Olympiad problems

Estonia Open Junior - geometry, 2007.1.2

The sides $AB, BC, CD$ and $DA$ of the convex quadrilateral $ABCD$ have midpoints $E, F, G$ and $H$. Prove that the triangles $EFB, FGC, GHD$ and $HEA$ can be put together into a parallelogram equal to $EFGH$.

2006 IMO Shortlist, 4

Tags: incenter , midpoint , triangle , 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.

2000 May Olympiad, 2

Tags: geometry , midpoint , right triangle , angle bisector

Let $ABC$ be a right triangle in $A$ , whose leg measures $1$ cm. The bisector of the angle $BAC$ cuts the hypotenuse in $R$, the perpendicular to $AR$ on $R$ , cuts the side $AB$ at its midpoint. Find the measurement of the side $AB$ .

Indonesia MO Shortlist - geometry, g5

Tags: midpoint , geometry , equal angles

Given a cyclic quadrilateral $ABCD$. Suppose $E, F, G, H$ are respectively the midpoint of the sides $AB, BC, CD, DA$. The line passing through $G$ and perpendicular on $AB$ intersects the line passing through $H$ and perpendicular on $BC$ at point $K$. Prove that $\angle EKF = \angle ABC$.

2022 Argentina National Olympiad, 3

Tags: midpoint , locus , equilateral , geometry

Given a square $ABCD$, let us consider an equilateral triangle $KLM$, whose vertices $K$, $L$ and $M$ belong to the sides $AB$, $BC$ and $CD$ respectively. Find the locus of the midpoints of the sides $KL$ for all possible equilateral triangles $KLM$. Note: The set of points that satisfy a property is called a locus.

1994 IMO Shortlist, 2

Tags: quadrilateral , midpoint , geometry

$ ABCD$ is a quadrilateral with $ BC$ parallel to $ AD$. $ M$ is the midpoint of $ CD$, $ P$ is the midpoint of $ MA$ and $ Q$ is the midpoint of $ MB$. The lines $ DP$ and $ CQ$ meet at $ N$. Prove that $ N$ is inside the quadrilateral $ ABCD$.

2010 Balkan MO Shortlist, G2

Tags: midpoint , cyclic , geometric inequality , circles , area , cyclic quadrilateral , geometry

Consider a cyclic quadrilateral such that the midpoints of its sides form another cyclic quadrilateral. Prove that the area of the smaller circle is less than or equal to half the area of the bigger circle

1988 All Soviet Union Mathematical Olympiad, 467

Tags: midpoint , fixed , cyclic , ratio , geometry

The quadrilateral $ABCD$ is inscribed in a fixed circle. It has $AB$ parallel to $CD$ and the length $AC$ is fixed, but it is otherwise allowed to vary. If $h$ is the distance between the midpoints of $AC$ and $BD$ and $k$ is the distance between the midpoints of $AB$ and $CD$, show that the ratio $h/k$ remains constant.

2010 Dutch IMO TST, 4

Tags: cyclic quadrilateral , midpoint , equal angles , geometry

Let $ABCD$ be a cyclic quadrilateral satisfying $\angle ABD = \angle DBC$. Let $E$ be the intersection of the diagonals $AC$ and $BD$. Let $M$ be the midpoint of $AE$, and $N$ be the midpoint of $DC$. Show that $MBCN$ is a cyclic quadrilateral.

2009 Dutch IMO TST, 2

Tags: geometry , midpoint , equal segments

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 $|AS| = |SP|$.

1996 Greece Junior Math Olympiad, 2

Tags: midpoint , area , geometry

In a triangle $ABC$ let $D,E,Z,H,G$ be the midpoints of $BC,AD,BD,ED,EZ$ respectively. Let $I$ be the intersection of $BE,AC$ and let $K$ be the intersection of $HG,AC$. Prove that: a) $AK=3CK$ b) $HK=3HG$ c) $BE=3EI$ d) $(EGH)=\frac{1}{32}(ABC)$ Notation $(...)$ stands for area of $....$

2011 Sharygin Geometry Olympiad, 21

Tags: concurrency , midpoint , chord , arc midpoint , arc , geometry

On a circle with diameter $AC$, let $B$ be an arbitrary point distinct from $A$ and $C$. Points $M, N$ are the midpoints of chords $AB, BC$, and points $P, Q$ are the midpoints of smaller arcs restricted by these chords. Lines $AQ$ and $BC$ meet at point $K$, and lines $CP$ and $AB$ meet at point $L$. Prove that lines $MQ, NP$ and $KL$ concur.

2010 Dutch IMO TST, 4

Tags: midpoint , equal angles , cyclic quadrilateral , geometry

Let $ABCD$ be a cyclic quadrilateral satisfying $\angle ABD = \angle DBC$. Let $E$ be the intersection of the diagonals $AC$ and $BD$. Let $M$ be the midpoint of $AE$, and $N$ be the midpoint of $DC$. Show that $MBCN$ is a cyclic quadrilateral.

May Olympiad L1 - geometry, 2016.4

Tags: midpoint , geometry

In a triangle $ABC$, let $D$ and $E$ point in the sides $BC$ and $AC$ respectively. The segments $AD$ and $BE$ intersects in $O$, let $r$ be line (parallel to $AB$) such that $r$ intersects $DE$ in your midpoint, show that the triangle $ABO$ and the quadrilateral $ODCE$ have the same area.

2012 Oral Moscow Geometry Olympiad, 1

Tags: midpoint , incenter , geometry

Is it true that the center of the inscribed circle of the triangle lies inside the triangle formed by the lines of connecting it's midpoints?

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 $AC$.

2014 Sharygin Geometry Olympiad, 16

Tags: midpoint , geometry , collinear

Given a triangle $ABC$ and an arbitrary point $D$.The lines passing through $D$ and perpendicular to segments $DA, DB, DC$ meet lines $BC, AC, AB$ at points $A_1, B_1, C_1$ respectively. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ are collinear.

1957 Moscow Mathematical Olympiad, 354

Tags: midpoint , equal segments , parallel , geometry

In a quadrilateral $ABCD$ points $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$, respectively. The line through $M$ and $N$ meets $AB$ and $CD$ at $M'$ and $N'$, respectively. Prove that if $MM' = NN'$, then $AD // BC$.

2015 Czech-Polish-Slovak Junior Match, 4

Tags: midpoint , right triangle , angle bisector , incenter , geometry

Let $ABC$ ne a right triangle with $\angle ACB=90^o$. Let $E, F$ be respecitvely the midpoints of the $BC, AC$ and $CD$ be it's altitude. Next, let $P$ be the intersection of the internal angle bisector from $A$ and the line $EF$. Prove that $P$ is the center of the circle inscribed in the triangle $CDE$ .

2005 Sharygin Geometry Olympiad, 9.3

Tags: midpoint , locus , arc , geometry

Given a circle and points $A, B$ on it. Draw the set of midpoints of the segments, one of the ends of which lies on one of the arcs $AB$, and the other on the second.

1983 IMO Shortlist, 23

Tags: circles , geometry , angle , midpoint

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$ and $M_2$ the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2=\angle M_1AM_2$.

2016 PAMO, 5

Tags: geometry , midpoint , trapezoid

Let $ABCD$ be a trapezium such that the sides $AB$ and $CD$ are parallel and the side $AB$ is longer than the side $CD$. Let $M$ and $N$ be on the segments $AB$ and $BC$ respectively, such that each of the segments $CM$ and $AN$ divides the trapezium in two parts of equal area. Prove that the segment $MN$ intersects the segment $BD$ at its midpoint.

1971 IMO Shortlist, 12

Tags: congruent triangles , geometry , midpoint , collinear

Two congruent equilateral triangles $ABC$ and $A'B'C'$ in the plane are given. Show that the midpoints of the segments $AA',BB', CC'$ either are collinear or form an equilateral triangle.

2008 Oral Moscow Geometry Olympiad, 3

Tags: parallelogram , equal segments , geometry , midpoint

Given a quadrilateral $ABCD$. $A ', B', C'$ and $D'$ are the midpoints of the sides $BC, CB, BA$ and $AB$, respectively. It is known that $AA'= CC'$, $BB'= DD'$. Is it true that $ABCD$ is a parallelogram? (M. Volchkevich)

2025 Philippine MO, P7

Tags: midpoint , geometry , circumcircle , arbitrary point , diameter , perpendicular , parallel

In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$. (a) Show that $AP$ and $BR$ are perpendicular. \\ (b) Show that $FM$ and $BM$ are perpendicular.