Olympiad problems

2012 JHMT, 3

Tags: geometry , trapezoid

In trapezoid $ABCD$, $BC \parallel AD$, $AB = 13$, $BC = 15$, $CD = 14$, and $DA = 30$. Find the area of $ABCD$.

2019 BMT Spring, 7

Tags: trapezoid , geometry , probability

Points $ A, B, C, D $ are vertices of an isosceles trapezoid, with $ \overline{AB} $ parallel to $ \overline{CD} $, $ AB = 1 $, $ CD = 2 $, and $ BC = 1 $. Point $ E $ is chosen uniformly and at random on $ \overline{CD} $, and let point $ F $ be the point on $ \overline{CD} $ such that $ EC = FD $. Let $ G $ denote the intersection of $ \overline{AE} $ and $ \overline{BF} $, not necessarily in the trapezoid. What is the probability that $ \angle AGB > 30^\circ $?

2002 Austria Beginners' Competition, 4

Tags: trapezoid , construction , geometry

In a trapezoid $ABCD$ with base $AB$ let $E$ be the midpoint of side $AD$. Suppose further that $2CD=EC=BC=b$. Let $\angle ECB=120^{\circ}$. Construct the trapezoid and determine its area based on $b$.

Indonesia MO Shortlist - geometry, g3.3

Tags: logarithm , geometry , geometric transformation , dilation , trapezoid , trigonometry , parallelogram

Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

1972 IMO Longlists, 43

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

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 $M$.

1963 Miklós Schweitzer, 9

Tags: real analysis , trapezoid , function , geometry

Let $ f(t)$ be a continuous function on the interval $ 0 \leq t \leq 1$, and define the two sets of points \[ A_t\equal{}\{(t,0): t\in[0,1]\} , B_t\equal{}\{(f(t),1): t\in [0,1]\}.\] Show that the union of all segments $ \overline{A_tB_t}$ is Lebesgue-measurable, and find the minimum of its measure with respect to all functions $ f$. [A. Csaszar]

1960 AMC 12/AHSME, 13

Tags: trapezoid , analytic geometry , graphing lines , geometry , slope

The polygon(s) formed by $y=3x+2$, $y=-3x+2$, and $y=-2$, is (are): $ \textbf{(A) }\text{An equilateral triangle}\qquad\textbf{(B) }\text{an isosceles triangle} \qquad\textbf{(C) }\text{a right triangle} \qquad$ $\textbf{(D) }\text{a triangle and a trapezoid}\qquad\textbf{(E) }\text{a quadrilateral} $

1990 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: perpendicular , trapezoid , geometry , collinear

Let $ABCD$ be a trapezium with bases $AB$ and $CD$ such that $AB = 2 CD$. From $A$ the line $r$ is drawn perpendicular to $BC$ and from $B$ the line $t$ is drawn perpendicular to $AD$. Let $P$ be the intersection point of $r$ and $t$. From $C$ the line $s$ is drawn perpendicular to $BC$ and from $D$ the line $u$ perpendicular to $AD$. Let $Q$ be the intersection point of $s$ and $u$. If $R$ is the intersection point of the diagonals of the trapezium, prove that points $P, Q$ and $R$ are collinear.

2014 IberoAmerican, 2

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

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.

2003 Iran MO (3rd Round), 26

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

2002 Bosnia Herzegovina Team Selection Test, 2

Tags: geometry , trapezoid

Triangle $ABC$ is given in a plane. Draw the bisectors of all three of its angles. Then draw the line that connects the points where the bisectors of angles $ABC$ and $ACB$ meet the opposite sides of the triangle. Through the point of intersection of this line and the bisector of angle $BAC$, draw another line parallel to $BC$. Let this line intersect $AB$ in $M$ and $AC$ in $N$. Prove that $2MN = BM+CN$.

2012 Iran Team Selection Test, 2

Tags: trapezoid , geometry , circumcircle

Points $A$ and $B$ are on a circle $\omega$ with center $O$ such that $\tfrac{\pi}{3}< \angle AOB <\tfrac{2\pi}{3}$. Let $C$ be the circumcenter of the triangle $AOB$. Let $l$ be a line passing through $C$ such that the angle between $l$ and the segment $OC$ is $\tfrac{\pi}{3}$. $l$ cuts tangents in $A$ and $B$ to $\omega$ in $M$ and $N$ respectively. Suppose circumcircles of triangles $CAM$ and $CBN$, cut $\omega$ again in $Q$ and $R$ respectively and theirselves in $P$ (other than $C$). Prove that $OP\perp QR$. [i]Proposed by Mehdi E'tesami Fard, Ali Khezeli[/i]

2025 AIME, 6

Tags: trapezoid , geometry

An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is $3$, and the area of the trapezoid is $72$. Let the parallel sides of the trapezoid have lengths $r$ and $s$, with $r \neq s$. Find $r^2+s^2$

1996 Moscow Mathematical Olympiad, 4

Tags: ratio , trapezoid , geometry

Consider an equilateral triangle $\triangle ABC$. The points $K$ and $L$ divide the leg $BC$ into three equal parts, the point $M$ divides the leg $AC$ in the ratio $1:2$, counting from the vertex $A$. Prove that $\angle AKM+\angle ALM=30^{\circ}$. Proposed by V. Proizvolov

2001 Junior Balkan Team Selection Tests - Romania, 2

Tags: geometry , circumcircle , trapezoid

Let $ABCDEF$ be a hexagon with $AB||DE,\ BC||EF,\ CD||FA$ and in which the diagonals $AD,BE$ and $CF$ are congruent. Prove that the hexagon can be inscribed in a circle.

2009 Sharygin Geometry Olympiad, 3

Tags: incenter , trapezoid , rectangle , geometry , parallelogram

The bisectors of trapezoid's angles form a quadrilateral with perpendicular diagonals. Prove that this trapezoid is isosceles.

2022 Adygea Teachers' Geometry Olympiad, 2

Tags: geometry , trapezoid , fixed , fixed point

An arbitrary point $P$ is chosen on the lateral side $AB$ of the trapezoid $ABCD$. Straight lines passing through it parallel to the diagonals of the trapezoid intersect the bases at points $Q$ and $R$. Prove that the sides $QR$ of all possible triangles $PQR$ pass through a fixed point.

2010 ITAMO, 4

Tags: trapezoid , geometry

In a trapezium $ABCD$, the sides $AB$ and $CD$ are parallel and the angles $\angle ABC$ and $\angle BAD$ are acute. Show that it is possible to divide the triangle $ABC$ into 4 disjoint triangle $X_1. . . , X_4$ and the triangle $ABD$ into 4 disjoint triangles $Y_1,. . . , Y_4$ such that the triangles $X_i$ and $Y_i$ are congruent for all $i$.

2003 Balkan MO, 2

Tags: trigonometry , calculus , geometric transformation , trapezoid , ratio , circumcircle , geometry

Let $ABC$ be a triangle, and let the tangent to the circumcircle of the triangle $ABC$ at $A$ meet the line $BC$ at $D$. The perpendicular to $BC$ at $B$ meets the perpendicular bisector of $AB$ at $E$. The perpendicular to $BC$ at $C$ meets the perpendicular bisector of $AC$ at $F$. Prove that the points $D$, $E$ and $F$ are collinear. [i]Valentin Vornicu[/i]

2010 Contests, 2

Tags: trapezoid , geometry , angle bisector

In trapezoid $ABCD$, $AD$ is parallel to $BC$. Knowing that $AB=AD+BC$, prove that the bisector of $\angle A$ also bisects $CD$.

2014 National Olympiad First Round, 21

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

Let $ABCD$ be a trapezoid such that side $[AB]$ and side $[CD]$ are perpendicular to side $[BC]$. Let $E$ be a point on side $[BC]$ such that $\triangle AED$ is equilateral. If $|AB|=7$ and $|CD|=5$, what is the area of trapezoid $ABCD$? $ \textbf{(A)}\ 27\sqrt{3} \qquad\textbf{(B)}\ 42 \qquad\textbf{(C)}\ 24\sqrt{3} \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ 36 $

2020 Tournament Of Towns, 5

Tags: equal segments , geometry , right angle , trapezoid , trapezium

Let $ABCD$ be an inscribed trapezoid. The base $AB$ is $3$ times longer than $CD$. Tangents to the circumscribed circle at the points $A$ and $C$ intersect at the point $K$. Prove that the angle $KDA$ is a right angle. Alexandr Yuran

2014 All-Russian Olympiad, 2

Tags: symmetry , geometry , trapezoid

Let $ABCD$ be a trapezoid with $AB\parallel CD$ and $ \Omega $ is a circle passing through $A,B,C,D$. Let $ \omega $ be the circle passing through $C,D$ and intersecting with $CA,CB$ at $A_1$, $B_1$ respectively. $A_2$ and $B_2$ are the points symmetric to $A_1$ and $B_1$ respectively, with respect to the midpoints of $CA$ and $CB$. Prove that the points $A,B,A_2,B_2$ are concyclic. [i]I. Bogdanov[/i]

2009 Sharygin Geometry Olympiad, 4

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

Let $ P$ and $ Q$ be the common points of two circles. The ray with origin $ Q$ reflects from the first circle in points $ A_1$, $ A_2$,$ \ldots$ according to the rule ''the angle of incidence is equal to the angle of reflection''. Another ray with origin $ Q$ reflects from the second circle in the points $ B_1$, $ B_2$,$ \ldots$ in the same manner. Points $ A_1$, $ B_1$ and $ P$ occurred to be collinear. Prove that all lines $ A_iB_i$ pass through P.

2012 China Western Mathematical Olympiad, 1

Tags: circumcircle , parallelogram , trapezoid , geometry , ratio

$O$ is the circumcenter of acute $\Delta ABC$, $H$ is the Orthocenter. $AD \bot BC$, $EF$ is the perpendicular bisector of $AO$,$D,E$ on the $BC$. Prove that the circumcircle of $\Delta ADE$ through the midpoint of $OH$.