Olympiad problems

2011 India National Olympiad, 5

Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\Gamma.$ Let $E,F,G,H$ be the midpoints of arcs $AB,BC,CD,AD$ of $\Gamma,$ respectively. Suppose that $AC\cdot BD=EG\cdot FH.$ Show that $AC,BD,EG,FH$ are all concurrent.

2000 Saint Petersburg Mathematical Olympiad, 10.2

Tags: midpoint , altitude , geometry , cyclic quadrilateral

Let $AA_1$ and $BB_1$ be the altitudes of acute angled triangle $ABC$. Points $K$ and $M$ are midpoints of $AB$ and $A_1B_1$ respectively. Segments $AA_1$ and $KM$ intersect at point $L$. Prove that points $A$, $K$, $L$ and $B_1$ are noncyclic. [I]Proposed by S. Berlov[/i]

2012 Korea Junior Math Olympiad, 5

Tags: geometry , equal angles , equal segments , tangent , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral inscirbed in a circle $O$ ($AB> AD$), and let $E$ be a point on segment $AB$ such that $AE = AD$. Let $AC \cap DE = F$, and $DE \cap O = K(\ne D)$. The tangent to the circle passing through $C,F,E$ at $E$ hits $AK$ at $L$. Prove that $AL = AD$ if and only if $\angle KCE = \angle ALE$.

2010 Philippine MO, 2

Tags: geometry , cyclic quadrilateral , perimeter

On a cyclic quadrilateral $ABCD$, there is a point $P$ on side $AD$ such that the triangle $CDP$ and the quadrilateral $ABCP$ have equal perimeters and equal areas. Prove that two sides of $ABCD$ have equal lengths.

1966 IMO Shortlist, 36

Tags: geometry , complex number , cyclic quadrilateral

Let $ABCD$ be a quadrilateral inscribed in a circle. Show that the centroids of triangles $ABC,$ $CDA,$ $BCD,$ $DAB$ lie on one circle.

1986 IMO Shortlist, 18

Tags: cyclic quadrilateral , cevian triangle , circumscribed quadrilateral , geometry

Let $AX,BY,CZ$ be three cevians concurrent at an interior point $D$ of a triangle $ABC$. Prove that if two of the quadrangles $DY AZ,DZBX,DXCY$ are circumscribable, so is the third.

2011 Balkan MO, 1

Tags: trapezoid , trigonometry , cyclic quadrilateral , similar triangles , geometry

Let $ABCD$ be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at $E$. The midpoints of $AB$ and $CD$ are $F$ and $G$ respectively, and $\ell$ is the line through $G$ parallel to $AB$. The feet of the perpendiculars from E onto the lines $\ell$ and $CD$ are $H$ and $K$, respectively. Prove that the lines $EF$ and $HK$ are perpendicular.

2010 All-Russian Olympiad Regional Round, 11.3

Tags: geometry , perpendicular bisector , cyclic quadrilateral

Quadrangle $ABCD$ is inscribed in a circle with diameter $AC$. Points $K$ and $M$ are projections of vertices $A$ and $C$, respectively, onto line $BD$. A line parallel to $BC$ is drawn through point $K$ and intersecting $AC$ at point $P$. Prove that angle $KPM$ is a right angle.

2019 Regional Olympiad of Mexico West, 4

Tags: concyclic , geometry , cyclic quadrilateral

Let $ABC$ be a triangle. $M$ the midpoint of $AB$ and $L$ the midpoint of $BC$. We denote by $G$ the intersection of $AL$ with $CM$ and we take $E$ a point such that $G$ is the midpoint of the segment $AE$. Prove that the quadrilateral $MCEB$ is cyclic if and only if $MB = BG$.

2011 Sharygin Geometry Olympiad, 7

Tags: altitude , geometry , cyclic quadrilateral , circumcenter , circles , circumcircle , concurrency

Point $O$ is the circumcenter of acute-angled triangle $ABC$, points $A_1,B_1, C_1$ are the bases of its altitudes. Points $A', B', C'$ lying on lines $OA_1, OB_1, OC_1$ respectively are such that quadrilaterals $AOBC', BOCA', COAB'$ are cyclic. Prove that the circumcircles of triangles $AA_1A', BB_1B', CC_1C'$ have a common point.

2013 Iran Team Selection Test, 1

Tags: incenter , trigonometry , cyclic quadrilateral , geometry

In acute-angled triangle $ABC$, let $H$ be the foot of perpendicular from $A$ to $BC$ and also suppose that $J$ and $I$ are excenters oposite to the side $AH$ in triangles $ABH$ and $ACH$. If $P$ is the point that incircle touches $BC$, prove that $I,J,P,H$ are concyclic.

2017 Taiwan TST Round 1, 2

Tags: cyclic quadrilateral , geometry , incenter , reflection , inversion

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

2006 India National Olympiad, 5

Tags: trigonometry , inequalities , function , circumcircle , cyclic quadrilateral , geometry

In a cyclic quadrilateral $ABCD$, $AB=a$, $BC=b$, $CD=c$, $\angle ABC = 120^\circ$ and $\angle ABD = 30^\circ$. Prove that (1) $c \ge a + b$; (2) $|\sqrt{c + a} - \sqrt{c + b} | = \sqrt{c - a - b}$.

2008 Mathcenter Contest, 1

Tags: geometry , angle bisector , ratio , cyclic quadrilateral

In a triangle $ABC$, the angle bisector at $A,B,C$ meet the opposite sides at $A_1,B_1,C_1$, respectively. Prove that if the quadrilateral $BA_1B_1C_1$ is cyclic, then $$\frac{AC}{AB+BC}=\frac{AB}{AC+CB}+\frac{BC}{BA+AC}.$$

2014 AIME Problems, 15

Tags: trigonometry , analytic geometry , geometry , cyclic quadrilateral , pythagorean theorem

In $ \triangle ABC $, $ AB = 3 $, $ BC = 4 $, and $ CA = 5 $. Circle $\omega$ intersects $\overline{AB}$ at $E$ and $B$, $\overline{BC}$ at $B$ and $D$, and $\overline{AC}$ at $F$ and $G$. Given that $EF=DF$ and $\tfrac{DG}{EG} = \tfrac{3}{4}$, length $DE=\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.

2019 Pan-African Shortlist, G3

Tags: geometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral with its diagonals intersecting at $E$. Let $M$ be the midpoint of $AB$. Suppose that $ME$ is perpendicular to $CD$. Show that either $AC$ is perpendicular to $BD$, or $AB$ is parallel to $CD$.

2023-IMOC, G3

Tags: geometry , cyclic quadrilateral

$ABCD$ is a cyclic quadrilateral with circumcenter $O$. The lines $AC, BD$ intersect at $E$ and $AD, BC$ intersect at $F$. $O_1$ and $O_2$ are the circumcenters of $\triangle ABE$ and $\triangle CDE$, respectively. Assume that $(ABCD)$ and $(OO_1O_2)$ intersect at two points $P, Q$. Prove that $P, Q, F$ are collinear.

1989 Bundeswettbewerb Mathematik, 3

Tags: geometry , rectangle , cyclic quadrilateral , cyclic

Over each side of a cyclic quadrilateral erect a rectangle whose height is equal to the length of the opposite side. Prove that the centers of these rectangles form another rectangle.

2023 Germany Team Selection Test, 1

Tags: geometry , cyclic quadrilateral

Let $ABC$ be a acute angled triangle and let $AD,BE,CF$ be its altitudes. $X \not=A,B$ and $Y \not=A,C$ lie on sides $AB$ and $AC$, respectively, so that $ADXY$ is a cyclic quadrilateral. Let $H$ be the orthocenter of triangle $AXY$. Prove that $H$ lies on line $EF$.

1999 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: geometry , distance , maximum , semicircle , cyclic quadrilateral

The quadrilateral $ABCD$ is inscribed in a circle of radius $1$, so that $AB$ is a diameter of the circumference and $CD = 1$. A variable point $X$ moves along the semicircle determined by $AB$ that does not contain $C$ or $D$. Determine the position of $X$ for which the sum of the distances from $X$ to lines $BC, CD$ and $DA$ is maximum.

2023 4th Memorial "Aleksandar Blazhevski-Cane", P3

Tags: geometry , cyclic quadrilateral , complete quadrilateral , angle bisector , tangent circle

Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\omega$ with center $O$. The lines $AD$ and $BC$ meet at $E$, while the lines $AB$ and $CD$ meet at $F$. Let $P$ be a point on the segment $EF$ such that $OP \perp EF$. The circle $\Gamma_{1}$ passes through $A$ and $E$ and is tangent to $\omega$ at $A$, while $\Gamma_{2}$ passes through $C$ and $F$ and is tangent to $\omega$ at $C$. If $\Gamma_{1}$ and $\Gamma_{2}$ meet at $X$ and $Y$, prove that $PO$ is the bisector of $\angle XPY$. [i]Proposed by Nikola Velov[/i]

2018 Romania Team Selection Tests, 1

Tags: cyclic quadrilateral , incenter , geometry

Let $ABCD$ be a cyclic quadrilateral and let its diagonals $AC$ and $BD$ cross at $X$. Let $I$ be the incenter of $XBC$, and let $J$ be the center of the circle tangent to the side $BC$ and the extensions of sides $AB$ and $DC$ beyond $B$ and $C$. Prove that the line $IJ$ bisects the arc $BC$ of circle $ABCD$, not containing the vertices $A$ and $D$ of the quadrilateral.

1995 APMO, 3

Tags: circumcircle , cyclic quadrilateral , power of a point , radical axis , geometry

Let $PQRS$ be a cyclic quadrilateral such that the segments $PQ$ and $RS$ are not parallel. Consider the set of circles through $P$ and $Q$, and the set of circles through $R$ and $S$. Determine the set $A$ of points of tangency of circles in these two sets.

2021 German National Olympiad, 2

Tags: perimeter , cyclic quadrilateral , interior , geometry

Let $P$ on $AB$, $Q$ on $BC$, $R$ on $CD$ and $S$ on $AD$ be points on the sides of a convex quadrilateral $ABCD$. Show that the following are equivalent: (1) There is a choice of $P,Q,R,S$, for which all of them are interior points of their side, such that $PQRS$ has minimal perimeter. (2) $ABCD$ is a cyclic quadrilateral with circumcenter in its interior.

2024 Ukraine National Mathematical Olympiad, Problem 6

Tags: geometry , cyclic quadrilateral

Cyclic quadrilateral $ABCD$ is such that $\angle BAD = 2\angle ADC$ and $CD = 2BC$. Let $H$ be the projection of $C$ onto $AD$. Prove that $BH \parallel CD$. [i]Proposed by Fedir Yudin, Anton Trygub[/i]