Olympiad problems

1984 IMO Shortlist, 14

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.

2020 BMT Fall, 23

Tags: geometry , ratio , diameter

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 $m + n$.

Kyiv City MO 1984-93 - geometry, 1993.8.4

Tags: geometry , sum , diameter

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.

1970 Vietnam National Olympiad, 4

Tags: geometry , locus , construction , tangent , diameter

$AB$ and $CD$ are perpendicular diameters of a circle. $L$ is the tangent to the circle at $A$. $M$ is a variable point on the minor arc $AC$. The ray $BM, DM$ meet the line $L$ at $P$ and $Q$ respectively. Show that $AP\cdot AQ = AB\cdot PQ$. Show how to construct the point $M$ which gives$ BQ$ parallel to $DP$. If the lines $OP$ and $BQ$ meet at $N$ find the locus of $N$. The lines $BP$ and $BQ$ meet the tangent at $D$ at $P'$ and $Q'$ respectively. Find the relation between $P'$ and $Q$'. The lines $D$P and $DQ$ meet the line $BC$ at $P"$ and $Q"$ respectively. Find the relation between $P"$ and $Q"$.

2005 Sharygin Geometry Olympiad, 11.4

Tags: geometry , angle , incircle , chord , diameter

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 $BC$.

1996 Akdeniz University MO, 4

Tags: geometry , diameter , point set

$25$ 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 $1$ $cm$.

2024 Yasinsky Geometry Olympiad, 3

Tags: geometry , orthocenter , diameter

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]

1984 IMO, 1

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

2025 Philippine MO, P7

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

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.

2005 Oral Moscow Geometry Olympiad, 2

Tags: geometry , collinear , diameter

On a circle with diameter $AB$, lie points $C$ and $D$. $XY$ is the diameter passing through the midpoint $K$ of the chord $CD$. Point $M$ is the projection of point $X$ onto line $AC$, and point $N$ is the projection of point $Y$ on line $BD$. Prove that points $M, N$ and $K$ are collinear. (A. Zaslavsky)

2019 Nigeria Senior MO Round 2, 3

Tags: diameter , tangent circle , geometry

Circles $\Omega_a$ and $\Omega_b$ are externally tangent at $D$, circles $\Omega_b$ and $\Omega_c$ are externally tangent at $E$, circles $\Omega_a$ and $\Omega_c$ are externally tangent at $F$. Let $P$ be an arbitrary point on $\Omega_a$ different from $D$ and $F$. Extend $PD$ to meet $\Omega_b$ again at $B$, extend $BE$ to meet $\Omega_c$ again at $C$ and extend $CF$ to meet $\Omega_a$ again at $A$. Show that $PA$ is a diameter of circle $\Omega_a$.

2022 Indonesia TST, G

Tags: geometry , diameter , circumcircle , concyclic

Let $AB$ be the diameter of circle $\Gamma$ centred at $O$. Point $C$ lies on ray $\overrightarrow{AB}$. The line through $C$ cuts circle $\Gamma$ at $D$ and $E$, with point $D$ being closer to $C$ than $E$ is. $OF$ is the diameter of the circumcircle of triangle $BOD$. Next, construct $CF$, cutting the circumcircle of triangle $BOD$ at $G$. Prove that $O,A,E,G$ are concyclic. (Possibly proposed by Pak Wono)

1965 IMO, 6

Tags: geometry , diameter , point set , 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 $n$.

2020 OMMock - Mexico National Olympiad Mock Exam, 4

Tags: geometry , diameter , perpendicular bisector , vector

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]

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.

2016 Romanian Master of Mathematics Shortlist, G1

Tags: tangent , geometry , circles , diameter

Two circles, $\omega_1$ and $\omega_2$, centred at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ meets $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1, O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.

2012 Bundeswettbewerb Mathematik, 3

Tags: geometry , incircle , equal segments , diameter

The incircle of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ in points $A_1, B_1$ and $C_1$ respectively. $C_1D$ is a diameter of the incircle. Finally, let $E$ be the intersection of the lines $B_1C_1$ and $A_1D$. Prove that the segments $CE$ and $CB_1$ have equal length.

2017 Yasinsky Geometry Olympiad, 5

Tags: geometry , angle chasing , diameter , angle

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.

1984 IMO Longlists, 50

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

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.

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.