Olympiad problems

2003 Iran MO (3rd Round), 26

Tags: vector , trigonometry , trapezoid , absolute value , trig identities , law of cosines , geometry

Circles $ C_1,C_2$ intersect at $ P$. A line $ \Delta$ is drawn arbitrarily from $ P$ and intersects with $ C_1,C_2$ at $ B,C$. What is locus of $ A$ such that the median of $ AM$ of triangle $ ABC$ has fixed length $ k$.

2007 India IMO Training Camp, 1

Tags: trapezoid , circumcircle , ratio , homothety , geometry

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

1960 IMO, 7

Tags: geometry , trapezoid , locus , locus problems

An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given. a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$; b) Calculate the distance of $p$ from either base; c) Determine under what conditions such points $P$ actually exist. Discuss various cases that might arise.

2016 PAMO, 5

Tags: geometry , midpoint , trapezoid

Let $ABCD$ be a trapezium such that the sides $AB$ and $CD$ are parallel and the side $AB$ is longer than the side $CD$. Let $M$ and $N$ be on the segments $AB$ and $BC$ respectively, such that each of the segments $CM$ and $AN$ divides the trapezium in two parts of equal area. Prove that the segment $MN$ intersects the segment $BD$ at its midpoint.

2008 China Team Selection Test, 3

Tags: analytic geometry , modular arithmetic , trapezoid , induction , trigonometry , number theory , geometry

Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)

2011 Indonesia TST, 3

Tags: asymptote , geometry , circumcircle , geometric transformation , reflection , trapezoid , perpendicular bisector

Let $ABC$ and $PQR$ be two triangles such that [list] [b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$. [b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$ [/list] Prove that $AB+AC=PQ+PR$.

2022 Moldova EGMO TST, 11

Tags: geometry , trapezoid

Let there be a trapezoid $ABCD$ with bases $AD$ and $BC$. Points $M$ and $P$ are on sides $AB$ and $CD$ such that $CM$ and $BP$ intersect in $N$ and the pentagon $AMNPD$ is cyclic. Prove that the triangle $ADN$ is isosceles.

1993 AIME Problems, 13

Tags: geometry , trapezoid , analytic geometry , quadratic , pythagorean theorem

Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let $t$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

1998 IMO Shortlist, 2

Tags: geometry , circumcircle , trapezoid , ratio , area

Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE:EB=CF:FD$. Let $P$ be the point on the segment $EF$ such that $PE:PF=AB:CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.

2014 Purple Comet Problems, 12

Tags: trigonometry , trapezoid , rectangle , geometry

The vertices of hexagon $ABCDEF$ lie on a circle. Sides $AB = CD = EF = 6$, and sides $BC = DE = F A = 10$. The area of the hexagon is $m\sqrt3$. Find $m$.

2000 Turkey Team Selection Test, 2

Tags: trapezoid , geometry

In a triangle $ABC,$ the internal and external bisectors of the angle $A$ intersect the line $BC$ at $D$ and $E$ respectively. The line $AC$ meets the circle with diameter $DE$ again at $F.$ The tangent line to the circle $ABF$ at $A$ meets the circle with diameter $DE$ again at $G.$ Show that $AF = AG.$

2023 Silk Road, 1

Tags: geometry , trapezoid

Let $ABCD$ be a trapezoid with $AD\parallel BC$. A point $M $ is chosen inside the trapezoid, and a point $N$ is chosen inside the triangle $BMC$ such that $AM\parallel CN$, $BM\parallel DN$. Prove that triangles $ABN$ and $CDM$ have equal areas.

2015 AIME Problems, 4

Tags: geometry , trapezoid , perimeter

In an isosceles trapezoid, the parallel bases have lengths $\log3$ and $\log192$, and the altitude to these bases has length $\log16$. The perimeter of the trapezoid can be written in the form $\log2^p3^q$, where $p$ and $q$ are positive integers. Find $p+q$.

2016 Israel Team Selection Test, 1

Tags: geometry , trapezoid

A square $ABCD$ is given. A point $P$ is chosen inside the triangle $ABC$ such that $\angle CAP = 15^\circ = \angle BCP$. A point $Q$ is chosen such that $APCQ$ is an isosceles trapezoid: $PC \parallel AQ$, and $AP=CQ, AP\nparallel CQ$. Denote by $N$ the midpoint of $PQ$. Find the angles of the triangle $CAN$.

2006 Harvard-MIT Mathematics Tournament, 3

Tags: probability , geometry , trapezoid

The train schedule in Hummut is hopelessly unreliable. Train $A$ will enter Intersection $X$ from the west at a random time between $9:00$ am and $2:30$ pm; each moment in that interval is equally likely. Train $B$ will enter the same intersection from the north at a random time between $9:30$ am and $12:30$ pm, independent of Train $A$; again, each moment in the interval is equally likely. If each train takes $45$ minutes to clear the intersection, what is the probability of a collision today?

2013 India PRMO, 8

Tags: geometry , trapezium , trapezoid

Let $AD$ and $BC$ be the parallel sides of a trapezium $ABCD$. Let $P$ and $Q$ be the midpoints of the diagonals $AC$ and $BD$. If $AD = 16$ and $BC = 20$, what is the length of $PQ$?

1984 Czech And Slovak Olympiad IIIA, 2

Tags: geometry , trigonometry , trapezoid

Let $\alpha, \beta, \gamma, \delta$ be the interior angles of a convex quadrilateral, If $$ \cos\alpha + \cos\beta + \cos\gamma, + \cos\delta = 0 , $$ then this quadrilateral is cyclic or a trapezium. Prove it.

2003 Baltic Way, 14

Tags: geometric transformation , reflection , parallelogram , trapezoid , geometry

Equilateral triangles $AMB,BNC,CKA$ are constructed on the exterior of a triangle $ABC$. The perpendiculars from the midpoints of $MN, NK, KM$ to the respective lines $CA, AB, BC$ are constructed. Prove that these three perpendiculars pass through a single point.

2000 AIME Problems, 11

Tags: geometry , analytic geometry , trapezoid , graphing lines , slope , number theory , relatively prime

The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107).$ The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum o f the absolute values of all possible slopes for $\overline{AB}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

May Olympiad L2 - geometry, 2001.2

Tags: geometry , trapezoid , right angle

On the trapezoid $ABCD$ , side $DA$ is perpendicular to the bases $AB$ and $CD$ . The base $AB$ measures $45$, the base $CD$ measures $20$ and the $BC$ side measures $65$. Let $P$ on the $BC$ side such that $BP$ measures $45$ and $M$ is the midpoint of $DA$. Calculate the measure of the $PM$ segment.

2018 Ecuador NMO (OMEC), 3

Tags: geometry , trapezoid

Let $ABCD$ be a convex quadrilateral with $AB\le CD$. Points $E ,F$ are chosen on segment $AB$ and points $G ,H$ are chosen on the segment $CD$, are chosen such that $AE = BF = CG = DH <\frac{AB}{2}$. Let $P, Q$, and $R$ be the midpoints of $EG$, $FH$, and $CD$, respectively. It is known that $PR$ is parallel to $AD$ and $QR$ is parallel to $BC$. a) Show that $ABCD$ is a trapezoid. b) Let $d$ be the difference of the lengths of the parallel sides. Show that $2PQ\le d$.

2011 Sharygin Geometry Olympiad, 1

Tags: trapezoid , perpendicular , isosceles , diagonal , geometry

The diagonals of a trapezoid are perpendicular, and its altitude is equal to the medial line. Prove that this trapezoid is isosceles

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.

2015 Azerbaijan IMO TST, 3

Tags: geometry , trapezoid

Consider a trapezoid $ABCD$ with $BC||AD$ and $BC<AD$. Let the lines $AB$ and $CD$ meet at $X$. Let $\omega_1$ be the incircle of the triangle $XBC$, and let $\omega_2$ be the excircle of the triangle $XAD$ which is tangent to the segment $AD$ . Denote by $a$ and $d$ the lines tangent to $\omega_1$ , distinct from $AB$ and $CD$, and passing through $A$ and $D$, respectively. Denote by $b$ and $c$ the lines tangent to $\omega_2$ , distinct from $AB$ and $CD$, passing through $B$ and $C$ respectively. Assume that the lines $a,b,c$ and $d$ are distinct. Prove that they form a parallelogram.

Durer Math Competition CD 1st Round - geometry, 2017.D+2

Tags: geometry , trapezoid , collinear

Let the trapezoids $A_iB_iC_iD_i$ ($i = 1, 2, 3$) be similar and have the same clockwise direction. Their angles at $A_i$ and $B_i$ are $60^o$ and the sides $A_1B_1$, $B_2C_2$ and $A_3D_3$ are parallel. The lines $B_iD_{i+1}$ and $C_iA_{i+1}$ intersect at the point $P_i$ (the indices are understood cyclically, i.e. $A_4 = A_1$ and $D_4 = D_1$). Prove that the points $P_1$, $P_2$ and $P_3$ lie on a line.