Olympiad problems

2011 IFYM, Sozopol, 5

A circle is inscribed in a quadrilateral $ABCD$, which is tangent to its sides $AB$, $BC$, $CD$, and $DC$ in points $M$, $N$, $P$, and $Q$ respectively. Prove that the lines $MP$, $NQ$, $AC$, and $BD$ intersect in one point.

2008 Germany Team Selection Test, 2

Tags: geometry , quadrilateral , triangle , incircle

Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear. [i]Author: Waldemar Pompe, Poland[/i]

2021 Junior Balkan Team Selection Tests - Romania, P3

Tags: quadrilateral , geometry

Let $ABCD$ be a convex quadrilateral with angles $\sphericalangle A, \sphericalangle C\geq90^{\circ}$. On sides $AB,BC,CD$ and $DA$, consider the points $K,L,M$ and $N$ respectively. Prove that the perimeter of $KLMN$ is greater than or equal to $2\cdot AC$.

2022 Switzerland Team Selection Test, 2

Tags: quadrilateral , tangency , collinear , geometry

Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ is tangent to the line $CD$, and the circle with diameter $CD$ is tangent to the line $AB$. Prove that the two intersection points of these circles and the point $AC \cap BD$ are collinear.

2014 ELMO Shortlist, 4

Tags: reflection , quadrilateral , rectangle , trigonometry , analytic geometry , geometry

Let $ABCD$ be a quadrilateral inscribed in circle $\omega$. Define $E = AA \cap CD$, $F = AA \cap BC$, $G = BE \cap \omega$, $H = BE \cap AD$, $I = DF \cap \omega$, and $J = DF \cap AB$. Prove that $GI$, $HJ$, and the $B$-symmedian are concurrent. [i]Proposed by Robin Park[/i]

2017 Portugal MO, 5

Tags: geometry , quadrilateral

Let $[ABCD]$ be a convex quadrilateral with $AB = 2, BC = 3, CD = 7$ and $\angle B = 90^o$, for which there is a inscribed circle. Determine the radius of this circle. [img]https://1.bp.blogspot.com/-sDKOdmceJlY/X4KaJxi8AoI/AAAAAAAAMk8/7UkTzaWqQSkdqb0N_-r0CZZjD-OGZknSACLcBGAsYHQ/s260/2017%2Bportugal%2Bp5.png[/img]

2008 IMO Shortlist, 3

Tags: quadrilateral , trigonometry , circumcircle , inversion , homothety , geometry

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

1967 IMO Shortlist, 1

Tags: area , maximization , quadrilateral , geometry

Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius $r$, there exist one (or more) with maximum area. If so, determine their shape and area.

2020 IMO Shortlist, C2

Tags: quadrilateral , coloring , partition , combinatorics

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

2008 Grigore Moisil Intercounty, 2

Tags: quadrilateral , locus , geometry

Given a convex quadrilateral $ ABCD, $ find the locus of points $ X $ that verify the qualities: $$ XA^2+XB^2+CD^2=XB^2+XC^2+DA^2=XC^2+XD^2+AB^2=XD^2+XA^2+BC^2 $$ [i]Maria Pop[/i]

1970 Putnam, B6

Tags: area , quadrilateral , geometry

Show that if a circumscribable quadrilateral of sides $a,b,c,d$ has area $A= \sqrt{abcd},$ then it is also inscribable.

2011 IFYM, Sozopol, 1

Tags: geometry , quadrilateral , tangency

Let $ABCD$ be a quadrilateral inscribed in a circle $k$. Let the lines $AC\cap BD=O$, $AD\cap BC=P$, and $AB\cap CD=Q$. Line $QO$ intersects $k$ in points $M$ and $N$. Prove that $PM$ and $PN$ are tangent to $k$.

2009 Germany Team Selection Test, 2

Tags: geometry , quadrilateral , inversion , trigonometry , homothety , circumcircle

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

1997 Nordic, 2

Tags: geometry , diagonal , area of triangle , area , quadrilateral

Let $ABCD$ be a convex quadrilateral. We assume that there exists a point $P$ inside the quadrilateral such that the areas of the triangles $ABP, BCP, CDP$, and $DAP$ are equal. Show that at least one of the diagonals of the quadrilateral bisects the other diagonal.

1994 IMO Shortlist, 2

Tags: quadrilateral , midpoint , geometry

$ ABCD$ is a quadrilateral with $ BC$ parallel to $ AD$. $ M$ is the midpoint of $ CD$, $ P$ is the midpoint of $ MA$ and $ Q$ is the midpoint of $ MB$. The lines $ DP$ and $ CQ$ meet at $ N$. Prove that $ N$ is inside the quadrilateral $ ABCD$.

2011 Indonesia TST, 3

Tags: quadrilateral , collinearity , collinear points , geometry

Circle $\omega$ is inscribed in quadrilateral $ABCD$ such that $AB$ and $CD$ are not parallel and intersect at point $O.$ Circle $\omega_1$ touches the side $BC$ at $K$ and touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD;$ circle $\omega_2$ touches side $AD$ at $L$ and touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD.$ If $O,K,$ and $L$ are collinear$,$ then show that the midpoint of side $BC,AD,$ and the center of circle $\omega$ are also collinear.

2021 Azerbaijan IMO TST, 2

Tags: quadrilateral , partition , coloring , combinatorics

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

2021 Germany Team Selection Test, 1

Tags: quadrilateral , coloring , partition , combinatorics

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

2019 AIME Problems, 6

Tags: quadrilateral , geometry

In convex quadrilateral $KLMN$ side $\overline{MN}$ is perpendicular to diagonal $\overline{KM}$, side $\overline{KL}$ is perpendicular to diagonal $\overline{LN}$, $MN = 65$, and $KL = 28$. The line through $L$ perpendicular to side $\overline{KN}$ intersects diagonal $\overline{KM}$ at $O$ with $KO = 8$. Find $MO$.

2022 SAFEST Olympiad, 5

Tags: tangency , quadrilateral , collinear , geometry

Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ is tangent to the line $CD$, and the circle with diameter $CD$ is tangent to the line $AB$. Prove that the two intersection points of these circles and the point $AC \cap BD$ are collinear.

2007 IMO Shortlist, 8

Tags: incircle , quadrilateral , triangle , geometry

Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear. [i]Author: Waldemar Pompe, Poland[/i]

2021 Belarusian National Olympiad, 10.7

Tags: quadrilateral , geometry

An inscribed into a circle quadraliteral $ABCD$ is given. Points $M$ and $N$ lie on sides $AB$ and $CD$ such that $AK:KB=DM:MC$ and points $L$ and $N$ lie on sides $BC$ and $DA$ such that $BL:LC=AN:ND$. The circumcircle of the triangle $CML$ intersects diagonal $AC$ for the second time in point $P$. The circumcircle of triangle $DNM$ intersects diagonal $BD$ for the second time in point $Q$. Circumcircles of triangles $AKN$ and $BLK$ intersect for the second time in point $R$. Prove that the circumcircle of $PQR$ passes through the intersection of $AC$ and $BD$

2009 Brazil Team Selection Test, 4

Tags: symmetry , circumcircle , quadrilateral , homothety , geometry

There is given a convex quadrilateral $ ABCD$. Prove that there exists a point $ P$ inside the quadrilateral such that \[ \angle PAB \plus{} \angle PDC \equal{} \angle PBC \plus{} \angle PAD \equal{} \angle PCD \plus{} \angle PBA \equal{} \angle PDA \plus{} \angle PCB = 90^{\circ} \] if and only if the diagonals $ AC$ and $ BD$ are perpendicular. [i]Proposed by Dusan Djukic, Serbia[/i]

2019 Nordic, 3

Tags: quadrilateral , geometry

The quadrilateral $ABCD$ satisfies $\angle ACD = 2\angle CAB, \angle ACB = 2\angle CAD $ and $CB = CD.$ Show that $$\angle CAB=\angle CAD.$$

Russian TST 2021, P1

Tags: quadrilateral , partition , coloring , combinatorics

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]