Olympiad problems

1968 German National Olympiad, 6

Prove the following two statements: (a) If a triangle is isosceles, then two of its bisectors are of equal length. (b) If two angle bisectors in a triangle are of equal length, then it is isosceles.

Estonia Open Junior - geometry, 1999.1.2

Tags: geometry , locus , isosceles triangle , isosceles

Two different points $X$ and $Y$ are chosen in the plane. Find all the points $Z$ in this plane for which the triangle $XYZ$ is isosceles.

2011 Tournament of Towns, 4

Tags: isosceles , geometry , square , diagonal

Each diagonal of a convex quadrilateral divides it into two isosceles triangles. The two diagonals of the same quadrilateral divide it into four isosceles triangles. Must this quadrilateral be a square?

Durer Math Competition CD Finals - geometry, 2021.C3

Tags: equal angles , geometry , isosceles

In the isosceles triangle $ABC$ we have $AC = BC$. Let $X$ be an arbitrary point of the segment $AB$. The line parallel to $BC$ and passing through $X$ intersects the segment $AC$ in $N$, and the line parallel to $AC$ and passing through $BC$ intersects the segment $BC$ in $M$. Let $k_1$ be the circle with center $N$ and radius $NA$. Similarly, let $k_2$ be the circle with center $M$ and radius $MB$. Let $T$ be the intersection of the circles $k_1$ and $k_2$ different from $X$. Show that the angles $\angle NCM$ and $\angle NTM$ are equal.

2005 JBMO Shortlist, 4

Tags: geometry , isosceles , equal angles

Let $ABC$ be an isosceles triangle $(AB=AC)$ so that $\angle A< 2 \angle B$ . Let $D,Z $ points on the extension of height $AM$ so that $\angle CBD = \angle A$ and $\angle ZBA = 90^\circ$. Let $E$ the orthogonal projection of $M$ on height $BF$, and let $K$ the orthogonal projection of $Z$ on $AE$. Prove that $ \angle KDZ = \angle KDB = \angle KZB$.

Estonia Open Junior - geometry, 2009.1.2

Tags: geometry , isosceles

The feet of the altitudes drawn from vertices $A$ and $B$ of an acute triangle $ABC$ are $K$ and $L$, respectively. Prove that if $|BK| = |KL|$ then the triangle $ABC$ is isosceles.

2023 Mexican Girls' Contest, 1

Tags: isosceles , geometry

Let $\triangle ABC$ such that $AB=AC$, $D$ and $E$ points on $AB$ and $BC$, respectively, with $DE\parallel AC$. Let $F$ on line $DE$ such that $CADF$ it´s a parallelogram. If $O$ is the circumcenter of $\triangle BDE$, prove that $O,F,A$ and $D$ lie on a circle.

Kyiv City MO Juniors Round2 2010+ geometry, 2018.8.31

Tags: geometry , isosceles , equal angles , equal segments

On the sides $AB$, $BC$ and $CA$ of the isosceles triangle $ABC$ with the vertex at the point $B$ marked the points $M$, $D$ and $K$ respectively so that $AM = 2DC$ and $\angle AMD = \angle KDC$. Prove that $MD = KD$.

1962 IMO, 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 \[ d=\sqrt{R(R-2r)} \]

Kyiv City MO Juniors 2003+ geometry, 2015.8.3

Tags: geometry , angle bisector , isosceles , angle chasing , angle

In the isosceles triangle $ABC$, $ (AB = BC)$ the bisector $AD$ was drawn, and in the triangle $ABD$ the bisector $DE$ was drawn. Find the values of the angles of the triangle $ABC$, if it is known that the bisectors of the angles $ABD$ and $AED$ intersect on the line $AD$. (Fedak Ivan)

2019 Novosibirsk Oral Olympiad in Geometry, 4

Tags: square , geometry , isosceles , collinear

Two squares and an isosceles triangle are positioned as shown in the figure (the up left vertex of the large square lies on the side of the triangle). Prove that points $A, B$ and $C$ are collinear. [img]https://cdn.artofproblemsolving.com/attachments/d/c/03515e40f74ced1f8243c11b3e610ef92137ac.png[/img]

2020 Ukrainian Geometry Olympiad - December, 4

Tags: geometry , isosceles , equal segments , angle

In an isosceles triangle $ABC$ with an angle $\angle A= 20^o$ and base $BC=12$ point $E$ on the side $AC$ is chosen such that $\angle ABE= 30^o$ , and point $F$ on the side $AB$ such that $EF = FC$ . Find the length of $FC$.

2022 Federal Competition For Advanced Students, P2, 5

Tags: collinear , geometry , isosceles

Let $ABC$ be an isosceles triangle with base $AB$. We choose a point $P$ inside the triangle on altitude through $C$. The circle with diameter $CP$ intersects the straight line through $B$ and $P$ again at the point $D_P$ and the Straight through $A$ and $C$ one more time at point $E_P$. Prove that there is a point $F$ such that for any choice of $P$ the points $D_P , E_P$ and $F$ lie on a straight line. [i](Walther Janous)[/i]

2017 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: geometry , angle bisector , isosceles

It is given isosceles triangle $ABC$ ($AB=AC$) such that $\angle BAC=108^{\circ}$. Angle bisector of angle $\angle ABC$ intersects side $AC$ in point $D$, and point $E$ is on side $BC$ such that $BE=AE$. If $AE=m$, find $ED$

2022 Novosibirsk Oral Olympiad in Geometry, 5

Tags: geometry , area , isosceles

Two isosceles triangles of the same area are located as shown in the figure. Find the angle $x$. [img]https://cdn.artofproblemsolving.com/attachments/a/6/f7dbfd267274781b67a5f3d5a9036fb2905156.png[/img]

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

2016 Thailand Mathematical Olympiad, 4

Tags: isosceles triangle , coloring , combinatorial geometry , isosceles , combinatorics

Each point on the plane is colored either red, green, or blue. Prove that there exists an isosceles triangle whose vertices all have the same color.

1949-56 Chisinau City MO, 42

Tags: geometry , inscribed , trapezoid , isosceles , equal area

A trapezoid and an isosceles triangle are inscribed in a circle. The larger base of the trapezoid is the diameter of the circle, and the sides of the triangle are parallel to the sides of the trapezoid. Show that the trapezoid and the triangle have equal areas.

2015 Singapore Junior Math Olympiad, 4

Tags: combinatorics , combinatorial geometry , sidelenghts , isosceles

Let $A$ be a set of numbers chosen from $1,2,..., 2015$ with the property that any two distinct numbers, say $x$ and $y$, in $A$ determine a unique isosceles triangle (which is non equilateral) whose sides are of length $x$ or $y$. What is the largest possible size of $A$?

2017 Yasinsky Geometry Olympiad, 1

Tags: geometry , isosceles , trapezoid , inradius

In the isosceles trapezoid with the area of $28$, a circle of radius $2$ is inscribed. Find the length of the side of the trapezoid.

1977 Chisinau City MO, 138

Tags: geometry , isosceles , angle

In an isosceles triangle $BAC$ ($| AC | = | AB |$) , point $D$ is marked on the side $AC$. Determine the angles of the triangle $BDC$ if $\angle A = 40^o$ and $|BC|: |AD|= \sqrt3$.

2017 All-Russian Olympiad, 2

Tags: tangent , geometry , isosceles , circumcircle

Let $ABC$ be an acute angled isosceles triangle with $AB=AC$ and circumcentre $O$. Lines $BO$ and $CO$ intersect $AC, AB$ respectively at $B', C'$. A straight line $l$ is drawn through $C'$ parallel to $AC$. Prove that the line $l$ is tangent to the circumcircle of $\triangle B'OC$.