Let $ ABC$ be a triangle, and let the angle bisectors of its angles $ CAB$ and $ ABC$ meet the sides $ BC$ and $ CA$ at the points $ D$ and $ F$, respectively. The lines $ AD$ and $ BF$ meet the line through the point $ C$ parallel to $ AB$ at the points $ E$ and $ G$ respectively, and we have $ FG \equal{} DE$. Prove that $ CA \equal{} CB$. [i]Original formulation:[/i] Let $ ABC$ be a triangle and $ L$ the line through $ C$ parallel to the side $ AB.$ Let the internal bisector of the angle at $ A$ meet the side $ BC$ at $ D$ and the line $ L$ at $ E$ and let the internal bisector of the angle at $ B$ meet the side $ AC$ at $ F$ and the line $ L$ at $ G.$ If $ GF \equal{} DE,$ prove that $ AC \equal{} BC.$

Each point of the 3-dimensional space is coloured with exactly one of the colours red, green and blue. Let $R$, $G$ and $B$, respectively, be the sets of the lengths of those segments in space whose both endpoints have the same colour (which means that both are red, both are green and both are blue, respectively). Prove that at least one of these three sets includes all non-negative reals.

Let $ABC$ be an acute, non-isosceles triangle with circumcenter $O$, incenter $I$ and $(I)$ tangent to $BC$, $CA$, $AB$ at $D, E, F$ respectively. Suppose that $EF$ cuts $(O)$ at $P, Q$. Prove that $(PQD)$ bisects segment $BC$.

Two circles, $\omega_1$ and $\omega_2$, are internally tangent at $A.$ Let $B$ be the point on $\omega_2$ opposite of $A.$ The radius of $\omega_1$ is $4$ times the radius of $\omega_2.$ Point $P$ is chosen on the circumference of $\omega_1$ such that the ratio $\tfrac{AP}{BP}=\tfrac{2\sqrt{3}}{\sqrt{7}}.$ Let $O$ denote the center of $\omega_2.$ Determine $\tfrac{OP}{AO}.$

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively. If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .

Let $ABC$ be a triangle and $H$ be its orthocenter. If it is given that $B$ is $(0,0)$, $C$ is $(1,2)$ and $H$ is $(5,0)$, find $A$.

Segment $AD$ is the diameter of the circumscribed circle of an acute-angled triangle $ABC$. Through the intersection of the altitudes of this triangle, a straight line was drawn parallel to the side $BC$, which intersects sides $AB$ and $AC$ at points $E$ and $F$, respectively. Prove that the perimeter of the triangle $DEF$ is two times larger than the side $BC$.

In a triangle $\triangle ABC$ with $\angle ABC < \angle BCA$, we define $K$ as the excenter with respect to $A$. The lines $AK$ and $BC$ intersect in a point $D$. Let $E$ be the circumcenter of $\triangle BKC$. Prove that \[\frac{1}{|KA|} = \frac{1}{|KD|} + \frac{1}{|KE|}.\]

Vertices $A$, $B$ and $C$ of a triangle are connected with points $A'$ , $B'$ and $C'$ lying in the opposite sides of the triangle (not at vertices). Can the midpoints of the segments $AA'$, $BB'$ and $CC'$ lie in a straight line? (Folklore)

Let $M$ be the midpoint of the side $AB$ of a triangle $ABC$. A circle through point $C$ that has a point of tangency to the line $AB$ at point $A$ and a circle through point $C$ that has a point of tangency to the line $AB$ at point $B$ intersect the second time at point $N$. Prove that $|CM|^2 + |CN|^2 - |MN|^2 = |CA|^2 + |CB|^2 - |AB|^2$.

On the diagonal $AC$ of cyclic quadrilateral $ABCD$ a point $E$ is chosen such that $\angle ABE = \angle CBD$. Points $O,O_1,O_2$ are the circumcircles of triangles $ABC, ABE$ and $CBE$ respectively. Prove that lines $DO,AO_{1}$ and $CO_{2}$ are concurrent.

Given an acute triangle $ABC$, mark $3$ points $X, Y, Z$ in the interior of the triangle. Let $X_1, X_2, X_3$ be the projections of $X$ to $BC, CA, AB$ respectively, and define the points $Y_i, Z_i$ similarly for $i=1, 2, 3$. a) Suppose that $X_iY_i<X_iZ_i$ for all $i=1,2,3$, prove that $XY<XZ$. b) Prove that this is not neccesarily true, if triangle $ABC$ is allowed to be obtuse. [i]Proposed by Ivan Chan Kai Chin[/i]

Let $ABC$ and $DEF$ be two triangles, such that $AB=DE=20$, $BC=EF=13$, and $\angle A = \angle D$. If $AC-DF=10$, determine the area of $\triangle ABC$. [i]Proposed by Lewis Chen[/i]

The circle $\Omega$ centered at $O$ is the circumcircle of the triangle $ABC$. Point $D$ is chosen so that $BD \perp BC$ and points $A$ and $D$ lie in different half-planes with respect to the line $BC$. Let $E$ be a point such that $\angle ADB=\angle BDE$ and $\angle EBD+\angle ACB=90$. Point $P$ is chosen on the line $AD$ so that $OP \perp BC$. Let $Q$ be an arbitrary point on $\Omega$, and $R$ be a point on the line $BQ$ such that $PQ \parallel DR$. Prove that $\angle ARB=\angle BRE$. (All angles are oriented in the same way)

$A$ graph $G$ arises from $G_{1}$ and $G_{2}$ by pasting them along $S$ if $G$ has induced subgraphs $G_{1}$, $G_{2}$ with $G=G_{1}\cup G_{2}$ and $S$ is such that $S=G_{1}\cap G_{2}.$ A is graph is called [i]chordal[/i] if it can be constructed recursively by pasting along complete subgraphs, starting from complete subgraphs. For a graph $G(V,E)$ define its Hilbert polynomial $H_{G}(x)$ to be $H_{G}(x)=1+Vx+Ex^2+c(K_{3})x^3+c(K_{4})x^4+\ldots+c(K_{w(G)})x^{w(G)},$ where $c(K_{i})$ is the number of $i$-cliques in $G$ and $w(G)$ is the clique number of $G$. Prove that $H_{G}(-1)=0$ if and only if $G$ is chordal or a tree.

On the sides $AB$ and $BC$ arbitrarily mark points $M$ and $N$, respectively. Let $P$ be the point of intersection of segments $AN$ and $BM$. In addition, we note the points $Q$ and $R$ such that quadrilaterals $MCNQ$ and $ACBR$ are parallelograms. Prove that the points $P,Q$ and $R$ lie on one line.

Proof that there exist constant $\lambda$, so that for any positive integer $m(\ge 2)$, and any lattice triangle $T$ in the Cartesian coordinate plane, if $T$ contains exactly one $m$-lattice point in its interior(not containing boundary), then $T$ has area $\le \lambda m^3$. PS. lattice triangles are triangles whose vertex are lattice points; $m$-lattice points are lattice points whose both coordinates are divisible by $m$.

For any convex polygon $ W $ with area 1, let us denote by $ f(W) $ the area of the convex polygon whose vertices are the centers of all sides of the polygon $ W $. For each natural number $ n \geq 3 $, determine the lower limit and the upper limit of the set of numbers $ f(W) $ when $ W $ runs through the set of all $ n $ convex angles with area 1.

Let $ABC$ be a scalene triangle with circumcenter $O$ and incenter $I$. The $A$-excircle, $B$-excircle, and $C$-excircle of triangle $ABC$ touch $BC$, $CA$, and $AB$ at points $A_1$, $B_1$, and $C_1$, respectively. Let $P$ be the orthocenter of $AB_1C_1$ and $H$ be the orthocenter of $ABC$. Show that if $M$ is the midpoint of $PA_1$, then lines $HM$ and $OI$ are parallel. [i]Michael Ren[/i]

Three circles are pairwise tangent, with none of them lying inside another. The centres of the circles are the corners of a triangle with circumference $1$. What is the smallest possible value for the sum of the areas of the circles?

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Omega$. Let $M$ be the midpoint of side $BC$. Point $D$ is chosen from the minor arc $BC$ on $\Gamma$ such that $\angle BAD = \angle MAC$. Let $E$ be a point on $\Gamma$ such that $DE$ is perpendicular to $AM$, and $F$ be a point on line $BC$ such that $DF$ is perpendicular to $BC$. Lines $HF$ and $AM$ intersect at point $N$, and point $R$ is the reflection point of $H$ with respect to $N$. Prove that $\angle AER + \angle DFR = 180^\circ$. [i]Proposed by Li4.[/i]

Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other

In $\triangle ABC$, $P$ is a point on $BC$. $F\in AB,E\in AC,PF//CA,PE//BA$. If $S_{\triangle ABC}=1$, prove that at least one of $S_{\triangle BPF},S_{\triangle PCE},S_{PEAF}$ is not less than $\frac{4}{9}$.

Let $K, T$ be the points of tangency of inscribed and exscribed circles to the side $BC$ triangle $ABC$, $M$ is the midpoint of the side $BC$. Using a compass and a ruler, construct triangle ABC given rays $AK$ and $AT$ (points $K, T$ are not marked on them) and point $M$.