Olympiad problems

2022 Irish Math Olympiad, 5

5. Let $\triangle$ABC be a triangle with circumcentre [i]O[/i]. The perpendicular line from [i]O[/i] to [i]BC[/i] intersects line [i]BC[/i] at [i]M[/i] and line [i]AC[/i] at [i]P[/i], and the perpendicular line from [i]O[/i] to [i]AC[/i] intersects line [i]AC[/i] at [i]N[/i] and line [i]BC[/i] at [i]Q[/i]. Let [i]D[/i] be the intersection point of lines [i]PQ[/i] and [i]MN[/i]. construct the parallelogram [i]PCQJ[/i] with [i]PJ[/i] || [i]CQ[/i] and [i]QJ[/i] || [i]CP[/i]. Prove the following: a) The points [i]A[/i], [i]B[/i], [i]O[/i], [i]P[/i], [i]Q[/i], [i]J[/i] are all on the same circle. b) line [i]OD[/i] is perpendicular to line [i]CJ[/i].

2017-IMOC, G7

Tags: concyclic , equal angles , circumcircle , geometry

Given $\vartriangle ABC$ with circumcenter $O$. Let $D$ be a point satisfying $\angle ABD = \angle DCA$ and $M$ be the midpoint of $AD$. Suppose that $BM,CM$ intersect circle $(O)$ at another points $E, F$, respectively. Let $P$ be a point on $EF$ so that $AP$ is tangent to circle $(O)$. Prove that $A, P,M,O$ are concyclic. [img]https://2.bp.blogspot.com/-gSgUG6oywAU/XnSKTnH1yqI/AAAAAAAALdw/3NuPFuouCUMO_6KbydE-KIt6gCJ4OgWdACK4BGAYYCw/s320/imoc2017%2Bg7.png[/img]

2009 Germany Team Selection Test, 3

Tags: reflection , circumcircle , geometry

In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$. [i]Proposed by Davood Vakili, Iran[/i]

2007 CHKMO, 3

Tags: symmetry , radical axis , power of a point , circumcircle , geometry

A convex quadrilateral $ABCD$ with $AC \neq BD$ is inscribed in a circle with center $O$. Let $E$ be the intersection of diagonals $AC$ and $BD$. If $P$ is a point inside $ABCD$ such that $\angle PAB+\angle PCB=\angle PBC+\angle PDC=90^\circ$, prove that $O$, $P$ and $E$ are collinear.

2006 Bulgaria Team Selection Test, 1

Tags: symmetry , power of a point , angle bisector , parallelogram , circumcircle , geometry

[b]Problem 1.[/b] Points $D$ and $E$ are chosen on the sides $AB$ and $AC$, respectively, of a triangle $\triangle ABC$ such that $DE\parallel BC$. The circumcircle $k$ of triangle $\triangle ADE$ intersects the lines $BE$ and $CD$ at the points $M$ and $N$ (different from $E$ and $D$). The lines $AM$ and $AN$ intersect the side $BC$ at points $P$ and $Q$ such that $BC=2\cdot PQ$ and the point $P$ lies between $B$ and $Q$. Prove that the circle $k$ passes through the point of intersection of the side $BC$ and the angle bisector of $\angle BAC$. [i]Nikolai Nikolov[/i]

2017 CCA Math Bonanza, I15

Tags: circumcircle , geometry

Let $ABC$, $AB<AC$ be an acute triangle inscribed in circle $\Gamma$ with center $O$. The altitude from $A$ to $BC$ intersects $\Gamma$ again at $A_1$. $OA_1$ intersects $BC$ at $A_2$ Similarly define $B_1$, $B_2$, $C_1$, and $C_2$. Then $B_2C_2=2\sqrt{2}$. If $B_2C_2$ intersects $AA_2$ at $X$ and $BC$ at $Y$, then $XB_2=2$ and $YB_2=k$. Find $k^2$. [i]2017 CCA Math Bonanza Individual Round #15[/i]

2019 Oral Moscow Geometry Olympiad, 3

Tags: concyclic , circumcenter , circumcircle , geometry , orthocenter

In the acute triangle $ABC, \angle ABC = 60^o , O$ is the center of the circumscribed circle and $H$ is the orthocenter. The angle bisector $BL$ intersects the circumscribed circle at the point $W, X$ is the intersection point of segments $WH$ and $AC$ . Prove that points $O, L, X$ and $H$ lie on the same circle.

2002 Bulgaria National Olympiad, 2

Tags: pythagorean theorem , circumcircle , inradius , incenter , geometry

Consider the orthogonal projections of the vertices $A$, $B$ and $C$ of triangle $ABC$ on external bisectors of $ \angle ACB$, $ \angle BAC$ and $ \angle ABC$, respectively. Prove that if $d$ is the diameter of the circumcircle of the triangle, which is formed by the feet of projections, while $r$ and $p$ are the inradius and the semiperimeter of triangle $ABC$, prove that $r^2+p^2=d^2$ [i]Proposed by Alexander Ivanov[/i]

2023 Indonesia TST, G

Tags: geometry , concurrency , circumcircle

Given an acute triangle $ABC$ with altitudes $AD$ and $BE$ intersecting at $H$, $M$ is the midpoint of $AB$. A nine-point circle of $ABC$ intersects with a circumcircle of $ABH$ on $P$ and $Q$ where $P$ lays on the same side of $A$ (with respect to $CH$). Prove that $ED, PH, MQ$ are concurrent on circumcircle $ABC$

2012 Dutch BxMO/EGMO TST, 2

Tags: geometry , circles , concyclic , circumcircle

Let $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.

2009 South East Mathematical Olympiad, 2

Tags: trapezoid , circumcircle , geometry

In the convex pentagon $ABCDE$ we know that $AB=DE, BC=EA$ but $AB \neq EA$. $B,C,D,E$ are concyclic . Prove that $A,B,C,D$ are concyclic if and only if $AC=AD.$

2004 Postal Coaching, 1

Tags: circumcircle , geometry , cyclic quadrilateral

Let $ABC$ and $DEF$ be two triangles such that $A+ D = 120^{\circ}$ and $B+E = 120^{\circ}$. Suppose they have the same circumradius. Prove that they have the same 'Fermat length'.

2012 Balkan MO Shortlist, G1

Tags: circumcircle , reflection , geometry

Let $A$, $B$ and $C$ be points lying on a circle $\Gamma$ with centre $O$. Assume that $\angle ABC > 90$. Let $D$ be the point of intersection of the line $AB$ with the line perpendicular to $AC$ at $C$. Let $l$ be the line through $D$ which is perpendicular to $AO$. Let $E$ be the point of intersection of $l$ with the line $AC$, and let $F$ be the point of intersection of $\Gamma$ with $l$ that lies between $D$ and $E$. Prove that the circumcircles of triangles $BFE$ and $CFD$ are tangent at $F$.

2013 Online Math Open Problems, 46

Tags: circumcircle , geometric transformation , ratio , rhombus , homothety , geometry

Let $ABC$ be a triangle with $\angle B - \angle C = 30^{\circ}$. Let $D$ be the point where the $A$-excircle touches line $BC$, $O$ the circumcenter of triangle $ABC$, and $X,Y$ the intersections of the altitude from $A$ with the incircle with $X$ in between $A$ and $Y$. Suppose points $A$, $O$ and $D$ are collinear. If the ratio $\frac{AO}{AX}$ can be expressed in the form $\frac{a+b\sqrt{c}}{d}$ for positive integers $a,b,c,d$ with $\gcd(a,b,d)=1$ and $c$ not divisible by the square of any prime, find $a+b+c+d$. [i]James Tao[/i]

2004 Pan African, 3

Tags: circumcircle , geometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral such that $AB$ is a diameter of it's circumcircle. Suppose that $AB$ and $CD$ intersect at $I$, $AD$ and $BC$ at $J$, $AC$ and $BD$ at $K$, and let $N$ be a point on $AB$. Show that $IK$ is perpendicular to $JN$ if and only if $N$ is the midpoint of $AB$.

2009 Singapore Team Selection Test, 2

Tags: geometry , circumcircle

Let $H$ be the orthocentre of $\triangle ABC$ and let $P$ be a point on the circumcircle of $\triangle ABC$, distinct from $A,B,C$. Let $E$ and $F$ be the feet of altitudes from $H$ onto $AC$ and $AB$ respectively. Let $PAQB$ and $PARC$ be parallelograms. Suppose $QA$ meets $RH$ at $X$ and $RA$ meets $QH$ at $Y$. Prove that $XE$ is parallel to $YF$.

2006 Bulgaria Team Selection Test, 1

Tags: circumcircle , geometry

[b]Problem 4.[/b] Let $k$ be the circumcircle of $\triangle ABC$, and $D$ the point on the arc $\overarc{AB},$ which do not pass through $C$. $I_A$ and $I_B$ are the centers of incircles of $\triangle ADC$ and $\triangle BDC$, respectively. Proove that the circumcircle of $\triangle I_AI_BC$ touches $k$ iff \[ \frac{AD}{BD}=\frac{AC+CD}{BC+CD}. \] [i] Stoyan Atanasov[/i]

2008 National Olympiad First Round, 1

Tags: geometry , circumcircle

Let $AD$ be a median of $\triangle ABC$ such that $m(\widehat{ADB})=45^{\circ}$ and $m(\widehat{ACB})=30^{\circ}$. What is the measure of $\widehat{ABC}$ in degrees? $ \textbf{(A)}\ 75 \qquad\textbf{(B)}\ 90 \qquad\textbf{(C)}\ 105 \qquad\textbf{(D)}\ 120 \qquad\textbf{(E)}\ 135 $

2022 Caucasus Mathematical Olympiad, 6

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle. Let $P$ be a point on the circle $(ABC)$, and $Q$ be a point on the segment $AC$ such that $AP\perp BC$ and $BQ\perp AC$. Lot $O$ be the circumcenter of triangle $APQ$. Find the angle $OBC$.

2012 Spain Mathematical Olympiad, 3

Tags: geometry , circumcircle

Let $ABC$ be an acute-angled triangle. Let $\omega$ be the inscribed circle with centre $I$, $\Omega$ be the circumscribed circle with centre $O$ and $M$ be the midpoint of the altitude $AH$ where $H$ lies on $BC$. The circle $\omega$ be tangent to the side $BC$ at the point $D$. The line $MD$ cuts $\omega$ at a second point $P$ and the perpendicular from $I$ to $MD$ cuts $BC$ at $N$. The lines $NR$ and $NS$ are tangent to the circle $\Omega$ at $R$ and $S$ respectively. Prove that the points $R,P,D$ and $S$ lie on the same circle.

2014 Online Math Open Problems, 17

Tags: circumcircle , ratio , analytic geometry , homothety , geometry , geometric transformation

Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$ [i]Proposed by Ray Li[/i]

2015 APMO, 1

Tags: circumcircle , geometry

Let $ABC$ be a triangle, and let $D$ be a point on side $BC$. A line through $D$ intersects side $AB$ at $X$ and ray $AC$ at $Y$ . The circumcircle of triangle $BXD$ intersects the circumcircle $\omega$ of triangle $ABC$ again at point $Z$ distinct from point $B$. The lines $ZD$ and $ZY$ intersect $\omega$ again at $V$ and $W$ respectively. Prove that $AB = V W$ [i]Proposed by Warut Suksompong, Thailand[/i]

2016 Middle European Mathematical Olympiad, 5

Tags: circumcircle , law of cosines , geometry

Let $ABC$ be an acute triangle for which $AB \neq AC$, and let $O$ be its circumcenter. Line $AO$ meets the circumcircle of $ABC$ again in $D$, and the line $BC$ in $E$. The circumcircle of $CDE$ meets the line $CA$ again in $P$. The lines $PE$ and $AB$ intersect in $Q$. Line passing through $O$ parallel to the line $PE$ intersects the $A$-altitude of $ABC$ in $F$. Prove that $FP = FQ$.

2011 Greece JBMO TST, 4

Tags: circles , circumcircle , geometry , collinear

Let $ABC$ be an acute and scalene triangle with $AB<AC$, inscribed in a circle $c(O,R)$ (with center $O$ and radius $R$). Circle $c_1(A,AB)$ intersects side $BC$ at point $E$ and circle $c$ at point $F$. $EF$ intersects for the second time circle $c$ at point $D$ and side $AC$ at point $M$. $AD$ intersects $BC$ at point $K$. Circumcircle of triangle $BKD$ intersects $AB$ at point $L$ . Prove that points $K,L,M$ lie on a line parallel to $BF$.

2011 CentroAmerican, 2

Tags: circumcircle , reflection , geometric transformation , geometry

In a scalene triangle $ABC$, $D$ is the foot of the altitude through $A$, $E$ is the intersection of $AC$ with the bisector of $\angle ABC$ and $F$ is a point on $AB$. Let $O$ the circumcenter of $ABC$ and $X=AD\cap BE$, $Y=BE\cap CF$, $Z=CF \cap AD$. If $XYZ$ is an equilateral triangle, prove that one of the triangles $OXY$, $OYZ$, $OZX$ must be equilateral.