geometry problems from Kyiv mathematical festival
with aops links in the names
2005 Kyiv mathematical festival VIII P4
Let $ M$ be the intersection point of medians of a triangle $ \triangle ABC.$ It is known that $ AC = 2BC$ and $ \angle ACM = \angle CBM.$ Find $ \angle ACB.$
2005 Kyiv mathematical festival IX P2
The quadrilateral $ ABCD$ is cyclic. Points $ E$ and $ F$ are chosen at the diagonals $ AC$ and $ BD$ in such a way that $ AF\bot CD$ and $ DE\bot AB.$ Prove that $ EF \parallel BC.$
2006 Kyiv mathematical festival VIII P4, IX P4, X P3
Let $O$ be the intersection point of altitudes $AD$ and $BE$ of equilateral triangle $ABC.$ Points $K$ and $L$ are chosen inside segments $AO$ and $BO$ respectively such that line $KL$ bisects the perimeter of triangle $ABC.$ Let $F$ be the intersection point of lines $EK$ and $DL.$ Prove that $O$ is the circumcenter of triangle $DEF.$
2016 Kyiv mathematical festival VIII P4
Let $H$ be the point of intersection of the altitudes $AD$ and $BE$ of acute triangle $ABC.$ The circles with diameters $AE$ and $BD$ touch at point $L$. Prove that $HL$ is the angle bisector of angle $\angle AHB.$
2016 Kyiv mathematical festival IX P5, X P4
Let $AD$ and $BE$ be the altitudes of acute triangle $ABC.$ The circles with diameters $AD$ and $BE$ intersect at points $S$ and $T$. Prove that $\angle ACS=\angle BCT.$
2017 Kyiv mathematical festival VIII P2
A triangle $ABC$ is given. Let $D$ be a point on the extension of the segment $AB$ beyond $A$ such that $AD=BC,$ and $E$ be a point on the extension of the segment $BC$ beyond $B$ such that $BE=AC.$ Prove that the circumcircle of the triangle $DEB$ passes through the incenter of the triangle $ABC.$
2017 Kyiv mathematical festival IX P3, X P3
A point $C$ is marked on a chord $AB$ of a circle $\omega.$ Let $D$ be the midpoint of $AC,$ and $O$ be the center of the circle $\omega.$ The circumcircle of the triangle $BOD$ intersects the circle $\omega$ again at point $E$ and the straight line $OC$ again at point $F.$ Prove that the circumcircle of the triangle $CEF$ touches $AB.$
2018 Kyiv mathematical festival VIII P2
Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$). It is known that the circumcircles of triangles $AEM$ and $CDM$ are tangent. Find the angle $\angle BMC.$
2018 Kyiv mathematical festival IX P2
Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$), $\omega_1$ and $\omega_2$ be the circumcircles of triangles $AEM$ and $CDM.$ It is known that the circles $\omega_1$ and $\omega_2$ are tangent. Find the ratio in which the circle $\omega_1$ divides $AB.$
2018 Kyiv mathematical festival X P2
Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$), $\omega_1$ and $\omega_2$ be the circumcircles of triangles $AEM$ and $CDM.$ It is known that the circles $\omega_1$ and $\omega_2$ are tangent. Find the ratio in which the circle $\omega_2$ divides $AC.$
2019 Kyiv mathematical festival VIII P3
2019 Kyiv mathematical festival IX P4, X P4
Let $D$ be the midpoint of the base $BC$ of an isosceles triangle $ABC,$ $E$ be the point at the side $AC$ such that $\angle CDE=60^\circ,$ and $M$ be the midpoint of $DE.$ Prove that $\angle AME=\angle BMD.$
with aops links in the names
inside aops 2020 problem set
2005 - 2021
Let $ M$ be the intersection point of medians of a triangle $ \triangle ABC.$ It is known that $ AC = 2BC$ and $ \angle ACM = \angle CBM.$ Find $ \angle ACB.$
The quadrilateral $ ABCD$ is cyclic. Points $ E$ and $ F$ are chosen at the diagonals $ AC$ and $ BD$ in such a way that $ AF\bot CD$ and $ DE\bot AB.$ Prove that $ EF \parallel BC.$
2006 Kyiv mathematical festival VIII P4, IX P4, X P3
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.
2006 Kyiv mathematical festival X P1
Triangle $ABC$ and straight line $l$ are given at the plane. Construct using a compass and a ruler the straightline which is parallel to $l$ and bisects the area of triangle $ABC.$
2007 Kyiv mathematical festival VIII P4, IX P2, X P2
2007 Kyiv mathematical festival VIII P4, IX P2, X P2
The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$
2008 Kyiv mathematical festival VIII P4
2008 Kyiv mathematical festival VIII P4
Let $ K,L,M$ and $ N$ be the midpoints of sides $ AB,$ $ BC,$ $ CD$ and $ AD$ of the convex quadrangle $ ABCD.$ Is it possible that points $ A,B,L,M,D$ lie on the same circle and points $ K,B,C,D,N$ lie on the same circle?
2008 Kyiv mathematical festival IX P4, X P3
2010 Kyiv mathematical festival VIII P3
2011 Kyiv mathematical festival VIII P3
Quadrilateral can be cut into two isosceles triangles in two different ways.
a) Can this quadrilateral be nonconvex?
b) If given quadrilateral is convex, is it necessarily a rhomb?
$ABC$ is right triangle with right angle near vertex $B, M$ is the middle point of $AC$. The square $BKLM$ is built on $BM$, such that segments $ML$ and $BC$ intersect. Segment $AL$ intersects $BC$ in point $E$. Prove that lines $AB,CL$ and$ KE$ intersect in one point.
2015 Kyiv mathematical festival VIII P4, IX P4, X P42008 Kyiv mathematical festival IX P4, X P3
Points $ K,L,M$ and $ N$ lie on sides $ AB,$ $ BC,$ $ CD$ and $ AD$ of the convex quadrangle $ ABCD.$ Let $ ABLMD$ and $ KBCDN$ be inscribed pentagons and $ KN = LM.$ Prove that $ \angle BAD= \angle BCD.$
2009 Kyiv mathematical festival VIII P5, IX P4, X P2
Assume that a triangle $ABC$ satisfies the following property: For any point from the triangle, the sum of distances from $D$ to the lines $AB,BC$ and $CA$ is less than $1$. Prove that the area of the triangle is less than or equal to $\frac{1}{\sqrt3}$
Assume that a triangle $ABC$ satisfies the following property: For any point from the triangle, the sum of distances from $D$ to the lines $AB,BC$ and $CA$ is less than $1$. Prove that the area of the triangle is less than or equal to $\frac{1}{\sqrt3}$
Let $O$ be the circumcenter and $I$ be the incenter of triangle $ABC.$ Prove that if $AI\perp OB$ and $BI\perp OC$ then $CI\parallel OA$.
2010 Kyiv mathematical festival IX P3, X P3
Let $AD$, $BE,$ $CF$ be the altitudes of triangle $ABC$ such that $\angle A=60^\circ$ and $\angle B=50^\circ$, $H$ be the orthocenter of triangle $DEF.$ Prove that $AH$ is the angle bisector of $\angle DAC$.
2011 Kyiv mathematical festival VIII P3
Quadrilateral can be cut into two isosceles triangles in two different ways.
a) Can this quadrilateral be nonconvex?
b) If given quadrilateral is convex, is it necessarily a rhomb?
$ABC$ is right triangle with right angle near vertex $B, M$ is the middle point of $AC$. The square $BKLM$ is built on $BM$, such that segments $ML$ and $BC$ intersect. Segment $AL$ intersects $BC$ in point $E$. Prove that lines $AB,CL$ and$ KE$ intersect in one point.
2012 Kyiv mathematical festival VIII P3
Let $O$ be the center and $R$ be the radius of circumcircle $\omega$ of triangle $ABC$. Circle $\omega_1$ with center $O_1$ and radius $R$ pass through points $A, O$ and intersects the side $AC$ at point $K$. Let $AF$ be the diameter of circle $\omega$ and points $F, K, O_1$ are collinear. Determine $\angle ABC$.
Let $O$ be the center and $R$ be the radius of circumcircle $\omega$ of triangle $ABC$. Circle $\omega_1$ with center $O_1$ and radius $R$ pass through points $A, O$ and intersects the side $AC$ at point $K$. Let $AF$ be the diameter of circle $\omega$ and points $F, K, O_1$ are collinear. Determine $\angle ABC$.
2012 Kyiv mathematical festival IX P3, X P3
Let $O$ be the circumcenter of triangle $ABC$: Points $D$ and $E$ are chosen at sides $AB$ and $AC$ respectively such that $\angle ADO = \angle AEO = 60^o$ and $BDEC$ is inscribed quadrangle. Prove or disprove that $ABC$ is isosceles triangle.
Let $O$ be the circumcenter of triangle $ABC$: Points $D$ and $E$ are chosen at sides $AB$ and $AC$ respectively such that $\angle ADO = \angle AEO = 60^o$ and $BDEC$ is inscribed quadrangle. Prove or disprove that $ABC$ is isosceles triangle.
Let $ABCD$ be a parallelogram ($AB < BC$). The bisector of the angle $BAD$ intersects the side $BC$ at the point K; and the bisector of the angle $ADC$ intersects the diagonal $AC$ at the point $F$. Suppose that $KD \perp BC$. Prove that $KF \perp BD$.
Let $AD, BE$ be the altitudes and $CF$ be the angle bissector of acute non-isosceles triangle $ABC$ and $AE+BD=AB.$ Denote by $I_A, I_B, I_C$ the incentres of triangles $AEF,$ $BDF,$ $CDE$ respectively. Prove that points $D, E, F, I_A, I_B$ and $I_C$ lie on the same circle.
Let $O$ be the intersection point of altitudes $AD$ and $BE$ of equilateral triangle $ABC.$ Points $K$ and $L$ are chosen inside segments $AO$ and $BO$ respectively such that line $KL$ bisects the perimeter of triangle $ABC.$ Let $F$ be the intersection point of lines $EK$ and $DL.$ Prove that $O$ is the circumcenter of triangle $DEF.$
2016 Kyiv mathematical festival VIII P4
Let $H$ be the point of intersection of the altitudes $AD$ and $BE$ of acute triangle $ABC.$ The circles with diameters $AE$ and $BD$ touch at point $L$. Prove that $HL$ is the angle bisector of angle $\angle AHB.$
2016 Kyiv mathematical festival IX P5, X P4
Let $AD$ and $BE$ be the altitudes of acute triangle $ABC.$ The circles with diameters $AD$ and $BE$ intersect at points $S$ and $T$. Prove that $\angle ACS=\angle BCT.$
2017 Kyiv mathematical festival VIII P2
A triangle $ABC$ is given. Let $D$ be a point on the extension of the segment $AB$ beyond $A$ such that $AD=BC,$ and $E$ be a point on the extension of the segment $BC$ beyond $B$ such that $BE=AC.$ Prove that the circumcircle of the triangle $DEB$ passes through the incenter of the triangle $ABC.$
2017 Kyiv mathematical festival IX P3, X P3
A point $C$ is marked on a chord $AB$ of a circle $\omega.$ Let $D$ be the midpoint of $AC,$ and $O$ be the center of the circle $\omega.$ The circumcircle of the triangle $BOD$ intersects the circle $\omega$ again at point $E$ and the straight line $OC$ again at point $F.$ Prove that the circumcircle of the triangle $CEF$ touches $AB.$
2018 Kyiv mathematical festival VIII P2
Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$). It is known that the circumcircles of triangles $AEM$ and $CDM$ are tangent. Find the angle $\angle BMC.$
2018 Kyiv mathematical festival IX P2
Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$), $\omega_1$ and $\omega_2$ be the circumcircles of triangles $AEM$ and $CDM.$ It is known that the circles $\omega_1$ and $\omega_2$ are tangent. Find the ratio in which the circle $\omega_1$ divides $AB.$
2018 Kyiv mathematical festival X P2
Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$), $\omega_1$ and $\omega_2$ be the circumcircles of triangles $AEM$ and $CDM.$ It is known that the circles $\omega_1$ and $\omega_2$ are tangent. Find the ratio in which the circle $\omega_2$ divides $AC.$
2019 Kyiv mathematical festival VIII P3
Let $ABC$ be an isosceles triangle in which $\angle BAC=120^\circ,$ $D$ be the midpoint of $BC,$ $DE$ be the altitude of triangle $ADC,$ and $M$ be the midpoint of $DE.$ Prove that $BM=3AM.$
2019 Kyiv mathematical festival IX P4, X P4
Let $D$ be the midpoint of the base $BC$ of an isosceles triangle $ABC,$ $E$ be the point at the side $AC$ such that $\angle CDE=60^\circ,$ and $M$ be the midpoint of $DE.$ Prove that $\angle AME=\angle BMD.$
Let $\omega$ be the circumcircle of a triangle $ABC$ ($AB>AC$), $E$ be the midpoint of the arc
$AC$ which does not contain point $B,$ аnd $F$ the midpoint of the arc $AB$ which does not contain
point $C.$ Lines $AF$ and $BE$ meet at point $P,$ line $CF$ and $AE$ meet at point $R,$ and the
tangent to $\omega$ at point $A$ meets line $BC$ at point $Q.$ Prove that points $P,Q,R$ are collinear.
2021 Kyiv mathematical festival IX P3
Let $AD$ be the altitude, $AE$ be the median, and $O$ be the circumcenter of a triangle $ABC.$
Points $X$ and $Y$ are selected inside the triangle such that $\angle BAX=\angle CAY,$
$OX\perp AX,$ and $OY\perp AY.$ Prove that points $D,E,X,Y$ are concyclic.
2021 Kyiv mathematical festival X P5
Let $\omega$ be the circumcircle of a triangle $ABC$ (${AB\ne AC}$), $I$ be the incenter, $P$ be
the point on $\omega$ for which $\angle API=90^\circ,$ $S$ be the intersection point of lines $AP$
and $BC,$ $W$ be the intersection point of line $AI$ and $\omega.$ Line which passes through point
$W$ orthogonally to $AW$ meets $AP$ and $BC$ at points $D$ and $E$ respectively. Prove that
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