Olympiad problems

1953 AMC 12/AHSME, 48

If the larger base of an isosceles trapezoid equals a diagonal and the smaller base equals the altitude, then the ratio of the smaller base to the larger base is: $ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{2}{3} \qquad\textbf{(C)}\ \frac{3}{4} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{2}{5}$

2013 Sharygin Geometry Olympiad, 6

Tags: trapezoid , circumcircle , asymptote , perpendicular bisector , geometry

Diagonals $AC$ and $BD$ of a trapezoid $ABCD$ meet at $P$. The circumcircles of triangles $ABP$ and $CDP$ intersect the line $AD$ for the second time at points $X$ and $Y$ respectively. Let $M$ be the midpoint of segment $XY$. Prove that $BM = CM$.

May Olympiad L2 - geometry, 2006.4

Tags: geometry , trapezoid , area

Let $ABCD$ be a trapezoid of bases $AB$ and $CD$ . Let $O$ be the intersection point of the diagonals $AC$ and $BD$. If the area of the triangle $ABC$ is $150$ and the area of the triangle $ACD$ is $120$, calculate the area of the triangle $BCO$.

2008 Hong Kong TST, 4

Tags: geometry , geometric transformation , homothety , ratio , circumcircle , trapezoid , projective geometry

Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$. (a) Show that all such lines $ AB$ are concurrent. (b) Find the locus of midpoints of all such segments $ AB$.

2024 Oral Moscow Geometry Olympiad, 1

Tags: geometry , trapezoid

In an isosceles trapezoid, the diagonals are perpendicular. Find the distance from the center of the circle described around the trapezoid to the point of intersection of its diagonals, if the lengths of the bases are equal to $a$ and $b$.

2008 AMC 12/AHSME, 15

Tags: geometry , trapezoid , trigonometry

On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $ R$ be the region formed by the union of the square and all the triangles, and $ S$ be the smallest convex polygon that contains $ R$. What is the area of the region that is inside $ S$ but outside $ R$? $ \textbf{(A)} \; \frac{1}{4} \qquad \textbf{(B)} \; \frac{\sqrt{2}}{4} \qquad \textbf{(C)} \; 1 \qquad \textbf{(D)} \; \sqrt{3} \qquad \textbf{(E)} \; 2 \sqrt{3}$

1994 China Team Selection Test, 3

Tags: symmetry , geometry , geometric transformation , reflection , trapezoid , combinatorics

Find the smallest $n \in \mathbb{N}$ such that if any 5 vertices of a regular $n$-gon are colored red, there exists a line of symmetry $l$ of the $n$-gon such that every red point is reflected across $l$ to a non-red point.

2021 Saint Petersburg Mathematical Olympiad, 6

Tags: geometry , trapezoid , circumcircle , angle bisector

Point $M$ is the midpoint of base $AD$ of an isosceles trapezoid $ABCD$ with circumcircle $\omega$. The angle bisector of $ABD$ intersects $\omega$ at $K$. Line $CM$ meets $\omega$ again at $N$. From point $B$, tangents $BP, BQ$ are drawn to $(KMN)$. Prove that $BK, MN, PQ$ are concurrent. [i]A. Kuznetsov[/i]

2014 Stanford Mathematics Tournament, 5

Tags: geometry , circumcircle , trapezoid

Let $ABC$ be a triangle where $\angle BAC = 30^\circ$. Construct $D$ in $\triangle ABC$ such that $\angle ABD = \angle ACD = 30^\circ$. Let the circumcircle of $\triangle ABD$ intersect $AC$ at $X$. Let the circumcircle of $\triangle ACD$ intersect $AB$ at $Y$. Given that $DB - DC = 10$ and $BC = 20$, find $AX \cdot AY$.

1994 China National Olympiad, 1

Tags: trapezoid , geometry

Let $ABCD$ be a trapezoid with $AB\parallel CD$. Points $E,F$ lie on segments $AB,CD$ respectively. Segments $CE,BF$ meet at $H$, and segments $ED,AF$ meet at $G$. Show that $S_{EHFG}\le \dfrac{1}{4}S_{ABCD}$. Determine, with proof, if the conclusion still holds when $ABCD$ is just any convex quadrilateral.

2014 National Olympiad First Round, 21

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

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 $

1997 Pre-Preparation Course Examination, 2

Tags: circumcircle , incenter , rectangle , parallelogram , ratio , trapezoid , geometry

Let $P$ be a variable point on arc $BC$ of the circumcircle of triangle $ABC$ not containing $A$. Let $I_1$ and $I_2$ be the incenters of the triangles $PAB$ and $PAC$, respectively. Prove that: [b](a)[/b] The circumcircle of $?PI_1I_2$ passes through a fixed point. [b](b)[/b] The circle with diameter $I_1I_2$ passes through a fixed point. [b](c)[/b] The midpoint of $I_1I_2$ lies on a fixed circle.

2017 Yasinsky Geometry Olympiad, 6

Tags: geometry , trapezoid , ratio , computational

Given a trapezoid $ABCD$ with bases $BC$ and $AD$, with $AD=2 BC$. Let $M$ be the midpoint of $AD, E$ be the intersection point of the sides $AB$ and $CD$, $O$ be the intersection point of $BM$ and $AC, N$ be the intersection point of $EO$ and $BC$. In what ratio, point $N$ divides the segment $BC$?

2002 All-Russian Olympiad, 2

Tags: geometry , trapezoid , trigonometry , parallelogram , geometric transformation , reflection , trig identities

A quadrilateral $ABCD$ is inscribed in a circle $\omega$. The tangent to $\omega$ at $A$ intersects the ray $CB$ at $K$, and the tangent to $\omega$ at $B$ intersects the ray $DA$ at $M$. Prove that if $AM=AD$ and $BK=BC$, then $ABCD$ is a trapezoid.

2002 China Team Selection Test, 1

Tags: trapezoid , geometry

Circle $ O$ is inscribed in a trapzoid $ ABCD$, $ \angle{A}$ and $ \angle{B}$ are all acute angles. A line through $ O$ intersects $ AD$ at $ E$ and $ BC$ at $ F$, and satisfies the following conditions: (1) $ \angle{DEF}$ and $ \angle{CFE}$ are acute angles. (2) $ AE\plus{}BF\equal{}DE\plus{}CF$. Let $ AB\equal{}a$, $ BC\equal{}b$, $ CD\equal{}c$, then use $ a,b,c$ to express $ AE$.

2010 IFYM, Sozopol, 8

Tags: geometry , trapezoid , equal segments , equal angles

In the trapezoid $ABCD, AB // CD$ and the diagonals intersect at $O$. The points $P, Q$ are on $AD, BC$ respectively such that $\angle AP B = \angle CP D$ and $\angle AQB = \angle CQD$. Show that $OP = OQ$.

2023 All-Russian Olympiad, 7

Tags: geometry , trapezoid

Given a trapezoid $ABCD$, in which $AD \parallel BC$, and rays $AB$ and $DC$ intersect at point $G$. The common external tangents to the circles $(ABC), (ACD)$ intersect at point $E$. The common external tangents to circles $(ABD), (CBD)$ meet at $F$. Prove that the points $E, F$ and $G$ are collinear.

2009 Hong kong National Olympiad, 3

Tags: circumcircle , trapezoid , geometry

$ABC$ is a right triangle with $\angle C=90$,$CD$ is perpendicular to $AB$,and $D$ is the foot,$\omega$ is the circumcircle of triangle $BCD$,$\omega_{1}$ is a circle inside triangle $ACD$,tangent to $AD$ and $AC$ at $M$ and $N$ respectively,and $\omega_{1}$ is also tangent to $\omega$.prove that: (1)$BD*CN+BC*DM=CD*BM$ (2)$BM=BC$

2008 AIME Problems, 5

Tags: geometry , trapezoid , ratio , circumcircle , analytic geometry , geometric transformation , homothety

In trapezoid $ ABCD$ with $ \overline{BC}\parallel\overline{AD}$, let $ BC\equal{}1000$ and $ AD\equal{}2008$. Let $ \angle A\equal{}37^\circ$, $ \angle D\equal{}53^\circ$, and $ m$ and $ n$ be the midpoints of $ \overline{BC}$ and $ \overline{AD}$, respectively. Find the length $ MN$.

2000 Romania Team Selection Test, 1

Tags: geometry , trapezoid , ceiling function , combinatorics

Let $P_1$ be a regular $n$-gon, where $n\in\mathbb{N}$. We construct $P_2$ as the regular $n$-gon whose vertices are the midpoints of the edges of $P_1$. Continuing analogously, we obtain regular $n$-gons $P_3,P_4,\ldots ,P_m$. For $m\ge n^2-n+1$, find the maximum number $k$ such that for any colouring of vertices of $P_1,\ldots ,P_m$ in $k$ colours there exists an isosceles trapezium $ABCD$ whose vertices $A,B,C,D$ have the same colour. [i]Radu Ignat[/i]

2001 Turkey MO (2nd round), 1

Tags: circumcircle , trapezoid , geometry

Let $ABCD$ be a convex quadrilateral. The perpendicular bisectors of the sides $[AD]$ and $[BC]$ intersect at a point $P$ inside the quadrilateral and the perpendicular bisectors of the sides $[AB]$ and $[CD]$ also intersect at a point $Q$ inside the quadrilateral. Show that, if $\angle APD = \angle BPC$ then $\angle AQB = \angle CQD$

2004 All-Russian Olympiad, 2

Tags: circumcircle , euler , trapezoid , incenter , rectangle , geometric transformation , geometry

Let $ I(A)$ and $ I(B)$ be the centers of the excircles of a triangle $ ABC,$ which touches the sides $ BC$ and $ CA$ in its interior. Furthermore let $ P$ a point on the circumcircle $ \omega$ of the triangle $ ABC.$ Show that the center of the segment which connects the circumcenters of the triangles $ I(A)CP$ and $ I(B)CP$ coincides with the center of the circle $ \omega.$

1960 IMO Shortlist, 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.

1996 Baltic Way, 4

Tags: trapezoid , geometry

$ABCD$ is a trapezium where $AD\parallel BC$. $P$ is the point on the line $AB$ such that $\angle CPD$ is maximal. $Q$ is the point on the line $CD$ such that $\angle BQA$ is maximal. Given that $P$ lies on the segment $AB$, prove that $\angle CPD=\angle BQA$.

2011 India IMO Training Camp, 1

Tags: trapezoid , trigonometry , cyclic quadrilateral , geometry

Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that: $a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area. $b) a\cdot AP=b\cdot BP=c\cdot PC.$