$ is perpendicular to $AB$. Is the following statements true? Write down your reasons. $$\frac{BD}{CE}=\frac{AB}{CG}+ \frac{AD}{CF}$$ [img]https://cdn.artofproblemsolving.com/attachments/b/0/dbd97b4c72bc4ebd45ed6fa213610d62f29459.png[/img] p4. Suppose $M=2014^{2014}$. If the sum of all the numbers (digits) that make up the number $M$ equals $A$ and the sum of all the digits that make up the number $A$ equals $B$, then find the sum of all the numbers that make up $B$. p5. Find all positive integers $n < 200$ so that $n^2 + (n + 1)^2$ is square of an integer.

Let $k$ be the incircle of the triangle $ABC$ with the center of the incircle $I$. The circle $k$ touches the sides $BC, CA$ and $AB$ in points $D, E$ and $F$. Let $G$ be the intersection of the straight line $AI$ and the circle $k$, which lies between $A$ and $I$. Assume $BE$ and $FG$ are parallel. Show that $BD = EF$.

In the plane we are given the circles $\Gamma$ and $\Delta$ tangent to each other and $\Gamma$ contains $\Delta$. The radius of $\Gamma$ is $R$ and of $\Delta$ is $\frac{R}{2}$. Prove that for each positive integer $n \geq 3$, the equation: \[ (p(1) - p(n))^2 = (n-1)^2 \cdot (2 \cdot (p(1) + p(n)) - (n-1)^2 - 8) \] is the necessary and sufficient condition for $n$ to exist $n$ distinct circles $\Upsilon_1, \Upsilon_2, \ldots, \Upsilon_n$ such that all these circles are tangent to $\Gamma$ and $\Delta$ and $\Upsilon_i$ is tangent to $\Upsilon_{i+1}$, and $\Upsilon_1$ has radius $\frac{R}{p(1)}$ and $\Upsilon_n$ has radius $\frac{R}{p(n)}$.

Let $\omega_a, \omega_b, \omega_c$ be the exscribed circles tangent to the sides $a, b, c$ of a triangle $ABC$, respectively, $ I_a, I_b, I_c$ be the centers of these circles, respectively, $T_a, T_b, T_c$ be the points of contact of these circles to the line $BC$, respectively. The lines $T_bI_c$ and $T_cI_b$ intersect at the point $Q$. Prove that the center of the circle inscribed in triangle $ABC$ lies on the line $T_aQ$.

Let us denote $\prod = \{(x, y) | y > 0\}$. We call a [i]semicircle[/i] in $\prod$ with center on the $x-\text{axis}$ a [i]semi-line[/i]. Two intersecting [i]semi-lines [/i]determine four [i]semi-angles[/i]. A bisector of a [i]semi-angle [/i]is a [i]semi-line [/i]that bisects the [i]semi-angle[/i]. Prove that in every [i]semi-triangle [/i](determined by three [i]semi-lines[/i]) the bisectors are concurrent.

Csongi bought a $12$-sided convex polygon-shaped pizza. The pizza has no interior point with three or more distinct diagonals passing through it. Áron wants to cut the pizza along $3$ diagonals so that exactly $6$ pieces of pizza are created. In how many ways can he do this? Two ways of slicing are different if one of them has a cut line that the other does not have.