Olympiad problems

2004 Bulgaria Team Selection Test, 2

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

Let $H$ be the orthocenter of $\triangle ABC$. The points $A_{1} \not= A$, $B_{1} \not= B$ and $C_{1} \not= C$ lie, respectively, on the circumcircles of $\triangle BCH$, $\triangle CAH$ and $\triangle ABH$ and satisfy $A_{1}H=B_{1}H=C_{1}H$. Denote by $H_{1}$, $H_{2}$ and $H_{3}$ the orthocenters of $\triangle A_{1}BC$, $\triangle B_{1}CA$ and $\triangle C_{1}AB$, respectively. Prove that $\triangle A_{1}B_{1}C_{1}$ and $\triangle H_{1}H_{2}H_{3}$ have the same orthocenter.

2000 Federal Competition For Advanced Students, Part 2, 2

Tags: trapezoid , geometry

A trapezoid $ABCD$ with $AB \parallel CD$ is inscribed in a circle $k$. Points $P$ and $Q$ are chose on the arc $ADCB$ in the order $A-P -Q-B$. Lines $CP$ and $AQ$ meet at $X$, and lines $BP$ and $DQ$ meet at $Y$. Show that points $P,Q,X, Y$ lie on a circle.

2017 All-Russian Olympiad, 2

Tags: geometry , trapezoid , tangent

$ABCD$ is an isosceles trapezoid with $BC || AD$. A circle $\omega$ passing through $B$ and $C$ intersects the side $AB$ and the diagonal $BD$ at points $X$ and $Y$ respectively. Tangent to $\omega$ at $C$ intersects the line $AD$ at $Z$. Prove that the points $X$, $Y$, and $Z$ are collinear.

Kyiv City MO Juniors 2003+ geometry, 2003.8.5

Tags: construction , geometry , trapezoid

Three segments $2$ cm, $5$ cm and $12$ cm long are constructed on the plane. Construct a trapezoid with bases of $2$ cm and $5$ cm, the sum of the sides of which is $12$ cm, and one of the angles is $60^o$. (Bogdan Rublev)

2008 AIME Problems, 11

Tags: geometry , trapezoid , quadratic , trigonometry , probability

In triangle $ ABC$, $ AB \equal{} AC \equal{} 100$, and $ BC \equal{} 56$. Circle $ P$ has radius $ 16$ and is tangent to $ \overline{AC}$ and $ \overline{BC}$. Circle $ Q$ is externally tangent to $ P$ and is tangent to $ \overline{AB}$ and $ \overline{BC}$. No point of circle $ Q$ lies outside of $ \triangle ABC$. The radius of circle $ Q$ can be expressed in the form $ m \minus{} n\sqrt {k}$, where $ m$, $ n$, and $ k$ are positive integers and $ k$ is the product of distinct primes. Find $ m \plus{} nk$.

2012 Hanoi Open Mathematics Competitions, 3

Tags: trapezoid , geometry

Let be given a trapezoidal $ABCD$ with the based edges $BC = 3$ cm, $DA = 6$ cm ($AD // BC$). Then the length of the line $EF$ ($E \in AB , F \in CD$ and $EF // AD$) through the common point $M$ of $AC$ and $BD$ is (A) $3,5$ cm (B): $4$ cm (C) $4,5$ cm (D) $5$ cm (E) None of the above

2009 AMC 10, 23

Tags: trapezoid , ratio , geometry , trigonometry , parallelogram

Convex quadrilateral $ ABCD$ has $ AB\equal{}9$ and $ CD\equal{}12$. Diagonals $ AC$ and $ BD$ intersect at $ E$, $ AC\equal{}14$, and $ \triangle AED$ and $ \triangle BEC$ have equal areas. What is $ AE$? $ \textbf{(A)}\ \frac{9}{2}\qquad \textbf{(B)}\ \frac{50}{11}\qquad \textbf{(C)}\ \frac{21}{4}\qquad \textbf{(D)}\ \frac{17}{3}\qquad \textbf{(E)}\ 6$

2019 Adygea Teachers' Geometry Olympiad, 4

Tags: tangential , right angle , trapezium , geometry , trapezoid

From which two statements about the trapezoid follows the third: 1) the trapezoid is tangential, 2) the trapezoid is right, 3) its area is equal to the product of the bases?

2016 Sharygin Geometry Olympiad, P1

Tags: geometry , midpoint , trapezoid , equal angles

A trapezoid $ABCD$ with bases $AD$ and $BC$ is such that $AB = BD$. Let $M$ be the midpoint of $DC$. Prove that $\angle MBC$ = $\angle BCA$.

2013 USAJMO, 5

Tags: trapezoid , geometry , trig identities , similar triangles , ratio , trigonometry , analytic geometry

Quadrilateral $XABY$ is inscribed in the semicircle $\omega$ with diameter $XY$. Segments $AY$ and $BX$ meet at $P$. Point $Z$ is the foot of the perpendicular from $P$ to line $XY$. Point $C$ lies on $\omega$ such that line $XC$ is perpendicular to line $AZ$. Let $Q$ be the intersection of segments $AY$ and $XC$. Prove that \[\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.\]

2014 Harvard-MIT Mathematics Tournament, 14

Tags: trapezoid , geometry

Let $ABCD$ be a trapezoid with $AB\parallel CD$ and $\angle D=90^\circ$. Suppose that there is a point $E$ on $CD$ such that $AE=BE$ and that triangles $AED$ and $CEB$ are similar, but not congruent. Given that $\tfrac{CD}{AB}=2014$, find $\tfrac{BC}{AD}$.

1986 Iran MO (2nd round), 2

Tags: trapezoid , geometry

In a trapezoid $ABCD$, the legs $AB$ and $CD$ meet in $M$ and the diagonals $AC$ and $BD$ meet in $N.$ Let $AC=a$ and $BC=b.$ Find the area of triangles $AMD$ and $AND$ in terms of $a$ and $b.$

2015 Iran MO (3rd round), 1

Tags: trapezoid , circumcircle , geometry

Let $ABCD$ be the trapezoid such that $AB\parallel CD$. Let $E$ be an arbitrary point on $AC$. point $F$ lies on $BD$ such that $BE\parallel CF$. Prove that circumcircles of $\triangle ABF,\triangle BED$ and the line $AC$ are concurrent.

2022-23 IOQM India, 12

Tags: trapezoid , geometry

Given $\triangle{ABC}$ with $\angle{B}=60^{\circ}$ and $\angle{C}=30^{\circ}$, let $P,Q,R$ be points on the sides $BA,AC,CB$ respectively such that $BPQR$ is an isosceles trapezium with $PQ \parallel BR$ and $BP=QR$.\\ Find the maximum possible value of $\frac{2[ABC]}{[BPQR]}$ where $[S]$ denotes the area of any polygon $S$.

2021 Indonesia TST, G

Tags: geometry , trapezoid

Given points $A$, $B$, $C$, and $D$ on circle $\omega$ such that lines $AB$ and $CD$ intersect on point $T$ where $A$ is between $B$ and $T$, moreover $D$ is between $C$ and $T$. It is known that the line passing through $D$ which is parallel to $AB$ intersects $\omega$ again on point $E$ and line $ET$ intersects $\omega$ again on point $F$. Let $CF$ and $AB$ intersect on point $G$, $X$ be the midpoint of segment $AB$, and $Y$ be the reflection of point $T$ to $G$. Prove that $X$, $Y$, $C$, and $D$ are concyclic.

2007 AMC 12/AHSME, 19

Tags: integration , trapezoid , geometry , calculus , vector , trigonometry , analytic geometry

Triangles $ ABC$ and $ ADE$ have areas $ 2007$ and $ 7002,$ respectively, with $ B \equal{} (0,0),$ $ C \equal{} (223,0),$ $ D \equal{} (680,380),$ and $ E \equal{} (689,389).$ What is the sum of all possible x-coordinates of $ A?$ $ \textbf{(A)}\ 282 \qquad \textbf{(B)}\ 300 \qquad \textbf{(C)}\ 600 \qquad \textbf{(D)}\ 900 \qquad \textbf{(E)}\ 1200$

2024 Kyiv City MO Round 1, Problem 2

Tags: trapezoid , geometry

$ABCD$ is a trapezoid with $BC\parallel AD$ and $BC = 2AD$. Point $M$ is chosen on the side $CD$ such that $AB = AM$. Prove that $BM \perp CD$. [i]Proposed by Bogdan Rublov[/i]

Estonia Open Senior - geometry, 2010.2.1

Tags: area , trapezoid , geometric inequality , geometry

The diagonals of trapezoid $ABCD$ with bases $AB$ and $CD$ meet at $P$. Prove the inequality $S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA}$, where $S_{XYZ}$ denotes the area of triangle $XYZ$.

2017 Adygea Teachers' Geometry Olympiad, 1

Tags: trapezoid , geometry , area

Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$, respectively.

2008 Germany Team Selection Test, 2

Tags: trapezoid , circumcircle , geometry

Let $ ABCD$ be an isosceles trapezium with $ AB \parallel{} CD$ and $ \bar{BC} \equal{} \bar{AD}.$ The parallel to $ AD$ through $ B$ meets the perpendicular to $ AD$ through $ D$ in point $ X.$ The line through $ A$ drawn which is parallel to $ BD$ meets the perpendicular to $ BD$ through $ D$ in point $ Y.$ Prove that points $ C,X,D$ and $ Y$ lie on a common circle.

Ukrainian TYM Qualifying - geometry, 2017.1

Tags: geometry , construction , trapezoid , isosceles

In an isosceles trapezoid $ABCD$ with bases $AD$ and $BC$, diagonals intersect at point $P$, and lines $AB$ and $CD$ intersect at point $Q$. $O_1$ and $O_2$ are the centers of the circles circumscribed around the triangles $ABP$ and $CDP$, $r$ is the radius of these circles. Construct the trapezoid ABCD given the segments $O_1O_2$, $PQ$ and radius $r$.

2019 Peru MO (ONEM), 3

Tags: concurrency , equal angles , trapezoid , geometry

In the trapezoid $ABCD$ , the base $AB$ is smaller than the $CD$ base. The point $K$ is chosen such that $AK$ is parallel to BC and $BK$ is parallel to $AD$. The points $P$ and $Q$ are chosen on the $AK$ and $BK$ rays respectively, such that $\angle ADP = \angle BCK$ and $\angle BCQ = \angle ADK$. (a) Show that the lines $AD, BC$ and $PQ$ go through the same point. (b) Assuming that the circumscribed circumferences of the $APD$ and $BCQ$ triangles intersect at two points, show that one of those points belongs to the line $PQ$.

2003 Tournament Of Towns, 3

Tags: trapezoid , geometry

Points $K$ and $L$ are chosen on the sides $AB$ and $BC$ of the isosceles $\triangle ABC$ ($AB = BC$) so that $AK +LC = KL$. A line parallel to $BC$ is drawn through midpoint $M$ of the segment $KL$, intersecting side $AC$ at point $N$. Find the value of $\angle KNL$.

Durer Math Competition CD Finals - geometry, 2023.D2

Tags: trapezoid , geometry

Let $ABCD$ be a isosceles trapezoid. Base $AD$ is $11$ cm long while the other three sides are each $5$ cm long. We draw the line that is perpendicular to $BD$ and contains $C$ and the line that is perpendicular to $AC$ and contains$ B$. We mark the intersection of these two lines with $E$. What is the distance between point $E$ and line $AD$?

2021 Iranian Geometry Olympiad, 4

Tags: trapezoid , concentric circles , reflection , geometry

In isosceles trapezoid $ABCD$ ($AB \parallel CD$) points $E$ and $F$ lie on the segment $CD$ in such a way that $D, E, F$ and $C$ are in that order and $DE = CF$. Let $X$ and $Y$ be the reflection of $E$ and $C$ with respect to $AD$ and $AF$. Prove that circumcircles of triangles $ADF$ and $BXY$ are concentric. [i]Proposed by Iman Maghsoudi - Iran[/i]