Found problems: 3882
2012 Online Math Open Problems, 11
Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$. Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$. If $\angle MPN = 40^\circ$, find the degree measure of $\angle BPC$.
[i]Ray Li.[/i]
2011 AIME Problems, 10
A circle with center $O$ has radius 25. Chord $\overline{AB}$ of length 30 and chord $\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder where $m+n$ is divided by 1000.
2017 Korea - Final Round, 1
A acute triangle $\triangle ABC$ has circumcenter $O$. The circumcircle of $OAB$, called $O_1$, and the circumcircle of $OAC$, called $O_2$, meets $BC$ again at $D ( \not=B )$ and $E ( \not= C )$ respectively. The perpendicular bisector of $BC$ hits $AC$ again at $F$. Prove that the circumcenter of $\triangle ADE$ lies on $AC$ if and only if the centers of $O_1, O_2$ and $F$ are colinear.
1981 IMO Shortlist, 14
Prove that a convex pentagon (a five-sided polygon) $ABCDE$ with equal sides and for which the interior angles satisfy the condition $\angle A \geq \angle B \geq \angle C \geq \angle D \geq \angle E$ is a regular pentagon.
2013 Korea National Olympiad, 1
Let $P$ be a point on segment $BC$. $Q, R$ are points on $AC, AB$ such that $PQ \parallel AB $ and $ PR \parallel AC$. $O, O_{1}, O_{2} $ are the circumcenters of triangle $ABC, BPR, PCQ$. The circumcircles of $BPR, PCQ $ meet at point $K (\ne P)$. Prove that $OO_{1} = KO_{2} $.
2006 Switzerland Team Selection Test, 2
Let $D$ be inside $\triangle ABC$ and $E$ on $AD$ different to $D$. Let $\omega_1$ and $\omega_2$ be the circumscribed circles of $\triangle BDE$ and $\triangle CDE$ respectively. $\omega_1$ and $\omega_2$ intersect $BC$ in the interior points $F$ and $G$ respectively. Let $X$ be the intersection between $DG$ and $AB$ and $Y$ the intersection between $DF$ and $AC$. Show that $XY$ is $\|$ to $BC$.
2013 Sharygin Geometry Olympiad, 6
The altitudes $AA_1, BB_1, CC_1$ of an acute triangle $ABC$ concur at $H$. The perpendicular lines from $H$ to $B_1C_1, A_1C_1$ meet rays $CA, CB$ at $P, Q$ respectively. Prove that the line from $C$ perpendicular to $A_1B_1$ passes through the midpoint of $PQ$.
1996 Moldova Team Selection Test, 10
Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$.
[b]I.[/b] Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$.
[b]II.[/b] Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.
2020 USA TSTST, 2
Let $ABC$ be a scalene triangle with incenter $I$. The incircle of $ABC$ touches $\overline{BC},\overline{CA},\overline{AB}$ at points $D,E,F$, respectively. Let $P$ be the foot of the altitude from $D$ to $\overline{EF}$, and let $M$ be the midpoint of $\overline{BC}$. The rays $AP$ and $IP$ intersect the circumcircle of triangle $ABC$ again at points $G$ and $Q$, respectively. Show that the incenter of triangle $GQM$ coincides with $D$.
[i]Zack Chroman and Daniel Liu[/i]
2016 China Girls Math Olympiad, 2
In $\triangle ABC, BC=a, CA=b, AB=c,$ and $\Gamma$ is its circumcircle.
$(1)$ Determine a necessary and sufficient condition on $a,b$ and $c$ if there exists a unique point $P(P\neq B, P\neq C)$ on the arc $BC$ of $\Gamma$ not passing through point $A$ such that $PA=PB+PC$.
$(2)$ Let $P$ be the unique point stated in $(1)$. If $AP$ bisects $BC$, prove that $\angle BAC<60^{\circ}$.
2008 Finnish National High School Mathematics Competition, 2
The incentre of the triangle $ABC$ is $I.$ The lines $AI, BI$ and $CI$ meet the circumcircle of the triangle $ABC$ also at points $D, E$ and $F,$ respectively.
Prove that $AD$ and $EF$ are perpendicular.
1998 Belarus Team Selection Test, 4
The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$
2013 Uzbekistan National Olympiad, 4
Let circles $ \Gamma $ and $ \omega $ are circumcircle and incircle of the triangle $ABC$, the incircle touches sides $BC,CA,AB$ at the points $A_1,B_1,C_1$. Let $A_2$ and $B_2$ lies the lines $A_1I$ and $B_1I$ ($A_1$ and $A_2$ lies different sides from $I$, $B_1$ and $B_2$ lies different sides from $I$) such that $IA_2=IB_2=R$. Prove that :
(a) $AA_2=BB_2=IO$;
(b) The lines $AA_2$ and $BB_2$ intersect on the circle $ \Gamma ;$
2006 Iran MO (3rd Round), 3
In triangle $ABC$, if $L,M,N$ are midpoints of $AB,AC,BC$. And $H$ is orthogonal center of triangle $ABC$, then prove that \[LH^{2}+MH^{2}+NH^{2}\leq\frac14(AB^{2}+AC^{2}+BC^{2})\]
2024 Saint Petersburg Mathematical Olympiad, 3
On the side $BC$ of acute triangle $ABC$ point $P$ was chosen. Point $E$ is symmetric to point $B$ onto line $AP$. Segment $PE$ meets circumcircle of triangle $ABP$ in point $D$. $M$ is midpoint of side $AC$. Prove that $DE+AC>2BM$.
2024 Korea Junior Math Olympiad, 3
Acute triangle $ABC$ satisfies $\angle A > \angle C$. Let $D, E, F$ be the points that the triangle's incircle intersects with $BC, CA, AB$, respectively, and $P$ some point on $AF$ different from $F$. The angle bisector of $\angle ABC$ meets $PQR$'s circumcircle $O$ at $L, R$. $L$ is the point closer to $B$ than $R$. $O$ meets $DF, DR$ at point $Q(\neq F, L), S(\neq R)$ respectively, and $PS$ hits segment $BC$ at $T$. Show that $T, Q, L$ are collinear.
2009 Indonesia TST, 4
Given triangle $ ABC$. Let the tangent lines of the circumcircle of $ AB$ at $ B$ and $ C$ meet at $ A_0$. Define $ B_0$ and $ C_0$ similarly.
a) Prove that $ AA_0,BB_0,CC_0$ are concurrent.
b) Let $ K$ be the point of concurrency. Prove that $ KG\parallel BC$ if and only if $ 2a^2\equal{}b^2\plus{}c^2$.
2017 Lusophon Mathematical Olympiad, 6
Let ABC be a scalene triangle. Consider points D, E, F on segments AB, BC, CA, respectively, such that $\overline{AF}$=$\overline{DF}$ and $\overline{BE}$=$\overline{DE}$.
Show that the circumcenter of ABC lies on the circumcircle of CEF.
2024 Turkey EGMO TST, 1
Let $ABC$ be a triangle and its circumcircle be $\omega$. Let $I$ be the incentre of the $ABC$. Let the line $BI$ meet $AC$ at $E$ and $\omega$ at $M$ for the second time. The line $CI$ meet $AB$ at $F$ and $\omega$ at $N$ for the second time. Let the circumcircles of $BFI$ and $CEI$ meet again at point $K$. Prove that the lines $BN$, $CM$, $AK$ are concurrent.
2007 All-Russian Olympiad Regional Round, 9.6
Given a triangle. A variable poin $ D$ is chosen on side $ BC$. Points $ K$ and $ L$ are the incenters of triangles $ ABD$ and $ ACD$, respectively. Prove that the second intersection point of the circumcircles of triangles $ BKD$ and $ CLD$ moves along on a fixed circle (while $ D$ moves along segment $ BC$).
Cono Sur Shortlist - geometry, 2012.G6.6
6. Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$. Let $M$ and $N$, respectively, be variable points of the sides $AB$ and $AC$, different from $A$, such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$. Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$.
2017 Iranian Geometry Olympiad, 5
Let $X,Y$ be two points on the side $BC$ of triangle $ABC$ such that $2XY=BC$ ($X$ is between $B,Y$). Let $AA'$ be the diameter of the circumcirle of triangle $AXY$. Let $P$ be the point where $AX$ meets the perpendicular from $B$ to $BC$, and $Q$ be the point where $AY$ meets the perpendicular from $C$ to $BC$. Prove that the tangent line from $A'$ to the circumcircle of $AXY$ passes through the circumcenter of triangle $APQ$.
[i]Proposed by Iman Maghsoudi[/i]
2025 Philippine MO, P7
In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$.
(a) Show that $AP$ and $BR$ are perpendicular. \\
(b) Show that $FM$ and $BM$ are perpendicular.
2015 Indonesia MO Shortlist, G6
Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.
2003 German National Olympiad, 2
There are four circles $k_1 , k_2 , k_3$ and $k_4$ of equal radius inside the triangle $ABC$. The circle $k_1$ touches the sides $AB, CA$ and the circle $k_4 $, $k_2$ touches the sides $AB,BC$ and $k_4$, and $k_3$ touches the sides $AC, BC$ and $k_4.$ Prove that the center of $k_4$ lies on the line connecting the incenter and circumcenter of $ABC.$