Olympiad problems

Novosibirsk Oral Geo Oly VII, 2020.4

The altitudes $AN$ and $BM$ are drawn in triangle $ABC$. Prove that the perpendicular bisector to the segment $NM$ divides the segment $AB$ in half.

2013 Switzerland - Final Round, 7

Tags: geometry , bisects segment , equal angles

Let $O$ be the center of the circle of the triangle $ABC$ with $AB \ne AC$. Furthermore, let $S$ and $T$ be points on the rays $AB$ and $AC$, such that $\angle ASO = \angle ACO$ and $\angle ATO = \angle ABO$. Show that $ST$ bisects the segment $BC$.

Swiss NMO - geometry, 2009.5

Tags: geometry , bisects segment , incenter , incircle

Let $ABC$ be a triangle with $AB \ne AC$ and incenter $I$. The incircle touches $BC$ at $D$. Let $M$ be the midpoint of $BC$ . Show that the line $IM$ bisects segment $AD$ .

Croatia MO (HMO) - geometry, 2019.7

Tags: ratio , geometry , cyclic , bisects segment , cyclic quadrilateral

On the side $AB$ of the cyclic quadrilateral $ABCD$ there is a point $X$ such that diagonal $AC$ bisects the segment $DX$, and the diagonal $BD$ bisects the segment $CX$. What is the smallest possible ratio $|AB | : |CD|$ in such a quadrilateral ?

Geometry Mathley 2011-12, 8.1

Tags: bisects segment , similar , geometry , rectangle

Let $ABC$ be a triangle and $ABDE, BCFZ, CAKL$ be three similar rectangles constructed externally of the triangle. Let $A'$ be the intersection of $EF$ and $ZK, B'$ the intersection of $KZ$ and $DL$, and $C'$ the intersection of $DL$ and $EF$. Prove that $AA'$ passes through the midpoint of the line segment $B'C'$. Kostas Vittas

2021 Sharygin Geometry Olympiad, 8.7

Tags: geometry , equal angles , bisects segment

Let $ABCDE$ be a convex pentagon such that angles $CAB$, $BCA$, $ECD$, $DEC$ and $AEC$ are equal. Prove that $CE$ bisects $BD$.

1955 Moscow Mathematical Olympiad, 288

Tags: bisects segment , geometry , area , right triangle , angle

We are given a right triangle $ABC$ and the median $BD$ drawn from the vertex $B$ of the right angle. Let the circle inscribed in $\vartriangle ABD$ be tangent to side $AD$ at $K$. Find the angles of $\vartriangle ABC$ if $K$ divides $AD$ in halves.

2022 New Zealand MO, 7

Tags: geometry , bisects segment

Let $M$ be the midpoint of side $BC$ of acute triangle $ABC$. The circle centered at $M$ passing through $A$ intersects the lines $AB$ and $AC$ again at $P$ and $Q$, respectively. The tangents to this circle at $P$ and $Q$ meet at $D$. Prove that the perpendicular bisector of $BC$ bisects segment $AD$.

Swiss NMO - geometry, 2014.10

Tags: geometry , bisects segment , circles

Let $k$ be a circle with diameter $AB$. Let $C$ be a point on the straight line $AB$, so that $B$ between $A$ and $C$ lies. Let $T$ be a point on $k$ such that $CT$ is a tangent to $k$. Let $l$ be the parallel to $CT$ through $A$ and $D$ the intersection of $l$ and the perpendicular to $AB$ through $T$. Show that the line $DB$ bisects segment $CT$.

Kyiv City MO Juniors Round2 2010+ geometry, 2020.9.2

Tags: circumcircle , bisects segment , altitude , geometry

In the acute-angled triangle $ABC$ is drawn the altitude $CH$. A ray beginning at point $C$ that lies inside the $\angle BCA$ and intersects for second time the circles circumscribed circles of $\vartriangle BCH$ and $\vartriangle ABC$ at points $X$ and $Y$ respectively. It turned out that $2CX = CY$. Prove that the line $HX$ bisects the segment $AC$. (Hilko Danilo)

VI Soros Olympiad 1999 - 2000 (Russia), 10.7

Tags: geometry , bisects segment

Let a line, perpendicular to side $AD$ of parallelogram $ABCD$ passing through point $B$, intersect line $CD$ at point $M$, and a line, passing through point $B$ and perpendicular to side $CD$, intersect line $AD$ at point $N$. Prove that the line passing through point $B$ perpendicular to the diagonal $AC$, passes through the midpoint of the segment $MN$.

2013 Junior Balkan Team Selection Tests - Romania, 4

Tags: bisects segment , geometry , simson line , orthocenter , circumcircle

Let $H$ be the orthocenter of an acute-angled triangle $ABC$ and $P$ a point on the circumcenter of triangle $ABC$. Prove that the Simson line of $P$ bisects the segment $[P H]$.

1953 Moscow Mathematical Olympiad, 235

Tags: bisects segment , right isosceles , geometric construction , geometry

Divide a segment in halves using a right triangle. (With a right triangle one can draw straight lines and erect perpendiculars but cannot draw perpendiculars.)