Three segments $2$ cm, $5$ cm and $12$ cm long are constructed on the plane. Construct a trapezoid with bases of $2$ cm and $5$ cm, the sum of the sides of which is $12$ cm, and one of the angles is $60^o$. (Bogdan Rublev)

The diagonals of a convex quadrilateral divide it into four triangles. The radii of the circles circumscribed around these triangles are equal. Can such a property have a quadrilateral other than: a) parallelogram, b) rhombus? (Sharygin Igor)

Given a right triangle $ABC$ ($\angle A <45^o$,$ \angle C = 90^o$), on the sides $AC$ and $AB$ which are selected points $D,E$ respectively, such that $BD = AD$ and $CB = CE$. Let the segments $BD$ and $CE$ intersect at the point $O$. Prove that $\angle DOE = 90^o$.

In an isosceles triangle $ABC$ with base $AC$, on side $BC$ is selected point $K$ so that $\angle BAK = 24^o$. On the segment $AK$ the point $M$ is chosen so that $\angle ABM = 90^o$, $AM=2BK$. Find the values of all angles of triangle $ABC$.

The board depicts the triangle $ABC$, the altitude $AH$ and the angle bisector $AL$ which intersectthe inscribed circle in the triangle at the points $M$ and $N, P$ and $Q$, respectively. After that, the figure was erased, leaving only the points $H, M$ and $Q$. Restore the triangle $ABC$. (Bogdan Rublev)

Let $ABCDEF $ be a regular hexagon. On the line $AF $ mark the point $X$so that $ \angle DCX = 45^o$ . Find the value of the angle $FXE$. (Vyacheslav Yasinsky)

On the legs $AC, BC$ of a right triangle $\vartriangle ABC$ select points $M$ and $N$, respectively, so that $\angle MBC = \angle NAC$. The perpendiculars from points $M$ and $C$ on the line $AN$ intersect $AB$ at points $K$ and $L$, respectively. Prove that $KL=LB$. (O. Clurman)

On the sides $AB$ and $CD$ of the parallelogram $ABCD$ mark points $E$ and $F$, respectively. On the diagonals $AC$ and $BD$ chose the points $M$ and $N$ so that $EM\parallel BD$ and $FN\parallel AC$. Prove that the lines $AF, DE$ and $MN$ intersect at one point. (B. Rublev)

On a straight line $4$ points are successively set , $A, P, Q,W $, which are the points of intersection of the bisector $AL $ of the triangle $ABC$ with the circumscribed and inscribed circle. Knowing only these points, construct a triangle $ABC $.

There are two triangles $ABC$ and $BKL$ on the plane so that the segment $AK$ is divided into three equal parts by the point of intersection of the medians $\vartriangle ABC$ and the point of intersection of the bisectors $ \vartriangle BKL $ ($AK $ - median $ \vartriangle ABC$, $KA$ - bisector $\vartriangle BKL $) and quadrilateral $KALC $ is trapezoid. Find the angles of the triangle $BKL$. (Bogdan Rublev)

In the triangle $ABC$ on the side $AC$ the points $F$ and $L$ are selected so that $AF = LC <\frac{1}{2} AC$. Find the angle $ \angle FBL $ if $A {{B} ^ {2}} + B {{C} ^ {2}} = A {{L} ^ {2}} + L {{C } ^ {2}}$ (Zhidkov Sergey)

A chord $AB$ is drawn in the circle, on which the point $P$ is selected in such a way that $AP = 2PB$. The chord $DE$ is perpendicular to the chord $AB $ and passes through the point $P$. Prove that the midpoint of the segment $AP$ is the orthocener of the triangle $AED$.

Point $O$ is the center of the circumcircle of the acute triangle $ABC$. The line $AO$ intersects the side $BC$ at point $D$ so that $OD = BD = 1/3 BC$ . Find the angles of the triangle $ABC$. Justify the answer.

In an acute-angled triangle $ABC$, the points $M$ and $N$ are the midpoints of the sides $AB$ and $AC$, respectively. For an arbitrary point $S$ lying on the side of $BC$ prove that the condition holds $(MB- MS)(NC-NS) \le 0$

In an acute-angled triangle $ABC$, the point $O$ is the center of the circumcircle, $CH$ is the height of the triangle, and the point $T$ is the foot of the perpendicular dropped from the vertex $C$ on the line $AO$. Prove that the line $TH$ passes through the midpoint of the side $BC$ .

The medians $AL, BM$, and $CN$ are drawn in the triangle $ABC$. Prove that $\angle ANC = \angle ALB$ if and only if $\angle ABM =\angle LAC$. (Veklich Bogdan)

Let $ABCD$ be an inscribed quadrilateral. Denote the midpoints of the sides $AB, BC, CD$ and $DA$ through $M, L, N$ and $K$, respectively. It turned out that $\angle BM N = \angle MNC$. Prove that: i) $\angle DKL = \angle CLK$. ii) in the quadrilateral $ABCD$ there is a pair of parallel sides.

The triangle $ABC$ is inscribed in a circle. At points $A$ and $B$ are tangents to this circle, which intersect at point $T$. A line drawn through the point $T$ parallel to the side $AC$ intersects the side $BC$ at the point $D$. Prove that $AD = CD$.

Given an isosceles triangle $ABC$ with a vertex at the point $B$. Based on $AC$, an arbitrary point $D $ is selected, different from the vertices $A$ and $C $. On the line $AC $ select the point $E $ outside the segment $AC$, for which $AE = CD$. Prove that the perimeter $\Delta BDE$ is larger than the perimeter $\Delta ABC$.

On the circle $\gamma$ the points $A$ and $B$ are selected. The circle $\omega$ touches the segment $AB$ at the point $K$ and intersects the circle $\gamma$ at the points $M$ and $N$. The points lie on the circle $\gamma$ in the following order: $A, \, \, M, \, \, N, \, \, B$. Prove that $\angle AMK = \angle KNB$. (Yuri Biletsky)

The triangle $ABC$ with $AB> AC$ is inscribed in a circle, the angle bisector of $\angle BAC$ intersects the side $BC$ of the triangle at the point $K$, and the circumscribed circle at the point $M$. The midline of $\Delta ABC$, which is parallel to the side $AB$, intersects $AM$ at the point $O$, the line $CO$ intersects the line $AB$ at the point $N$. Prove that a circle can be circumscribed around the quadrilateral $BNKM$. (Nagel Igor)

Let $ABCD$ be a convex quadrilateral. Prove that the circles inscribed in the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a common point if and only if $ABCD$ is a rhombus.

The two circles ${{w} _ {1}}, \, \, {{w} _ {2}}$ touch externally at the point $Q$. The common external tangent of these circles is tangent to ${{w} _ {1}}$ at the point $B$, $BA$ is the diameter of this circle. A tangent to the circle ${{w} _ {2}} $ is drawn through the point $A$, which touches this circle at the point $C$, such that the points $B$ and $C$ lie in one half-plane relative to the line $AQ$. Prove that the circle ${{w} _ {1}}$ bisects the segment $C $. (Igor Nagel)

In the quadrilateral $ABCD$ the condition $AD = AB + CD$ is fulfilled. The bisectors of the angles $BAD$ and $ADC$ intersect at the point $P $, as shown in Fig. Prove that $BP = CP$. (Maria Rozhkova)

The sides of triangles $ABC$ and $ACD$ satisfy the following conditions: $AB = AD = 3$ cm, $BC = 7$ cm, $DC = 11$ cm. What values can the side length $AC$ take if it is an integer number of centimeters, is the average in $\Delta ACD$ and the largest in $\Delta ABC$?

Given an equilateral $\Delta ABC$, in which ${{A} _ {1}}, {{B} _ {1}}, {{C} _ {1}}$ are the midpoint of the sides $ BC, \, \, AC, \, \, AB$ respectively. The line $l$ passes through the vertex $A$, we denote by $P, Q$ the projection of the points $B, C$ on the line $l$, respectively (the line $ l $ and the point $Q, \, \, A, \, \, P$ are located as shown in fig.). Denote by $T $ the intersection point of the lines ${{B} _ {1}} P$ and ${{C} _ {1}} Q$. Prove that the line ${{A} _ {1}} T$ is perpendicular to the line $l$. (Serdyuk Nazar)

On the side $AB$ of the triangle $ABC$ mark the point $K$. The segment $CK$ intersects the median $AM$ at the point $F$. It is known that $AK = AF$. Find the ratio $MF: BK$.

Two circles ${{c} _ {1}}, \, \, {{c} _ {2}}$ pass through the center $O$ of the circle $c$ and touch it internally in points $A$ and $B$, respectively. Prove that the line $AB$ passes though a common point of circles ${{c} _ {1}}, \, \, {{c} _ {2}} $.

In the isosceles triangle $ABC$, $ (AB = BC)$ the bisector $AD$ was drawn, and in the triangle $ABD$ the bisector $DE$ was drawn. Find the values of the angles of the triangle $ABC$, if it is known that the bisectors of the angles $ABD$ and $AED$ intersect on the line $AD$. (Fedak Ivan)

It is known that a square can be inscribed in a given right trapezoid so that each of its vertices lies on the corresponding side of the trapezoid (none of the vertices of the square coincides with the vertex of the trapezoid). Construct this inscribed square with a compass and a ruler. (Maria Rozhkova)

In the triangle $ABC$ the angle bisectors $AD$ and $BE$ are drawn. Prove that $\angle ACB = 60 {} ^ \circ$ if and only if $AE + BD = AB$. (Hilko Danilo)

In the quadrilateral $ABCD$, shown in fig. , the equations are true: $\angle ABC = \angle BCD$ and $2AB = CD$. On the side $BC$, a point $X$ is selected such that $\angle BAX = \angle CDA$. Prove that $AX = AD$.

On the sides $BC$ and $AB$ of the triangle $ABC$ the points ${{A} _ {1}}$ and ${{C} _ {1}} $ are selected accordingly so that the segments $A {{A} _ {1}}$ and $C {{C} _ {1}}$ are equal and perpendicular. Prove that if $\angle ABC = 45 {} ^ \circ$, then $AC = A {{A} _ {1}} $. (Gogolev Andrew)

On the sides $AB$ and $AD$ of the square $ABCD$, the points $N$ and $P$ are selected, respectively, so that $PN = NC$, the point $Q$ Is a point on the segment $AN$ for which $\angle NCB = \angle QPN$. Prove that $\angle BCQ = \tfrac {1} {2} \angle PQA$.

On the sides $BC$ and $CD$ of the square $ABCD$, the points $M$ and $N$ are selected in such a way that $\angle MAN= 45^o$. Using the segment $MN$, as the diameter, we constructed a circle $w$, which intersects the segments $AM$ and $AN$ at points $P$ and $Q$, respectively. Prove that the points $B, P$ and $Q$ lie on the same line.

In a trapezoid $ABCD$ with bases $AD$ and $BC$, the bisector of the angle $\angle DAB$ intersects the bisectors of the angles $\angle ABC$ and $\angle CDA$ at the points $P$ and $S$, respectively, and the bisector of the angle $\angle BCD$ intersects the bisectors of the angles $\angle ABC$ and $\angle CDA$ at the points $Q$ and $R$, respectively. Prove that if $PS\parallel RQ$, then $AB = CD$.

Let $I$ be the center of the inscribed circle of $ABC$ and let $I_A$ be the center of the exscribed circle touching the side $BC$. Let $M$ be the midpoint of the side $BC$, and $N$ be the midpoint of the arc $BAC$ of the circumscribed circle of $ABC$ . The point $T$ is symmetric to the point $N$ wrt point $A$. Prove that the points $I_A,M,I,T$ lie on the same circle. (Danilo Hilko)

In the triangle $ABC$, the medians $BB_1$ and $CC_1$, which intersect at the point $M$, are drawn. Prove that a circle can be inscribed in the quadrilateral $AC_1MB_1$ if and only if $AB = AC$.

Inside the triangle $ABC $, the point $P $ is selected so that $BC = AP $ and $\angle APC = 180 {} ^ \circ - \angle ABC $. On the side $AB $ there is a point $K $, for which $AK = KB + PC $. Prove that $\angle AKC = 90 {} ^ \circ $. (Danilo Hilko)

In the quadrilateral $ABCD$ point $E$ - the midpoint of the side $AB$, point $F$ - the midpoint of the side $BC$, point $G$ - the midpoint $AD$ . It turned out that the segment $GE$ is perpendicular to $AB$, and the segment $GF$ is perpendicular to the segment $BC$. Find the value of the angle $GCD$, if it is known that $\angle ADC = 70 {} ^ \circ$.

In the isosceles triangle $ABC$ with the vertex at the point $B$, the altitudes $BH$ and $CL$ are drawn. The point $D$ is such that $BDCH$ is a rectangle. Find the value of the angle $DLH$. (Bogdan Rublev)

Given a triangle $ABC$, the perpendicular bisector of the side $AC$ intersects the angle bisector of the triangle $AK$ at the point $P$, $M$ - such a point that $\angle MAC = \angle PCB$, $\angle MPA = \angle CPK$, and points $M$ and $K$ lie on opposite sides of the line $AC$. Prove that the line $AK$ bisects the segment $BM$. (Anton Trygub)

Given a circle $\Gamma$ with center at point $O$ and diameter $AB$. $OBDE$ is square, $F$ is the second intersection point of the line $AD$ and the circle $\Gamma$, $C$ is the midpoint of the segment $AF$. Find the value of the angle $OCB$.

In the triangle $ABC$ it is known that $2AC=AB$ and $\angle A = 2\angle B$. In this triangle draw the angle bisector $AL$, and mark point $M$, the midpoint of the side $AB$. It turned out that $CL = ML$. Prove that $\angle B= 30^o$. (Hilko Danilo)

In a right triangle $ABC$, the lengths of the legs satisfy the condition: $BC =\sqrt2 AC$. Prove that the medians $AN$ and $CM$ are perpendicular. (Hilko Danilo)

Given a square $ABCD$ with side $10$. On sides BC and $AD$ of this square are selected respectively points $E$ and $F$ such that formed a rectangle $ABEF$. Rectangle $KLMN$ is located so that its the vertices $K, L, M$ and $N$ lie one on each segments $CD, DF, FE$ and $EC$, respectively. It turned out that the rectangles $ABEF$ and $KLMN$ are equal with $AB = MN$. Find the length of segment $AL$.

In the quadrilateral $ABCD$, $AB = BC$ . The point $E$ lies on the line $AB$ is such that $BD= BE$ and $AD \perp DE$. Prove that the perpendicular bisectors to segments $AD, CD$ and $CE$ intersect at one point.

Given a triangle $ABC, O$ is the center of the circumcircle, $M$ is the midpoint of $BC, W$ is the second intersection of the bisector of the angle $C$ with this circle. A line parallel to $BC$ passing through $W$, intersects$ AB$ at the point $K$ so that $BK = BO$. Find the measure of angle $WMB$. (Anton Trygub)

Let $ABCDEF$ be a hexagon inscribed in a circle in which $AB = BC, CD = DE$ and $EF = FA$. Prove that the lines $AD, BE$ and $CF$ intersect at one point.

Let the point $D$ lie on the arc $AC$ of the circumcircle of the triangle $ABC$ ($AB < BC$), which does not contain the point $B$. On the side $AC$ are selected an arbitrary point $X$ and a point $X'$ for which $\angle ABX= \angle CBX'$. Prove that regardless of the choice of the point $X$, the circle circumscribed around $\vartriangle DXX'$, passes through a fixed point, which is different from point $D$. (Nikolaev Arseniy)

The points $A, B, C, D$ are selected on the circle as followed so that $AB = BC = CD$. Bisectors of $\angle ABD$ and $\angle ACD$ intersect at point $E$. Find $\angle ABC$, if it is known that $AE \parallel CD$.

Let $BM$ be the median of the triangle $ABC$, in which $AB> BC$. Point $P$ is chosen so that $AB \parallel PC$ and $PM \perp BM$. Prove that $\angle ABM = \angle MBP$. (Mikhail Standenko)

On the sides $AB$ and $BC$ of the triangle $ABC$, the points $K$ and $M$ are chosen so that $KM \parallel AC$. The segments $AM$ and $KC$ intersect at the point $O$. It is known that $AK =AO$ and $KM =MC$. Prove that $AM=KB$.

Let $BM$ be the median of the triangle $ABC$, in which $AB> BC$. Point $P$ is chosen so that $AB \parallel PC$ and$ PM \perp BM$. The point $Q$ is chosen on the line $BP$ so that $\angle AQC = 90^o$, and the points $B$ and $Q$ lie on opposite sides of the line $AC$. Prove that $AB = BQ$. (Mikhail Standenko)

Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A line passing through point $B$ intersects $\omega_1$ for the second time at point $C$ and $\omega_2$ at point $D$. The line $AC$ intersects circle $\omega_2$ for the second time at point $F$, and the line $AD$ intersects the circle $\omega_1$ for the second time at point $E$ . Let point $O$ be the center of the circle circumscribed around $\vartriangle AEF$. Prove that $OB \perp CD$.

There are $n$ sticks which have distinct integer length. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What's the largest possible value of $n$ under such conditions? (Proposed by Bogdan Rublov)

In triangle $ABC$ $\angle B > 90^\circ$. Tangents to this circle in points $A$ and $B$ meet at point $P$, and the line passing through $B$ perpendicular to $BC$ meets the line $AC$ at point $K$. Prove that $PA = PK$. (Proposed by Danylo Khilko)

Let $AL$ be the inner bisector of triangle $ABC$. The circle centered at $B$ with radius $BL$ meets the ray $AL$ at points $L$ and $E$, and the circle centered at $C$ with radius $CL$ meets the ray $AL$ at points $L$ and $D$. Show that $AL^2 = AE\times AD$. (Proposed by Mykola Moroz)