Olympiad problems

2003 Junior Balkan Team Selection Tests - Moldova, 7

The triangle $ABC$ is isosceles with $AB=BC$. The point F on the side $[BC]$ and the point $D$ on the side $AC$ are the feets of the the internals bisectors drawn from $A$ and altitude drawn from $B$ respectively so that $AF=2BD$. Fine the measure of the angle $ABC$.

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 $\frac{FB}{BD}$.

2011 Saudi Arabia Pre-TST, 3.3

Tags: isosceles , angle , geometry

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.

2000 Estonia National Olympiad, 4

Tags: parallelogram , geometry , isosceles , midpoint

Let $E$ be the midpoint of the side $AB$ of the parallelogram $ABCD$. Let $F$ be the projection of $B$ on $AC$. Prove that the triangle $ABF$ is isosceles

Ukraine Correspondence MO - geometry, 2004.10.

Tags: geometry , isosceles

In an isosceles triangle $ABC$ ($AB = AC$), the bisector of the angle $B$ intersects $AC$ at point $D$ such that $BC = BD + AD$. Find $\angle A$.

2020 Czech and Slovak Olympiad III A, 5

Tags: geometry , equal angles , isosceles , concurrency

Given an isosceles triangle $ABC$ with base $BC$. Inside the side $BC$ is given a point $D$. Let $E, F$ be respectively points on the sides $AB, AC$ that $|\angle BED | = |\angle DF C| > 90^o$ . Prove that the circles circumscribed around the triangles $ABF$ and $AEC$ intersect on the line $AD$ at a point different from point $A$. (Patrik Bak, Michal Rolínek)

2013 NZMOC Camp Selection Problems, 9

Tags: parallelogram , geometry , right triangle , isosceles

Let $ABC$ be a triangle with $\angle CAB > 45^o$ and $\angle CBA > 45^o$. Construct an isosceles right angled triangle $RAB$ with $AB$ as its hypotenuse and $R$ inside $ABC$. Also construct isosceles right angled triangles $ACQ$ and $BCP$ having $AC$ and $BC$ respectively as their hypotenuses and lying entirely outside $ABC$. Show that $CQRP$ is a parallelogram.

2021 Irish Math Olympiad, 2

Tags: ratio , right triangle , isosceles , geometry

An isosceles triangle $ABC$ is inscribed in a circle with $\angle ACB = 90^o$ and $EF$ is a chord of the circle such that neither E nor $F$ coincide with $C$. Lines $CE$ and $CF$ meet $AB$ at $D$ and $G$ respectively. Prove that $|CE|\cdot |DG| = |EF| \cdot |CG|$.

1951 Moscow Mathematical Olympiad, 191

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

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.

2014 Contests, 1

Tags: geometry , perpendicular , isosceles

As shown in the figure, given $\vartriangle ABC$ with $\angle B$, $\angle C$ acute angles, $AD \perp BC$, $DE \perp AC$, $M$ midpoint of $DE$, $AM \perp BE$. Prove that $\vartriangle ABC$ is isosceles. [img]https://cdn.artofproblemsolving.com/attachments/a/8/f553c33557979f6f7b799935c3bde743edcc3c.png[/img]

2008 BAMO, 3

Tags: geometry , combinatorics , combinatorial geometry , isosceles

A triangle is constructed with the lengths of the sides chosen from the set $\{2, 3, 5, 8, 13, 21, 34, 55, 89, 144\}$. Show that this triangle must be isosceles. (A triangle is isosceles if it has at least two sides the same length.)

2003 All-Russian Olympiad Regional Round, 9.3

Tags: geometry , incircle , isosceles , concurrency

In an isosceles triangle $ABC$ ($AB = BC$), the midline parallel to side $BC$ intersects the incircle at a point $F$ that does not lie on the base $AC$. Prove that the tangent to the circle at point $F$ intersects the bisector of angle $C$ on side $AB$.

Brazil L2 Finals (OBM) - geometry, 2002.5

Tags: circumcircle , isosceles , inscribed , perpendicular , geometry

Let $ABC$ be a triangle inscribed in a circle of center $O$ and $P$ be a point on the arc $AB$, that does not contain $C$. The perpendicular drawn fom $P$ on line $BO$ intersects $AB$ at $S$ and $BC$ at $T$. The perpendicular drawn from $P$ on line $AO$ intersects $AB$ at $Q$ and $AC$ at $R$. Prove that: a) $PQS$ is an isosceles triangle b) $PQ^2=QR= ST$

2007 Cuba MO, 6

Tags: geometry , isosceles

Let the triangle $ABC$ be acute. Let us take in the segment $BC$ two points $F$ and $G$ such that $BG > BF = GC$ and an interior point$ P$ to the triangle on the bisector of $\angle BAC$. Then are drawn through $P$, $PD\parallel AB$ and $PE \parallel AC$, $D \in AC$ and $E \in AB$, $\angle FEP = \angle PDG$. prove that $\vartriangle ABC$ is isosceles.

2010 District Olympiad, 3

Tags: geometry , isosceles , equal segments , angle

Consider triangle $ABC$ with $AB = AC$ and $\angle A = 40 ^o$. The points $S$ and $T$ are on the sides $AB$ and $BC$, respectively, so that $\angle BAT = \angle BCS= 10 ^o$. The lines $AT$ and $CS$ intersect at point $P$. Prove that $BT = 2PT$.

1997 Tournament Of Towns, (560) 1

Tags: geometry , isosceles , equal segments , angle bisector

$M$ and $N$ are the midpoints of the sides $AB$ and $AC$ of a triangle ABC respectively. $P$ and $Q$ are points on the sides $AB$ and $AC$ respectively such that the bisector of the angle $ACB$ also bisects the angle $MCP$, and the bisector of the angle $ABC$ also bisects the angle $NBQ$. If $AP = AQ$, does it follow that $ABC$ is isosceles? (V Senderov)

2017 Hanoi Open Mathematics Competitions, 15

Tags: isosceles triangle , isosceles , quadrilateral , geometry

Show that an arbitrary quadrilateral can be divided into nine isosceles triangles.

2021 Novosibirsk Oral Olympiad in Geometry, 5

Tags: geometry , right triangle , isosceles , equal segments

On the legs $AC$ and $BC$ of an isosceles right-angled triangle with a right angle $C$, points $D$ and $E$ are taken, respectively, so that $CD = CE$. Perpendiculars on line $AE$ from points $C$ and $D$ intersect segment $AB$ at points $P$ and $Q$, respectively. Prove that $BP = PQ$.

2013 Grand Duchy of Lithuania, 2

Tags: geometry , isosceles , tangent , equal angles

Let $ABC$ be an isosceles triangle with $AB = AC$. The points $D, E$ and $F$ are taken on the sides $BC, CA$ and $AB$, respectively, so that $\angle F DE = \angle ABC$ and $FE$ is not parallel to $BC$. Prove that the line $BC$ is tangent to the circumcircle of $\vartriangle DEF$ if and only if $D$ is the midpoint of the side $BC$.

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 $|AF| = |EC|$.

Ukrainian From Tasks to Tasks - geometry, 2011.8

Tags: geometry , angle bisector , parallel , isosceles

On the median $AD$ of the isosceles triangle $ABC$, point $E$ is marked. Point $F$ is the projection of point $E$ on the line $BC$, point $M$ lies on the segment $EF$, points $N$ and $P$ are projections of point $M$ on the lines $AC$ and $AB$, respectively. Prove that the bisectors of the angles $PMN$ and $PEN$ are parallel.