Olympiad problems

2016 Romania National Olympiad, 4

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.

2024 Regional Olympiad of Mexico West, 4

Tags: isosceles , parallelogram , geometry

Let $\triangle ABC$ be a triangle and $\omega$ its circumcircle. The tangent to $\omega$ through $B$ cuts the parallel to $BC$ through $A$ at $P$. The line $CP$ cuts the circumcircle of $\triangle ABP$ again in $Q$ and line $AQ$ cuts $\omega$ at $R$. Prove that $BQCR$ is parallelogram if and only if $AC=BC$.

2011 Saudi Arabia Pre-TST, 3.3

Tags: angle , geometry , isosceles

In the isosceles triangle $ABC$, with $AB = AC$, the angle bisector of $\angle B$ intersects side $AC$ at $B'$. Suppose that $ B B' + B'A = BC$. Find the angles of the triangle.

Denmark (Mohr) - geometry, 2002.4

Tags: right triangle , geometry , isosceles

In triangle $ABC$ we have $\angle C = 90^o$ and $AC = BC$. Furthermore $M$ is an interior pont in the triangle so that $MC = 1 , MA = 2$ and $MB =\sqrt2$. Determine $AB$

2020 Yasinsky Geometry Olympiad, 4

Tags: geometry , circumcircle , trapezoid , incenter , isosceles

In an isosceles trapezoid $ABCD$, the base $AB$ is twice as large as the base $CD$. Point $M$ is the midpoint of $AB$. It is known that the center of the circle inscribed in the triangle $MCB$ lies on the circle circumscribed around the triangle $MDC$. Find the angle $\angle MBC$. [img]https://cdn.artofproblemsolving.com/attachments/8/a/7af6a1d32c4e2affa49cb3eed9c10ba1e7ab71.png[/img]

2013 Saudi Arabia BMO TST, 1

Tags: geometry , right triangle , isosceles , cyclic quadrilateral

In triangle $ABC$, $AB = AC = 3$ and $\angle A = 90^o$. Let $M$ be the midpoint of side $BC$. Points $D$ and $E$ lie on sides $AC$ and $AB$ respectively such that $AD > AE$ and $ADME$ is a cyclic quadrilateral. Given that triangle $EMD$ has area $2$, find the length of segment $CD$.

Estonia Open Senior - geometry, 2008.1.2

Tags: concyclic , isosceles , geometry

Let $O$ be the circumcentre of triangle $ABC$. Lines $AO$ and $BC$ intersect at point $D$. Let $S$ be a point on line $BO$ such that $DS \parallel AB$ and lines $AS$ and $BC$ intersect at point $T$. Prove that if $O, D, S$ and $T$ lie on the same circle, then $ABC$ is an isosceles triangle.

2007 Sharygin Geometry Olympiad, 2

Tags: trapezoid , isosceles , geometry

Points $E$ and $F$ are chosen on the base side $AD$ and the lateral side $AB$ of an isosceles trapezoid $ABCD$, respectively. Quadrilateral $CDEF$ is an isosceles trapezoid as well. Prove that $AE \cdot ED = AF \cdot FB$.

Durer Math Competition CD 1st Round - geometry, 2018.C3

Tags: midpoint , isosceles , geometry

In the isosceles triangle $ABC$, $AB = AC$. Let $E$ be on side $AB$ such that $\angle ACE = \angle ECB = 18^o$, and let $D$ be the midpoint of side $CB$. If we know the length of $AD$ is $3$ units, what is the length of $CE$?

1951 Moscow Mathematical Olympiad, 191

Tags: geometry , geometric construction , isosceles trapezoid , trapezoid , isosceles

Given an isosceles trapezoid $ABCD$ and a point $P$. Prove that a quadrilateral can be constructed from segments $PA, PB, PC, PD$. Note: It is allowed that the vertices of a quadrilateral lie not only not only on the sides of the trapezoid, but also on their extensions.

2001 All-Russian Olympiad Regional Round, 8.8

Tags: isosceles , geometry , combinatorial geometry

Prove that any triangle can be cut by at most into $3$ parts, from which an isosceles triangle is formed.

Indonesia MO Shortlist - geometry, g10

Tags: geometry , equal segments , isosceles

Given a triangle $ABC$ with $AB = AC$, angle $\angle A = 100^o$ and $BD$ bisector of angle $\angle B$. Prove that $$BC = BD + DA.$$

1997 Tournament Of Towns, (534) 6

Tags: isosceles , angle , isosceles triangle , geometry

Let $P$ be a point inside the triangle $ABC$ such that $AB = BC$, $\angle ABC = 80^o$, $\angle PAC = 40^o$ and $\angle ACP = 30^o$. Find $\angle BPC$. (G Galperin)

2023 Iranian Geometry Olympiad, 1

Tags: geometry , trapezoid , isosceles

All of the polygons in the figure below are regular. Prove that $ABCD$ is an isosceles trapezoid. [img]https://cdn.artofproblemsolving.com/attachments/e/a/3f4de32becf4a90bf0f0b002fb4d8e724e8844.png[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

2012 Estonia Team Selection Test, 4

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

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 $|AP| = |CQ|$.

2017 Estonia Team Selection Test, 4

Tags: fixed , geometry , isosceles

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 $\angle BAC$

2008 Federal Competition For Advanced Students, P1, 4

Tags: equal segments , isosceles , midpoint , isosceles triangle , geometry

In a triangle $ABC$ let $E$ be the midpoint of the side $AC$ and $F$ the midpoint of the side $BC$. Let $G$ be the foot of the perpendicular from $C$ to $ AB$. Show that $\vartriangle EFG$ is isosceles if and only if $\vartriangle ABC$ is isosceles.

2014 Hanoi Open Mathematics Competitions, 8

Tags: right triangle , isosceles triangle , geometry , isosceles

Let $ABC$ be a triangle. Let $D,E$ be the points outside of the triangle so that $AD=AB,AC=AE$ and $\angle DAB =\angle EAC =90^o$. Let $F$ be at the same side of the line $BC$ as $A$ such that $FB = FC$ and $\angle BFC=90^o$. Prove that the triangle $DEF$ is a right- isosceles triangle.

2022 Austrian MO National Competition, 5

Tags: geometry , collinear , 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]

2003 Cuba MO, 9

Tags: geometry , concurrency , isosceles

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.

Ukrainian TYM Qualifying - geometry, 2019.9

Tags: construction , isosceles , geometry , inradius

On the base $BC$ of the isosceles triangle $ABC$ chose a point $D$ and in each of the triangles $ABD$ and $ACD$ inscribe a circle. Then everything was wiped, leaving only two circles. It is known from which side of their line of centers the apex $A$ is located . Use a compass and ruler to restore the triangle $ABC$ , if we know that : a) $AD$ is angle bisector, b) $AD$ is median.

2022 Azerbaijan EGMO/CMO TST, G1

Tags: tangent , geometry , isosceles

Let $ABC$ be an isosceles triangle with $AC = BC$ and circumcircle $k$. The point $D$ lies on the shorter arc of $k$ over the chord $BC$ and is different from $B$ and $C$. Let $E$ denote the intersection of $CD$ and $AB$. Prove that the line through $B$ and $C$ is a tangent of the circumcircle of the triangle $BDE$. (Karl Czakler)

Mathley 2014-15, 4

Tags: trigonometry , circumcenter , equal segments , orthocenter , isosceles

Points $E, F$ are in the plane of triangle $ABC$ so that triangles $ABE$ and $ACF$ are the opposite directed, and the two triangles are isosceles in that $BE = AE, AF = CF$. Let $H, K$ be the orthocenter of triangle $ABE, ACF$ respectively. Points $M, N$ are the intersections of $BE$ and $CF, CK$ and $CH$. Prove that $MN$ passes through the center of the circumcircle of triangle $ABC$. Nguyen Minh Ha, High School for Education, Hanoi Pedagogical University

2023 Novosibirsk Oral Olympiad in Geometry, 6

Tags: equal segments , geometry , angle , isosceles

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 $\angle CFA$.

2009 Cuba MO, 8

Tags: geometry , angle , isosceles

Let $ABC$ be an isosceles triangle with base $BC$ and $\angle BAC = 20^o$. Let $D$ a point on side $AB$ such that $AD = BC$. Determine $\angle DCA$.