Olympiad problems

1984 IMO, 1

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.

2009 Postal Coaching, 1

Tags: fixed , fixed point , diameter , circles

A circle $\Gamma$ and a line $\ell$ which does not intersect $\Gamma$ are given. Suppose $P, Q,R, S$ are variable points on circle $\Gamma$ such that the points $A = PQ\cap RS$ and $B = PS \cap QR$ lie on $\ell$. Prove that the circle on $AB$ as a diameter passes through two fixed points.

1984 IMO Shortlist, 14

Tags: convex quadrilateral , diameter , parallel , geometry , circles

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.

1965 IMO, 6

Tags: diameter , combinatorial geometry , point set , geometry

In a plane a set of $n\geq 3$ points is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.

2020 BMT Fall, 23

Tags: diameter , ratio , geometry

Circle $\Gamma$ has radius $10$, center $O$, and diameter $AB$. Point $C$ lies on $\Gamma$ such that $AC = 12$. Let $P$ be the circumcenter of $\vartriangle AOC$. Line $AP$ intersects $\Gamma$ at $Q$, where $Q$ is different from $A$. Then the value of $\frac{AP}{AQ}$ can be expressed in the form $\frac{m}{n}$, where m and n are relatively prime positive integers. Compute $m + n$.

2018 Stars of Mathematics, 4

Tags: convex polygon , diameter , disc , convex , geometry , geometric inequality

Given an integer $n \ge 3$, prove that the diameter of a convex $n$-gon (interior and boundary) containing a disc of radius $r$ is (strictly) greater than $r(1 + 1/ \cos( \pi /n))$. The Editors

2014 Junior Balkan Team Selection Tests - Romania, 3

Tags: midpoint , diameter , equal angles , geometry

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 $\angle FOG=\angle BAC$

Brazil L2 Finals (OBM) - geometry, 2003.3

Tags: diameter , perpendicular , circles , geometry

The triangle $ABC$ is inscribed in the circle $S$ and $AB <AC$. The line containing $A$ and is perpendicular to $BC$ meets $S$ in $P$ ($P \ne A$). Point $X$ is on the segment $AC$ and the line $BX$ intersects $S$ in $Q$ ($Q \ne B$). Show that $BX = CX$ if, and only if, $PQ$ is a diameter of $S$.

2017 Yasinsky Geometry Olympiad, 5

Tags: angle , geometry , angle chasing , diameter

The four points of a circle are in the following order: $A, B, C, D$. Extensions of chord $AB$ beyond point $B$ and of chord $CD$ beyond point $C$ intersect at point $E$, with $\angle AED= 60^o$. If $\angle ABD =3 \angle BAC$ , prove that $AD$ is the diameter of the circle.

1996 Akdeniz University MO, 4

Tags: diameter , point set , geometry

$25$ point in a plane and for all $3$ points, we find $2$ points such that this $2$ points' distance less than $1$ $cm$ . Prove that at least $13$ points in a circle of radius $1$ $cm$.

2009 Oral Moscow Geometry Olympiad, 4

Tags: concurrency , diameter , median , geometry

Three circles are constructed on the medians of a triangle as on diameters. It is known that they intersect in pairs. Let $C_1$ be the intersection point of the circles built on the medians $AM_1$ and $BM_2$, which is more distant from the vertex $C$. Points $A_1$ and $B_1$ are defined similarly. Prove that the lines $AA_1, BB_1$ and $CC_1$ intersect at one point. (D. Tereshin)

1982 All Soviet Union Mathematical Olympiad, 333

Tags: point , diameter , combinatorics , combinatorial geometry

$3k$ points are marked on the circumference. They divide it onto $3k$ arcs. Some $k$ of them have length $1$, other $k$ of them have length $2$, the rest $k$ of them have length $3$. Prove that some two of the marked points are the ends of one diameter.

Kyiv City MO Seniors 2003+ geometry, 2013.11.3

Tags: bisects segment , diameter , perpendicular , geometry

The segment $AB$ is the diameter of the circle. The points $M$ and $C$ belong to this circle and are located in different half-planes relative to the line $AB$. From the point $M$ the perpendiculars $MN$ and $MK$ are drawn on the lines $AB$ and $AC$, respectively. Prove that the line $KN$ intersects the segment $CM$ in its midpoint. (Igor Nagel)

2020 OMMock - Mexico National Olympiad Mock Exam, 4

Tags: diameter , perpendicular bisector , vector , geometry

Let $ABC$ be a triangle. Suppose that the perpendicular bisector of $BC$ meets the circle of diameter $AB$ at a point $D$ at the opposite side of $BC$ with respect to $A$, and meets the circle through $A, C, D$ again at $E$. Prove that $\angle ACE=\angle BCD$. [i]Proposed by José Manuel Guerra and Victor Domínguez[/i]

2015 China Northern MO, 2

Tags: geometry , angle , circumcircle , diameter

It is known that $\odot O$ is the circumcircle of $\vartriangle ABC$ wwith diameter $AB$. The tangents of $\odot O$ at points $B$ and $C$ intersect at $P$ . The line perpendicular to $PA$ at point $A$ intersects the extension of $BC$ at point $D$. Extend $DP$ at length $PE = PB$. If $\angle ADP = 40^o$ , find the measure of $\angle E$.

2017 Finnish National High School Mathematics Comp, 5

Tags: circumcenter , diameter , circles , geometry

Let $A$ and $B$ be two arbitrary points on the circumference of the circle such that $AB$ is not the diameter of the circle. The tangents to the circle drawn at points $A$ and $B$ meet at $T$. Next, choose the diameter $XY$ so that the segments $AX$ and $BY$ intersect. Let this be the intersection of $Q$. Prove that the points $A, B$, and $Q$ lie on a circle with center $T$.

1965 IMO Shortlist, 6

Tags: point set , geometry , diameter , combinatorial geometry

In a plane a set of $n\geq 3$ points is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.

2024 Yasinsky Geometry Olympiad, 3

Tags: diameter , orthocenter , geometry

Let $ H $ be the orthocenter of an acute triangle $ ABC $, and let $ AT $ be the diameter of the circumcircle of this triangle. Points $ X $ and $ Y $ are chosen on sides $ AC $ and $ AB $, respectively, such that $ TX = TY $ and $ \angle XTY + \angle XAY = 90^\circ $. Prove that $ \angle XHY = 90^\circ $. [i] Proposed by Matthew Kurskyi[/i]

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.

1995 Singapore Team Selection Test, 2

Tags: altitude , geometry , concyclic , diameter

Let $ABC$ be an acute-angled triangle. Suppose that the altitude of $\vartriangle ABC$ at $B$ intersects the circle with diameter $AC$ at $P$ and $Q$, and the altitude at $C$ intersects the circle with diameter $AB$ at $M$ and $N$. Prove that $P, Q, M$ and $N$ lie on a circle.

Estonia Open Senior - geometry, 2017.2.5

Tags: diameter , geometry , angle bisector

The bisector of the exterior angle at vertex $C$ of the triangle $ABC$ intersects the bisector of the interior angle at vertex $B$ in point $K$. Consider the diameter of the circumcircle of the triangle $BCK$ whose one endpoint is $K$. Prove that $A$ lies on this diameter.

Cono Sur Shortlist - geometry, 2003.G2

Tags: diameter , tangent circle , geometry

The circles $C_1, C_2$ and $C_3$ are externally tangent in pairs (each tangent to other two externally). Let $M$ the common point of $C_1$ and $C_2, N$ the common point of $C_2$ and $C_3$ and $P$ the common point of $C_3$ and $C_1$. Let $A$ be an arbitrary point of $C_1$. Line $AM$ cuts $C_2$ in $B$, line $BN$ cuts $C_3$ in $C$ and line $CP$ cuts $C_1$ in $D$. Prove that $AD$ is diameter of $C_1$.

Kyiv City MO 1984-93 - geometry, 1993.8.4

Tags: sum , diameter , geometry

The diameter of a circle of radius $R$ is divided into $4$ equal parts. The point $M$ is taken on the circle. Prove that the sum of the squares of the distances from the point $M$ to the points of division (together with the ends of the diameter) does not depend on the choice of the point $M$. Calculate this sum.

2005 Sharygin Geometry Olympiad, 11.4

Tags: incircle , angle , chord , diameter , geometry

In the triangle $ABC , \angle A = \alpha, BC = a$. The inscribed circle touches the lines $AB$ and $AC$ at points $M$ and $P$. Find the length of the chord cut by the line $MP$ in a circle with diameter $BC$.

2006 Sharygin Geometry Olympiad, 14

Tags: geometry , diameter , locus , locus problems , orthocenter

Given a circle and a fixed point $P$ not lying on it. Find the geometrical locus of the orthocenters of the triangles $ABP$, where $AB$ is the diameter of the circle.