Olympiad problems

1967 IMO Shortlist, 2

Tags: geometry , 3d geometry , sphere , tetrahedron , circumcircle

Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.

1984 IMO Longlists, 67

Tags: inequalities , circumcircle , geometry

With the medians of an acute-angled triangle another triangle is constructed. If $R$ and $R_m$ are the radii of the circles circumscribed about the first and the second triangle, respectively, prove that \[R_m>\frac{5}{6}R\]

1996 India Regional Mathematical Olympiad, 1

Tags: circumcircle , inradius , perimeter , geometry

The sides of a triangle are three consecutive integers and its inradius is $4$. Find the circumradius.

2008 Brazil National Olympiad, 1

Tags: cyclic quadrilateral , reflection , geometric transformation , circumcircle , homothety , incenter , geometry

Let $ ABCD$ be a cyclic quadrilateral and $ r$ and $ s$ the lines obtained reflecting $ AB$ with respect to the internal bisectors of $ \angle CAD$ and $ \angle CBD$, respectively. If $ P$ is the intersection of $ r$ and $ s$ and $ O$ is the center of the circumscribed circle of $ ABCD$, prove that $ OP$ is perpendicular to $ CD$.

2012 India National Olympiad, 5

Tags: trigonometry , geometric transformation , reflection , geometry , circumcircle

Let $ABC$ be an acute angled triangle. Let $D,E,F$ be points on $BC, CA, AB$ such that $AD$ is the median, $BE$ is the internal bisector and $CF$ is the altitude. Suppose that $\angle FDE=\angle C, \angle DEF=\angle A$ and $\angle EFD=\angle B.$ Show that $ABC$ is equilateral.

2024 Bulgarian Autumn Math Competition, 10.2

Tags: homothety , circumcircle , geometry

Let $ABC$ be a scalene acute triangle, where $AL$ $(L \in BC)$ is the internal bisector of $\angle BAC$ and $M$ is the midpoint of $BC$. Let the internal bisectors of $\angle AMB$ and $\angle CMA$ intersect $AB$ and $AC$ in $P$ and $Q$, respectively. Prove that the circumcircle of $APQ$ is tangent to $BC$ if and only if $L$ belongs to it.

1996 IberoAmerican, 2

Tags: circumcircle , geometry

Let $\triangle{ABC}$ be a triangle, $D$ the midpoint of $BC$, and $M$ be the midpoint of $AD$. The line $BM$ intersects the side $AC$ on the point $N$. Show that $AB$ is tangent to the circuncircle to the triangle $\triangle{NBC}$ if and only if the following equality is true: \[\frac{{BM}}{{MN}} =\frac{({BC})^2}{({BN})^2}.\]

1985 All Soviet Union Mathematical Olympiad, 412

Tags: circumradius , circumcircle , equal circles , geometry

One of two circumferences of radius $R$ comes through $A$ and $B$ vertices of the $ABCD$ parallelogram. Another comes through $B$ and $D$. Let $M$ be another point of circumferences intersection. Prove that the circle circumscribed around $AMD$ triangle has radius $R$.

2013 Brazil Team Selection Test, 3

Tags: geometry , triangle , reflection , circumcircle , trigonometry

Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.

1993 Moldova Team Selection Test, 8

Tags: geometry , circumcircle , parallelogram

Inside the parallelogram $ABCD$ points $M, N, K$ and $L{}$ are on sides $AB, BC, CD{}$ and $DA$, respectively. Let $O_1, O_2, O_3$ and $O_4$ be the circumcenters of triangles repesctively $MBN, NCK, KDL$ and $LAM{}$. Prove that the quadrilateral $O_1O_2O_3O_4$ is a parallelogram.

KoMaL A Problems 2023/2024, A. 878

Tags: circumcenter , circumcircle , geometry

Let point $A$ be one of the intersections of circles $c$ and $k$. Let $X_1$ and $X_2$ be arbitrary points on circle $c$. Let $Y_i$ denote the intersection of line $AX_i$ and circle $k$ for $i=1,2$. Let $P_1$, $P_2$ and $P_3$ be arbitrary points on circle $k$, and let $O$ denote the center of circle $k$. Let $K_{ij}$ denote the center of circle $(X_iY_iP_j)$ for $i=1,2$ and $j=1,2,3$. Let $L_j$ denote the center of circle $(OK_{1j}K_{2j})$ for $j=1,2,3$. Prove that points $L_1$, $L_2$ and $L_3$ are collinear. Proposed by [i]Vilmos Molnár-Szabó[/i], Budapest

2020 Sharygin Geometry Olympiad, 14

Tags: geometry , circumcircle

A non-isosceles triangle is given. Prove that one of the circles touching internally its incircle and circumcircle and externally one of its excircles passes through a vertex of the triangle.

2009 Math Prize For Girls Problems, 19

Tags: circumcircle , geometry

Let $ S$ be a set of $ 100$ points in the plane. The distance between every pair of points in $ S$ is different, with the largest distance being $ 30$. Let $ A$ be one of the points in $ S$, let $ B$ be the point in $ S$ farthest from $ A$, and let $ C$ be the point in $ S$ farthest from $ B$. Let $ d$ be the distance between $ B$ and $ C$ rounded to the nearest integer. What is the smallest possible value of $ d$?

2008 IMO Shortlist, 3

Tags: quadrilateral , trigonometry , circumcircle , inversion , homothety , geometry

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

2015 Balkan MO Shortlist, G1

Tags: geometry , trigonometry , circumcircle

In an acute angled triangle $ABC$ , let $BB' $ and $CC'$ be the altitudes. Ray $C'B'$ intersects the circumcircle at $B''$ andl let $\alpha_A$ be the angle $\widehat{ABB''}$. Similarly are defined the angles $\alpha_B$ and $\alpha_C$. Prove that $$\displaystyle\sin \alpha _A \sin \alpha _B \sin \alpha _C\leq \frac{3\sqrt{6}}{32}$$ (Romania)

2022 German National Olympiad, 3

Tags: incircle , circumcircle , touching circles , sidelengths , geometry

Let $M$ and $N$ be the midpoints of segments $BC$ and $AC$ of a triangle $ABC$, respectively. Let $Q$ be a point on the line through $N$ parallel to $BC$ such that $Q$ and $C$ are on opposite sides of $AB$ and $\vert QN\vert \cdot \vert BC\vert=\vert AB\vert \cdot \vert AC\vert$. Suppose that the circumcircle of triangle $AQN$ intersects the segment $MN$ a second time in a point $T \ne N$. Prove that there is a circle through points $T$ and $N$ touching both the side $BC$ and the incircle of triangle $ABC$.

2008 USA Team Selection Test, 6

Tags: geometry , symmetry , circumcircle , ratio

Determine the smallest positive real number $ k$ with the following property. Let $ ABCD$ be a convex quadrilateral, and let points $ A_1$, $ B_1$, $ C_1$, and $ D_1$ lie on sides $ AB$, $ BC$, $ CD$, and $ DA$, respectively. Consider the areas of triangles $ AA_1D_1$, $ BB_1A_1$, $ CC_1B_1$ and $ DD_1C_1$; let $ S$ be the sum of the two smallest ones, and let $ S_1$ be the area of quadrilateral $ A_1B_1C_1D_1$. Then we always have $ kS_1\ge S$. [i]Author: Zuming Feng and Oleg Golberg, USA[/i]

2012 NIMO Problems, 4

Tags: trigonometry , circumcircle , geometry

In $\triangle ABC$, $AB = AC$. Its circumcircle, $\Gamma$, has a radius of 2. Circle $\Omega$ has a radius of 1 and is tangent to $\Gamma$, $\overline{AB}$, and $\overline{AC}$. The area of $\triangle ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$, where $b$ is squarefree and $\gcd (a, c) = 1$. Compute $a + b + c$. [i]Proposed by Aaron Lin[/i]

2023 All-Russian Olympiad, 1

Tags: rotation , circumcircle , geometry

Sidelines of an acute-angled triangle $T$ are colored in red, green, and blue. These lines were rotated about the circumcenter of $T$ clockwise by $120^\circ$ (we assume that the line has the same color after rotation). Prove that three points of pairs of lines of the same color are the vertices of a triangle which is congruent to $T$.

2016 Tournament Of Towns, 3

Tags: circumcircle , geometry

The quadrilateral $ABCD$ is inscribed in circle $\Omega$ with center $O$, not lying on either of the diagonals. Suppose that the circumcircle of triangle $AOC$ passes through the midpoint of the diagonal $BD$. Prove that the circumcircle of triangle $BOD$ passes through the midpoint of diagonal $AC$. [i](A. Zaslavsky)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

2008 Sharygin Geometry Olympiad, 9

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

(A.Zaslavsky, 9--10) The reflections of diagonal $ BD$ of a quadrilateral $ ABCD$ in the bisectors of angles $ B$ and $ D$ pass through the midpoint of diagonal $ AC$. Prove that the reflections of diagonal $ AC$ in the bisectors of angles $ A$ and $ C$ pass through the midpoint of diagonal $ BD$ (There was an error in published condition of this problem).

2013 Kazakhstan National Olympiad, 1

Tags: geometry , circumcircle , geometric transformation , incenter

Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.

2005 Slovenia National Olympiad, Problem 3

Tags: circumcircle , square , geometry

Let $T$ be a point inside a square $ABCD$. The lines $TA,TB,TC,TD$ meet the circumcircle of $ABCD$ again at $A',B',C',D'$, respectively. Prove that $A'B'\cdot C'D'=A'D'\cdot B'C'$.

2014 Sharygin Geometry Olympiad, 20

Tags: concurrency , circumcircle , circles , geometry

A quadrilateral $KLMN$ is given. A circle with center $O$ meets its side $KL$ at points $A$ and $A_1$, side $LM$ at points $B$ and $B_1$, etc. Prove that if the circumcircles of triangles $KDA, LAB, MBC$ and $NCD$ concur at point $P$, then a) the circumcircles of triangles $KD_1A_1, LA_1B_1, MB_1C_1$ and $NC1D1$ also concur at some point $Q$; b) point $O$ lies on the perpendicular bisector to $PQ$.

2010 Oral Moscow Geometry Olympiad, 6

Tags: circumcircle , fixed , circles , perpendicular bisector , geometry

Perpendicular bisectors of the sides $BC$ and $AC$ of an acute-angled triangle $ABC$ intersect lines $AC$ and $BC$ at points $M$ and $N$. Let point $C$ move along the circumscribed circle of triangle $ABC$, remaining in the same half-plane relative to $AB$ (while points $A$ and $B$ are fixed). Prove that line $MN$ touches a fixed circle.