Olympiad problems

2004 Iran MO (3rd Round), 16

Tags: parallelogram , rectangle , geometric transformation , homothety , circumcircle , cyclic quadrilateral , geometry

Let $ABC$ be a triangle . Let point $X$ be in the triangle and $AX$ intersects $BC$ in $Y$ . Draw the perpendiculars $YP,YQ,YR,YS$ to lines $CA,CX,BX,BA$ respectively. Find the necessary and sufficient condition for $X$ such that $PQRS$ be cyclic .

2014 IberoAmerican, 2

Tags: geometric transformation , reflection , trapezoid , circumcircle , cyclic quadrilateral , geometry

Let $ABC$ be an acute triangle and $H$ its orthocenter. Let $D$ be the intersection of the altitude from $A$ to $BC$. Let $M$ and $N$ be the midpoints of $BH$ and $CH$, respectively. Let the lines $DM$ and $DN$ intersect $AB$ and $AC$ at points $X$ and $Y$ respectively. If $P$ is the intersection of $XY$ with $BH$ and $Q$ the intersection of $XY$ with $CH$, show that $H, P, D, Q$ lie on a circumference.

2014 Germany Team Selection Test, 2

Tags: circumcircle , incenter , cyclic quadrilateral , geometry

Let $ABCD$ be a convex cyclic quadrilateral with $AD=BD$. The diagonals $AC$ and $BD$ intersect in $E$. Let the incenter of triangle $\triangle BCE$ be $I$. The circumcircle of triangle $\triangle BIE$ intersects side $AE$ in $N$. Prove \[ AN \cdot NC = CD \cdot BN. \]

2018 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: cyclic quadrilateral , incircle , tangent , parallel , geometry

Let $ABCD$ be a cyclic quadrilateral and let $k_1$ and $k_2$ be circles inscribed in triangles $ABC$ and $ABD$. Prove that external common tangent of those circles (different from $AB$) is parallel with $CD$

2012 Indonesia TST, 3

Tags: angle chasing , complex number , cyclic quadrilateral , geometry

The incircle of a triangle $ABC$ is tangent to the sides $AB,AC$ at $M,N$ respectively. Suppose $P$ is the intersection between $MN$ and the bisector of $\angle ABC$. Prove that $BP$ and $CP$ are perpendicular.

1993 Tournament Of Towns, (370) 2

Tags: geometry , cyclic quadrilateral , equal segments

Quadrilateral $ABCD$ is inscribed in a circle, $M$ is the intersection point of the lines $AB$ and $CD$ and $N$ is the intersection point of the lines $BC$ and $AD$. It is known that $BM = DN$. Prove that $CM = CN$. (F Nazarov)

2000 Polish MO Finals, 2

Tags: geometry , trigonometry , circumcircle , analytic geometry , geometric transformation , reflection , cyclic quadrilateral

Let a triangle $ABC$ satisfy $AC = BC$; in other words, let $ABC$ be an isosceles triangle with base $AB$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$. Denote by $M$ the midpoint of the segment $AB$. Show that $\angle APM + \angle BPC = 180^{\circ}$.

Croatia MO (HMO) - geometry, 2013.3

Tags: geometry , parallelogram , bisects , cyclic quadrilateral , concyclic , bisects segment

Given a pointed triangle $ABC$ with orthocenter $H$. Let $D$ be the point such that the quadrilateral $AHCD$ is parallelogram. Let $p$ be the perpendicular to the direction $AB$ through the midpoint $A_1$ of the side $BC$. Denote the intersection of the lines $p$ and $AB$ with $E$, and the midpoint of the length $A_1E$ with $F$. The point where the parallel to the line $BD$ through point $A$ intersects $p$ denote by $G$. Prove that the quadrilateral $AFA_1C$ is cyclic if and only if the lines $BF$ passes through the midpoint of the length $CG$.

1986 IMO Longlists, 69

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

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.

2021 Korea Junior Math Olympiad, 3

Tags: geometry , cyclic quadrilateral , circumcircle , perpendicular bisector , tangent

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\Omega$ and let diagonals $AC$ and $BD$ intersect at $X$. Suppose that $AEFB$ is inscribed in a circumcircle of triangle $ABX$ such that $EF$ and $AB$ are parallel. $FX$ meets the circumcircle of triangle $CDX$ again at $G$. Let $EX$ meets $AB$ at $P$, and $XG$ meets $CD$ at $Q$. Denote by $S$ the intersection of the perpendicular bisector of $\overline{EG}$ and $\Omega$ such that $S$ is closer to $A$ than $B$. Prove that line through $S$ parallel to $PQ$ is tangent to $\Omega$.

2013 Saudi Arabia BMO TST, 1

Tags: concyclic , cyclic quadrilateral , geometry

$ABCD$ is a cyclic quadrilateral and $\omega$ its circumcircle. The perpendicular line to $AC$ at $D$ intersects $AC$ at $E$ and $\omega$ at F. Denote by $\ell$ the perpendicular line to $BC$ at $F$. The perpendicular line to $\ell$ at A intersects $\ell$ at $G$ and $\omega$ at $H$. Line $GE$ intersects $FH$ at $I$ and $CD$ at $J$. Prove that points $C, F, I$ and $J$ are concyclic

2020 Federal Competition For Advanced Students, P2, 1

Tags: geometry , concurrency , cyclic quadrilateral

Let $ABCD$ be a convex cyclic quadrilateral with the diagonal intersection $S$. Let further be $P$ the circumcenter of the triangle $ABS$ and $Q$ the circumcenter of the triangle $BCS$. The parallel to $AD$ through $P$ and the parallel to $CD$ through $Q$ intersect at point $R$. Prove that $R$ is on $BD$. (Karl Czakler)

Russian TST 2016, P3

Tags: geometry , cyclic quadrilateral , harmonic division

The diagonals of a cyclic quadrilateral $ABCD$ intersect at $P$, and there exist a circle $\Gamma$ tangent to the extensions of $AB,BC,AD,DC$ at $X,Y,Z,T$ respectively. Circle $\Omega$ passes through points $A,B$, and is externally tangent to circle $\Gamma$ at $S$. Prove that $SP\perp ST$.

1986 IMO Longlists, 51

Tags: inequalities , rectangle , trigonometry , cyclic quadrilateral , geometry

Let $a, b, c, d$ be the lengths of the sides of a quadrilateral circumscribed about a circle and let $S$ be its area. Prove that $S \leq \sqrt{abcd}$ and find conditions for equality.

1966 IMO Longlists, 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.

2008 Brazil National Olympiad, 1

Tags: geometry , geometric transformation , reflection , circumcircle , incenter , homothety , cyclic quadrilateral

Let $ ABCD$ be a cyclic quadrilateral and $ r$ and $ s$ the lines obtained reflecting $ AB$ with respect to the internal bisectors of $ \angle CAD$ and $ \angle CBD$, respectively. If $ P$ is the intersection of $ r$ and $ s$ and $ O$ is the center of the circumscribed circle of $ ABCD$, prove that $ OP$ is perpendicular to $ CD$.

2005 China Team Selection Test, 2

Tags: perimeter , cyclic quadrilateral , pythagorean theorem , geometry

Cyclic quadrilateral $ABCD$ has positive integer side lengths $AB$, $BC$, $CA$, $AD$. It is known that $AD=2005$, $\angle{ABC}=\angle{ADC} = 90^o$, and $\max \{ AB,BC,CD \} < 2005$. Determine the maximum and minimum possible values for the perimeter of $ABCD$.

2016 China Western Mathematical Olympiad, 7

Tags: geometry , cyclic quadrilateral

$ABCD$ is a cyclic quadrilateral, and $\angle BAC = \angle DAC$. $\astrosun I_1$ and $\astrosun I_2$ are the incircles of $\triangle ABD$ and $\triangle ADC$ respectively. Prove that one of the common external tangents of $\astrosun I_1$ and $\astrosun I_2$ is parallel to $BD$

2021 Oral Moscow Geometry Olympiad, 1

Tags: geometry , circumcircle , cyclic quadrilateral

Quadrilateral $ABCD$ is inscribed in a circle, $E$ is an arbitrary point of this circle. It is known that distances from point $E$ to lines $AB, AC, BD$ and $CD$ are equal to $a, b, c$ and $d$ respectively. Prove that $ad= bc$.

2014 USA Team Selection Test, 2

Tags: analytic geometry , cyclic quadrilateral , orthocenter , area , geometry

Let $ABCD$ be a cyclic quadrilateral, and let $E$, $F$, $G$, and $H$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. Let $W$, $X$, $Y$ and $Z$ be the orthocenters of triangles $AHE$, $BEF$, $CFG$ and $DGH$, respectively. Prove that the quadrilaterals $ABCD$ and $WXYZ$ have the same area.

2008 Oral Moscow Geometry Olympiad, 4

Tags: projection , geometry , concyclic , cyclic quadrilateral

A circle can be circumscribed around the quadrilateral $ABCD$. Point $P$ is the foot of the perpendicular drawn from point $A$ on line $BC$, and respectively $Q$ from $A$ on $DC$, $R$ from $D$ on $AB$ and $T$ from $D$ on $BC$ . Prove that points $P,Q,R$ and $T$ lie on the same circle. (A. Myakishev)

Croatia MO (HMO) - geometry, 2019.7

Tags: ratio , geometry , cyclic , bisects segment , cyclic quadrilateral

On the side $AB$ of the cyclic quadrilateral $ABCD$ there is a point $X$ such that diagonal $AC$ bisects the segment $DX$, and the diagonal $BD$ bisects the segment $CX$. What is the smallest possible ratio $|AB | : |CD|$ in such a quadrilateral ?

2015 Turkey MO (2nd round), 5

Tags: cyclic quadrilateral , geometry

In a cyclic quadrilateral $ABCD$ whose largest interior angle is $D$, lines $BC$ and $AD$ intersect at point $E$, while lines $AB$ and $CD$ intersect at point $F$. A point $P$ is taken in the interior of quadrilateral $ABCD$ for which $\angle EPD=\angle FPD=\angle BAD$. $O$ is the circumcenter of quadrilateral $ABCD$. Line $FO$ intersects the lines $AD$, $EP$, $BC$ at $X$, $Q$, $Y$, respectively. If $\angle DQX = \angle CQY$, show that $\angle AEB=90^\circ$.

IV Soros Olympiad 1997 - 98 (Russia), 10.1

Tags: geometry , cyclic quadrilateral

Two sides of the cyclic quadrilateral $ABCD$ are known: $AB = a$, $BC = b$. A point $K$ is taken on the side $CD$ so that $CK = m$. A circle passing through $B$, $K$ and $D$ intersects line $DA$ at a point $M$, different from $D$. Find $AM$.

2007 National Olympiad First Round, 13

Tags: geometry , incenter , trigonometry , cyclic quadrilateral

Let $ABCD$ be an circumscribed quadrilateral such that $m(\widehat{A})=m(\widehat{B})=120^\circ$, $m(\widehat{C})=30^\circ$, and $|BC|=2$. What is $|AD|$? $ \textbf{(A)}\ \sqrt 3 - 1 \qquad\textbf{(B)}\ \sqrt 2 - 3 \qquad\textbf{(C)}\ \sqrt 6 - \sqrt 2 \qquad\textbf{(D)}\ 2 - \sqrt 2 \qquad\textbf{(E)}\ 3 - \sqrt 3 $