Olympiad problems

2001 Switzerland Team Selection Test, 7

Let $ABC$ be an acute-angled triangle with circumcenter $O$. The circle $S$ through $A,B$, and $O$ intersects $AC$ and $BC$ again at points $P$ and $Q$ respectively. Prove that $CO \perp PQ$.

2022 Kyiv City MO Round 1, Problem 3

Tags: geometry , orthocenter , circumcenter , circumcircle

Let $H$ and $O$ be the orthocenter and the circumcenter of the triangle $ABC$. Line $OH$ intersects the sides $AB, AC$ at points $X, Y$ correspondingly, so that $H$ belongs to the segment $OX$. It turned out that $XH = HO = OY$. Find $\angle BAC$. [i](Proposed by Oleksii Masalitin)[/i]

2007 Korea Junior Math Olympiad, 7

Tags: geometry , incenter , circumcenter , isosceles , isosceles triangle

Let the incircle of $\triangle ABC$ meet $BC,CA,AB$ at $J,K,L$. Let $D(\ne B, J),E(\ne C,K), F(\ne A,L)$ be points on $BJ,CK,AL$. If the incenter of $\triangle ABC$ is the circumcenter of $\triangle DEF$ and $\angle BAC = \angle DEF$, prove that $\triangle ABC$ and $\triangle DEF$ are isosceles triangles.

2020 SAFEST Olympiad, 4

Tags: geometry , equal angles , circumcenter , orthocenter

Let $O$ be the circumcenter and $H$ the orthocenter of an acute-triangle $ABC$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$. Let $L$ be the midpoint of $OH$. Prove that $\angle OAH = \angle LSA$.

2007 Sharygin Geometry Olympiad, 2

Tags: geometry , circumcenter , concurrency , altitude

Points $A', B', C'$ are the feet of the altitudes $AA', BB'$ and $CC'$ of an acute triangle $ABC$. A circle with center $B$ and radius $BB'$ meets line $A'C'$ at points $K$ and $L$ (points $K$ and $A$ are on the same side of line $BB'$). Prove that the intersection point of lines $AK$ and $CL$ belongs to line $BO$ ($O$ is the circumcenter of triangle $ABC$).

2021 Sharygin Geometry Olympiad, 9.5

Tags: concyclic , circumcenter , geometry

Let $O$ be the clrcumcenter of triangle $ABC$. Points $X$ and $Y$ on side $BC$ are such that $AX = BX$ and $AY = CY$. Prove that the circumcircle of triangle $AXY$ passes through the circumceuters of triangles $AOB$ and $AOC$.

2016 Croatia Team Selection Test, Problem 3

Tags: circumcenter , midpoint , circumcircle , geometry

Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.

OIFMAT II 2012, 4

Tags: geometry , circumcenter , orthocenter , equal segments

Given a $ \vartriangle ABC $ with $ AB> AC $ and $ \angle BAC = 60^o$. Denote the circumcenter and orthocenter as $ O $ and $ H $ respectively. We also have that $ OH $ intersects $ AB $ in $ P $ and $ AC $ in $ Q $. Prove that $ PO = HQ $.

1940 Moscow Mathematical Olympiad, 068

Tags: triangle construction , geometric construction , circumcenter , geometry

The center of the circle circumscribing $\vartriangle ABC$ is mirrored through each side of the triangle and three points are obtained: $O_1, O_2, O_3$. Reconstruct $\vartriangle ABC$ from $O_1, O_2, O_3$ if everything else is erased.

2020 China Northern MO, P2

Tags: geometry , circumcenter , incenter

In $\triangle ABC$, $AB>AC$. Let $O$ and $I$ be the circumcenter and incenter respectively. Prove that if $\angle AIO = 30^{\circ}$, then $\angle ABC = 60^{\circ}$.

2004 India IMO Training Camp, 1

Tags: excenter , circumcircle , circumcenter , triangle , geometry

Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$. [i]Proposed by C.R. Pranesachar, India [/i]

2001 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: equal segments , circumcenter , symmetry , perpendicular , geometry

Let $ABC$ be an acute triangle and $A_1, B_1$ and $C_1$, points on the sides $BC, CA$ and $AB$, respectively, such that $CB_1 = A_1B_1$ and $BC_1 = A_1C_1$. Let $D$ be the symmetric of $A_1$ with respect to $B_1C_1, O$ and $O_1$ are the circumcenters of triangles $ABC$ and $A_1B_1C_1$, respectively. If $A \ne D, O \ne O_1$ and $AD$ is perpendicular to $OO_1$, prove that $AB = AC$.

2009 Postal Coaching, 3

Tags: altitude , concyclic , circumcenter , incenter , geometry

Let $ABC$ be a triangle with circumcentre $O$ and incentre $I$ such that $O$ is different from $I$. Let $AK, BL, CM$ be the altitudes of $ABC$, let $U, V , W$ be the mid-points of $AK, BL, CM$ respectively. Let $D, E, F$ be the points at which the in-circle of $ABC$ respectively touches the sides $BC, CA, AB$. Prove that the lines $UD, VE, WF$ and $OI$ are concurrent.

2023 Yasinsky Geometry Olympiad, 5

Tags: geometry , circumcenter , construction

Let $ABC$ be a scalene triangle. Given the center $I$ of the inscribe circle and the points $K_1$, $K_2$ and $K_3$ where the inscribed circle is tangent to the sides $BC$, $AC$ and $AB$. Using only a ruler, construct the center of the circumscribed circle of triangle $ABC$. (Hryhorii Filippovskyi)

2021-IMOC qualification, G1

Tags: projection , geometry , circumcenter , incenter , circumcircle

Let $O$ be the circumcenter and $I$ be the incenter of $\vartriangle$, $P$ is the reflection from $I$ through $O$, the foot of perpendicular from $P$ to $BC,CA,AB$ is $X,Y,Z$, respectively. Prove that $AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2$.

2006 Sharygin Geometry Olympiad, 22

Tags: geometry , fixed , circumcenter , locus

Given points $A, B$ on a circle and a point $P$ not lying on the circle. $X$ is an arbitrary point of the circle, $Y$ is the intersection point of lines $AX$ and $BP$. Find the locus of the centers of the circles circumscribed around the triangles $PXY$.