Olympiad problems

1993 All-Russian Olympiad Regional Round, 9.7

On the diagonal $AC$ of the rhombus $ABCD$, a point $E$ is taken, which is different from points $A$ and $C$, and on the lines $AB$ and $BC$ are points $N$ and $M$, respectively, with $AE = NE$ and $CE = ME$. Let $K$ be the intersection point of lines $AM$ and $CN$. Prove that points $K, E$ and $D$ are collinear.

Denmark (Mohr) - geometry, 2013.5

Tags: geometry , ratio , angle bisector , equal segments

The angle bisector of $A$ in triangle $ABC$ intersects $BC$ in the point $D$. The point $E$ lies on the side $AC$, and the lines $AD$ and $BE$ intersect in the point $F$. Furthermore, $\frac{|AF|}{|F D|}= 3$ and $\frac{|BF|}{|F E|}=\frac{5}{3}$. Prove that $|AB| = |AC|$. [img]https://1.bp.blogspot.com/-evofDCeJWPY/XzT9dmxXzVI/AAAAAAAAMVY/ZN87X3Cg8iMiULwvMhgFrXbdd_f1f-JWwCLcBGAsYHQ/s0/2013%2BMohr%2Bp5.png[/img]

2015 Indonesia MO Shortlist, G6

Tags: geometry , equal segments , circumcircle , perpendicular

Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.

2003 All-Russian Olympiad Regional Round, 8.7

Tags: geometry , equal segments , equal angles

In triangle $ABC$, angle $C$ is a right angle. Found on the side $AC$ point $D$, and on the segment $BD$, point $K$ such that $\angle ABC = \angle KAD =\angle AKD$. Prove that $BK = 2DC$.

2007 Oral Moscow Geometry Olympiad, 2

Tags: geometry , equal segments , right triangle , isosceles

An isosceles right-angled triangle $ABC$ is given. On the extensions of sides $AB$ and $AC$, behind vertices $B$ and $C$ equal segments $BK$ and $CL$ were laid. $E$ and F are the points of intersection of the segment $KL$ and the lines perpendicular to the $KC$ , passing through the points $B$ and $A$, respectively. Prove that $EF = FL$.

1990 Swedish Mathematical Competition, 4

Tags: geometry , angle bisector , equal segments

$ABCD$ is a quadrilateral. The bisectors of $\angle A$ and $\angle B$ meet at $E$. The line through $E$ parallel to $CD$ meets $AD$ at $L$ and $BC$ at $M$. Show that $LM = AL + BM$.

2006 Thailand Mathematical Olympiad, 4

Tags: geometry , trigonometry , equal segments , angle

Let $P$ be a point outside a circle centered at $O$. From $P$, tangent lines are drawn to the circle, touching the circle at points $A$ and $B$. Ray $\overrightarrow{BO}$ is drawn intersecting the circle again at $C$ and intersecting ray $\overrightarrow{PA}$ at $Q$. If $3QA = 2AP$, what is the value of $\sin \angle CAQ$?

Estonia Open Junior - geometry, 2010.1.2

Tags: geometry , equal angles , equal segments , angle

Given a convex quadrangle $ABCD$ with $|AD| = |BD| = |CD|$ and $\angle ADB = \angle DCA$, $\angle CBD = \angle BAC$, find the sizes of the angles of the quadrangle.

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 $BK$.

2009 Dutch IMO TST, 2

Tags: geometry , equal segments , midpoint

Let $ABC$ be a triangle, $P$ the midpoint of $BC$, and $Q$ a point on segment $CA$ such that $|CQ| = 2|QA|$. Let $S$ be the intersection of $BQ$ and $AP$. Prove that $|AS| = |SP|$.

1997 Tournament Of Towns, (549) 3

Tags: geometry , equal segments , square

In a square $ABCD$, $K$ is a point on the side $BC$ and the bisector of $\angle KAD$ cuts the side $CD$ at the point $M$. Prove that the length of segment $AK$ is equal to the sum of the lengths of segments $DM$ and $BK$. (Folklore)

2002 Singapore Team Selection Test, 1

Tags: geometry , parallel , equal segments

Let $A, B, C, D, E$ be five distinct points on a circle $\Gamma$ in the clockwise order and let the extensions of $CD$ and $AE$ meet at a point $Y$ outside $\Gamma$. Suppose $X$ is a point on the extension of $AC$ such that $XB$ is tangent to $\Gamma$ at $B$. Prove that $XY = XB$ if and only if $XY$ is parallel $DE$.

2004 Oral Moscow Geometry Olympiad, 4

Tags: centroid , circumcenter , equal segments , geometry

In triangle $ABC$, $M$ is the intersection point of the medians, $O$ is the center of the inscribed circle. Prove that if the line $OM$ is parallel to the side $BC$, then the point $O$ is equidistant from the sides $AB$ and $AC$.

2020 Yasinsky Geometry Olympiad, 4

Tags: equal angles , equal segments , median , geometry

The median $AM$ is drawn in the triangle $ABC$ ($AB \ne AC$). The point $P$ is the foot of the perpendicular drawn on the segment $AM$ from the point $B$. On the segment $AM$ we chose such a point $Q$ that $AQ = 2PM$. Prove that $\angle CQM = \angle BAM$.

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 $AE = EG$.

2014 Denmark MO - Mohr Contest, 3

Tags: equal angles , equal segments , geometry

The points $C$ and $D$ lie on a halfline from the midpoint $M$ of a segment $AB$, so that $|AC| = |BD|$. Prove that the angles $u = \angle ACM$ and $v = \angle BDM$ are equal. [img]https://1.bp.blogspot.com/-tQEJ1VBCa8U/XzT7IhwlZHI/AAAAAAAAMVI/xpRdlj5Rl64VUt_tCRsQ1UxIsv_SGrMlACLcBGAsYHQ/s0/2014%2BMohr%2Bp3.png[/img]

1996 Singapore Team Selection Test, 1

Tags: geometry , incenter , equal segments , right angle

Let $C, B, E$ be three points on a straight line $\ell$ in that order. Suppose that $A$ and $D$ are two points on the same side of $\ell$ such that (i) $\angle ACE = \angle CDE = 90^o$ and (ii) $CA = CB = CD$. Let $F$ be the point of intersection of the segment $AB$ and the circumcircle of $\vartriangle ADC$. Prove that $F$ is the incentre of $\vartriangle CDE$.

Denmark (Mohr) - geometry, 2009.4

Tags: geometry , square , equal segments

Let $E$ be an arbitrary point different from $A$ and $B$ on the side $AB$ of a square $ABCD$, and let $F$ and $G$ be points on the segment $CE$ so that $BF$ and $DG$ are perpendicular to $CE$. Prove that $DF = AG$.

2019 Novosibirsk Oral Olympiad in Geometry, 3

Tags: geometry , equal segments , angle

Equal line segments are marked in triangle $ABC$. Find its angles. [img]https://cdn.artofproblemsolving.com/attachments/0/2/bcb756bba15ba57013f1b6c4cbe9cc74171543.png[/img]

Kyiv City MO Seniors 2003+ geometry, 2021.10.3

Tags: geometry , circles , equal segments , parallelogram

Circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. A point $C$ is constructed such that $AO_2CO_1$ is a parallelogram. An arbitrary line is drawn through point $A$, which intersects the circles $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y$, respectively. Prove that $CX = CY$. (Oleksii Masalitin)

2009 Dutch IMO TST, 5

Tags: geometry , rational , polygon , equal segments , analytic geometry

Suppose that we are given an $n$-gon of which all sides have the same length, and of which all the vertices have rational coordinates. Prove that $n$ is even.

1995 All-Russian Olympiad Regional Round, 11.7

Tags: geometry , equal segments

Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ respectively intersect at $A$ and $B$. Ray $O_1B$ meets $S_2$ again at $F$, and ray $O_2B$ meets $ S_1$ again at $E$. The line through $B$ parallel to $ EF$ intersects $S_1$ and $S_2$ again at $M$ and $N$, respectively. Prove that $MN = AE +AF$.

2010 Greece Junior Math Olympiad, 2

Tags: rectangle , equilateral , equal segments , perpendicular , geometry

Let $ABCD$ be a rectangle with sides $AB=a$ and $BC=b$. Let $O$ be the intersection point of it's diagonals. Extent side $BA$ towards $A$ at a segment $AE=AO$, and diagonal $DB$ towards $B$ at a segment $BZ=BO$. If the triangle $EZC$ is an equilateral, then prove that: i) $b=a\sqrt3$ ii) $AZ=EO$ iii) $EO \perp ZD$

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 $|RS| = \sqrt2 |SQ|$.

Kyiv City MO Seniors 2003+ geometry, 2015.11.4.1

Tags: geometry , angle bisector , equal segments , perpendicular

On the bisector of the angle $ BAC $ of the triangle $ ABC $ we choose the points $ {{B} _ {1}}, \, \, {{C} _ {1}} $ for which $ B {{B} _ {1 }}\perp AB $, $ C {{C} _ {1}} \perp AC $. The point $ M $ is the midpoint of the segment $ {{B} _ {1}} {{C} _ {1}} $. Prove that $ MB = MC $.