In triangle $ABC$, choose point $A_1$ on side $BC$, point $B_1$ on side $CA$, and point $C_1$ on side $AB$ in such a way that the three segments $AA_1, BB_1$, and $CC_1$ intersect in one point $P$. Prove that $P$ is the centroid of triangle $ABC$ if and only if $P$ is the centroid of triangle $A_1B_1C_1$. Note: A median in a triangle is a segment connecting a vertex of the triangle with the midpoint of the opposite side. The centroid of a triangle is the intersection point of the three medians of the triangle. The centroid of a triangle is also known by the names ”center of mass” and ”medicenter” of the triangle.

Given points $A, B, C$ on a line, equilateral triangles $ABC_1$ and $BCA_1$ constructed on segments $AB$ and $BC$, and midpoints $M$ and $N$ of $AA_1$ and $CC_1$, respectively. Prove that $\vartriangle BMN$ is equilateral. (We assume that $B$ lies between $A$ and $C$, and points $A_1$ and $C_1$ lie on the same side of line $AB$)

Construct a right-angled triangle along its two medians, starting from the acute angles.

The diagram below shows some small squares each with area $3$ enclosed inside a larger square. Squares that touch each other do so with the corner of one square coinciding with the midpoint of a side of the other square. Find integer $n$ such that the area of the shaded region inside the larger square but outside the smaller squares is $\sqrt{n}$. [asy] size(150); real r=1/(2sqrt(2)+1); path square=(0,1)--(r,1)--(r,1-r)--(0,1-r)--cycle; path square2=(0,.5)--(r/sqrt(2),.5+r/sqrt(2))--(r*sqrt(2),.5)--(r/sqrt(2),.5-r/sqrt(2))--cycle; defaultpen(linewidth(0.8)); filldraw(unitsquare,gray); filldraw(square2,white); filldraw(shift((0.5-r/sqrt(2),0.5-r/sqrt(2)))*square2,white); filldraw(shift(1-r*sqrt(2),0)*square2,white); filldraw(shift((0.5-r/sqrt(2),-0.5+r/sqrt(2)))*square2,white); filldraw(shift(0.5-r/sqrt(2)-r,-(0.5-r/sqrt(2)-r))*square,white); filldraw(shift(0.5-r/sqrt(2)-r,-(0.5+r/sqrt(2)))*square,white); filldraw(shift(0.5+r/sqrt(2),-(0.5+r/sqrt(2)))*square,white); filldraw(shift(0.5+r/sqrt(2),-(0.5-r/sqrt(2)-r))*square,white); filldraw(shift(0.5-r/2,-0.5+r/2)*square,white); [/asy]

In triangle $ABC$, angle $A$ is equal to $a$ and the altitude drawn to side $BC$ is equal to $h$. The inscribed circle of the triangle touches the sides of the triangle at points $K$, $M$ and $P$, where $P$ lies on side $BC$. Find the distance from $P$ to $KM$.

In triangle $ABC$, $I$ is the incenter. We have chosen points $P,Q,R$ on segments $IA,IB,IC$ respectively such that $IP\cdot IA=IQ \cdot IB=IR\cdot IC$. Prove that the points $I$ and $O$ belong to Euler line of triangle $PQR$ where $O$ is circumcenter of $ABC$.

A pentagon has four interior angles each equal to $110^o$. Find the degree measure of the fifth interior angle.

Given a cube and two colours. Two players paint in turn a triple of arbitrary unpainted edges with his colour. (Everyone makes two moves.) The first wins if he has painted all the edges of some face with his colour. Can he always win?

Let $\vartriangle ABC$ be a triangle with $\angle BAC = 45^o, \angle BCA = 30^o$, and $AB = 1$. Point $D$ lies on segment $AC$ such that $AB = BD$. Find the square of the length of the common external tangent to the circumcircles of triangles $\vartriangle BDC$ and $\vartriangle ABC$.

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

In triangle $ABC$, arbitrary points $P,Q$ lie on side $BC$ such that $BP=CQ$ and $P$ lies between $B,Q$.The circumcircle of triangle $APQ$ intersects sides $AB,AC$ at $E,F$ respectively.The point $T$ is the intersection of $EP,FQ$.Two lines passing through the midpoint of $BC$ and parallel to $AB$ and $AC$, intersect $EP$ and $FQ$ at points $X,Y$ respectively. Prove that the circumcircle of triangle $TXY$ and triangle $APQ$ are tangent to each other. [i]Proposed by Iman Maghsoudi[/i]

Consider on the coordinate plane all rectangles whose (i) vertices have integer coordinates; (ii) edges are parallel to coordinate axes; (iii) area is $2^k$, where $k = 0,1,2....$ Is it possible to color all points with integer coordinates in two colors so that no such rectangle has all its vertices of the same color?

Consider a triangle $ABC$ with $BC = 3$. Choose a point $D$ on $BC$ such that $BD = 2$. Find the value of \[AB^2 + 2AC^2 - 3AD^2.\]

Let $ABC$ be a triangle with circumcentre $O$ and circumcircle $\Omega$. $\Gamma$ is the circle passing through $O,B$ and tangent to $AB$ at $B$. Let $\Gamma$ intersect $\Omega$ a second time at $P \neq B$. The circle passing through $P,C$ and tangent to $AC$ at $C$ intersects with $\Gamma$ at $M$. Prove that $|MP|=|MC|$.

Let $A_1A_2A_3$ be a triangle and let $\omega_1$ be a circle in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\omega_2, \omega_3, \dots, \omega_7$ such that for $k = 2, 3, \dots, 7,$ $\omega_k$ is externally tangent to $\omega_{k-1}$ and passes through $A_k$ and $A_{k+1},$ where $A_{n+3} = A_{n}$ for all $n \ge 1$. Prove that $\omega_7 = \omega_1.$

Altitudes $AH_A, BH_B, CH_C$ of triangle $ABC$ intersect at $H$, and let $M$ be the midpoint of the side $AC$. The bisector $BL$ of $\triangle ABC$ intersects $H_AH_C$ at point $K$. The line through $L$ parallel to $HM$ intersects $BH_B$ in point $T$. Prove that $TK = TL$. [i]Proposed by Anton Trygub[/i]

In the hexagon $ABCDEF$, all angles are equally large. The side lengths satisfy $AB = CD = EF = 3$ and $BC = DE = F A = 2$. The diagonals $AD$ and $CF$ intersect each other in the point $G$. The point $H$ lies on the side $CD$ so that $DH = 1$. Prove that triangle $EGH$ is equilateral.

We are given a triangle $ABC$. Let $M$ be the mid-point of its side $AB$. Let $P$ be an interior point of the triangle. We let $Q$ denote the point symmetric to $P$ with respect to $M$. Furthermore, let $D$ and $E$ be the common points of $AP$ and $BP$ with sides $BC$ and $AC$, respectively. Prove that points $A$, $B$, $D$, and $E$ lie on a common circle if and only if $\angle ACP = \angle QCB$ holds. (Karl Czakler)

Let $ABC$ be a triangle with $\angle BAC = 90^{\circ}$ and $\angle ACB = 54^{\circ}.$ We construct bisector $BD (D \in AC)$ of angle $ABC$ and consider point $E \in (BD)$ such that $DE = DC.$ Show that $BE = 2 \cdot AD.$

Quadrilateral $ABCD$ is inscribed on circle $O$. $AC\cap BD=P$. Circumcenters of $\triangle ABP,\triangle BCP,\triangle CDP,\triangle DAP$ are $O_1,O_2,O_3,O_4$. Prove that $OP,O_1O_3,O_2O_4$ share one point.

Let $\alpha_1,\alpha_2,\alpha_3$ be three congruent circles that are tangent to each other. A third circle $\beta$ is tangent to them at points $A_1,A_2,A_3$ respectively. Let $P$ be a point on $\beta$ which is different from $A_1,A_2,A_3$. For $i=1,2,3$, let $B_i$ be the second intersection point of the line $PA_i$ with circle $\alpha_i$. Prove that $\Delta B_1B_2B_3$ is equilateral.

Let $ABC$ be a triangle with $AB < AC$. Its incircle with centre $I$ touches the sides $BC, CA,$ and $AB$ in the points $D, E,$ and $F$ respectively. The angle bisector $AI$ intersects the lines $DE$ and $DF$ in the points $X$ and $Y$ respectively. Let $Z$ be the foot of the altitude through $A$ with respect to $BC$. Prove that $D$ is the incentre of the triangle $XYZ$.

Points $A,B,C$ are on a plane such that $AB=BC=CA=6$. At any step, you may choose any three existing points and draw that triangle's circumcentre. Prove that you can draw a point such that its distance from an previously drawn point is: $(a)$ greater than 7 $(b)$ greater than 2019

A square $ABCD$ is given. A point $P$ is chosen inside the triangle $ABC$ such that $\angle CAP = 15^\circ = \angle BCP$. A point $Q$ is chosen such that $APCQ$ is an isosceles trapezoid: $PC \parallel AQ$, and $AP=CQ, AP\nparallel CQ$. Denote by $N$ the midpoint of $PQ$. Find the angles of the triangle $CAN$.

Let $ABC$ be a triangle and $D, E, F$ be the foots of altitudes drawn from $A,B,C$ respectively. Let $H$ be the orthocenter of $ABC$. Lines $EF$ and $AD$ intersect at $G$. Let $K$ the point on circumcircle of $ABC$ such that $AK$ is a diameter of this circle. $AK$ cuts $BC$ in $M$. Prove that $GM$ and $HK$ are parallel.