Olympiad problems

1990 IMO Shortlist, 9

Tags: geometry , incenter , trigonometry , angle , midpoint

The incenter of the triangle $ ABC$ is $ K.$ The midpoint of $ AB$ is $ C_1$ and that of $ AC$ is $ B_1.$ The lines $ C_1K$ and $ AC$ meet at $ B_2,$ the lines $ B_1K$ and $ AB$ at $ C_2.$ If the areas of the triangles $ AB_2C_2$ and $ ABC$ are equal, what is the measure of angle $ \angle CAB?$

2016 Novosibirsk Oral Olympiad in Geometry, 4

Tags: geometry , midpoint , square

The two angles of the squares are adjacent, and the extension of the diagonals of one square intersect the diagonal of another square at point $O$ (see figure). Prove that $O$ is the midpoint of $AB$. [img]https://cdn.artofproblemsolving.com/attachments/7/8/8daaaa55c38e15c4a8ac7492c38707f05475cc.png[/img]

1990 All Soviet Union Mathematical Olympiad, 512

Tags: midpoint , diagonal , geometry , equal segments , equal angles

The line joining the midpoints of two opposite sides of a convex quadrilateral makes equal angles with the diagonals. Show that the diagonals are equal.

2018 Ukraine Team Selection Test, 10

Tags: midpoint , geometry , circumcircle , altitude , equal angles

Let $ABC$ be a triangle with $AH$ altitude. The point $K$ is chosen on the segment $AH$ as follows such that $AH =3KH$. Let $O$ be the center of the circle circumscribed around by triangle $ABC, M$ and $N$ be the midpoints of $AC$ and AB respectively. Lines $KO$ and $MN$ intersect at the point $Z$, a perpendicular to $OK$ passing through point $Z$ intersects lines $AC$ and $AB$ at points $X$ and $Y$ respectively. Prove that $\angle XKY =\angle CKB$.

2013 Saudi Arabia BMO TST, 6

Tags: midpoint , geometry , incenter , equal segments

Let $ABC$ be a triangle with incenter $I,$ and let $D,E,F$ be the midpoints of sides $BC, CA, AB$, respectively. Lines $BI$ and $DE$ meet at $P $ and lines $CI$ and $DF$ meet at $Q$. Line $PQ$ meets sides $AB$ and $AC$ at $T$ and $S$, respectively. Prove that $AS = AT$

2018 Oral Moscow Geometry Olympiad, 2

Tags: geometry , trapezoid , angle bisector , midpoint

Bisectors of angle $C$ and externalangle $A$ of trapezoid $ABCD$ with bases $BC$ and $AD$ intersect at point $M$, and the bisector of angle $B$ and external angle $D$ intersect at point $N$. Prove that the midpoint of the segment $MN$ is equidistant from the lines $AB$ and $CD$.

2016 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: midpoint , geometry , perpendicular , circumcenter , orthocenter

Let $A B C$ be an acute-angled triangle of circumcenter $O$ and orthocenter $H$. Let $M$ be the midpoint of $BC, N$ be the symmetric of $H$ with respect to $A, P$ be the midpoint of $NM$ and $X$ be a point on the line A H such that $MX$ is parallel to $CH$. Prove that $BX$ and $OP$ are perpendicular.

2015 Sharygin Geometry Olympiad, P13

Tags: midpoint , geometry , concurrency

Let $AH_1, BH_2$ and $CH_3$ be the altitudes of a triangle $ABC$. Point $M$ is the midpoint of $H_2H_3$. Line $AM$ meets $H_2H_1$ at point $K$. Prove that $K$ lies on the medial line of $ABC$ parallel to $AC$.

2013 Ukraine Team Selection Test, 1

Tags: midpoint , isosceles , circumcircle , geometry

Let $ABC$ be an isosceles triangle $ABC$ with base $BC$ insribed in a circle. The segment $AD$ is the diameter of the circle, and point $P$ lies on the smaller arc $BD$. Line $DP$ intersects rays $AB$ and $AC$ at points $M$ and $N$, and the lines $BP$ and $CP$ intersects the line $AD$ at points $Q$ and $R$. Prove that the midpoint of the segment $MN$ lies on the circumscribed circle of triangle $PQR$.

2019 Adygea Teachers' Geometry Olympiad, 1

Tags: geometry , midpoint , area

Inside the quadrangle, a point is taken and connected with the midpoint of all sides. Areas of the three out of four formed quadrangles are $S_1, S_2, S_3$. Find the area of the fourth quadrangle.

Kyiv City MO Juniors 2003+ geometry, 2011.89.4

Tags: midpoint , geometry , cyclic , equal angles

Let $ABCD$ be an inscribed quadrilateral. Denote the midpoints of the sides $AB, BC, CD$ and $DA$ through $M, L, N$ and $K$, respectively. It turned out that $\angle BM N = \angle MNC$. Prove that: i) $\angle DKL = \angle CLK$. ii) in the quadrilateral $ABCD$ there is a pair of parallel sides.

2006 IMO Shortlist, 4

Tags: incenter , triangle , geometry , midpoint , congruent triangles

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

2024 Brazil Team Selection Test, 2

Tags: geometry , circumcenter , midpoint , tangent , isosceles

Let $ ABC $ be an acute-angled scalene triangle with circumcenter $ O $. Denote by $ M $, $ N $, and $ P $ the midpoints of sides $ BC $, $ CA $, and $ AB $, respectively. Let $ \omega $ be the circle passing through $ A $ and tangent to $ OM $ at $ O $. The circle $ \omega $ intersects $ AB $ and $ AC $ at points $ E $ and $ F $, respectively (where $ E $ and $ F $ are distinct from $ A $). Let $ I $ be the midpoint of segment $ EF $, and let $ K $ be the intersection of lines $ EF $ and $ NP $. Prove that $ AO = 2IK $ and that triangle $ IMO $ is isosceles.

2007 France Team Selection Test, 3

Tags: incenter , triangle , midpoint , congruent triangles , geometry

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

2014 Cuba MO, 6

Tags: tangent , midpoint , isosceles , geometry

Let $ABC$ be an isosceles triangle with $AB = AC$. Points $D$, $E$ and $F$ are on sides $BC$, $CA $ and $AB$ respectively, such that $\angle FDE =\angle ABC$ and $FE$ is not parallel to $BC$. Prove that $BC$ is tangent to the circumcircle of the triangle $DEF$, if and only if, $D$ is the midpoint of $BC$.

2018 Junior Balkan Team Selection Tests - Romania, 2

Tags: geometry , circumcircle , midpoint , parallel

Let $ABC$ be an acute triangle, with $AB \ne AC$. Let $D$ be the midpoint of the line segment $BC$, and let $E$ and $F$ be the projections of $D$ onto the sides $AB$ and $AC$, respectively. If $M$ is the midpoint of the line segment $EF$, and $O$ is the circumcenter of triangle $ABC$, prove that the lines $DM$ and $AO$ are parallel. [hide=PS] As source was given [url=https://artofproblemsolving.com/community/c629086_caucasus_mathematical_olympiad]Caucasus MO[/url], but I was unable to find this problem in the contest collections [/hide]

2009 Federal Competition For Advanced Students, P1, 4

Tags: midpoint , altitude , geometry , concurrency

Let $D, E$, and $F$ be respectively the midpoints of the sides $BC, CA$, and $AB$ of $\vartriangle ABC$. Let $H_a, H_b, H_c$ be the feet of perpendiculars from $A, B, C$ to the opposite sides, respectively. Let $P, Q, R$ be the midpoints of the $H_bH_c, H_cH_a$, and $H_aH_b$ respectively. Prove that $PD, QE$, and $RF$ are concurrent.

1982 IMO Longlists, 36

Tags: geometry , triangle , midpoint , incircle , concurrency

A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.

1998 Belarus Team Selection Test, 1

Tags: midpoint , geometry , perpendicular bisector , angle bisector , angle

Let $O$ be a point inside an acute angle with the vertex $A$ and $H, N$ be the feet of the perpendiculars drawn from $O$ onto the sides of the angle. Let point $B$ belong to the bisector of the angle, $K$ be the foot of the perpendicular from $B$ onto either side of the angle. Denote by $P,F$ the midpoints of the segments $AK,HN$ respectively. Known that $ON + OH = BK$, prove that $PF$ is perpendicular to $AB$. Ya. Konstantinovski

1988 Tournament Of Towns, (165) 2

Tags: equal area , geometry , midpoint , area

We are given convex quadrilateral $ABCD$. The midpoints of $BC$ and $DA$ are $M$ and $N$ respectively. The diagonal $AC$ divides $MN$ in half. Prove that the areas of triangles $ABC$ and $ACD$ are equal .

2007 Sharygin Geometry Olympiad, 4

Tags: midpoint , geometry , circumcircle , locus , locus problems

Given a triangle $ABC$. An arbitrary point $P$ is chosen on the circumcircle of triangle $ABH$ ($H$ is the orthocenter of triangle $ABC$). Lines $AP$ and $BP$ meet the opposite sidelines of the triangle at points $A' $ and $B'$, respectively. Determine the locus of midpoints of segments $A'B'$.

Kharkiv City MO Seniors - geometry, 2017.10.4

Tags: geometry , midpoint , angle

In the quadrangle $ABCD$, the angle at the vertex $A$ is right. Point $M$ is the midpoint of the side $BC$. It turned out that $\angle ADC = \angle BAM$. Prove that $\angle ADB = \angle CAM$.

2017 Switzerland - Final Round, 8

Tags: midpoint , isosceles , equal segments , geometry

Let $ABC$ be an isosceles triangle with vertex $A$ and $AB> BC$. Let $k$ be the circle with center $A$ passsing through $B$ and $C$. Let $H$ be the second intersection of $k$ with the altitude of the triangle $ABC$ through $B$. Further let $G$ be the second intersection of $k$ with the median through $B$ in triangle $ABC$. Let $X$ be the intersection of the lines $AC$ and $GH$. Show that $C$ is the midpoint of $AX$.

Kharkiv City MO Seniors - geometry, 2012.10.4

Tags: geometry , altitude , midpoint , concyclic

In the acute-angled triangle $ABC$ on the sides $AC$ and $BC$, points $D$ and $E$ are chosen such that points $A, B, E$, and $D$ lie on one circle. The circumcircle of triangle $DEC$ intersects side $AB$ at points $X$ and $Y$. Prove that the midpoint of segment $XY$ is the foot of the altitude of the triangle, drawn from point $C$.

2008 Singapore Junior Math Olympiad, 3

Tags: midpoint , geometry , right

In the quadrilateral $PQRS, A, B, C$ and $D$ are the midpoints of the sides $PQ, QR, RS$ and $SP$ respectively, and $M$ is the midpoint of $CD$. Suppose $H$ is the point on the line $AM$ such that $HC = BC$. Prove that $\angle BHM = 90^o$.