Olympiad problems

2014 China Girls Math Olympiad, 6

Tags: geometry , circumcircle , trapezoid , geometric transformation , reflection , analytic geometry , vector

In acute triangle $ABC$, $AB > AC$. $D$ and $E$ are the midpoints of $AB$, $AC$ respectively. The circumcircle of $ADE$ intersects the circumcircle of $BCE$ again at $P$. The circumcircle of $ADE$ intersects the circumcircle $BCD$ again at $Q$. Prove that $AP = AQ$.

2022 Puerto Rico Team Selection Test, 5

Tags: geometry , trapezoid

Let $ABCD$ be a trapezoid of bases $AB$ and $CD$, and non-parallel sides $BC$ and $DA$. The angles $\angle BCD$ and $\angle CDA$ are acute. The lines $BC$ and $DA$ are cut at a point $E$. It is known that $AE = 2$, $AC = 6$, $CD =\sqrt{72}$ and area $( \vartriangle BCD)= 18$. (a) Find the height of the trapezoid $ABCD$. (b) Find the area of $\vartriangle ABC$.

2022 Oral Moscow Geometry Olympiad, 1

Tags: trapezoid , circumcircle , concyclic , geometry

Given an isosceles trapezoid $ABCD$. The bisector of angle $B$ intersects the base $AD$ at point $L$. Prove that the center of the circle circumscribed around triangle $BLD$ lies on the circle circumscribed around the trapezoid. (Yu. Blinkov)

2017 Iran Team Selection Test, 1

Tags: geometry , trapezoid , circumcircle

$ABCD$ is a trapezoid with $AB \parallel CD$. The diagonals intersect at $P$. Let $\omega _1$ be a circle passing through $B$ and tangent to $AC$ at $A$. Let $\omega _2$ be a circle passing through $C$ and tangent to $BD$ at $D$. $\omega _3$ is the circumcircle of triangle $BPC$. Prove that the common chord of circles $\omega _1,\omega _3$ and the common chord of circles $\omega _2, \omega _3$ intersect each other on $AD$. [i]Proposed by Kasra Ahmadi[/i]

2007 Moldova Team Selection Test, 3

Tags: ratio , trigonometry , circumcircle , trapezoid , radical axis , projective geometry , geometry

Let $M, N$ be points inside the angle $\angle BAC$ usch that $\angle MAB\equiv \angle NAC$. If $M_{1}, M_{2}$ and $N_{1}, N_{2}$ are the projections of $M$ and $N$ on $AB, AC$ respectively then prove that $M, N$ and $P$ the intersection of $M_{1}N_{2}$ with $N_{1}M_{2}$ are collinear.

2009 Italy TST, 2

Tags: geometry , circumcircle , trapezoid , geometric transformation , reflection , power of a point , radical axis

Two circles $O_1$ and $O_2$ intersect at $M,N$. The common tangent line nearer to $M$ of the two circles touches $O_1,O_2$ at $A,B$ respectively. Let $C,D$ be the symmetric points of $A,B$ with respect to $M$ respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ and $O_2$ at points $E,F$ respectively which are distinct from $M$. Prove that the circumradii of the triangles $MEF$ and $NEF$ are equal.

2013 NIMO Summer Contest, 13

Tags: geometry , trapezoid , trigonometry

In trapezoid $ABCD$, $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$. Compute $AB^2$ given that $AB \cdot \tan(\angle BCD) = 20$ and $CD = 13$. [i]Proposed by Lewis Chen[/i]

2011 China Girls Math Olympiad, 2

Tags: geometry , circumcircle , trapezoid , rectangle , projective geometry , perpendicular bisector , geometric transformation

The diagonals $AC,BD$ of the quadrilateral $ABCD$ intersect at $E$. Let $M,N$ be the midpoints of $AB,CD$ respectively. Let the perpendicular bisectors of the segments $AB,CD$ meet at $F$. Suppose that $EF$ meets $BC,AD$ at $P,Q$ respectively. If $MF\cdot CD=NF\cdot AB$ and $DQ\cdot BP=AQ\cdot CP$, prove that $PQ\perp BC$.

2009 Balkan MO Shortlist, G6

Tags: geometry , geometric transformation , reflection , circumcircle , rotation , trapezoid , trigonometry

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

2001 National Olympiad First Round, 29

Tags: geometry , trapezoid , trigonometry , power of a point

Let $ABCD$ be a isosceles trapezoid such that $AB || CD$ and all of its sides are tangent to a circle. $[AD]$ touches this circle at $N$. $NC$ and $NB$ meet the circle again at $K$ and $L$, respectively. What is $\dfrac {|BN|}{|BL|} + \dfrac {|CN|}{|CK|}$? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 10 $

May Olympiad L1 - geometry, 2001.2

Tags: geometry , trapezoid , rectangle

Let's take a $ABCD$ rectangle of paper; the side $AB$ measures $5$ cm and the side $BC$ measures $9$ cm. We do three folds: 1.We take the $AB$ side on the $BC$ side and call $P$ the point on the $BC$ side that coincides with $A$. A right trapezoid $BCDQ$ is then formed. 2. We fold so that $B$ and $Q$ coincide. A $5$-sided polygon $RPCDQ$ is formed. 3. We fold again by matching $D$ with $C$ and $Q$ with $P$. A new right trapezoid $RPCS$. After making these folds, we make a cut perpendicular to $SC$ by its midpoint $T$, obtaining the right trapezoid $RUTS$. Calculate the area of the figure that appears as we unfold the last trapezoid $RUTS$.

2012 China Girls Math Olympiad, 4

Tags: geometry , trapezoid , pigeonhole principle , inequalities , combinatorics

There is a stone at each vertex of a given regular $13$-gon, and the color of each stone is black or white. Prove that we may exchange the position of two stones such that the coloring of these stones are symmetric with respect to some symmetric axis of the $13$-gon.

2024 Korea Junior Math Olympiad (First Round), 16.

Tags: geometry , trapezoid

There is an Equilateral trapezoid $ ABCD. $ $ \bar{AB} =60, \bar{BC}=\bar{DA}= 36, \bar{CD}=108. $ $ M $ is the middle point of $ \bar {AB} $, and point $P$ on $ \bar{AM} $ follows that $ \bar {AP} $ =10. The foot of perpendicular dropped from $P$ to $ \bar {BD} $ is $E$. $ \bar{AC} \cap \bar{BD} $ is $ F $. Point $X$ is on $ \bar {AF} $ which follows $ \bar{MX}=\bar{ME} $ Find $ \bar{AX} \times \bar{AF} $

2005 Harvard-MIT Mathematics Tournament, 6

Tags: geometry , trapezoid , quadratic , similar triangles

A triangular piece of paper of area $1$ is folded along a line parallel to one of the sides and pressed ﬂat. What is the minimum possible area of the resulting ﬁgure?

2007 Romania Team Selection Test, 2

Tags: parallelogram , trapezoid , geometry

Let $ A_{1}A_{2}A_{3}A_{4}A_{5}$ be a convex pentagon, such that \[ [A_{1}A_{2}A_{3}] \equal{} [A_{2}A_{3}A_{4}] \equal{} [A_{3}A_{4}A_{5}] \equal{} [A_{4}A_{5}A_{1}] \equal{} [A_{5}A_{1}A_{2}].\] Prove that there exists a point $ M$ in the plane of the pentagon such that \[ [A_{1}MA_{2}] \equal{} [A_{2}MA_{3}] \equal{} [A_{3}MA_{4}] \equal{} [A_{4}MA_{5}] \equal{} [A_{5}MA_{1}].\] Here $ [XYZ]$ stands for the area of the triangle $ \Delta XYZ$.

2002 AMC 10, 17

Tags: geometry , trapezoid

A regular octagon $ ABCDEFGH$ has sides of length two. Find the area of $ \triangle{ADG}$. $ \textbf{(A)}\ 4 \plus{} 2 \sqrt{2} \qquad \textbf{(B)}\ 6 \plus{} \sqrt{2} \qquad \textbf{(C)}\ 4 \plus{} 3 \sqrt{2} \qquad \textbf{(D)}\ 3 \plus{} 4 \sqrt{2} \qquad \textbf{(E)}\ 8 \plus{} \sqrt{2}$

2013 Tuymaada Olympiad, 2

Tags: geometry , rhombus , geometric transformation , reflection , symmetry , trapezoid , trigonometry

Points $X$ and $Y$ inside the rhombus $ABCD$ are such that $Y$ is inside the convex quadrilateral $BXDC$ and $2\angle XBY = 2\angle XDY = \angle ABC$. Prove that the lines $AX$ and $CY$ are parallel. [i]S. Berlov[/i]

2003 Costa Rica - Final Round, 4

Tags: geometry , rectangle , trapezoid

$S_{1}$ and $S_{2}$ are two circles that intersect at distinct points $P$ and $Q$. $\ell_{1}$ and $\ell_{2}$ are two parallel lines through $P$ and $Q$. $\ell_{1}$ intersects $S_{1}$ and $S_{2}$ at points $A_{1}$ and $A_{2}$, different from $P$, respectively. $\ell_{2}$ intersects $S_{1}$ and $S_{2}$ at points $B_{1}$ and $B_{2}$, different from $Q$, respectively. Show that the perimeters of the triangles $A_{1}QA_{2}$ and $B_{1}PB_{2}$ are equal.

1972 IMO Longlists, 43

Tags: parallelogram , rectangle , trapezoid , pythagorean theorem , perpendicular bisector , geometry

A fixed point $A$ inside a circle is given. Consider all chords $XY$ of the circle such that $\angle XAY$ is a right angle, and for all such chords construct the point $M$ symmetric to $A$ with respect to $XY$ . Find the locus of points $M$.

2007 Sharygin Geometry Olympiad, 14

Tags: diagonal , geometry , equal angles , trapezoid

In a trapezium with bases $AD$ and $BC$, let $P$ and $Q$ be the middles of diagonals $AC$ and $BD$ respectively. Prove that if $\angle DAQ = \angle CAB$ then $\angle PBA = \angle DBC$.

2017 Iran MO (3rd round), 2

Tags: geometry , trapezoid , angle chasing

Let $ABCD$ be a trapezoid ($AB<CD,AB\parallel CD$) and $P\equiv AD\cap BC$. Suppose that $Q$ be a point inside $ABCD$ such that $\angle QAB=\angle QDC=90-\angle BQC$. Prove that $\angle PQA=2\angle QCD$.

2002 National Olympiad First Round, 17

Tags: geometry , trapezoid , incenter , symmetry

Let $ABCD$ be a trapezoid and a tangential quadrilateral such that $AD || BC$ and $|AB|=|CD|$. The incircle touches $[CD]$ at $N$. $[AN]$ and $[BN]$ meet the incircle again at $K$ and $L$, respectively. What is $\dfrac {|AN|}{|AK|} + \dfrac {|BN|}{|BL|}$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 16 $

1980 IMO, 14

Tags: trigonometry , trapezoid , geometry

Let $A$ be a fixed point in the interior of a circle $\omega$ with center $O$ and radius $r$, where $0<OA<r$. Draw two perpendicular chords $BC,DE$ such that they pass through $A$. For which position of these cords does $BC+DE$ maximize?

2000 Dutch Mathematical Olympiad, 3

Tags: geometry , parallelogram , trapezoid , similar triangles

Isosceles, similar triangles $QPA$ and $SPB$ are constructed (outwards) on the sides of parallelogram $PQRS$ (where $PQ = AQ$ and $PS = BS$). Prove that triangles $RAB$, $QPA$ and $SPB$ are similar.

2013 ELMO Shortlist, 2

Tags: geometry , circumcircle , incenter , geometric transformation , reflection , trapezoid

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$, and let $D$,$E$,$F$ be the points where its incircle meets $BC$, $AC$, $AB$ respectively. Let the circumcircles of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$ meet $\Gamma$ a second time at $X,Y,Z$ respectively. Prove that the perpendiculars from $A,B,C$ to $AX,BY,CZ$ respectively are concurrent. [i]Proposed by Michael Kural[/i]