Olympiad problems

2020 Bulgaria Team Selection Test, 1

Tags: geometry , circumcircle

In acute triangle $\triangle ABC$, $BC>AC$, $\Gamma$ is its circumcircle, $D$ is a point on segment $AC$ and $E$ is the intersection of the circle with diameter $CD$ and $\Gamma$. $M$ is the midpoint of $AB$ and $CM$ meets $\Gamma$ again at $Q$. The tangents to $\Gamma$ at $A,B$ meet at $P$, and $H$ is the foot of perpendicular from $P$ to $BQ$. $K$ is a point on line $HQ$ such that $Q$ lies between $H$ and $K$. Prove that $\angle HKP=\angle ACE$ if and only if $\frac{KQ}{QH}=\frac{CD}{DA}$.

2002 Iran MO (3rd Round), 24

Tags: geometry , incenter , circumcircle , geometric transformation , homothety , ratio , trapezoid

$A,B,C$ are on circle $\mathcal C$. $I$ is incenter of $ABC$ , $D$ is midpoint of arc $BAC$. $W$ is a circle that is tangent to $AB$ and $AC$ and tangent to $\mathcal C$ at $P$. ($W$ is in $\mathcal C$) Prove that $P$ and $I$ and $D$ are on a line.

2002 Iran Team Selection Test, 4

Tags: circumcircle , perimeter , geometry

$O$ is a point in triangle $ABC$. We draw perpendicular from $O$ to $BC,AC,AB$ which intersect $BC,AC,AB$ at $A_{1},B_{1},C_{1}$. Prove that $O$ is circumcenter of triangle $ABC$ iff perimeter of $ABC$ is not less than perimeter of triangles $AB_{1}C_{1},BC_{1}A_{1},CB_{1}A_{1}$.

2019 Saudi Arabia BMO TST, 3

Tags: circumcircle , incenter , arc midpoint , excenter , geometry

The triangle $ABC$ ($AB > BC$) is inscribed in the circle $\Omega$. On the sides $AB$ and $BC$, the points $M$ and $N$ are chosen, respectively, so that $AM = CN$, The lines $MN$ and $AC$ intersect at point $K$. Let $P$ be the center of the inscribed circle of triangle $AMK$, and $Q$ the center of the excircle of the triangle $CNK$ tangent to side $CN$. Prove that the midpoint of the arc $ABC$ of the circle $\Omega$ is equidistant from the $P$ and $Q$.

2013 Saudi Arabia IMO TST, 3

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle, $M$ be the midpoint of $BC$ and $P$ be a point on line segment $AM$. Lines $BP$ and $CP$ meet the circumcircle of $ABC$ again at $X$ and $Y$ , respectively, and sides $AC$ at $D$ and $AB$ at $E$, respectively. Prove that the circumcircles of $AXD$ and $AYE$ have a common point $T \ne A$ on line $AM$.

2011 Serbia National Math Olympiad, 3

Tags: circumcircle , geometry

Let $H$ be orthocenter and $O$ circumcenter of an acuted angled triangle $ABC$. $D$ and $E$ are feets of perpendiculars from $A$ and $B$ on $BC$ and $AC$ respectively. Let $OD$ and $OE$ intersect $BE$ and $AD$ in $K$ and $L$, respectively. Let $X$ be intersection of circumcircles of $HKD$ and $HLE$ different than $H$, and $M$ is midpoint of $AB$. Prove that $K, L, M$ are collinear iff $X$ is circumcenter of $EOD$.

1999 IberoAmerican, 2

Tags: circumcircle , symmetry , geometry

An acute triangle $\triangle{ABC}$ is inscribed in a circle with centre $O$. The altitudes of the triangle are $AD,BE$ and $CF$. The line $EF$ cut the circumference on $P$ and $Q$. a) Show that $OA$ is perpendicular to $PQ$. b) If $M$ is the midpoint of $BC$, show that $AP^2=2AD\cdot{OM}$.

2017 Poland - Second Round, 4

Tags: geometry , circumcircle

Incircle of a triangle $ABC$ touches $AB$ and $AC$ at $D$ and $E$, respectively. Point $J$ is the excenter of $A$. Points $M$ and $N$ are midpoints of $JD$ and $JE$. Lines $BM$ and $CN$ cross at point $P$. Prove that $P$ lies on the circumcircle of $ABC$.

2021 Dutch IMO TST, 3

Tags: geometry , orthocenter , collinear , circumcircle

Let $ABC$ be an acute-angled and non-isosceles triangle with orthocenter $H$. Let $O$ be the center of the circumscribed circle of triangle $ABC$ and let $K$ be center of the circumscribed circle of triangle $AHO$. Prove that the reflection of $K$ wrt $OH$ lies on $BC$.

2017 Azerbaijan EGMO TST, 1

Tags: geometry , circumcircle , equilateral

Given an equilateral triangle $ABC$ and a point $P$ so that the distances $P$ to $A$ and to $C$ are not farther than the distances $P$ to $B$. Prove that $PB = PA + PC$ if and only if $P$ lies on the circumcircle of $\vartriangle ABC$.

2023 Kyiv City MO Round 1, Problem 3

Tags: geometry , circumcircle

You are given a right triangle $ABC$ with $\angle ACB = 90^\circ$. Let $W_A , W_B$ respectively be the midpoints of the smaller arcs $BC$ and $AC$ of the circumcircle of $\triangle ABC$, and $N_A , N_B$ respectively be the midpoints of the larger arcs $BC$ and $AC$ of this circle. Denote by $P$ and $Q$ the points of intersection of segment $AB$ with the lines $N_AW_B$ and $N_BW_A$, respectively. Prove that $AP = BQ$. [i]Proposed by Oleksiy Masalitin[/i]

1990 IMO Longlists, 96

Tags: geometry , circumcircle , orthocenter , circumcenter

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 $ABC.$

1999 Finnish National High School Mathematics Competition, 4

Tags: circumcircle , ratio , geometry

Three unit circles have a common point $O.$ The other points of (pairwise) intersection are $A, B$ and $C$. Show that the points $A, B$ and $C$ are located on some unit circle.

2015 India IMO Training Camp, 1

Tags: geometry , circumcircle , geometric transformation , reflection

Let $ABC$ be a triangle in which $CA>BC>AB$. Let $H$ be its orthocentre and $O$ its circumcentre. Let $D$ and $E$ be respectively the midpoints of the arc $AB$ not containing $C$ and arc $AC$ not containing $B$. Let $D'$ and $E'$ be respectively the reflections of $D$ in $AB$ and $E$ in $AC$. Prove that $O, H, D', E'$ lie on a circle if and only if $A, D', E'$ are collinear.

2005 Moldova Team Selection Test, 1

Tags: geometry , circumcircle , inradius , inequalities , trigonometry , trig identities , law of sines

Let $ABC$ and $A_{1}B_{1}C_{1}$ be two triangles. Prove that $\frac{a}{a_{1}}+\frac{b}{b_{1}}+\frac{c}{c_{1}}\leq\frac{3R}{2r_{1}}$, where $a = BC$, $b = CA$, $c = AB$ are the sidelengths of triangle $ABC$, where $a_{1}=B_{1}C_{1}$, $b_{1}=C_{1}A_{1}$, $c_{1}=A_{1}B_{1}$ are the sidelengths of triangle $A_{1}B_{1}C_{1}$, where $R$ is the circumradius of triangle $ABC$ and $r_{1}$ is the inradius of triangle $A_{1}B_{1}C_{1}$.

2014 PUMaC Geometry A, 8

Tags: geometry , circumcircle , trigonometry , cyclic quadrilateral

$ABCD$ is a cyclic quadrilateral with circumcenter $O$ and circumradius $7$. $AB$ intersects $CD$ at $E$, $DA$ intersects $CB$ at $F$. $OE=13$, $OF=14$. Let $\cos\angle FOE=\dfrac pq$, with $p$, $q$ coprime. Find $p+q$.

2023 4th Memorial "Aleksandar Blazhevski-Cane", P4

Tags: cyclic quadrilateral , segment sum , angle bisector , circumcenter , circumcircle , geometry

Let $ABCD$ be a cyclic quadrilateral such that $AB = AD + BC$ and $CD < AB$. The diagonals $AC$ and $BD$ intersect at $P$, while the lines $AD$ and $BC$ intersect at $Q$. The angle bisector of $\angle APB$ meets $AB$ at $T$. Show that the circumcenter of the triangle $CTD$ lies on the circumcircle of the triangle $CQD$. [i]Proposed by Nikola Velov[/i]

2020 Australian Maths Olympiad, 3

Tags: geometry , right triangle , perpendicular bisector , tangent , midpoint , circumcircle , parallel

Let $ABC$ be a triangle with $\angle ACB=90^{\circ}$. Suppose that the tangent line at $C$ to the circle passing through $A,B,C$ intersects the line $AB$ at $D$. Let $E$ be the midpoint of $CD$ and let $F$ be a point on $EB$ such that $AF$ is parallel to $CD$. Prove that the lines $AB$ and $CF$ are perpendicular.

2010 India IMO Training Camp, 7

Tags: geometry , circumcircle , geometric transformation , dilation , reflection , parallelogram , angle bisector

Let $ABCD$ be a cyclic quadrilaterla and let $E$ be the point of intersection of its diagonals $AC$ and $BD$. Suppose $AD$ and $BC$ meet in $F$. Let the midpoints of $AB$ and $CD$ be $G$ and $H$ respectively. If $\Gamma $ is the circumcircle of triangle $EGH$, prove that $FE$ is tangent to $\Gamma $.

2017 Sharygin Geometry Olympiad, P12

Tags: geometry , circumcircle

Let $AA_1 , CC_1$ be the altitudes of triangle $ABC, B_0$ the common point of the altitude from $B$ and the circumcircle of $ABC$; and $Q$ the common point of the circumcircles of $ABC$ and $A_1C_1B_0$, distinct from $B_0$. Prove that $BQ$ is the symmedian of $ABC$. [i]Proposed by D.Shvetsov[/i]

2019 Pan-African Shortlist, G4

Tags: geometry , circumcenter , equilateral triangle , circumcircle

Let $ABC$ be a triangle, and $D$, $E$, $F$ points on the segments $BC$, $CA$, and $AB$ respectively such that $$ \frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}. $$ Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral triangle.

2006 China Team Selection Test, 2

Tags: geometry , circumcircle , geometric transformation , reflection , parallelogram , ratio , analytic geometry

Let $\omega$ be the circumcircle of $\triangle{ABC}$. $P$ is an interior point of $\triangle{ABC}$. $A_{1}, B_{1}, C_{1}$ are the intersections of $AP, BP, CP$ respectively and $A_{2}, B_{2}, C_{2}$ are the symmetrical points of $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of side $BC, CA, AB$. Show that the circumcircle of $\triangle{A_{2}B_{2}C_{2}}$ passes through the orthocentre of $\triangle{ABC}$.

2009 Romania Team Selection Test, 2

Tags: euler , circumcircle , geometry

Prove that the circumcircle of a triangle contains exactly 3 points whose Simson lines are tangent to the triangle's Euler circle and these points are the vertices of an equilateral triangle.

2013 Online Math Open Problems, 32

Tags: geometry , incenter , circumcircle , inradius , trigonometry , trapezoid

In $\triangle ABC$ with incenter $I$, $AB = 61$, $AC = 51$, and $BC=71$. The circumcircles of triangles $AIB$ and $AIC$ meet line $BC$ at points $D$ ($D \neq B$) and $E$ ($E \neq C$), respectively. Determine the length of segment $DE$. [i]James Tao[/i]

2002 ITAMO, 3

Tags: circumcircle , trigonometry , perpendicular bisector , geometry

Let $A$ and $B$ are two points on a plane, and let $M$ be the midpoint of $AB$. Let $r$ be a line and let $R$ and $S$ be the projections of $A$ and $B$ onto $r$. Assuming that $A$, $M$, and $R$ are not collinear, prove that the circumcircle of triangle $AMR$ has the same radius as the circumcircle of $BSM$.