Let $O$ be the circumcenter of triangle $ABC$ and let $A_1,B_1,C_1$ be the midpoints of arcs $BC, CA,AB$ respectively. If $I$ is the incenter of triangle $ABC$, prove that $$\overrightarrow{OI}= \overrightarrow{OA_1}+ \overrightarrow{OB_1}+ \overrightarrow{OC_1}.$$

Show that the triangle whose angles satisfy the equality \[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\] is right angled

In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.

Mr Green measures his rectangular garden by walking two of the sides and finds that it is 15 steps by 20 steps. Each or Mr Green's steps is two feet long. Mr Green expect half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr Green expect from his garden? $ \textbf{(A) }600\qquad\textbf{(B) }800\qquad\textbf{(C) }1000\qquad\textbf{(D) }1200\qquad\textbf{(E) }1400 $

In a triangle $ABC$ it is given that $2AB=AC+BC$. Prove that the incentre of $\triangle ABC$, the circumcentre of $\triangle ABC$, and the midpoints of $AC$ and $BC$ are concyclic.

Prove that the line segment, joining the orthocenter and the intersection point of the medians of the acute-angled triangle $ABC$ is parallel to the side $AB$ iff $\tan \angle A \cdot \tan \angle B = 3$.

Given a point $A$ that lies on circle $c(o,R)$ (with center $O$ and radius $R$). Let $(e)$ be the tangent of the circle $c$ at point $A$ and a line $(d)$ that passes through point $O$ and intersects $(e)$ at point $M$ and the circle at points $B,C$ (let $B$ lie between $O$ and $A$). If $AM = R\sqrt3$ , prove that a) triangle $AMC$ is isosceles. b) circumcenter of triangle $AMC$ lies on circle $c$ .

Three circles, $\omega_1$, $\omega_2$, $\omega_3$ are drawn, with $\omega_3$ externally tangent to $\omega_1$ at $C$ and internally tangent to $\omega_2$ at $D$. Say also that $\omega_1$, $\omega_2$ intersect at points $A, B$. Suppose the radius of $\omega_1$ is $20$, the radius of $\omega_2$ is $15$, and the radius of $\omega_3$ is $6$. Draw line $CD$, and suppose it meets $AB$ at point $X$. If $AB = 24$, then $CX$ can be written in the form $\frac{a \sqrt{b}}{c}$, where$ a, b, c$ are positive integers where $b$ is square-free, and $a, c$ are relatively prime. Find $a + b + c$.

Triangle $\vartriangle ABC$ has $AB = 8$, $AC = 10$, and $AD =\sqrt{33}$, where $D$ is the midpoint of $BC$. Perpendiculars are drawn from $D$ to meet $AB$ and $AC$ at $E$ and $F$, respectively. The length of $EF$ can be expressed as $\frac{a\sqrt{b}}{c}$ , where $a, c$ are relatively prime and $b$ is square-free. Compute $a + b + c$.

The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]

There are 2003 points chosen randomly in the plane in such a way that no three of them lie on a straight line. Prove that there exists a circle which contains at least three of the given points on its circumference, and no other given points inside.

A finite number of circles are placed into a $1 \times 1$ square. Let $C$ be the sum of the perimeters of the circles. For how many $C$s from $C=\dfrac {43}5$, $9$, $\dfrac{91}{10}$, $\dfrac{19}{2}$, $10$, we can definitely say there exists a line cutting four of the circles? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

Let $D$ the touchpoint of the inscribed circle of triangle $ABC$ be with side $AB$ . From $A$ the perpendicular lines on the angle bisectors of vertices $B$ and $C$ intersect them at points $A_1$ and $A_2$ respectively . Prove that $A_1A_2 = AD$.

With $21$ pieces, some white and some black, a rectangle is formed of $3 \times 7$. Prove that there are always four pieces of the same color located at the vertices of a rectangle.

Three 12 cm $\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in $\text{cm}^3$) of this polyhedron? [asy] defaultpen(fontsize(10)); size(250); draw(shift(0, sqrt(3)+1)*scale(2)*rotate(45)*polygon(4)); draw(shift(-sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(165)*polygon(4)); draw(shift(sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(285)*polygon(4)); filldraw(scale(2)*polygon(6), white, black); pair X=(2,0)+sqrt(2)*dir(75), Y=(-2,0)+sqrt(2)*dir(105), Z=(2*dir(300))+sqrt(2)*dir(225); pair[] roots={2*dir(0), 2*dir(60), 2*dir(120), 2*dir(180), 2*dir(240), 2*dir(300)}; draw(roots[0]--X--roots[1]); label("$B$", centroid(roots[0],X,roots[1])); draw(roots[2]--Y--roots[3]); label("$B$", centroid(roots[2],Y,roots[3])); draw(roots[4]--Z--roots[5]); label("$B$", centroid(roots[4],Z,roots[5])); label("$A$", (1+sqrt(3))*dir(90)); label("$A$", (1+sqrt(3))*dir(210)); label("$A$", (1+sqrt(3))*dir(330)); draw(shift(-10,0)*scale(2)*polygon(4)); draw((sqrt(2)-10,0)--(-10,sqrt(2))); label("$A$", (-10,0)); label("$B$", centroid((sqrt(2)-10,0),(-10,sqrt(2)),(sqrt(2)-10, sqrt(2))));[/asy]

In square $ABCD$ with side length $2$, let $M$ be the midpoint of $AB$. Let $N$ be a point on $AD$ such that $AN = 2ND$. Let point $P$ be the intersection of segment $MN$ and diagonal $AC$. Find the area of triangle $BPM$. [i]Proposed by Jacob Xu[/i]

Let $O$ be the centre of the circle touching the side $AC$ of triangle $ABC$ and the continuations of the sides $BA$ and $BC$. $D$ is the centre of the circle passing through the points $A$, $B$ and $O$. Prove that the points $A$, $B$, $C$ and $D$ lie on a circle. (YF Akurlich)

Given is an acute triangle $ABC$ $\left(AB<AC<BC\right)$,inscribed in circle $c(O,R)$.The perpendicular bisector of the angle bisector $AD$ $\left(D\in BC\right)$ intersects $c$ at $K,L$ ($K$ lies on the small arc $\overarc{AB}$).The circle $c_1(K,KA)$ intersects $c$ at $T$ and the circle $c_2(L,LA)$ intersects $c$ at $S$.Prove that $\angle{BAT}=\angle{CAS}$. [hide=Diagram][asy]import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -6.94236331697463, xmax = 15.849400903703716, ymin = -5.002235438802758, ymax = 7.893104843949444; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); pen qqqqtt = rgb(0.,0.,0.2); draw((1.8318261909633622,3.572783369254345)--(0.,0.)--(6.,0.)--cycle, aqaqaq); draw(arc((1.8318261909633622,3.572783369254345),0.6426249310341638,-117.14497824050169,-101.88970202103212)--(1.8318261909633622,3.572783369254345)--cycle, qqqqtt); draw(arc((1.8318261909633622,3.572783369254345),0.6426249310341638,-55.85706977865775,-40.60179355918817)--(1.8318261909633622,3.572783369254345)--cycle, qqqqtt); /* draw figures */ draw((1.8318261909633622,3.572783369254345)--(0.,0.), uququq); draw((0.,0.)--(6.,0.), uququq); draw((6.,0.)--(1.8318261909633622,3.572783369254345), uququq); draw(circle((3.,0.7178452373968209), 3.0846882800136055)); draw((2.5345020274407277,0.)--(1.8318261909633622,3.572783369254345)); draw(circle((-0.01850947366601585,1.3533783539547308), 2.889550258039566)); draw(circle((5.553011501106743,2.4491551634556963), 3.887127532933951)); draw((-0.01850947366601585,1.3533783539547308)--(5.553011501106743,2.4491551634556963), linetype("2 2")); draw((1.8318261909633622,3.572783369254345)--(0.7798408954511686,-1.423695174396108)); draw((1.8318261909633622,3.572783369254345)--(5.22015910454883,-1.4236951743961088)); /* dots and labels */ dot((1.8318261909633622,3.572783369254345),linewidth(3.pt) + dotstyle); label("$A$", (1.5831274347452782,3.951671933606579), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("$B$", (-0.6,0.05), NE * labelscalefactor); dot((6.,0.),linewidth(3.pt) + dotstyle); label("$C$", (6.188606107156787,0.07450151636712989), NE * labelscalefactor); dot((2.5345020274407277,0.),linewidth(3.pt) + dotstyle); label("$D$", (2.3,-0.7), NE * labelscalefactor); dot((-0.01850947366601585,1.3533783539547308),linewidth(3.pt) + dotstyle); label("$K$", (-0.3447473583572136,1.6382221818835927), NE * labelscalefactor); dot((5.553011501106743,2.4491551634556963),linewidth(3.pt) + dotstyle); label("$L$", (5.631664500260511,2.580738747400365), NE * labelscalefactor); dot((0.7798408954511686,-1.423695174396108),linewidth(3.pt) + dotstyle); label("$T$", (0.5977692071595602,-1.960477431907719), NE * labelscalefactor); dot((5.22015910454883,-1.4236951743961088),linewidth(3.pt) + dotstyle); label("$S$", (5.160406217502124,-1.8747941077698307), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/hide]

Let $m$ and $n$ be positive integers. A sequence of points $(A_0,A_1,\ldots,A_n)$ on the Cartesian plane is called [i]interesting[/i] if $A_i$ are all lattice points, the slopes of $OA_0,OA_1,\cdots,OA_n$ are strictly increasing ($O$ is the origin) and the area of triangle $OA_iA_{i+1}$ is equal to $\frac{1}{2}$ for $i=0,1,\ldots,n-1$. Let $(B_0,B_1,\cdots,B_n)$ be a sequence of points. We may insert a point $B$ between $B_i$ and $B_{i+1}$ if $\overrightarrow{OB}=\overrightarrow{OB_i}+\overrightarrow{OB_{i+1}}$, and the resulting sequence $(B_0,B_1,\ldots,B_i,B,B_{i+1},\ldots,B_n)$ is called an [i]extension[/i] of the original sequence. Given two [i]interesting[/i] sequences $(C_0,C_1,\ldots,C_n)$ and $(D_0,D_1,\ldots,D_m)$, prove that if $C_0=D_0$ and $C_n=D_m$, then we may perform finitely many [i]extensions[/i] on each sequence until the resulting two sequences become identical.

Let $ABC$ be a triangle and let $P$ be an interior point of $ABC$. We assume that a line $l$, which passes through $P$, but not through $A$, intersects $AB$ and $AC$ (or their extensions over $B$ or $C$) at $Q$ and $R$, respectively. Find $l$ such that the perimeter of the triangle $AQR$ is as small as possible.

The incircle of triangle $ABC$ touches sides $AB$;$BC$;$CA$ at $M$;$N$;$K$, respectively. The line through $A$ parallel to $NK$ meets $MN$ at $D$. The line through $A$ parallel to $MN$ meets $NK$ at $E$. Show that the line $DE$ bisects sides $AB$ and $AC$ of triangle $ABC$. [i]M. Sonkin[/i]

Let $P$ be a convex $2006$-gon. The $1003$ diagonals connecting opposite vertices and the $1003$ lines connecting the midpoints of opposite sides are concurrent, that is, all $2006$ lines have a common point. Prove that the opposite sides of $P$ are parallel and congruent.

The diagonals of a convex quadrilateral $ABCD$ are orthogonal and intersect at a point $E$. Prove that the projections of $E$ on $AB,BC,CD,DA$ are concyclic.

Lines \( FD \) and \( BE \) intersect at point \( O \). Rays \( OA \) and \( OC \) are drawn from point \( O \). You are given the following information about the angles: \[ \angle DOC = 36^\circ, \quad \angle AOC = 90^\circ, \quad \angle AOB = 4x, \quad \angle FOE = 5x, \] as shown in the figure below. What is the degree measure of \( x \)? [img]https://i.ibb.co/m5rwmXm/Kyiv-MO-2025-R1-7.png[/img]

In $\triangle ABC$ with circumcenter $O$, $\measuredangle A = 45^\circ$. Denote by $X$ the second intersection of $\overrightarrow{AO}$ with the circumcircle of $\triangle BOC$. Compute the area of quadrilateral $ABXC$ if $BX = 8$ and $CX = 15$. [i]Proposed by Aaron Lin[/i]