Olympiad problems

2014 Sharygin Geometry Olympiad, 8

A convex polygon $P$ lies on a flat wooden table. You are allowed to drive some nails into the table. The nails must not go through $P$, but they may touch its boundary. We say that a set of nails blocks $P$ if the nails make it impossible to move $P$ without lifting it off the table. What is the minimum number of nails that suffices to block any convex polygon $P$? (N. Beluhov, S. Gerdgikov)

Brazil L2 Finals (OBM) - geometry, 2010.2

Tags: geometry , parallelogram , circumcircle , circumcenter

Let $ABCD$ be a parallelogram and $\omega$ be the circumcircle of the triangle $ABD$. Let $E ,F$ be the intersections of $\omega$ with lines $BC ,CD$ respectively . Prove that the circumcenter of the triangle $CEF$ lies on $\omega$.

2025 Kyiv City MO Round 1, Problem 3

Tags: geometry , tangent

Point $ H $ is the orthocenter of the acute triangle $ ABC $, and $ AD $ is its altitude. Tangents are drawn from points $ B $ and $ C $ to the circle with center $ A $ and radius $ AD $, which do not coincide with the line $ BC $. These tangents intersect at point $ P $. Prove that the radius of the incircle of $ \triangle BCP $ is equal to $ HD $. [i]Proposed by Danylo Khilko[/i]

2013 Iran MO (3rd Round), 2

Tags: circumcircle , incenter , geometric transformation , geometry

Let $ABC$ be a triangle with circumcircle $(O)$. Let $M,N$ be the midpoint of arc $AB,AC$ which does not contain $C,B$ and let $M',N'$ be the point of tangency of incircle of $\triangle ABC$ with $AB,AC$. Suppose that $X,Y$ are foot of perpendicular of $A$ to $MM',NN'$. If $I$ is the incenter of $\triangle ABC$ then prove that quadrilateral $AXIY$ is cyclic if and only if $b+c=2a$.

2006 Sharygin Geometry Olympiad, 9.3

Tags: equal ratio , similar triangles , collinear , geometry

Triangles $ABC$ and $A_1B_1C_1$ are similar and differently oriented. On the segment $AA_1$, a point $A'$ is taken such that $AA' / A_1A'= BC / B_1C_1$. We similarly construct $B'$ and $C'$. Prove that $A', B',C'$ lie on one straight line.

2001 National High School Mathematics League, 14

Tags: analytic geometry , geometry

$C_1:\frac{x^2}{a^2}+y^2=1(a>0),C_2:y^2=2(x+m)$, one intersection of $C_1$ and $C_2$ is $P$, and $P$ is above the $x$-axis. [b](a)[/b] Find the range value of $m$ (express with $a$). [b](b)[/b] $O(0,0),A(-a,0)$. If $0<a<\frac{1}{2}$, find the maximum value of $S_{\triangle OAP}$.

1980 All Soviet Union Mathematical Olympiad, 298

Tags: geometry , equilateral , angle

Given equilateral triangle $ABC$. Some line, parallel to $[AC]$ crosses $[AB]$ and $[BC]$ in $M$ and $P$ points respectively. Let $D$ be the centre of $PMB$ triangle, $E$ be the midpoint of the $[AP]$ segment. Find the angles of triangle $DEC$ .

2015 239 Open Mathematical Olympiad, 5

Tags: geometry , 3d geometry

The nodes of a three dimensional unit cube lattice with all three coordinates even are coloured red and blue otherwise. A convex polyhedron with all vertices red is given. Assuming the number of red points on its border is $n$. How many blue vertices can be on its border?

1994 Argentina National Olympiad, 5

Tags: combinatorics , combinatorial geometry , analytic geometry , geometry

Let $A$ be an infinite set of points in the plane such that inside each circle there are only a finite number of points of $A$, with the following properties: $\bullet$ $(0, 0)$ belongs to $A$. $\bullet$ If $(a, b)$ and $(c, d)$ belong to $A$, then $(a-c, b-d)$ belongs to $A$. $\bullet$ There is a value of $\alpha$ such that by rotating the set $A$ with center at $(0, 0)$ and angle $\alpha$, the set $A$ is obtained again. Prove that $\alpha$ must be equal to $\pm 60^{\circ}$ or $\pm 90^{\circ}$ or $\pm 120^{\circ}$ or $\pm 180^{\circ}$.

2015 Sharygin Geometry Olympiad, P8

Tags: geometry , trapezoid , isosceles , perpendicular , equal angles

Diagonals of an isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ are perpendicular. Let $DE$ be the perpendicular from $D$ to $AB$, and let $CF$ be the perpendicular from $C$ to $DE$. Prove that angle $DBF$ is equal to half of angle $FCD$.

2023-24 IOQM India, 5

Tags: geometry , area , area of a triangle , barycentric coordinates , barycentric

In a triangle $A B C$, let $E$ be the midpoint of $A C$ and $F$ be the midpoint of $A B$. The medians $B E$ and $C F$ intersect at $G$. Let $Y$ and $Z$ be the midpoints of $B E$ and $C F$ respectively. If the area of triangle $A B C$ is 480 , find the area of triangle $G Y Z$.

2019 Sharygin Geometry Olympiad, 7

Tags: geometry

Let $AH_A$, $BH_B$, $CH_C$ be the altitudes of the acute-angled $\Delta ABC$. Let $X$ be an arbitrary point of segment $CH_C$, and $P$ be the common point of circles with diameters $H_CX$ and BC, distinct from $H_C$. The lines $CP$ and $AH_A$ meet at point $Q$, and the lines $XP$ and $AB$ meet at point $R$. Prove that $A, P, Q, R, H_B$ are concyclic.

2021 Argentina National Olympiad, 3

Tags: geometry , angle chasing

Let $ABCD$ be a quadrilateral inscribed in a circle such that $\angle ABC=60^{\circ}.$ a) Prove that if $BC=CD$ then $AB= CD+DA.$ b) Is it true that if $AB= CD+DA$ then $BC=CD$?

2018 Oral Moscow Geometry Olympiad, 5

Tags: geometry , 3d geometry , tetrahedron , inequalities , geometric inequality

Two ants sit on the surface of a tetrahedron. Prove that they can meet by breaking the sum of a distance not exceeding the diameter of a circle is circumscribed around the edge of a tetrahedron.

2008 China Northern MO, 1A

Tags: geometry , trapezoid , equal angles , tangential

As shown in figure , $\odot O$ is the inscribed circle of trapezoid $ABCD$, and the tangent points are $E, F, G, H$, $AB \parallel CD$. The line passing through$ B$, parallel to $AD$ intersects extension of $DC$ at point $P$. The extension of $AO$ intersects $CP$ at point $Q$. If $AE=BE$ , prove that $\angle CBQ = \angle PBQ$. [img]https://cdn.artofproblemsolving.com/attachments/d/2/7c3a04bb1c59bc6d448204fd78f553ea53cb9e.png[/img]

2004 Estonia Team Selection Test, 6

Tags: pentagon , hexagon , combinatorics , combinatorial geometry , 3d geometry , geometry , polyhedron

Call a convex polyhedron a [i]footballoid [/i] if it has the following properties. (1) Any face is either a regular pentagon or a regular hexagon. (2) All neighbours of a pentagonal face are hexagonal (a [i]neighbour [/i] of a face is a face that has a common edge with it). Find all possibilities for the number of pentagonal and hexagonal faces of a footballoid.

1962 All Russian Mathematical Olympiad, 022

Tags: geometry , isosceles , perpendicular

The $M$ point is the midpoint of the base $[AC]$ of an isosceles triangle $ABC$. $[MH]$ is orthogonal to $[BC]$ side. Point $P$ is the midpoint of the segment $[MH]$. Prove that $[AH]$ is orthogonal to $[BP]$.

2012 Baltic Way, 14

Tags: similar triangles , geometry

Given a triangle $ABC$, let its incircle touch the sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Let $G$ be the midpoint of the segment $DE$. Prove that $\angle EFC = \angle GFD$.

2017 QEDMO 15th, 8

Tags: median , geometry , triangle inequality

Let $ABC$ be a triangle of area $1$ with medians $s_a, s_b,s_c$. Show that there is a triangle whose sides are the same length as $s_a, s_b$, and $s_c$, and determine its area.

1961 AMC 12/AHSME, 11

Tags: geometry , perimeter

Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of triangle $APR$ is ${{ \textbf{(A)}\ 42\qquad\textbf{(B)}\ 40.5 \qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 39\frac{7}{8} }\qquad\textbf{(E)}\ \text{not determined by the given information} } $

2008 AMC 12/AHSME, 8

Tags: 3d geometry , geometry

What is the volume of a cube whose surface area is twice that of a cube with volume $ 1$? $ \textbf{(A)}\ \sqrt{2} \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 2\sqrt{2} \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

2006 Moldova National Olympiad, 10.6

Tags: geometry , trigonometry , circumcircle , analytic geometry , geometric transformation , reflection , cyclic quadrilateral

Let a triangle $ABC$ satisfy $AC = BC$; in other words, let $ABC$ be an isosceles triangle with base $AB$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$. Denote by $M$ the midpoint of the segment $AB$. Show that $\angle APM + \angle BPC = 180^{\circ}$.

2014 ELMO Shortlist, 2

Tags: geometry , circumcircle , asymptote , cyclic quadrilateral , power of a point , radical axis , projective geometry

$ABCD$ is a cyclic quadrilateral inscribed in the circle $\omega$. Let $AB \cap CD = E$, $AD \cap BC = F$. Let $\omega_1, \omega_2$ be the circumcircles of $AEF, CEF$, respectively. Let $\omega \cap \omega_1 = G$, $\omega \cap \omega_2 = H$. Show that $AC, BD, GH$ are concurrent. [i]Proposed by Yang Liu[/i]

Geometry Mathley 2011-12, 13.3

Tags: concyclic , cyclic quadrilateral , concurrency , geometry

Let $ABCD$ be a quadrilateral inscribed in circle $(O)$. Let $M,N$ be the midpoints of $AD,BC$. A line through the intersection $P$ of the two diagonals $AC,BD$ meets $AD,BC$ at $S, T$ respectively. Let $BS$ meet $AT$ at $Q$. Prove that three lines $AD,BC,PQ$ are concurrent if and only if $M, S, T,N$ are on the same circle. Đỗ Thanh Sơn

2022 IMO Shortlist, G5

Tags: geometry

Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.