Let $ABC$ be an acute triangle with orthocentre $H$. Let $D$ be a point outside the circumcircle of triangle $ABC$ such that $\angle ABD=\angle DCA$. The reflection of $AB$ in $BD$ intersects $CD$ at $X$. The reflection of $AC$ in $CD$ intersects $BD$ at $Y$. The lines through $X$ and $Y$ perpendicular to $AC$ and $AB$, respectively, intersect at $P$. Prove that points $D$, $P$ and $H$ are collinear.

Given is an acute angled triangle $ \triangle ABC$ with side lengths $ a$, $ b$ and $ c$ (in an usual way) and circumcenter $ O$. Angle bisector of angle $ \angle BAC$ intersects circumcircle at points $ A$ and $ A_{1}$. Let $ D$ be projection of point $ A_{1}$ onto line $ AB$, $ L$ and $ M$ be midpoints of $ AC$ and $ AB$ , respectively. (i) Prove that $ AD\equal{}\frac{1}{2}(b\plus{}c)$ (ii) If triangle $ \triangle ABC$ is an acute angled prove that $ A_{1}D\equal{}OM\plus{}OL$

Let $p, q$, and $r$ be the angles of a triangle, and let $a = \sin2p, b = \sin2q$, and $c = \sin2r$. If $s = \frac{(a + b + c)}2$, show that \[s(s - a)(s - b)(s -c) \geq 0.\] When does equality hold?

Let $ABCD$ be a convex quadrilateral such that $AC = BD$. Equilateral triangles are constructed on the sides of the quadrilateral. Let $O_1,O_2,O_3,O_4$ be the centers of the triangles constructed on $AB,BC,CD,DA$ respectively. Show that $O_1O_3$ is perpendicular to $O_2O_4.$

In isosceles triangle $ABC(AC=BC)$ the point $M$ is in the segment $AB$ such that $AM=2MB,$ $F$ is the midpoint of $BC$ and $H$ is the orthogonal projection of $M$ in $AF.$ Prove that $\angle BHF=\angle ABC.$

In a scalene triangle $ABC$,the bisectors of angle $A,B$ intersect their corresponding sides at $K,L$ respectively.$I,O,H$ denote respectively the incenter,circumcenter and orthocenter of triangle $ABC$. Prove that $A,B,K,L,O$ are concyclic iff $KL$ is the common tangent line of the circumcircles of the three triangles $ALI,BHI$ and $BKI$.

Let any point $D$ be chosen on the side $BC$ of the triangle $ABC$. Let the radii of the incircles of the triangles $ABC, ABD$ and $ACD$ be $r_1, r_2$ and $r_3$. Prove that $r_1 <r_2 + r_3$.

Consider an acute-angled triangle $ABC$, with $AC>AB$, and let $\Gamma$ be its circumcircle. Let $E$ and $F$ be the midpoints of the sides $AC$ and $AB$, respectively. The circumcircle of the triangle $CEF$ and $\Gamma$ meet at $X$ and $C$, with $X\neq C$. The line $BX$ and the tangent to $\Gamma$ through $A$ meet at $Y$. Let $P$ be the point on segment $AB$ so that $YP = YA$, with $P\neq A$, and let $Q$ be the point where $AB$ and the parallel to $BC$ through $Y$ meet each other. Show that $F$ is the midpoint of $PQ$.

Two circles centered at $O_1$ and $O_2$ have radii $2$ and $3$ and are externally tangent at $P$. The common external tangent of the two circles intersects the line $O_1O_2$ at $Q$. What is the length of $PQ$ ?

Let $ABC$ be an acute non-isosceles triangle with circumcircle $\omega$, circumcenter $O$ and orthocenter $H$. We draw a line perpendicular to $AH$ through $O$ and a line perpendicular to $AO$ through $H$. Prove that the points of intersection of these lines with sides $AB$ and $AC$ lie on a circle, which is tangent to $\omega$. [i]Proposed by A. Kuznetsov[/i]

Let $ t$ be a positive number. Draw two tangent lines from the point $ (t, \minus{} 1)$ to the parabpla $ y \equal{} x^2$. Denote $ S(t)$ the area bounded by the tangents line and the parabola. Find the minimum value of $ \frac {S(t)}{\sqrt {t}}$.

Is there a triangle with sides of integer lengths such that the length of the shortest side is $ 2007$ and that the largest angle is twice the smallest?

Let $A_1A_2 \dots A_n$ a cyclic $n$-agon with center $O$ and $P$, $Q$ being two isogonal conjugates of it (i.e, $\angle PA_{i+1}A_i = \angle QA_{i+1}A_{i+2}$ for all $i$). Let $P_i$ be the circumcenter of $\triangle PA_iA_{i+1}$ and $Q_i$ the circumcenter of $\triangle QA_iA_{i+1}$ for all $i$. Prove that: $a) ~P_1P_2 \dots P_n$ and $Q_1Q_2 \dots Q_n$ are cyclic, with centers $O_P$ and $O_Q$, respectively. $b)~O, O_P$ and $O_Q$ are collinears. $c)~O_PO_Q \mid \mid PQ.$ Remark: indices are taken modulo $n$.

In a polygon, every external angle is one sixth of its corresponding internal angle. How many sides does the polygon have?

Prove that for every positive integer $m$ we can find a finite set $S$ of points in the plane, such that given any point $A$ of $S$, there are exactly $m$ points in $S$ at unit distance from $A$.

Let $ABC$ be a triangle with $AB < AC$. Let $D$ and $E$ be on the lines $CA$ and $BA$, respectively, such that $CD = AB$, $BE = AC$, and $A$, $D$ and $E$ lie on the same side of $BC$. Let $I$ be the incenter of triangle $ABC$, and let $H$ be the orthocenter of triangle $BCI$. Show that $D$, $E$, and $H$ are collinear.

In a half-plane, bounded by a line \(r\), equilateral triangles \(S_1, S_2, \ldots, S_n\) are placed, each with one side parallel to \(r\), and their opposite vertex is the point of the triangle farthest from \(r\). For each triangle \(S_i\), let \(T_i\) be its medial triangle. Let \(S\) be the region covered by triangles \(S_1, S_2, \ldots, S_n\), and let \(T\) be the region covered by triangles \(T_1, T_2, \ldots, T_n\). Prove that \[\text{area}(S) \leq 4 \cdot \text{area}(T).\]

Let $ABCD$ be a parallelogram with $\angle BAD < 90^{\circ}$. A circle tangent to sides $\overline{DA}$, $\overline{AB}$, and $\overline{BC}$ intersects diagonal $\overline{AC}$ at points $P$ and $Q$ with $AP < AQ$, as shown. Suppose that $AP = 3$, $PQ = 9$, and $QC = 16$. Then the area of $ABCD$ can be expressed in the form $m\sqrt n$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$. [asy] defaultpen(linewidth(0.6)+fontsize(11)); size(8cm); pair A,B,C,D,P,Q; A=(0,0); label("$A$", A, SW); B=(6,15); label("$B$", B, NW); C=(30,15); label("$C$", C, NE); D=(24,0); label("$D$", D, SE); P=(5.2,2.6); label("$P$", (5.8,2.6), N); Q=(18.3,9.1); label("$Q$", (18.1,9.7), W); draw(A--B--C--D--cycle); draw(C--A); draw(Circle((10.95,7.45), 7.45)); dot(A^^B^^C^^D^^P^^Q); [/asy]

Let $ABC$ be a triangle and a circle $\Gamma'$ be drawn lying outside the triangle, touching its incircle $\Gamma$ externally, and also the two sides $AB$ and $AC$. Show that the ratio of the radii of the circles $\Gamma'$ and $\Gamma$ is equal to $\tan^ 2 { \left( \dfrac{ \pi - A }{4} \right) }.$

In the coordinate plane a rectangle with vertices $ (0, 0),$ $ (m, 0),$ $ (0, n),$ $ (m, n)$ is given where both $ m$ and $ n$ are odd integers. The rectangle is partitioned into triangles in such a way that [i](i)[/i] each triangle in the partition has at least one side (to be called a “good” side) that lies on a line of the form $ x \equal{} j$ or $ y \equal{} k,$ where $ j$ and $ k$ are integers, and the altitude on this side has length 1; [i](ii)[/i] each “bad” side (i.e., a side of any triangle in the partition that is not a “good” one) is a common side of two triangles in the partition. Prove that there exist at least two triangles in the partition each of which has two good sides.

$\triangle ABC$, $\angle A=23^{\circ},\angle B=46^{\circ}$. Let $\Gamma$ be a circle with center $C$, radius $AC$. Let the external angle bisector of $\angle B$ intersects $\Gamma$ at $M,N$. Find $\angle MAN$.

$P$ is any point inside a triangle $ABC$. The perimeter of the triangle $AB + BC + Ca = 2s$. Prove that $s < AP +BP +CP < 2s$.

A solid pyramid $TABCD$, with a quadrilateral base $ABCD$ is to be coloured on each of the five faces such that no two faces with a common edge will have the same colour. If five different colours are available, what is the number of ways to colour the pyramid?

The parallelepiped contains a sphere of radius $r$ and is contained within a sphere of radius $R$. Prove that $ \frac{R}{r} \geq \sqrt{3} $.

Prove that the line joining the centroid and the incenter of a non-isosceles triangle is perpendicular to the base if and only if the sum of the other two sides is thrice the base.