Olympiad problems

2014 Oral Moscow Geometry Olympiad, 1

In triangle $ABC, \angle A= 45^o, BH$ is the altitude, the point $K$ lies on the $AC$ side, and $BC = CK$. Prove that the center of the circumscribed circle of triangle $ABK$ coincides with the center of an excircle of triangle $BCH$.

2007 Sharygin Geometry Olympiad, 3

Tags: geometry , Circumcenter , Locus , Locus problems , circumcircle , circles

Given two circles intersecting at points $P$ and $Q$. Let C be an arbitrary point distinct from $P$ and $Q$ on the former circle. Let lines $CP$ and $CQ$ intersect again the latter circle at points A and B, respectively. Determine the locus of the circumcenters of triangles $ABC$.

Ukrainian From Tasks to Tasks - geometry, 2013.13

Tags: angles , Circumcenter , geometry

In the quadrilateral $ABCD$ it is known that $ABC + DBC = 180^o$ and $ADC + BDC = 180^o$. Prove that the center of the circle circumscribed around the triangle $BCD$ lies on the diagonal $AC$.

2001 Switzerland Team Selection Test, 7

Tags: geometry , Circumcenter , perpendicular

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 $CO \perp PQ$.

2016 Saudi Arabia GMO TST, 1

Tags: geometry , orthocenter , midpoint , parallel , Circumcenter

Let $ABC$ be an acute, non-isosceles triangle which is inscribed in a circle $(O)$. A point $I$ belongs to the segment $BC$. Denote by $H$ and $K$ the projections of $I$ on $AB$ and $AC$, respectively. Suppose that the line $HK $ intersects $(O)$ at $M, N$ ($H$ is between $M, K$ and $K$ is between $H, N$). Let $X, Y$ be the centers of the circles $(ABK),(ACH)$ respectively. Prove the following assertions: a) If $I$ is the projection of $A$ on $BC$, then $A$ is the center of circle $(IMN)$. b) If $XY\parallel BC$, then the orthocenter of $XOY$ is the midpoint of $IO$.

2010 Bundeswettbewerb Mathematik, 3

Tags: geometry , similar triangles , Circumcenter , circumcircle

On the sides of a triangle $XYZ$ to the outside construct similar triangles $YDZ, EXZ ,YXF$ with circumcenters $K, L ,M$ respectively. Here are $\angle ZDY = \angle ZXE = \angle FXY$ and $\angle YZD = \angle EZX = \angle YFX$. Show that the triangle $KLM$ is similar to the triangles . [img]https://cdn.artofproblemsolving.com/attachments/e/f/fe0d0d941015d32007b6c00b76b253e9b45ca5.png[/img]

2004 Estonia Team Selection Test, 2

Tags: geometry , parallel , equal segments , Circumcenter

Let $O$ be the circumcentre of the acute triangle $ABC$ and let lines $AO$ and $BC$ intersect at point $K$. On sides $AB$ and $AC$, points $L$ and $M$ are chosen such that $|KL|= |KB|$ and $|KM| = |KC|$. Prove that segments $LM$ and $BC$ are parallel.

1940 Moscow Mathematical Olympiad, 068

Tags: geometry , Triangle construction , geometric construction , Circumcenter

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.

2019 Girls in Mathematics Tournament, 2

Tags: geometry , incenter , right triangle , Circumcenter

Let $ABC$ be a right triangle with hypotenuse $BC$ and center $I$. Let bisectors of the angles $\angle B$ and $\angle C$ intersect the sides $AC$ and $AB$ in$ D$ and $E$, respectively. Let $P$ and $Q$ be the feet of the perpendiculars of the points $D$ and $E$ on the side $BC$. Prove that $I$ is the circumcenter of $APQ$.

2013 Korea Junior Math Olympiad, 5

Tags: geometry , circumcircle , acute , Circumcenter , perpendicular , midpoint

In an acute triangle $\triangle ABC, \angle A > \angle B$. Let the midpoint of $AB$ be $D$, and let the foot of the perpendicular from $A$ to $BC$ be $E$, and $B$ from $CA$ be $F$. Let the circumcenter of $\triangle DEF$ be $O$. A point $J$ on segment $BE$ satisfies $\angle ODC = \angle EAJ$. Prove that $AJ \cap DC$ lies on the circumcircle of $\triangle BDE$.

2017 Brazil National Olympiad, 5.

Tags: geometry , incenter , orthocenter , Circumcenter , angle bisector , Brazilian Math Olympiad , Brazilian Math Olympiad 2017

[b]5.[/b] In triangle $ABC$, let $r_A$ be the line that passes through the midpoint of $BC$ and is perpendicular to the internal bisector of $\angle{BAC}$. Define $r_B$ and $r_C$ similarly. Let $H$ and $I$ be the orthocenter and incenter of $ABC$, respectively. Suppose that the three lines $r_A$, $r_B$, $r_C$ define a triangle. Prove that the circumcenter of this triangle is the midpoint of $HI$.

2024 Brazil Team Selection Test, 2

Tags: geometry , Circumcenter , midpoint , tangent , isosceles

Let $ ABC $ be an acute-angled scalene triangle with circumcenter $ O $. Denote by $ M $, $ N $, and $ P $ the midpoints of sides $ BC $, $ CA $, and $ AB $, respectively. Let $ \omega $ be the circle passing through $ A $ and tangent to $ OM $ at $ O $. The circle $ \omega $ intersects $ AB $ and $ AC $ at points $ E $ and $ F $, respectively (where $ E $ and $ F $ are distinct from $ A $). Let $ I $ be the midpoint of segment $ EF $, and let $ K $ be the intersection of lines $ EF $ and $ NP $. Prove that $ AO = 2IK $ and that triangle $ IMO $ is isosceles.

2000 Estonia National Olympiad, 3

Tags: geometry , angle bisector , orthocenter , Circumcenter

Let $ABC$ be an acute-angled triangle with $\angle ACB = 60^o$ , and its heights $AD$ and $BE$ intersect at point $H$. Prove that the circumcenter of triangle $ABC$ lies on a line bisecting the angles $AHE$ and $BHD$.

2019 Tuymaada Olympiad, 2

Tags: geometry , trapezoid , Circumcenter , midpoint , distance

A trapezoid $ABCD$ with $BC // AD$ is given. The points $B'$ and $C'$ are symmetrical to $B$ and $C$ with respect to $CD$ and $AB$, respectively. Prove that the midpoint of the segment joining the circumcentres of $ABC'$ and $B'CD$ is equidistant from $A$ and $D$.

1992 IMO Longlists, 81

Tags: geometry , circumcircle , orthocenter , Circumcenter , IMO Shortlist , IMO Longlist

Suppose that points $X, Y,Z$ are located on sides $BC, CA$, and $AB$, respectively, of triangle $ABC$ in such a way that triangle $XY Z$ is similar to triangle $ABC$. Prove that the orthocenter of triangle $XY Z$ is the circumcenter of triangle $ABC.$

2019 Irish Math Olympiad, 5

Tags: geometry , reflection , Concyclic , Circumcenter

Let $M$ be a point on the side $BC$ of triangle $ABC$ and let $P$ and $Q$ denote the circumcentres of triangles $ABM$ and $ACM$ respectively. Let $L$ denote the point of intersection of the extended lines $BP$ and $CQ$ and let $K$ denote the reflection of $L$ through the line $PQ$. Prove that $M, P, Q$ and $K$ all lie on the same circle.

2000 Tournament Of Towns, 2

Tags: geometry , Chords , equal segments , Circumcenter , circumcircle

The chords $AC$ and $BD$ of a, circle with centre $O$ intersect at the point $K$. The circumcentres of triangles $AKB$ and $CKD$ are $M$ and $N$ respectively. Prove that $OM = KN$. (A Zaslavsky )

2015 IMO, 4

Tags: geometry , IMO Shortlist , IMO , IMO 2015 , circumcircle , Circumcenter , Triangle

Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$. Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$. [i]Proposed by Greece[/i]

2008 Greece JBMO TST, 1

Tags: geometry , Circumcenter , tangent , isosceles , circle

Given a point $A$ that lies on circle $c(o,R)$ (with center $O$ and radius $R$). Let $(e)$ be the tangent of the circle $c$ at point $A$ and a line $(d)$ that passes through point $O$ and intersects $(e)$ at point $M$ and the circle at points $B,C$ (let $B$ lie between $O$ and $A$). If $AM = R\sqrt3$ , prove that a) triangle $AMC$ is isosceles. b) circumcenter of triangle $AMC$ lies on circle $c$ .

2009 Ukraine Team Selection Test, 5

Tags: geometry , circumcircle , Circumcenter , right triangle , orthocenter

Let $A,B,C,D,E$ be consecutive points on a circle with center $O$ such that $AC=BD=CE=DO$. Let $H_1,H_2,H_3$ be the orthocenters triangles $ACD,BCD,BCE$ respectively. Prove that the triangle $H_1H_2H_3$ is right.

2016 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: geometry , perpendicular , midpoints , Circumcenter , orthocenter

Let $A B C$ be an acute-angled triangle of circumcenter $O$ and orthocenter $H$. Let $M$ be the midpoint of $BC, N$ be the symmetric of $H$ with respect to $A, P$ be the midpoint of $NM$ and $X$ be a point on the line A H such that $MX$ is parallel to $CH$. Prove that $BX$ and $OP$ are perpendicular.

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 $\angle ABC = 60^{\circ}$.

2019 Saudi Arabia Pre-TST + Training Tests, 2.3

Tags: geometry , circumcircle , Euler Circle , Circumcenter , orthocenter

Let $ABC$ be an acute, non isosceles triangle with $O,H$ are circumcenter and orthocenter, respectively. Prove that the nine-point circles of $AHO,BHO,CHO$ has two common points.

2018 Saudi Arabia GMO TST, 3

Tags: geometry , incenter , Circumcenter , orthocenter

Let $I, O$ be the incenter, circumcenter of triangle $ABC$ and $A_1, B_1, C_1 $be arbitrary points on the segments $AI, BI, CI$ respectively. The perpendicular bisectors of $AA_1, BB_1, CC_1$ intersect each other at $X, Y$ and $Z$. Prove that the circumcenter of triangle $XYZ$ coincides with $O$ if and only if $I$ is the orthocenter of triangle $A_1B_1C_1$

2022 MMATHS, 10

Tags: geometry , circumcircle , Circumcenter , MMATHS

Suppose that $A_1A_2A_3$ is a triangle with $A_1A_2 = 16$ and $A_1A_3 = A_2A_3 = 10$. For each integer $n \ge 4$, set An to be the circumcenter of triangle $A_{n-1}A_{n-2}A_{n-3}$. There exists a unique point $Z$ lying in the interiors of the circumcircles of triangles $A_kA_{k+1}A_{k+2}$ for all integers $k \ge 1$. If $ZA^2_1+ ZA^2_2+ ZA^2_3+ ZA^2_4$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, find $a + b$.