Olympiad problems

2008 Singapore Junior Math Olympiad, 3

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 $\angle BHM = 90^o$.

1949-56 Chisinau City MO, 31

Tags: geometry , Locus , midpoints , chord

Find the locus of the points that are the midpoints of the chords of the secant to the given circle and passing through a given point.

2014 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , equal angles , midpoints , diameter

Let $ABC$ be an acute triangle and let $O$ be its circumcentre. Now, let the diameter $PQ$ of circle $ABC$ intersects sides $AB$ and $AC$ in their interior at$ D$ and $E$, respectively. Now, let $F$ and $G$ be the midpoints of $CD$ and $BE$. Prove that $\angle FOG=\angle BAC$

2011 Sharygin Geometry Olympiad, 2

Tags: geometry , cyclic quadrilateral , tangential quadrilateral , midpoints

Quadrilateral $ABCD$ is circumscribed. Its incircle touches sides $AB, BC, CD, DA$ in points $K, L, M, N$ respectively. Points $A', B', C', D'$ are the midpoints of segments $LM, MN, NK, KL$. Prove that the quadrilateral formed by lines $AA', BB', CC', DD'$ is cyclic.

2016 May Olympiad, 3

Tags: geometry , midpoints

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.

2014 Danube Mathematical Competition, 3

Tags: geometry , right angle , orthocenter , midpoints

Let $ABC$ be a triangle with $\angle A<90^o, AB \ne AC$. Denote $H$ the orthocenter of triangle $ABC$, $N$ the midpoint of segment $[AH]$, $M$ the midpoint of segment $[BC]$ and $D$ the intersection point of the angle bisector of $\angle BAC$ with the segment $[MN]$. Prove that $<ADH=90^o$

2010 Balkan MO Shortlist, G2

Tags: geometry , circles , areas , geometric inequality , cyclic quadrilateral , midpoints , Cyclic

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

2008 Oral Moscow Geometry Olympiad, 3

Tags: geometry , parallelogram , midpoints , equal segments

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)

1948 Moscow Mathematical Olympiad, 151

Tags: parallel , distance , geometry , convex , midpoints

The distance between the midpoints of the opposite sides of a convex quadrilateral is equal to a half sum of lengths of the other two sides. Prove that the first pair of sides is parallel.

2017 Singapore Senior Math Olympiad, 3

Tags: midpoints , combinatorics , combinatorial geometry

There are $2017$ distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points?

1998 Tournament Of Towns, 3

Tags: ratio , geometry , midpoints , angle

$AB$ and $CD$ are segments lying on the two sides of an angle whose vertex is $O$. $A$ is between $O$ and $B$, and $C$ is between $O$ and $D$ . The line connecting the midpoints of the segments $AD$ and $BC$ intersects $AB$ at $M$ and $CD$ at $N$. Prove that $\frac{OM}{ON}=\frac{AB}{CD}$ (V Senderov)

Kyiv City MO Juniors 2003+ geometry, 2010.8.5

Tags: geometry , midpoints , geometric inequality

In an acute-angled triangle $ABC$, the points $M$ and $N$ are the midpoints of the sides $AB$ and $AC$, respectively. For an arbitrary point $S$ lying on the side of $BC$ prove that the condition holds $(MB- MS)(NC-NS) \le 0$

1988 All Soviet Union Mathematical Olympiad, 467

Tags: ratio , fixed , geometry , midpoints , Cyclic

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.

2007 Sharygin Geometry Olympiad, 4

Tags: geometry , circumcircle , Locus , Locus problems , midpoints

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 $A'B'$.

2015 Caucasus Mathematical Olympiad, 5

Tags: geometry , altitudes , midpoints , equal angles , equal segments

Let $AA_1$ and $CC_1$ be the altitudes of the acute-angled triangle $ABC$. Let $K,L$ and $M$ be the midpoints of the sides $AB,BC$ and $CA$ respectively. Prove that if $\angle C_1MA_1 =\angle ABC$, then $C_1 K = A_1L$.

2017 Sharygin Geometry Olympiad, P19

Tags: geometry , circumcircle , concurrency , Cevian , midpoints

Let cevians $AA', BB'$ and $CC'$ of triangle $ABC$ concur at point $P.$ The circumcircle of triangle $PA'B'$ meets $AC$ and $BC$ at points $M$ and $N$ respectively, and the circumcircles of triangles $PC'B'$ and $PA'C'$ meet $AC$ and $BC$ for the second time respectively at points $K$ and $L$. The line $c$ passes through the midpoints of segments $MN$ and $KL$. The lines $a$ and $b$ are defined similarly. Prove that $a$, $b$ and $c$ concur.