On the line $g$ we are given the segment $AB$ and a point $C$ not on $AB$. Prove that on $g$, there exists at least one pair of points $P,Q$ symmetrical with respect to $C$, which divide the segment $AB$ internally and externally in the same ratios, i.e $$\frac{PA}{PB}=\frac{QA}{QB}\qquad(1)$$ If $A,B,P,Q$ are such points from the line $g$ satisfying $(1)$, prove that the midpoint $C$ of the segment $PQ$ is the external point for the segment $AB$. [i]K. Petrov[/i]

Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle ABC =\angle BCD > 90^o$. The circle circumscribed around the triangle $ABC$ intersects the sides $AD$ and $CD$ at points $K$ and $L$, respectively, different from any vertex of the quadrilateral $ABCD$ . Segments $AL$ and $CK$ intersect at point $P$. Prove that $\angle ADB =\angle PDC$.

The point $M$ , inside $\vartriangle ABC$, satisfies the conditions that $\angle BMC = 90^o +\frac12 \angle BAC$ and that the line $AM$ contains the centre of the circumscribed circle of $\vartriangle BMC$. Prove that $M$ is the centre of the inscribed circle of $\vartriangle ABC$.

Given a triangle $ABC$ , a circle centered at some point $O$ meets the segments $BC$ , $CA$ , $AB$ in the pairs of points $X$ and $X^{'}$ , $Y$ and $Y^{'}$ , $Z$ and $Z^{'}$ , respectively ,labelled in circular order : $X,X^{'},Y,Y^{'},Z,Z^{'}$. Let $M$ be the Miquel point of the triangle $XYZ$ and let $M^{'}$ be that of the triangle $X^{'}Y^{'}Z^{'}$ . Prove that the segments $OM$ and $OM^{'}$ have equal lehgths.

Let $ABCD$ be a tangential and cyclic quadrilateral. Let $S$ be the intersection point of diagonals $AC$ and $BD$ of the quadrilateral. Let $I$, $I_1$, and $I_2$ be the incenters of quadrilateral $ABCD$ and triangles $ACD$ and $BCS$, respectively. Let the ray $II_2$ intersect the circumcircle of quadrilateral $ABCD$ at point $E$. Prove that the points $D$, $E$, $I_1$, and $I_2$ are collinear or concyclic. [i]Proposed by Teodor von Burg[/i]

Suppose $n>2$ and let $A_1,\dots,A_n$ be points on the plane such that no three are collinear. [b](a)[/b] Suppose $M_1,\dots,M_n$ be points on segments $A_1A_2,A_2A_3,\dots ,A_nA_1$ respectively. Prove that if $B_1,\dots,B_n$ are points in triangles $M_2A_2M_1,M_3A_3M_2,\dots ,M_1A_1M_n$ respectively then \[|B_1B_2|+|B_2B_3|+\dots+|B_nB_1| \leq |A_1A_2|+|A_2A_3|+\dots+|A_nA_1|\] Where $|XY|$ means the length of line segment between $X$ and $Y$. [b](b)[/b] If $X$, $Y$ and $Z$ are three points on the plane then by $H_{XYZ}$ we mean the half-plane that it's boundary is the exterior angle bisector of angle $\hat{XYZ}$ and doesn't contain $X$ and $Z$ ,having $Y$ crossed out. Prove that if $C_1,\dots ,C_n$ are points in ${H_{A_nA_1A_2},H_{A_1A_2A_3},\dots,H_{A_{n-1}A_nA_1}}$ then \[|A_1A_2|+|A_2A_3|+\dots +|A_nA_1| \leq |C_1C_2|+|C_2C_3|+\dots+|C_nC_1|\] Time allowed for this problem was 2 hours.

Let $D, E, F$, be the touchpoints of the incircle in triangle $ABC$ with sides $BC, CA, AB$, respectively, . Also, let $AD$ and $EF$ intersect at $P$. Prove that $$\frac{AP}{AD} \ge 1 - \frac{BC}{AB + CA}$$.

Let $A,B,C$ be different points on a circle such that $AB = AC$. Point $E$ lies on the segment $BC$, and $D \ne A$ is the intersection point of the circle and line $AE$. Show that the product $AE \cdot AD$ is independent of the choice of $E$.

Suppose that $A,B,C,D$ are points on a circle, $AB$ is the diameter, $CD$ is perpendicular to $AB$ and meets $AB$ and meets $AB$ at $E , AB$ and $CD$ are integers and $AE - EB=\sqrt{3}$. Find $AE$?

$ ABC$ is a triangle with $ \angle BAC \equal{} 10{}^\circ$, $ \angle ABC \equal{} 150{}^\circ$. Let $ X$ be a point on $ \left[AC\right]$ such that $ \left|AX\right| \equal{} \left|BC\right|$. Find $ \angle BXC$. $\textbf{(A)}\ 15^\circ \qquad\textbf{(B)}\ 20^\circ \qquad\textbf{(C)}\ 25^\circ \qquad\textbf{(D)}\ 30^\circ \qquad\textbf{(E)}\ 35^\circ$

Draw a circle that has a given radius $R$ and is tangent to a given line and a given circle. How many solutions does this problem have?

Prove that there exists a positive integer $ n_0$ with the following property: for each integer $ n \geq n_0$ it is possible to partition a cube into $ n$ smaller cubes.

Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\epsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.

Let $w$-circumcircle of triangle $ABC$ with an obtuse angle $C$ and $C '$symmetric point of point $C$ with respect to $AB$. $M$ midpoint of $AB$. $C'M$ intersects $w$ at $N$ ($C '$ between $M$ and $N$). Let $BC'$ second crossing point $w$ in $F$, and $AC'$ again crosses the $w$ at point $E$. $K$-midpoint $EF$. Prove that the lines $AB, CN$ and$ KC'$are concurrent.

The area of a circle is doubled when its radius $ r$ is increased by $ n$. Then $ r$ equals: $ \textbf{(A)}\ n(\sqrt{2} \plus{} 1)\qquad \textbf{(B)}\ n(\sqrt{2} \minus{} 1)\qquad \textbf{(C)}\ n\qquad \textbf{(D)}\ n(2 \minus{} \sqrt{2})\qquad \textbf{(E)}\ \frac{n\pi}{\sqrt{2} \plus{} 1}$

Consider a regular heptagon ( polygon of $7$ equal sides and angles) $ABCDEFG$ as in the figure below:- $(a).$ Prove $\frac{1}{\sin\frac{\pi}{7}}=\frac{1}{\sin\frac{2\pi}{7}}+\frac{1}{\sin\frac{3\pi}{7}}$ $(b).$ Using $(a)$ or otherwise, show that $\frac{1}{AG}=\frac{1}{AF}+\frac{1}{AE}$ [asy] draw(dir(360/7)..dir(2*360/7),blue); draw(dir(2*360/7)..dir(3*360/7),blue); draw(dir(3*360/7)..dir(4*360/7),blue); draw(dir(4*360/7)..dir(5*360/7),blue); draw(dir(5*360/7)..dir(6*360/7),blue); draw(dir(6*360/7)..dir(7*360/7),blue); draw(dir(7*360/7)..dir(360/7),blue); draw(dir(2*360/7)..dir(4*360/7),blue); draw(dir(4*360/7)..dir(1*360/7),blue); label("$A$",dir(4*360/7),W); label("$B$",dir(5*360/7),S); label("$C$",dir(6*360/7),S); label("$D$",dir(7*360/7),E); label("$E$",dir(1*360/7),E); label("$F$",dir(2*360/7),N); label("$G$",dir(3*360/7),W); [/asy]

Given a triangle $ABC$ with $AB>BC$, $ \Omega $ is circumcircle. Let $M$, $N$ are lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. $K(.)=MN\cap AC$ and $P$ is incenter of the triangle $AMK$, $Q$ is K-excenter of the triangle $CNK$ (opposite to $K$ and tangents to $CN$). If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$. M. Kungodjin

One of the excircles of triangle $ABC$ is tangent to the side $AB$ and to the extensions of sides $CA$ and $CB$ at points $C_1$, $B_1$ and $A_1$ respectively. Another excircle is tangent to side $AB$ and to the extensions of sides $BA$ and $BC$ at points $B_2$, $C_2$ and $A_2$ respectively. Line $A_1B_1$ and $A_2B_2$ intersect at point $P$,. lines $A_1C_1$ and $A_2C_2$ intersect at point $Q$. Prove that the points $A$, $P$, $Q$ are collinear [I]Proposed by S. Berlov[/i]

We call a triangle consisting of three vertices of a pentagon [i]big[/i] if it's area is larger than half of the pentagon's area. Find the maximum number of [i]big[/i] triangles that can be in a convex pentagon. [i]Proposed by Gonchigdorj Sandag[/i]

Let point $H$ be the orthocenter of a scalene triangle $ABC$. Line $AH$ intersects with the circumcircle $\Omega$ of triangle $ABC$ again at point $P$. Line $BH, CH$ meets with $AC,AB$ at point $E$ and $F$, respectively. Let $PE, PF$ meet $\Omega$ again at point $Q,R$, respectively. Point $Y$ lies on $\Omega$ so that lines $AY,QR$ and $EF$ are concurrent. Prove that $PY$ bisects $EF$.

Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.

Determine whether there exist $1976$ nonsimilar triangles with angles $\alpha, \beta, \gamma,$ each of them satisfying the relations \[\frac{\sin \alpha + \sin\beta + \sin\gamma}{\cos \alpha + \cos \beta + \cos \gamma}=\frac{12}{7}\text{ and }\sin \alpha \sin \beta \sin \gamma =\frac{12}{25}\]

An integer sequence $\{a_{n}\}_{n \ge 1}$ is given such that \[2^{n}=\sum^{}_{d \vert n}a_{d}\] for all $n \in \mathbb{N}$. Show that $a_{n}$ is divisible by $n$ for all $n \in \mathbb{N}$.

In triangle $ABC$, the angle $\angle BCA$ is obtuse and $\angle BAC = 2\angle ABC\,.$ The line through $B$ and perpendicular to $BC$ intersects line $AC$ in $D$. Let $M$ be the midpoint of $AB$. Prove that $\angle AMC=\angle BMD$. source : http://cage.ugent.be/~hvernaev/Olympiade/PMO982.pdf

Suppose $\triangle ABC$ is scalene. $O$ is the circumcenter and $A'$ is a point on the extension of segment $AO$ such that $\angle BA'A = \angle CA'A$. Let point $A_1$ and $A_2$ be foot of perpendicular from $A'$ onto $AB$ and $AC$. $H_{A}$ is the foot of perpendicular from $A$ onto $BC$. Denote $R_{A}$ to be the radius of circumcircle of $\triangle H_{A}A_1A_2$. Similiarly we can define $R_{B}$ and $R_{C}$. Show that: \[\frac{1}{R_{A}} + \frac{1}{R_{B}} + \frac{1}{R_{C}} = \frac{2}{R}\] where R is the radius of circumcircle of $\triangle ABC$.