This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 45

2011 NZMOC Camp Selection Problems, 2
Tags: geometry , diameter
Let an acute angled triangle $ABC$ be given. Prove that the circles whose diameters are $AB$ and $AC$ have a point of intersection on $BC$.

2019 Swedish Mathematical Competition, 2
Tags: geometry , concyclic , diameter , tangent
Segment $AB$ is the diameter of a circle. Points $C$ and $D$ lie on the circle. The rays $AC$ and $AD$ intersect the tangent to the circle at point $B$ at points $P$ and $Q$, respectively. Show that points $C, D, P$ and $Q$ lie on a circle.

2014 Rioplatense Mathematical Olympiad, Level 3, 5
Tags: geometry , circles , diameter
In the segment $A C$ a point $B$ is taken. Construct circles $T_1, T_2$ and $T_3$ of diameters $A B, BC$ and $AC$ respectively. A line that passes through $B$ cuts $T_3$ in the points $P$ and $Q$, and the circles $T_1$ and $T_2$ respectively at points $R$ and $S$. Prove that $PR = Q S$.

2005 Bosnia and Herzegovina Team Selection Test, 4
Tags: geometry , diameter , tangent , collinear
On the line which contains diameter $PQ$ of circle $k(S,r)$, point $A$ is chosen outside the circle such that tangent $t$ from point $A$ touches the circle in point $T$. Tangents on circle $k$ in points $P$ and $Q$ are $p$ and $q$, respectively. If $PT \cap q={N}$ and $QT \cap p={M}$, prove that points $A$, $M$ and $N$ are collinear.

2015 China Northern MO, 2
Tags: geometry , diameter , circumcircle , angle
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 $\angle E$.

1958 November Putnam, B3
Tags: square , diameter
Show that if a unit square is partitioned into two sets, then the diameter (least upper bound of the distances between pairs of points) of one of the sets is not less than $\sqrt{5} \slash 2.$ Show also that no larger number will do.

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 $S$.

1965 IMO Shortlist, 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 $n$.

2009 Oral Moscow Geometry Olympiad, 4
Tags: median , geometry , diameter , concurrency
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)

2014 Junior Balkan Team Selection Tests - Romania, 3
Tags: midpoint , equal angles , diameter , 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 $\angle FOG=\angle BAC$

1995 Singapore Team Selection Test, 2
Tags: concyclic , geometry , altitude , 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 , angle bisector , geometry
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 , geometry , tangent circle
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 $C_1$.

2018 Stars of Mathematics, 4
Tags: convex polygon , disc , convex , diameter , geometric inequality , geometry
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

Kyiv City MO Seniors 2003+ geometry, 2013.11.3
Tags: geometry , bisects segment , diameter , perpendicular
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)

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.

1999 Tournament Of Towns, 4
Tags: combinatorics , combinatorial geometry , diameter , disc , coloring
$n$ diameters divide a disk into $2n$ equal sectors. $n$ of the sectors are coloured blue , and the other $n$ are coloured red (in arbitrary order) . Blue sectors are numbered from $1$ to $n$ in the anticlockwise direction, starting from an arbitrary blue sector, and red sectors are numbered from $1$ to $n$ in the clockwise direction, starting from an arbitrary red sector. Prove that there is a semi-disk containing sectors with all numbers from $1$ to $n$. (V Proizvolov)

2020 Bundeswettbewerb Mathematik, 3
Tags: diameter , distance , geometry
Let $AB$ be the diameter of a circle $k$ and let $E$ be a point in the interior of $k$. The line $AE$ intersects $k$ a second time in $C \ne A$ and the line $BE$ intersects $k$ a second time in $D \ne B$. Show that the value of $AC \cdot AE+BD\cdot BE$ is independent of the choice of $E$.

1936 Moscow Mathematical Olympiad, 025
Tags: geometric construction , diameter , geometry
Consider a circle and a point $P$ outside the circle. The angle of given measure with vertex at $P$ subtends a diameter of the circle. Construct the circle’s diameter with ruler and compass.

Estonia Open Senior - geometry, 1998.2.5
Tags: diameter , chord , geometry , circumference , area
The plane has a semicircle with center $O$ and diameter $AB$. Chord $CD$ is parallel to the diameter $AB$ and $\angle AOC = \angle DOB = \frac{7}{16}$ (radians). Which of the two parts it divides into a semicircle is larger area?

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 $T$.

2015 Swedish Mathematical Competition, 1
Tags: geometry , circumcircle , diameter , trigonometry
Given the acute triangle $ABC$. A diameter of the circumscribed circle of the triangle intersects the sides $AC$ and $BC$, dividing the side $BC$ in half. Show that the same diameter divides the side $AC$ in a ratio of $1: 3$, calculated from $A$, if and only if $\tan B = 2 \tan C$.

2019 Oral Moscow Geometry Olympiad, 1
Tags: geometry , diameter , incircle , incenter , distance
In the triangle $ABC, I$ is the center of the inscribed circle, point $M$ lies on the side of $BC$, with $\angle BIM = 90^o$. Prove that the distance from point $M$ to line $AB$ is equal to the diameter of the circle inscribed in triangle $ABC$

2012 Swedish Mathematical Competition, 6
Tags: geometry , trapezoid , diameter , tangential
A circle is inscribed in an trapezoid. Show that the diagonals of the trapezoid intersect at a point on the diameter of the circle perpendicular to the two parallel sides.

Kyiv City MO Seniors Round2 2010+ geometry, 2017.10.3
Tags: geometry , circumcircle , diameter , circles
Circles $w_1$ and $w_2$ with centers at points $O_1$ and $O_2$ respectively, intersect at points $A$ and $B$. A line passing through point $B$, intersects the circles $w_1$ and $w_2$ at points $C$ and $D$ other than $B$. Tangents to the circles $w_1$ and $w_2$ at points $C$ and $D$ intersect at point $E$. Line $EA$ intersects the circumscribed circle $w$ of triangle $AO_1O_2$ at point $F$. Prove that the length of the segment is $EF$ is equal to the diameter of the circle $w$. (Vovchenko V., Plotnikov M.)