Olympiad problems

2019 Girls in Mathematics Tournament, 5

Tags: geometry , midpoint , isosceles , equal segments , angle bisector , projective geometry

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $X$ and $K$ points over $AC$ and $AB$, respectively, such that $KX = CX$. Bisector of $\angle AKX$ intersects line $BC$ at $Z$. Show that $XZ$ passes through the midpoint of $BK$.

Indonesia MO Shortlist - geometry, g7

Tags: equal angles , trapezoid , isosceles , geometry

Given an isosceles trapezoid $ABCD$ with base $AB$. The diagonals $AC$ and $BD$ intersect at point $S$. Let $M$ the midpoint of $BC$ and the bisector of the angle $BSC$ intersect $BC$ at $N$. Prove that $\angle AMD = \angle AND$.

2017 Portugal MO, 2

Tags: geometry , symmetric , isosceles , circumcircle

In triangle $[ABC]$, the bisector in $C$ and the altitude passing through $B$ intersect at point $D$. Point $E$ is the symmetric of point $D$ wrt $BC$ and lies on the circle circumscribed to the triangle $[ABC]$. Prove that the triangle is $[ABC]$ isosceles.

2012 Sharygin Geometry Olympiad, 4

Tags: angle chasing , isosceles , geometry

Let $ABC$ be an isosceles triangle with $\angle B = 120^o$ . Points $P$ and $Q$ are chosen on the prolongations of segments $AB$ and $CB$ beyond point $B$ so that the rays $AQ$ and $CP$ intersect and are perpendicular to each other. Prove that $\angle PQB = 2\angle PCQ$. (A.Akopyan, D.Shvetsov)

1992 Poland - Second Round, 3

Tags: geometry , isosceles

Through the center of gravity of the acute-angled triangle $ ABC $, lines are drawn perpendicular to the sides $ BC $, $ CA $, $ AB $, intersecting them at the points $ P $, $ Q $, $ R $, respectively. Prove that if $ |BP|\cdot |CQ| \cdot |AR| = |PC| \cdot |QA| \cdot |RB| $, then the triangle $ ABC $ is isosceles. Note: According to Ceva's theorem, the assumed equality of products is equivalent to the fact that the lines $ AP $, $ BQ $, $ CR $ have a common point.

2020 Ukrainian Geometry Olympiad - December, 2

Tags: geometry , combinatorics , combinatorial geometry , isosceles

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?

2020 Argentina National Olympiad, 3

Tags: right triangle , isosceles , angle , geometry

Let $ABC$ be a right isosceles triangle with right angle at $A$. Let $E$ and $F$ be points on A$B$ and $AC$ respectively such that $\angle ECB = 30^o$ and $\angle FBC = 15^o$. Lines $CE$ and $BF$ intersect at $P$ and line $AP$ intersects side $BC$ at $D$. Calculate the measure of angle $\angle FDC$.

2015 Indonesia MO Shortlist, G4

Tags: geometry , equal segments , isosceles , circumcircle

Given an isosceles triangle $ABC$ with $AB = AC$, suppose $D$ is the midpoint of the $AC$. The circumcircle of the $DBC$ triangle intersects the altitude from $A$ at point $E$ inside the triangle $ABC$, and the circumcircle of the triangle $AEB$ cuts the side $BD$ at point $F$. If $CF$ cuts $AE$ at point $G$, prove that $AE = EG$.

2007 Regional Olympiad of Mexico Northeast, 2

Tags: geometry , parallel , isosceles

In the isosceles triangle $ABC$, with $AB=AC$, $D$ is a point on the extension of $CA$ such that $DB$ is perpendicular to $BC$, $E$ is a point on the extension of $BC$ such that $CE=2BC$, and $F$ is a point on $ED$ such that $FC$ is parallel to $AB$. Prove that $FA$ is parallel to $BC$.

2009 May Olympiad, 2

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

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 $\angle CED$.

Indonesia MO Shortlist - geometry, g2

Tags: isosceles , equal segments , geometry

Let $ABC$ be an isosceles triangle right at $C$ and $P$ any point on $CB$. Let also $Q$ be the midpoint of $AB$ and $R, S$ be the points on $AP$ such that $CR$ is perpendicular to $AP$ and $|AS|=|CR|$. Prove that the $|RS| = \sqrt2 |SQ|$.

1952 Moscow Mathematical Olympiad, 229

Tags: geometry , isosceles , angle , angle chasing

In an isosceles triangle $\vartriangle ABC, \angle ABC = 20^o$ and $BC = AB$. Points $P$ and $Q$ are chosen on sides $BC$ and $AB$, respectively, so that $\angle PAC = 50^o$ and $\angle QCA = 60^o$ . Prove that $\angle PQC = 30^o$ .

2017 Estonia Team Selection Test, 4

Tags: isosceles , fixed , geometry

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 $\angle BAC$

2015 Hanoi Open Mathematics Competitions, 12

Tags: geometry , perpendicular , isosceles

Give an isosceles triangle $ABC$ at $A$. Draw ray $Cx$ being perpendicular to $CA, BE$ perpendicular to $Cx$ ($E \in Cx$).Let $M$ be the midpoint of $BE$, and $D$ be the intersection point of $AM$ and $Cx$. Prove that $BD \perp BC$.

2018 Yasinsky Geometry Olympiad, 5

Tags: geometry , equal angles , incircle , isosceles , isosceles triangle

The inscribed circle of the triangle $ABC$ touches its sides $AB, BC, CA$, at points $K,N, M$ respectively. It is known that $\angle ANM = \angle CKM$. Prove that the triangle $ABC$ is isosceles. (Vyacheslav Yasinsky)

Estonia Open Senior - geometry, 2008.1.2

Tags: geometry , isosceles , concyclic

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.

Ukrainian TYM Qualifying - geometry, 2019.8

Tags: integer , geometry , isosceles

Hannusya, Petrus and Mykolka drew independently one isosceles triangle $ABC$, all angles of which are measured as a integer number of degrees. It turned out that the bases $AC$ of these triangles are equals and for each of them on the ray $BC$ there is a point $E$ such that $BE=AC$, and the angle $AEC$ is also measured by an integer number of degrees. Is it in necessary that: a) all three drawn triangles are equal to each other? b) among them there are at least two equal triangles?

Champions Tournament Seniors - geometry, 2006.3

Tags: isosceles , equal angles , geometry

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $D$ be a point on the base $BC$ such that $BD:DC = 2: 1$. Note on the segment $AD$ a point $P$ such that $\angle BAC= \angle BPD $. Prove that $\angle BPD = 2 \angle CPD$.

2005 Sharygin Geometry Olympiad, 9.2

Tags: isosceles , cut , isosceles triangle , geometry

Find all isosceles triangles that cannot be cut into three isosceles triangles with the same sides.

2021 Regional Competition For Advanced Students, 2

Tags: geometry , tangent , 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)

Kyiv City MO Seniors 2003+ geometry, 2017.11.5

Tags: geometry , concyclic , isosceles , orthocenter

In the acute isosceles triangle $ABC$ the altitudes $BB_1$ and $CC_1$ are drawn, which intersect at the point $H$. Let $L_1$ and $L_2$ be the feet of the angle bisectors of the triangles $B_1AC_1$ and $B_1HC_1$ drawn from vertices $A$ and $H$, respectively. The circumscribed circles of triangles $AHL_1$ and $AHL_2$ intersects the line $B_1C_1$ for the second time at points $P$ and $Q$, respectively. Prove that points $B, C, P$ and $Q$ lie on the same circle. (M. Plotnikov, D. Hilko)

2021 Adygea Teachers' Geometry Olympiad, 2

Tags: geometry , incircle , isosceles , excircle

In triangle $ABC$, the incircle touches the side $AC$ at point $B_1$ and one excircle is touching the same side at point $B_2$. It is known that the segments $BB_1$ and $BB_2$ are equal. Is it true that $\vartriangle ABC$ is isosceles?

2022 Grand Duchy of Lithuania, 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 $\Omega$.

2017 Balkan MO Shortlist, G6

Tags: geometry , perpendicular , isosceles , concurrency

Construct outside the acute-angled triangle $ABC$ the isosceles triangles $ABA_B, ABB_A , ACA_C,ACC_A ,BCB_C$ and $BCC_B$, so that $$AB = AB_A = BA_B, AC = AC_A=CA_C, BC = BC_B = CB_C$$ and $$\angle BAB_A = \angle ABA_B =\angle CAC_A=\angle ACA_C= \angle BCB_C =\angle CBC_B = a < 90^o$$. Prove that the perpendiculars from $A$ to $B_AC_A$, from $B$ to $A_BC_B$ and from $C$ to $A_CB_C$ are concurrent

2015 Sharygin Geometry Olympiad, P3

Tags: geometry , isosceles , square

The side $AD$ of a square $ABCD$ is the base of an obtuse-angled isosceles triangle $AED$ with vertex $E$ lying inside the square. Let $AF$ be a diameter of the circumcircle of this triangle, and $G$ be a point on $CD$ such that $CG = DF$. Prove that angle $BGE$ is less than half of angle $AED$.