Olympiad problems

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 $MQ \perp CF$.

2012 Swedish Mathematical Competition, 5

Tags: combinatorial geometry , coloring , isosceles

The vertices of a regular $13$-gon are colored in three different colors. Show that there are three vertices which have the same color and are also the vertices of an isosceles triangle.

2021 Bolivia Ibero TST, 4

Tags: isosceles , geometry

On a isosceles triangle $\triangle ABC$ with $AB=BC$ let $K,M$ be the midpoints of $AB,AC$ respectivily. Let $(CKB)$ intersect $BM$ at $N \ne M$, the line through $N$ parallel to $AC$ intersects $(ABC)$ at $A_1,C_1$. Show that $\triangle A_1BC_1$ is equilateral.

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 Regional Olympiad - Republic of Srpska, 3

Tags: geometry , isosceles , angle

Let $ABC$ be an isosceles triangle with $\angle A=\angle B=80^\circ$. A straight line passes through $B$ and through the circumcenter of the triangle and intersects the side $AC$ at $D$. Prove that $AB=CD$.

2009 Moldova National Olympiad, 10.4

Tags: geometry , perpendicular , isosceles

Let the isosceles triangle $ABC$ with $| AB | = | AC |$. The point $M$ is the midpoint of the base $[BC]$, the point $N$ is the orthogonal projection of the point $M$ on the line $AC$, and the point $P$ is located on the segment $(MC)$ such that $| MP | = | P C | \sin^2 C$. Prove that the lines $AP$ and $BN$ are perpendicular.

1996 Spain Mathematical Olympiad, 2

Tags: geometry , isosceles , centroid

Let $G$ be the centroid of a triangle $ABC$. Prove that if $AB+GC = AC+GB$, then the triangle is isosceles

2003 Estonia National Olympiad, 3

Tags: geometry , rectangle , right triangle , isosceles

In the rectangle $ABCD$ with $|AB|<2 |AD|$, let $E$ be the midpoint of $AB$ and $F$ a point on the chord $CE$ such that $\angle CFD = 90^o$. Prove that $FAD$ is an isosceles triangle.

2023 Dutch IMO TST, 3

Tags: geometry , concyclic , isosceles

The center $O$ of the circle $\omega$ passing through the vertex $C$ of the isosceles triangle $ABC$ ($AB = AC$) is the interior point of the triangle $ABC$. This circle intersects segments $BC$ and $AC$ at points $D \ne C$ and $E \ne C$, respectively, and the circumscribed circle $\Omega$ of the triangle $AEO$ at the point $F \ne E$. Prove that the center of the circumcircle of the triangle $BDF$ lies on the circle $\Omega$.

2009 Tournament Of Towns, 6

Tags: isosceles , geometry , area

Angle $C$ of an isosceles triangle $ABC$ equals $120^o$. Each of two rays emitting from vertex $C$ (inwards the triangle) meets $AB$ at some point ($P_i$) reflects according to the rule the angle of incidence equals the angle of reflection" and meets lateral side of triangle $ABC$ at point $Q_i$ ($i = 1,2$). Given that angle between the rays equals $60^o$, prove that area of triangle $P_1CP_2$ equals the sum of areas of triangles $AQ_1P_1$ and $BQ_2P_2$ ($AP_1 < AP_2$).

2002 Estonia Team Selection Test, 2

Tags: angle bisector , isosceles , trigonometry , geometry

Consider an isosceles triangle $KL_1L_2$ with $|KL_1|=|KL_2|$ and let $KA, L_1B_1,L_2B_2$ be its angle bisectors. Prove that $\cos \angle B_1AB_2 < \frac35$

Denmark (Mohr) - geometry, 1994.5

Tags: right triangle , isosceles , area , equal area , geometry

In a right-angled and isosceles triangle, the two catheti are both length $1$. Find the length of the shortest line segment dividing the triangle into two figures with the same area, and specify the location of this line segment

2007 Sharygin Geometry Olympiad, 2

Tags: geometry , quadrilateral , isosceles , isosceles triangle , diagonal

Each diagonal of a quadrangle divides it into two isosceles triangles. Is it true that the quadrangle is a diamond?

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 $\angle CFA$.

2005 Denmark MO - Mohr Contest, 3

Tags: geometry , isosceles , right triangle

The point $P$ lies inside $\vartriangle ABC$ so that $\vartriangle BPC$ is isosceles, and angle $P$ is a right angle. Furthermore both $\vartriangle BAN$ and $\vartriangle CAM$ are isosceles with a right angle at $A$, and both are outside $\vartriangle ABC$. Show that $\vartriangle MNP$ is isosceles and right-angled. [img]https://1.bp.blogspot.com/-i9twOChu774/XzcBLP-RIXI/AAAAAAAAMXA/n5TJCOJypeMVW28-9GDG4st5C47yhvTCgCLcBGAsYHQ/s0/2005%2BMohr%2Bp3.png[/img]

2002 Nordic, 1

Tags: isosceles , trapezoid , geometry

The trapezium ${ABCD}$, where ${AB}$ and ${CD}$ are parallel and ${AD < CD}$, is inscribed in the circle ${c}$. Let ${DP}$ be a chord of the circle, parallel to ${AC}$. Assume that the tangent to ${c}$ at ${D}$ meets the line ${AB}$ at ${E}$ and that ${PB}$ and ${DC}$ meet at ${Q}$. Show that ${EQ = AC}$.

2022 Puerto Rico Team Selection Test, 3

Tags: ratio , geometry , isosceles

Let $\omega$ be a circle with center $O$ and diameter $AB$. A circle with center at $B$ intersects $\omega$ at C and $AB$ at $D$. The line $CD$ intersects $\omega$ at a point $E$ ($E\ne C$). The intersection of lines $OE$ and $BC$ is $F$. (a) Prove that triangle $OBF$ is isosceles. (b) If $D$ is the midpoint of $OB$, find the value of the ratio $\frac{FB}{BD}$.

2016 Romania National Olympiad, 4

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

Consider the isosceles right triangle $ABC$, with $\angle A = 90^o$ and the point $M \in (BC)$ such that $\angle AMB = 75^o$. On the inner bisector of the angle $MAC$ take a point $F$ such that $BF = AB$. Prove that: a) the lines $AM$ and $BF$ are perpendicular; b) the triangle $CFM$ is isosceles.

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.

Ukraine Correspondence MO - geometry, 2006.10

Tags: radii , geometry , mixtilinear excircle , fixed , isosceles

Let $ABC$ be an isosceles triangle ($AB=AC$). An arbitrary point $M$ is chosen on the extension of the $BC$ beyond point $B$. Prove that the sum of the radius of the circle inscribed in the triangle $AMB$ and the radius of the circle tangent to the side $AC$ and the extensions of the sides $AM, CM$ of the triangle $AMC$ does not depend on the choice of point $M$.

2019 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , circumcircle , orthocenter , isosceles

Let $d$ be the tangent at $B$ to the circumcircle of the acute scalene triangle $ABC$. Let $K$ be the orthogonal projection of the orthocenter, $H$, of triangle $ABC$ to the line $d$ and $L$ the midpoint of the side $AC$. Prove that the triangle $BKL$ is isosceles.

2020 Ukrainian Geometry Olympiad - December, 2

Tags: combinatorics , combinatorial geometry , isosceles , geometry

On a straight line lie $100$ points and another point outside the line. Which is the biggest the number of isosceles triangles can be formed from the vertices of these $101$ points?

2019 Yasinsky Geometry Olympiad, p4

Tags: geometry , circumcircle , isosceles , isosceles triangle

Let $ABC$ be a triangle, $O$ is the center of the circle circumscribed around it, $AD$ the diameter of this circle. It is known that the lines $CO$ and $DB$ are parallel. Prove that the triangle $ABC$ is isosceles. (Andrey Mostovy)

1990 Chile National Olympiad, 1

Tags: isosceles triangle , isosceles , geometry

Show that any triangle can be subdivided into isosceles triangles.

2015 Lusophon Mathematical Olympiad, 5

Tags: geometry , isosceles , tangent circle

Two circles of radius $R$ and $r$, with $R>r$, are tangent to each other externally. The sides adjacent to the base of an isosceles triangle are common tangents to these circles. The base of the triangle is tangent to the circle of the greater radius. Determine the length of the base of the triangle.