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: 24

2003 IMO Shortlist, 4
Tags: geometry , parallelogram , homothety , Inversion , IMO Shortlist , geometry solved , lengths
Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that \[ \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}. \]

2010 Contests, 2
Tags: geometry , parallelogram , homothety , Inversion , IMO Shortlist , geometry solved , lengths
Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that \[ \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}. \]

1965 Spain Mathematical Olympiad, 3
Tags: lengths , geometry
A disk in a record turntable makes $100$ revolutions per minute and it plays during $24$ minutes and $30$ seconds. The recorded line over the disk is a spiral with a diameter that decreases uniformly from $29$cm to $11.5$cm. Compute the length of the recorded line.

2020 Spain Mathematical Olympiad, 5
Tags: geometry , Spain , lengths
In an acute-angled triangle $ABC$, let $M$ be the midpoint of $AB$ and $P$ the foot of the altitude to $BC$. Prove that if $AC+BC = \sqrt{2}AB$, then the circumcircle of triangle $BMP$ is tangent to $AC$.

2024 CAPS Match, 3
Tags: geometry , international competitions , circumcircle , interior , same , lengths , incenter
Let $ABC$ be a triangle and $D$ a point on its side $BC.$ Points $E, F$ lie on the lines $AB, AC$ beyond vertices $B, C,$ respectively, such that $BE = BD$ and $CF = CD.$ Let $P$ be a point such that $D$ is the incenter of triangle $P EF.$ Prove that $P$ lies inside the circumcircle $\Omega$ of triangle $ABC$ or on it.

2010 Indonesia TST, 2
Tags: geometry , parallelogram , homothety , Inversion , IMO Shortlist , geometry solved , lengths
Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that \[ \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}. \]

2021 Argentina National Olympiad Level 2, 3
Tags: geometry , arcs , lengths , contests
A circle is divided into $2n$ equal arcs by $2n$ points. Find all $n>1$ such that these points can be joined in pairs using $n$ segments, all of different lengths and such that each point is the endpoint of exactly one segment.

2010 German National Olympiad, 1
Tags: geometry , circles , chord , Tangents , lengths
Given two circles $k$ and $l$ which intersect at two points. One of their common tangents touches $k$ at point $K$, while the other common tangent touches $l$ at $L.$ Let $A$ and $B$ be the intersections of the line $KL$ with the circles $k$ and $l$, respectively. Prove that $\overline{AK} = \overline{BL}.$

2021 German National Olympiad, 4
Tags: geometry , similar triangles , parallelogram , geometry proposed , lengths
Let $OFT$ and $NOT$ be two similar triangles (with the same orientation) and let $FANO$ be a parallelogram. Show that \[\vert OF\vert \cdot \vert ON\vert=\vert OA\vert \cdot \vert OT\vert.\]

2019 Poland - Second Round, 6
Tags: geometry , lengths , angles
Let $X$ be a point lying in the interior of the acute triangle $ABC$ such that \begin{align*} \sphericalangle BAX = 2\sphericalangle XBA \ \ \ \ \hbox{and} \ \ \ \ \sphericalangle XAC = 2\sphericalangle ACX. \end{align*} Denote by $M$ the midpoint of the arc $BC$ of the circumcircle $(ABC)$ containing $A$. Prove that $XM=XA$.

2022/2023 Tournament of Towns, P4
Tags: geometry , hexagon , lengths
The triangles $AB'C, CA'B$ and $BC'A$ are constructed on the sides of the equilateral triangle $ABC.$ In the resulting hexagon $AB'CA'BC'$ each of the angles $\angle A'BC',\angle C'AB'$ and $\angle B'CA'$ is greater than $120^\circ$ and the sides satisfy the equalities $AB' = AC',BC' = BA'$ and $CA' = CB'.$ Prove that the segments $AB',BC'$ and $CA'$ can form a triangle. [i]David Brodsky[/i]

1971 IMO Shortlist, 4
Tags: geometry , lengths , circles , Locus problems , IMO Shortlist
We are given two mutually tangent circles in the plane, with radii $r_1, r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_1$ and $r_2$ and the condition for the solvability of the problem.

2022 Bulgarian Spring Math Competition, Problem 11.2
Tags: geometry , lengths
A circle through the vertices $A$ and $B$ of $\triangle ABC$ intersects segments $AC$ and $BC$ at points $P$ and $Q$ respectively. If $AQ=AC$, $\angle BAQ=\angle CBP$ and $AB=(\sqrt{3}+1)PQ$, find the measures of the angles of $\triangle ABC$.

2021/2022 Tournament of Towns, P3
Tags: geometry , lengths , Tournament of Towns
The hypotenuse of a right triangle has length 1. Consider the line passing through the points of tangency of the incircle with the legs of the triangle. The circumcircle of the triangle cuts out a segment of this line. What is the possible length of this segment? [i]Maxim Volchkevich[/i]

Kvant 2023, M2761
Tags: geometry , lengths
Is it possible to fit a regular polygon into a circle of radius one so that among the lengths of its diagonals there are 2023 different values whose product is equal to one? [i]Proposed by A. Kuznetsov[/i]

2022 German National Olympiad, 2
Tags: planar geometry , planametry , geometry proposed , lengths , geometry
As everyone knows, the people of [i]Plane Land[/i] love Planimetrics. Therefore, they imagine their country as completely planar, every city in the country as a geometric point and every road as the line segment connecting two points. Additionally to the existing cities, it is possible to build [i]roundabouts[/i], i.e. points in the road network from where at least two roads emanate. All road crossings or junctions are build as roundabouts. Via this route network, every two cities should be connected by a sequence of roads and possibly roundabouts. In Plane Land, the length of a road is taken as the geometric length of the corresponding line segment. The ingenious road engineer Armin Asphalt presents a new road map, of which it is known that there is no road network with a smaller total length of all roads. Moreover, there is no road map with the same total length of all roads and fewer roundabouts. Prove that in the road map of Armin Asphalt, at most three roads emanate from each city, and exactly three from each roundabout.

2021 Argentina National Olympiad, 3
Tags: geometry , arcs , lengths , contests
A circle is divided into $2n$ equal arcs by $2n$ points. Find all $n>1$ such that these points can be joined in pairs using $n$ segments, all of different lengths and such that each point is the endpoint of exactly one segment.

Kvant 2023, M2767
Tags: geometry , lengths , incircle
It is easy to prove that in a right triangle the sum of the radii of the incircle and three excircles is equal to the perimeter. Prove that the opposite statement is also true. [i]Proposed by I. Weinstein[/i]

2015 BAMO, 4
Tags: geometry , quadrilateral , midpoints , lengths , Olympiad , B8 , prove
In a quadrilateral, the two segments connecting the midpoints of its opposite sides are equal in length. Prove that the diagonals of the quadrilateral are perpendicular. (In other words, let $M,N,P,$ and $Q$ be the midpoints of sides $AB,BC,CD,$ and $DA$ in quadrilateral $ABCD$. It is known that segments $MP$ and $NQ$ are equal in length. Prove that $AC$ and $BD$ are perpendicular.)

1971 IMO Longlists, 17
Tags: geometry , lengths , circles , Locus problems , IMO Shortlist
We are given two mutually tangent circles in the plane, with radii $r_1, r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_1$ and $r_2$ and the condition for the solvability of the problem.

2012 USA Team Selection Test, 1
Tags: geometry , circumcircle , trigonometry , geometry solved , Spiral Similarity , lengths , Fixed point
In acute triangle $ABC$, $\angle{A}<\angle{B}$ and $\angle{A}<\angle{C}$. Let $P$ be a variable point on side $BC$. Points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $BP=PD$ and $CP=PE$. Prove that as $P$ moves along side $BC$, the circumcircle of triangle $ADE$ passes through a fixed point other than $A$.

2021 Bulgaria EGMO TST, 1
Tags: geometry , areas , Menelaus , lengths
On the side $AB$ of a triangle $ABC$ is chosen a point $P$. Let $Q$ be the midpoint of $BC$ and let $CP$ and $AQ$ intersect at $R$. If $AB + AP = CP$, prove that $CR = AB$.

2010 Indonesia TST, 2
Tags: geometry , parallelogram , homothety , Inversion , IMO Shortlist , geometry solved , lengths
Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that \[ \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}. \]

2019 IMEO, 1
Tags: geometry , geometry proposed , geometry solved , parallelogram , humpty point , humpty points , lengths
Let $ABC$ be a scalene triangle with circumcircle $\omega$. The tangent to $\omega$ at $A$ meets $BC$ at $D$. The $A$-median of triangle $ABC$ intersects $BC$ and $\omega$ at $M$ and $N$, respectively. Suppose that $K$ is a point such that $ADMK$ is a parallelogram. Prove that $KA = KN$. [i]Proposed by Alexandru Lopotenco (Moldova)[/i]