Let $O$ be a point (in the plane) and $T$ be an infinite set of points such that $|P_1P_2| \le 2012$ for every two distinct points $P_1,P_2\in T$. Let $S(T)$ be the set of points $Q$ in the plane satisfying $|QP| \le 2013$ for at least one point $P\in T$. Now let $L$ be the set of lines containing exactly one point of $S(T)$. Call a line $\ell_0$ passing through $O$ [i]bad[/i] if there does not exist a line $\ell\in L$ parallel to (or coinciding with) $\ell_0$. (a) Prove that $L$ is nonempty. (b) Prove that one can assign a line $\ell(i)$ to each positive integer $i$ so that for every bad line $\ell_0$ passing through $O$, there exists a positive integer $n$ with $\ell(n) = \ell_0$. [i]Proposed by David Yang[/i]

Segments $AA'$, $BB'$, and $CC'$ are the bisectrices of triangle $ABC$. It is known that these lines are also the bisectrices of triangle $A'B'C'$. Is it true that triangle $ABC$ is regular?

In space, $n$ wire triangles are situated so that any two of them have a common vertex and each vertex is the vertex of $k$ triangles. Find all $n$ and $k$ for which this is possible.

In triangle $ABC$, ($AB \ne AC$), let the orthocenter be $H$, circumcenter be $O$, and the midpoint of $BC$ be $M$. Let $HM \cap AO = D$. Let $P,Q,R,S$ be the midpoints of $AB,CD,AC,BD$. Let $X = PQ\cap RS$. Find $AH/OX$.

Let $G$ be the centroid of a triangle $ABC$. Prove that if $AB+GC = AC+GB$, then the triangle is isosceles

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?

Find the area of the set of all points $ z $ in the complex plane that satisfy $ \left| z - 3i \right| + \left| z - 4 \right| \leq 5\sqrt{2} $.

Let $ A,B,C,M$ points in the plane and no three of them are on a line. And let $ A',B',C'$ points such that $ MAC'B, MBA'C$ and $ MCB'A$ are parallelograms: (a) Show that \[ \overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} < \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}.\] (b) Assume segments $ AA', BB'$ and $ CC'$ have the same length. Show that $ 2 \left(\overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} \right) \leq \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}.$ When do we have equality?

Let $A,B,C,D$ be four points in space, and $M$ and $N$ be the midpoints of $AC$ and $BD$, respectively. Show that $$AB^2+BC^2+CD^2+DA^2 = AC^2+BD^2+4MN^2$$

The triangle $ABC$ lies on the coordinate plane. The midpoint of $\overline{AB}$ has coordinates $(-16, -63)$, the midpoint of $\overline{AC}$ has coordinates $(13, 50)$, and the midpoint of $\overline{BC}$ has coordinates $(6, -85)$. What are the coordinates of point $A$?

Two circles with radii $71$ and $100$ are externally tangent. Compute the largest possible area of a right triangle whose vertices are each on at least one of the circles.

An $ABC$ triangle divide by a $d$ line such that, new two pieces' areas and perimeters are equal. Prove that $ABC$'s incenter lies $d$

Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.

A circle $ \omega$ with center $ O$ is tangent to the rays of an angle $ BAC$ at $ B$ and $ C$. Point $ Q$ is taken inside the angle $ BAC$. Assume that point $ P$ on the segment $ AQ$ is such that $ AQ\perp OP$. The line $ OP$ intersects the circumcircles $ \omega_{1}$ and $ \omega_{2}$ of triangles $ BPQ$ and $ CPQ$ again at points $ M$ and $ N$. Prove that $ OM \equal{} ON$.

Let $G$ be the barycenter of triangle $ABC$. Let $D$ be a point such that $AGDB$ is a parallelogram. Show that $BG \parallel CD$.

in a triangle $ABC$, $I$ is the incenter. $BI$ and $CI$ cut the circumcircle of $ABC$ at $E$ and $F$ respectively. $M$ is the midpoint of $EF$. $C$ is a circle with diameter $EF$. $IM$ cuts $C$ at two points $L$ and $K$ and the arc $BC$ of circumcircle of $ABC$ (not containing $A$) at $D$. prove that $\frac{DL}{IL}=\frac{DK}{IK}$.(25 points)

Let $\vartriangle ABC$ be an acute triangle with $AB < AC$. Let $H$ be its orthocentre and $M$ be the midpoint of arc $BAC$ on the circumcircle. It is given that $B$, $H$, $M$ are collinear, the length of the altitude from $M$ to $AB$ is $1$, and the length of the altitude from $M$ to $BC$ is $6$. Determine all possible areas for $\vartriangle ABC$ .

Let $\triangle{ABC}$ be a triangle with incenter $I,$ and let $M$ be the midpoint of $\overline{BC}.$ Line $AM$ intersects the circumcircle of triangle $\triangle{IBC}$ at points $P$ and $Q.$ Suppose that $AP=13, AQ=83,$ and $BC=56.$ Find the perimeter of $\triangle{ABC}.$

Fix the triangle $ABC$ on the plane. 1. Denote by $S_L,S_M$ and $S_K$ the areas of triangles whose vertices are, respectively, the bases of bisectors, medians and points of tangency of the inscribed circle of a given triangle $ABC$. Prove that $S_K\le S_L\le S_M$. 2. For the point $X$, which is inside the triangle $ABC$, consider the triangle $T_X$, the vertices of which are the points of intersection of the lines $AX, BX, CX$ with the lines $BC, AC, AB$, respectively. 2.1. Find the position of the point $X$ for which the area of ​​the triangle $T_x$ is the largest possible. 2.2. Suggest an effective criterion for comparing the areas of triangles $T_x$ for different positions of the point $X$. 2.3. Find the positions of the point $X$ for which the perimeter of the triangle $T_x$ is the smallest possible and the largest possible. 2.4. Propose an effective criterion for comparing the perimeters of triangles $T_x$ for different positions of point $X$. 2.5. Suggest and solve similar problems with respect to the extreme values ​​of other parameters (for example, the radius of the circumscribed circle, the length of the greatest height) of triangles $T_x$. 3. For the point $Y$, which is inside the circle $\omega$, circumscribed around the triangle $ABC$, consider the triangle $\Delta_Y$, the vertices of which are the points of intersection $AY, BX, CX$ with the circle $\omega$. Suggest and solve similar problems for triangles $\Delta_Y$ for different positions of point $Y$. 4. Suggest and solve similar problems for convex polygons. 5. For the point $Z$, which is inside the circle $\omega$, circumscribed around the triangle $ABC$, consider the triangle $F_Z$, the vertices of which are orthogonal projections of the point $Z$ on the lines $BC$, $AC$ and $AB$. Suggest and solve similar problems for triangles $F_Z$ for different positions of the point $Z$.

Are there such pairwise distinct natural numbers $ m, n, p, q$ satisfying $ m \plus{} n \equal{} p \plus{} q$ and $ \sqrt{m} \plus{} \sqrt[3]{n} \equal{} \sqrt{p} \plus{} \sqrt[3]{q} > 2004$ ?

Given a triangle $ABC$, let $A',B',C'$ be the perpendicular feet dropped from the centroid $G$ of the triangle $ABC$ onto the sides $BC,CA,AB$ respectively. Reflect $A',B',C'$ through $G$ to $A'',B'',C''$ respectively. Prove that the lines $AA'',BB'',CC''$ are concurrent.

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

In triangle $ ABC$ the bisector of $ \angle ACB$ intersects $ AB$ at $ D$. Consider an arbitrary circle $ O$ passing through $ C$ and $ D$, so that it is not tangent to $ BC$ or $ CA$. Let $ O\cap BC \equal{} \{M\}$ and $ O\cap CA \equal{} \{N\}$. a) Prove that there is a circle $ S$ so that $ DM$ and $ DN$ are tangent to $ S$ in $ M$ and $ N$, respectively. b) Circle $ S$ intersects lines $ BC$ and $ CA$ in $ P$ and $ Q$ respectively. Prove that the lengths of $ MP$ and $ NQ$ do not depend on the choice of circle $ O$.

A square and an equilateral triangle are inscribed in a same circle. The seven vertices form a convex heptagon $S$ inscribed in the circle ($S$ might be a hexagon if two vertices coincide). For which positions of the triangle relative to the square does $S$ have the largest and smallest area, respectively?

Find all real numbers with the following property: the difference of its cube and its square is equal to the square of the difference of its square and the number itself.