In triangle $ABC$, side $AB$ is $1$. It is known that one of the angle bisectors of triangle $ABC$ is perpendicular to one of its medians, and some other angle bisector is perpendicular to the other median. What can be the perimeter of triangle $ABC$?

Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar. [i]Robin Son[/i]

The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\tfrac{3}{2}$ and center $(0,\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata? [asy] import cse5;pathpen=black;pointpen=black; size(1.5inch); D(MP("x",(3.5,0),S)--(0,0)--MP("\frac{3}{2}",(0,3/2),W)--MP("y",(0,3.5),W)); path P=(0,0)--MP("3",(3,0),S)..(3*dir(45))..MP("3",(0,3),W)--(0,3)..(3/2,3/2)..cycle; draw(P,linewidth(2)); fill(P,gray); [/asy] $\textbf{(A) } \dfrac{4\pi}{5} \qquad\textbf{(B) } \dfrac{9\pi}{8} \qquad\textbf{(C) } \dfrac{4\pi}{3} \qquad\textbf{(D) } \dfrac{7\pi}{5} \qquad\textbf{(E) } \dfrac{3\pi}{2} $

A trapezium $ ABCD$, in which $ AB$ is parallel to $ CD$, is inscribed in a circle with centre $ O$. Suppose the diagonals $ AC$ and $ BD$ of the trapezium intersect at $ M$, and $ OM \equal{} 2$. [b](a)[/b] If $ \angle AMB$ is $ 60^\circ ,$ find, with proof, the difference between the lengths of the parallel sides. [b](b)[/b] If $ \angle AMD$ is $ 60^\circ ,$ find, with proof, the difference between the lengths of the parallel sides. [b][Weightage 17/100][/b]

Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$.Points $X,Y$ lie on $AB,AC$,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$.Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$.

Let $T$ be a point outside circle $\omega$ centered at $O$. Tangents from $T$ to $\omega$ touch $\omega$ at $A;B$. Line $TO$ intersects bigger $AB$ arc at $C$.The line drawn from $T$ parallel to $AC$ intersects $CB$ at $E$. Ray $TE$ intersects small $BC$ arc at $F$. Prove that the circumcircle of $OEF$ is tangent to $\omega$.

Let $\omega_1$ and $\omega_2$ be circles that intersect at points $A$ and $B$ and let $O_1$ and $O_2$ be their respective centers. We take $M$ in $\omega_1$ and $N$ in $\omega_2$ on the same side as $B$ with respect to segment $O_1O_2$, such that $MO_1\parallel BO_2$ and $BO_1 \parallel NO_2$. Draw the tangents to $\omega_1$ and $\omega_2$ through $M$ and $N$ respectively, which intersect at $K$. Show that $A$, $B$ and $K$ are collinear.

Let $ABC$ be an acute triange with $B,C$ fixed. Let $D$ be the midpoint of $BC$ and $E,F$ be the foot of the perpendiculars from $D$ to $AB,AC$, respectively. a) Let $O$ be the circumcenter of triangle $ABC$ and $M=EF\cap AO, N=EF\cap BC$. Prove that the circumcircle of triangle $AMN$ passes through a fixed point; b) Assume that tangents of the circumcircle of triangle $AEF$ at $E,F$ are intersecting at $T$. Prove that $T$ is on a fixed line.

Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

Given an isosceles triangle $ABC$ with $AB = AC$. Let $O$ be the circumcenter of $ABC$, $D$ the midpoint of $AB$ and $E$ the centroid of $ACD$. Prove that $CD \perp EO$.

The circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$, point $M$ is the midpoint of $AB$. On line $AB$ select points $S_1$ and $S_2$. Let $S_1X_1$ and $S_1Y_1$ be tangents drawn from $S_1$ to circle $\omega_1$, similarly $S_2X_2$ and $S_2Y_2$ are tangents drawn from $S_2$ to circles $\omega_2$. Prove that if the point $M$ lies on the line $X_1X_2$, then it also lies on the line $Y_1Y_2$.

Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.

A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is $ \textbf{(A)}\ 36\pi\minus{}34 \qquad \textbf{(B)}\ 30\pi \minus{} 15 \qquad \textbf{(C)}\ 36\pi \minus{} 33 \qquad$ $ \textbf{(D)}\ 35\pi \minus{} 9\sqrt3 \qquad \textbf{(E)}\ 30\pi \minus{} 9\sqrt3$

On the Cartesian plane, all points with both integer coordinates are painted blue. Two blue points are said to be [i]mutually visible[/i] if the line segment that connects them has no other blue points . Prove that there is a set of $ 2019$ blue points that are mutually visible two by two. [hide=official wording]No plano cartesiano, todos os pontos com ambas coordenadas inteiras são pintados de azul. Dois pontos azuis são ditos mutuamente visíveis se o segmento de reta que os conecta não possui outros pontos azuis. Prove que existe um conjunto de 2019 pontos azuis que são mutuamente visíveis dois a dois.[/hide] PS. There is a comment about problem being wrong / incorrect [url=https://artofproblemsolving.com/community/c6h1957974p14780265]here[/url]

Let $I, O$ be the incenter and circumcenter of the triangle $ABC$ respectively. Let the excircle $\omega_A$ of $ABC$ be tangent to the side $BC$ on $N$, and tangent to the extensions of the sides $AB, AC$ on $K, M$ respectively. If the midpoint of $KM$ lies on the circumcircle of $ABC$, prove that $O, I, N$ are collinear.

9.6, 10.6 Construct for each vertex of the trapezium a symmetric point wrt to the diagonal, which doesn't contain this vertex. Prove that if four new points form a quadrilateral then it is a trapezium. ([i]L. Emel'yanov[/i])

In a given tedrahedron $ ABCD$ let $ K$ and $ L$ be the centres of edges $ AB$ and $ CD$ respectively. Prove that every plane that contains the line $ KL$ divides the tedrahedron into two parts of equal volume.

Is it possible to build a $ 4\times 4\times4$ cube from blocks of the following shape consisting of $ 4$ unit cubes?

Points $A,B$ lies on the circle $S$. Tangent lines to $S$ at $A$ and $B$ intersects at $C$. $M$ -midpoint of $AB$. Circle $S_1$ goes through $M,C$ and intersects $AB$ at $D$ and $S$ at $K$ and $L$. Prove, that tangent lines to $S$ at $K$ and $L$ intersects at point on the segment $CD$.

Let $ABCD$ be the circumscribed quadrilateral with the incircle $(I)$. The circle $(I)$ touches $AB, BC, C D, DA$ at $M, N, P,Q$ respectively. Let $K$ and $L$ be the circumcenters of the triangles $AMN$ and $APQ$ respectively. The line $KL$ cuts the line $BD$ at $R$. The line $AI$ cuts the line $MQ$ at $J$. Prove that $RA = RJ$.

The cricles $\Gamma_1$ and $\Gamma_2$ intersect in $ M $ and $N.$ A line that goes through $ M $ intersects the cricles $\Gamma_1$ and $\Gamma_2$ in $ A$ and $B$, such that $M\in(AB)$. The bisector of angle $ AMN $ intersects the circle $\Gamma_1$ in $D,$ and the bisector of angle $BMN$ intersects the circle $\Gamma_2$ in $C.$ Prove that the circle with diameter $CD$ splits the segment $AB$ in half.

Let $AB$ be a diameter of a circle; let $t_1$ and $t_2$ be the tangents at $A$ and $B$, respectively; let $C$ be any point other than $A$ on $t_1$; and let $D_1D_2. E_1E_2$ be arcs on the circle determined by two lines through $C$. Prove that the lines $AD_1$ and $AD_2$ determine a segment on $t_2$ equal in length to that of the segment on $t_2$ determined by $AE_1$ and $AE_2.$

Provided a convex equilateral pentagon. On every side of the pentagon We construct equilateral triangles which run through the interior of the pentagon. Prove that at least one of the triangles does not protrude the pentagon's boundary.

Find an acutangle triangle such that its sides and altitudes have integer length.

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.