Olympiad problems

2017 Oral Moscow Geometry Olympiad, 2

Tags: geometry , trapezoid , isosceles , perpendicular , incircle

An isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ is given. Circles with centers $O_1$ and $O_2$ are inscribed in triangles $ABC$ and $ABD$. Prove that line $O_1O_2$ is perpendicular on $BC$.

Durer Math Competition CD 1st Round - geometry, 2018.D2

Tags: similar , isosceles , geometry

In an isosceles triangle, we drew one of the angle bisectors. At least one of the resulting two smaller ones triangles is similar to the original. What can be the leg of the original triangle if the length of its base is $1$ unit?

1985 Swedish Mathematical Competition, 3

Tags: geometry , isosceles , equilateral

Points $A,B,C$ with $AB = BC$ are given on a circle with radius $r$, and $D$ is a point inside the circle such that the triangle $BCD$ is equilateral. The line $AD$ meets the circle again at $E$. Show that $DE = r$.

Estonia Open Junior - geometry, 2013.2.3

Tags: geometry , right triangle , isosceles , equal angles

In an isosceles right triangle $ABC$ the right angle is at vertex $C$. On the side $AC$ points $K, L$ and on the side $BC$ points $M, N$ are chosen so that they divide the corresponding side into three equal segments. Prove that there is exactly one point $P$ inside the triangle $ABC$ such that $\angle KPL = \angle MPN = 45^o$.

2014 Romania National Olympiad, 4

Tags: geometry , square , perpendicular , isosceles , right triangle

Outside the square $ABCD$ is constructed the right isosceles triangle $ABD$ with hypotenuse $[AB]$. Let $N$ be the midpoint of the side $[AD]$ and ${M} = CE \cap AB$, ${P} = CN \cap AB$ , ${F} = PE \cap MN$. On the line $FP$ the point $Q$ is considered such that the $[CE$ is the bisector of the angle $QCB$. Prove that $MQ \perp CF$.

2018 Yasinsky Geometry Olympiad, 3

Tags: geometry , construction , circumcenter , circumcircle , angle bisector , isosceles

Point $O$ is the center of circumcircle $\omega$ of the isosceles triangle $ABC$ ($AB = AC$). Bisector of the angle $\angle C$ intersects $\omega$ at the point $W$. Point $Q$ is the center of the circumcircle of the triangle $OWB$. Construct the triangle $ABC$ given the points $Q,W, B$. (Andrey Mostovy)

2003 Cuba MO, 9

Tags: geometry , isosceles , concurrency

Let $D$ be the midpoint of the base $AB$ of the isosceles and acute angle triangle $ABC$, $E$ is a point on $AB$ and $O$ circumcenter of the triangle $ACE$. Prove that the line that passes through $D$ perpendicular to $DO$, the line that passes through $E$ perpendicular to $BC$ and the line that passes through$ B$ parallel to $AC$, they intersect at a point.

Ukraine Correspondence MO - geometry, 2014.7

Tags: geometry , isosceles , angle bisector

Let $ABC$ be an isosceles triangle ($AB = AC$). The points $D$ and $E$ were marked on the ray $AC$ so that $AC = 2AD$ and $AE = 2AC$. Prove that $BC$ is the bisector of the angle $\angle DBE$.

2011 Oral Moscow Geometry Olympiad, 5

Tags: geometry , circumcircle , collinear , isosceles

Let $AA _1$ and $BB_1$ be the altitudes of an isosceles acute-angled triangle $ABC, M$ the midpoint of $AB$. The circles circumscribed around the triangles $AMA_1$ and $BMB_1$ intersect the lines $AC$ and $BC$ at points $K$ and $L$, respectively. Prove that $K, M$, and $L$ lie on the same line.

Cono Sur Shortlist - geometry, 2021.G7

Tags: geometry , mixtilinear incircle , tangent , isosceles

Given an triangle $ABC$ isosceles at the vertex $A$, let $P$ and $Q$ be the touchpoints with $AB$ and $AC$, respectively with the circle $T$, which is tangent to both and is internally tangent to the circumcircle of $ABC$. Let $R$ and $S$ be the points of the circumscribed circle of $ABC$ such that $AP = AR = AS$ . Prove that $RS$ is tangent to $T$ .

2016 Dutch IMO TST, 3

Tags: geometry , equal segments , isosceles , angle bisector

Let $\vartriangle ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $|BF| = |BE|$ and such that $ED$ is the internal angle bisector of $\angle BEC$. Prove that $|BD|= |EF|$ if and only if $|AF| = |EC|$.

2004 Greece Junior Math Olympiad, 2

Tags: geometry , trapezoid , isosceles , area

Let $ABCD$ be a rectangle. Let $K,L$ be the midpoints of $BC, AD$ respectively. From point $B$ the perpendicular line on $AK$, intersects $AK$ at point $E$ and $CL$ at point $Z$. a) Prove that the quadrilateral $AKZL$ is an isosceles trapezoid b) Prove that $2S_{ABKZ}=S_{ABCD}$ c) If quadrilateral $ABCD$ is a square of side $a$, calculate the area of the isosceles trapezoid $AKZL$ in terms of side $BC=a$

2012 Bundeswettbewerb Mathematik, 4

Tags: combinatorial geometry , regular polygon , isosceles , trapezoid , isosceles triangle , combinatorics , geometry

From the vertices of a regular 27-gon, seven are chosen arbitrarily. Prove that among these seven points there are three points that form an isosceles triangle or four points that form an isosceles trapezoid.

2017 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: geometry , isosceles , cosine , angle

Let $ABC$ be an isosceles triangle such that $AB=AC$. Find angles of triangle $ABC$ if $\frac{AB}{BC}=1+2\cos{\frac{2\pi}{7}}$

Mathley 2014-15, 3

Tags: combinatorics , combinatorial geometry , isosceles

Given a regular $2013$-sided polygon, how many isosceles triangles are there whose vertices are vertices vertex of given polygon and haave an angle greater than $120^o$? Nguyen Tien Lam, High School for Natural Science,Hanoi National University.

1962 IMO Shortlist, 6

Tags: geometry , circumcircle , trigonometry , circles , isosceles

Consider an isosceles triangle. let $R$ be the radius of its circumscribed circle and $r$ be the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circle is \[ d=\sqrt{R(R-2r)} \]

Champions Tournament Seniors - geometry, 2011.2

Tags: right angle , isosceles , geometry

Let $ABC$ be an isosceles triangle in which $AB = AC$. On its sides $BC$ and $AC$ respectively are marked points $P$ and $Q$ so that $PQ\parallel AB$. Let $F$ be the center of the circle circumscribed about the triangle $PQC$, and $E$ the midpoint of the segment $BQ$. Prove that $\angle AEF = 90^o $.

2014 Costa Rica - Final Round, 4

Tags: equilateral , semicircle , isosceles , geometry

Consider the isosceles triangle $ABC$ inscribed in the semicircle of radius $ r$. If the $\vartriangle BCD$ and $\vartriangle CAE$ are equilateral, determine the altitude of $\vartriangle DEC$ on the side $DE$ in terms of $ r$. [img]https://cdn.artofproblemsolving.com/attachments/6/3/772ff9a1fd91e9fa7a7e45ef788eec7a1ba48e.png[/img]

2012 Estonia Team Selection Test, 4

Tags: circumcircle , circumcenter , geometry , isosceles , equal segments

Let $ABC$ be a triangle where $|AB| = |AC|$. Points $P$ and $Q$ are different from the vertices of the triangle and lie on the sides $AB$ and $AC$, respectively. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of $ABC$ if and only if $|AP| = |CQ|$.

2007 Oral Moscow Geometry Olympiad, 3

Tags: geometry , trapezoid , isosceles

In a trapezoid, the sum of the lengths of the side and the diagonal is equal to the sum of the lengths of the other side and the other diagonal. Prove that the trapezoid is isosceles.

2020 Ukrainian Geometry Olympiad - April, 1

Tags: geometry , isosceles , angle bisector , perpendicular

In triangle $ABC$, bisectors are drawn $AA_1$ and $CC_1$. Prove that if the length of the perpendiculars drawn from the vertex $B$ on lines $AA1$ and $CC_1$ are equal, then $\vartriangle ABC$ is isosceles.

Novosibirsk Oral Geo Oly VII, 2023.6

Tags: geometry , equal segments , isosceles , angle

An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.

2017 Estonia Team Selection Test, 4

Tags: isosceles , fixed , geometry

Let $ABC$ be an isosceles triangle with apex $A$ and altitude $AD$. On $AB$, choose a point $F$ distinct from $B$ such that $CF$ is tangent to the incircle of $ABD$. Suppose that $\vartriangle BCF$ is isosceles. Show that those conditions uniquely determine: a) which vertex of $BCF$ is its apex, b) the size of $\angle BAC$

2017 Singapore Junior Math Olympiad, 3

Tags: isosceles , equal segments , equal angles , angle

Let $ABC$ be a triangle with $AB=AC$. Let $D$ be a point on $BC$, and $E$ a point on $AD$ such that $\angle BED=\angle BAC=2\angle CED$. Prove that $BD=2CD$.

2019 Durer Math Competition Finals, 15

Tags: isosceles , angle , geometry

$ABC$ is an isosceles triangle such that $AB = AC$ and $\angle BAC = 96^o$. $D$ is the point for which $\angle ACD = 48^o$, $AD = BC$ and triangle $DAC$ is obtuse-angled. Find $\angle DAC$.