Olympiad problems

Denmark (Mohr) - geometry, 1994.5

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

V Soros Olympiad 1998 - 99 (Russia), 9.5

Tags: locus , geometry , isosceles

An angle with vertex $A$ is given on the plane. Points $K$ and $P$ are selected on its sides so that $AK + AP = a$, where $a$ is a given segment. Let $M$ be a point on the plane such that the triangle $KPM$ is isosceles with the base $KP$ and the angle at the vertex $M$ equal to the given angle. Find the locus of points $M$ (as $K$ and $P$ move).

2009 May Olympiad, 2

Tags: geometry , angle chasing , equilateral triangle , convex quadrilateral , isosceles

Let $ABCD$ be a convex quadrilateral such that the triangle $ABD$ is equilateral and the triangle $BCD$ is isosceles, with $\angle C = 90^o$. If $E$ is the midpoint of the side $AD$, determine the measure of the angle $\angle CED$.

1977 Chisinau City MO, 138

Tags: angle , isosceles , geometry

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 $|BC|: |AD|= \sqrt3$.

2010 Junior Balkan Team Selection Tests - Romania, 4

Tags: isosceles , equal segments , angle , equal angles , geometry

Let $ABC$ be an isosceles triangle with $AB = AC$ and let $n$ be a natural number, $n>1$. On the side $AB$ we consider the point $M$ such that $n \cdot AM = AB$. On the side $BC$ we consider the points $P_1, P_2, ....., P_ {n-1}$ such that $BP_1 = P_1P_2 = .... = P_ {n-1} C = \frac{1}{n} BC$. Show that: $\angle {MP_1A} + \angle {MP_2A} + .... + \angle {MP_ {n-1} A} = \frac{1} {2} \angle {BAC}$.

1975 Czech and Slovak Olympiad III A, 5

Tags: triangle , locus , isosceles , geometry , square

Let a square $\mathbf P=P_1P_2P_3P_4$ be given in the plane. Determine the locus of all vertices $A$ of isosceles triangles $ABC,AB=BC$ such that the vertices $B,C$ are points of the square $\mathbf P.$

2019 Israel National Olympiad, 3

Tags: isosceles , geometry

Six congruent isosceles triangles have been put together as described in the picture below. Prove that points M, F, C lie on one line. [img]https://i.imgur.com/1LU5Zmb.png[/img]

2018 Yasinsky Geometry Olympiad, 6

Tags: perpendicular , isosceles triangle , geometry , perpendicularity , circumcircle , incenter , isosceles

Given a triangle $ABC$, in which $AB = BC$. Point $O$ is the center of the circumcircle, point $I$ is the center of the incircle. Point $D$ lies on the side $BC$, such that the lines $DI$ and $AB$ parallel. Prove that the lines $DO$ and $CI$ are perpendicular. (Vyacheslav Yasinsky)

Durer Math Competition CD Finals - geometry, 2017.C2

Tags: line , right triangle , geometry , isosceles

The triangle $ABC$ is isosceles and has a right angle at the vertex $A$. Construct all points that simultaneously satisfy the following two conditions: (i) are equidistant from points $A$ and $B$ (ii) heve distance exactly three times from point $C$ as far as from point $B$.

Kyiv City MO Juniors Round2 2010+ geometry, 2018.8.31

Tags: equal angles , equal segments , isosceles , geometry

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 $MD = KD$.

2008 Chile National Olympiad, 2

Tags: isosceles , right triangle , triangle , geometry

Let $ABC$ be right isosceles triangle with right angle in $A$. Given a point $P$ inside the triangle, denote by $a, b$ and $c$ the lengths of $PA, PB$ and $PC$, respectively. Prove that there is a triangle whose sides have a length of $a\sqrt2 , b$ and $c$.

1994 Spain Mathematical Olympiad, 4

Tags: isosceles , trigonometry , geometry

In a triangle $ABC$ with $ \angle A = 36^o$ and $AB = AC$, the bisector of the angle at $C$ meets the oposite side at $D$. Compute the angles of $\triangle BCD$. Express the length of side $BC$ in terms of the length $b$ of side $AC$ without using trigonometric functions.

2007 BAMO, 3

Tags: isosceles , geometry

In $\vartriangle ABC, D$ and $E$ are two points on segment $BC$ such that $BD = CE$ and $\angle BAD = \angle CAE$. Prove that $\vartriangle ABC$ is isosceles

2003 All-Russian Olympiad Regional Round, 9.3

Tags: concurrency , isosceles , geometry , incircle

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 $AB$.

Estonia Open Junior - geometry, 2016.1.5

Tags: isosceles , right triangle , geometry

A right triangle $ABC$ has the right angle at vertex $A$. Circle $c$ passes through vertices $A$ and $B$ of the triangle $ABC$ and intersects the sides $AC$ and $BC$ correspondingly at points $D$ and $E$. The line segment $CD$ has the same length as the diameter of the circle $c$. Prove that the triangle $ABE$ is isosceles.

2014 Sharygin Geometry Olympiad, 2

Tags: isosceles , geometry

Let $AH_a$ and $BH_b$ be altitudes, $AL_a$ and $BL_b$ be angle bisectors of a triangle $ABC$. It is known that $H_aH_b // L_aL_b$. Is it necessarily true that $AC = BC$? (B. Frenkin)

1979 All Soviet Union Mathematical Olympiad, 269

Tags: area , inscribed , geometry , isosceles

What is the least possible ratio of two isosceles triangles areas, if three vertices of the first one belong to three different sides of the second one?

Brazil L2 Finals (OBM) - geometry, 2002.5

Tags: inscribed , isosceles , perpendicular , geometry , circumcircle

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) $PQ^2=QR= ST$

2012 Belarus Team Selection Test, 2

Tags: geometry , isosceles , inscirbed , equal segments

$A, B, C, D, E$ are five points on the same circle, so that $ABCDE$ is convex and we have $AB = BC$ and $CD = DE$. Suppose that the lines $(AD)$ and $(BE)$ intersect at $P$, and that the line $(BD)$ meets line $(CA)$ at $Q$ and line $(CE)$ at $T$. Prove that the triangle $PQT$ is isosceles. (I. Voronovich)

2020 Yasinsky Geometry Olympiad, 6

Tags: isosceles , geometry , concurrency , incircle , circumcircle

In an isosceles triangle $ABC, I$ is the center of the inscribed circle, $M_1$ is the midpoint of the side $BC, K_2, K_3$ are the points of contact of the inscribed circle of the triangle with segments $AC$ and $AB$, respectively. The point $P$ lies on the circumcircle of the triangle $BCI$, and the angle $M_1PI$ is right. Prove that the lines $BC, PI, K_2K_3$ intersect at one point. (Mikhail Plotnikov)

2002 Estonia Team Selection Test, 2

Tags: trigonometry , isosceles , angle bisector , 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 $\cos \angle B_1AB_2 < \frac35$

1997 Tournament Of Towns, (560) 1

Tags: geometry , isosceles , angle bisector , equal segments

$M$ 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)

2004 District Olympiad, 4

Tags: geometry , right triangle , isosceles , angle , equal angles

Consider the isosceles right triangle $ABC$ ($AB = AC$) and the points $M, P \in [AB]$ so that $AM = BP$. Let $D$ be the midpoint of the side $BC$ and $R, Q$ the intersections of the perpendicular from $A$ on$ CM$ with $CM$ and $BC$ respectively. Prove that a) $\angle AQC = \angle PQB$ b) $\angle DRQ = 45^o$

2002 May Olympiad, 3

Tags: geometry , right triangle , isosceles , square

In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.

2021 Yasinsky Geometry Olympiad, 2

Tags: isosceles , geometry

In the triangle $ABC$, it is known that $AB = BC = 20$ cm, and $AC = 24$ cm. The point $M$ lies on the side $BC$ and is equidistant from sides $AB$ and $AC$. Find this distance. (Alexander Shkolny)