Found problems: 361
2015 Caucasus Mathematical Olympiad, 2
In the convex quadrilateral $ABCD$, point $K$ is the midpoint of $AB$, point $L$ is the midpoint of $BC$, point $M$ is the midpoint of CD, and point $N$ is the midpoint of $DA$. Let $S$ be a point lying inside the quadrilateral $ABCD$ such that $KS = LS$ and $NS = MS$ .Prove that $\angle KSN = \angle MSL$.
Durer Math Competition CD 1st Round - geometry, 2015.C2
Given a rectangle $ABCD$, side $AB$ is longer than side $BC$. Find all the points $P$ of the side line $AB$ from which the sides $AD$ and $DC$ are seen from the point $P$ at an equal angle (i.e. $\angle APD = \angle DPC$)
Cono Sur Shortlist - geometry, 2021.G5
Let $\vartriangle ABC$ be a triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\omega$. $AA'$, $BB'$ and $CC'$ are altitudes of $\vartriangle ABC$ with $A'$ in $BC$, $B'$ in $AC$ and $C'$ in $AB$. $P$ is a point on the segment $AA'$. The perpenicular line to $B'C'$ from $P$ intersects $BC$ at $K$. $AA'$ intersects $\omega$ at $M \ne A$. The lines $MK$ and $AO$ intersect at $Q$. Prove that $\angle CBQ = \angle PBA$.
2002 Estonia Team Selection Test, 4
Let $ABCD$ be a cyclic quadrilateral such that $\angle ACB = 2\angle CAD$ and $\angle ACD = 2\angle BAC$. Prove that $|CA| = |CB| + |CD|$.
2017 Ukrainian Geometry Olympiad, 2
Point $M$ is the midpoint of the base $BC$ of trapezoid $ABCD$. On base $AD$, point $P$ is selected. Line $PM$ intersects line $DC$ at point $Q$, and the perpendicular from $P$ on the bases intersects line $BQ$ at point $K$. Prove that $\angle QBC = \angle KDA$.
Ukraine Correspondence MO - geometry, 2019.7
Given a triangle $ABC$. Construct a point $D$ on the side $AB$ and point $E$ on the side $AC$ so that $BD = CE$ and $\angle ADC = \angle BEC$
2010 Grand Duchy of Lithuania, 4
In the triangle $ABC$ angle $C$ is a right angle. On the side $AC$ point $D$ has been found, and on the segment $BD$ point K has been found such that $\angle ABC = \angle KAD = \angle AKD$. Prove that $BK = 2DC$.
2002 Switzerland Team Selection Test, 2
A point$ O$ inside a parallelogram $ABCD$ satisfies $\angle AOB + \angle COD = \pi$. Prove that $\angle CBO = \angle CDO$.
1997 Tournament Of Towns, (554) 4
Two circles intersect at points $A$ and $B$. A common tangent touches the first circle at point $C$ and the second at point $D$. Let $\angle CBD > \angle CAD$. Let the line $CB$ intersect the second circle again at point $E$. Prove that $AD$ bisects the angle $\angle CAE$.
(P Kozhevnikov)
1997 Swedish Mathematical Competition, 2
Let $D$ be the point on side $AC$ of a triangle $ABC$ such that $BD$ bisects $\angle B$, and $E$ be the point on side $AB$ such that $3\angle ACE = 2\angle BCE$. Suppose that $BD$ and $CE$ intersect at a point $P$ with $ED = DC = CP$. Determine the angles of the triangle.
Champions Tournament Seniors - geometry, 2018.3
The vertex $F$ of the parallelogram $ACEF$ lies on the side $BC$ of parallelogram $ABCD$. It is known that $AC = AD$ and $AE = 2CD$. Prove that $\angle CDE = \angle BEF$.
2011 Sharygin Geometry Olympiad, 12
Let $AP$ and $BQ$ be the altitudes of acute-angled triangle $ABC$. Using a compass and a ruler, construct a point $M$ on side $AB$ such that $\angle AQM = \angle BPM$.
Indonesia MO Shortlist - geometry, g5
Given a cyclic quadrilateral $ABCD$. Suppose $E, F, G, H$ are respectively the midpoint of the sides $AB, BC, CD, DA$. The line passing through $G$ and perpendicular on $AB$ intersects the line passing through $H$ and perpendicular on $BC$ at point $K$. Prove that $\angle EKF = \angle ABC$.
2021 Yasinsky Geometry Olympiad, 3
The segments $AC$ and $BD$ are perpendicular, and $AC$ is twice as large as $BD$ and intersects $BD$ in it in the midpoint. Find the value of the angle $BAD$, if we know that $\angle CAD = \angle CDB$.
(Gregory Filippovsky)
Kyiv City MO Juniors Round2 2010+ geometry, 2015.8.41
On the sides $AB, \, \, BC, \, \, CA$ of the triangle $ABC$ the points ${{C} _ {1}}, \, \, {{A} _ { 1}},\, \, {{B} _ {1}}$ are selected respectively, that are different from the vertices. It turned out that $\Delta {{A} _ {1}} {{B} _ {1}} {{C} _ {1}}$ is equilateral, $\angle B{{C}_{1}}{{A}_{1}}=\angle {{C}_{1}}{{B}_{1}}A$ and $\angle B{{A}_{1}}{{C}_{1}}=\angle {{A}_{1}}{{B}_{1}}C$ . Is $ \Delta ABC$ equilateral?
Kyiv City MO Seniors 2003+ geometry, 2007.11.5
The points $A$ and $P$ are marked on the plane. Consider all such points $B, C $ of this plane that $\angle ABP = \angle MAB$ and $\angle ACP = \angle MAC $, where $M$ is the midpoint of the segment $BC$. Prove that all the circumscribed circles around the triangle $ABC$ for different points $B$ and $C$ pass through some fixed point other than the point $A$.
(Alexei Klurman)
2015 Oral Moscow Geometry Olympiad, 4
In triangle $ABC$, point $M$ is the midpoint of $BC, P$ is the intersection point of the tangents at points $B$ and $C$ of the circumscribed circle, $N$ is the midpoint of the segment $MP$. The segment $AN$ intersects the circumscribed circle at point $Q$. Prove that $\angle PMQ = \angle MAQ$.
Oliforum Contest III 2012, 3
Show that if equiangular hexagon has sides $a, b, c, d, e, f$ in order then $a - d = e - b = c - f$.
2015 Sharygin Geometry Olympiad, P17
Let $O$ be the circumcenter of a triangle $ABC$. The projections of points $D$ and $X$ to the sidelines of the triangle lie on lines $\ell $ and $L $ such that $\ell // XO$. Prove that the angles formed by $L$ and by the diagonals of quadrilateral $ABCD$ are equal.
1995 Singapore Team Selection Test, 2
$ABC$ is a triangle with $\angle A > 90^o$ . On the side $BC$, two distinct points $P$ and $Q$ are chosen such that $\angle BAP = \angle PAQ$ and $BP \cdot CQ = BC \cdot PQ$. Calculate the size of $\angle PAC$.
2020 Durer Math Competition Finals, 5
The hexagon $ABCDEF$ has all angles equal . We know that four consecutive sides of the hexagon have lengths $7, 6, 3$ and $5$ in this order. What is the sum of the lengths of the two remaining sides?
2015 Costa Rica - Final Round, 6
Let $\vartriangle ABC$ be a triangle with circumcenter $O$. Let $ P$ and $Q$ be internal points on the sides $AB$ and $AC$ respectively such that $\angle POB = \angle ABC$ and $\angle QOC = \angle ACB$. Show that the reflection of line $BC$ over line $PQ$ is tangent to the circumcircle of triangle $\vartriangle APQ$.
2022 China Northern MO, 1
As shown in the figure, given $\vartriangle ABC$ with $AB \perp AC$, $AB=BC$, $D$ is the midpoint of the side $AB$, $DF\perp DE$, $DE=DF$ and $BE \perp EC$. Prove that $\angle AFD= \angle CEF$.
[img]https://cdn.artofproblemsolving.com/attachments/9/2/f16a8c8c463874f3ccb333d91cdef913c34189.png[/img]
2011 Balkan MO Shortlist, G2
Let $ABC$ be a triangle and let $O$ be its circumcentre. The internal and external bisectrices of the angle $BAC$ meet the line $BC$ at points $D$ and $E$, respectively. Let further $M$ and $L$ respectively denote the midpoints of the segments $BC$ and $DE$. The circles $ABC$ and $ALO$ meet again at point $N$. Show that the angles $BAN$ and $CAM$ are equal.
2018 Irish Math Olympiad, 8
Let $M$ be the midpoint of side $BC$ of an equilateral triangle $ABC$. The point $D$ is on $CA$ extended such that $A$ is between $D$ and $C$. The point $E$ is on $AB$ extended such that $B$ is between $A$ and $E$, and $|MD| = |ME|$. The point $F$ is the intersection of $MD$ and $AB$. Prove that $\angle BFM = \angle BME$.