Olympiad problems

2022 Iranian Geometry Olympiad, 3

Tags: geometry

In triangle $ABC$ $(\angle A\neq 90^\circ)$, let $O$, $H$ be the circumcenter and the foot of the altitude from $A$ respectively. Suppose $M$, $N$ are the midpoints of $BC$, $AH$ respectively. Let $D$ be the intersection of $AO$ and $BC$ and let $H'$ be the reflection of $H$ about $M$. Suppose that the circumcircle of $OH'D$ intersects the circumcircle of $BOC$ at $E$. Prove that $NO$ and $AE$ are concurrent on the circumcircle of $BOC$. [i]Proposed by Mehran Talaei[/i]

1955 Moscow Mathematical Olympiad, 296

Tags: geometry , circles , cyclic , concyclic

There are four points $A, B, C, D$ on a circle. Circles are drawn through each pair of neighboring points. Denote the intersection points of neighboring circles by $A_1, B_1, C_1, D_1$. (Some of these points may coincide with previously given ones.) Prove that points $A_1, B_1, C_1, D_1$ lie on one circle.

2017 USA Team Selection Test, 2

Tags: rhombus , circumcircle , projective geometry , geometry

Let $ABC$ be a triangle with altitude $\overline{AE}$. The $A$-excircle touches $\overline{BC}$ at $D$, and intersects the circumcircle at two points $F$ and $G$. Prove that one can select points $V$ and $N$ on lines $DG$ and $DF$ such that quadrilateral $EVAN$ is a rhombus. [i]Danielle Wang and Evan Chen[/i]

1996 China National Olympiad, 3

Tags: geometry

In the triangle $ABC$, $\angle{C}=90^{\circ},\angle {A}=30^{\circ}$ and $BC=1$. Find the minimum value of the longest side of all inscribed triangles (i.e. triangles with vertices on each of three sides) of the triangle $ABC$.

2025 Benelux, 3

Tags: geometry , incenter , circumcircle

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$. Let $D, E, F$ be the midpoints of the arcs $\stackrel{\frown}{BC}, \stackrel{\frown}{CA}, \stackrel{\frown}{AB}$ of $\Omega$ not containing $A, B, C$ respectively. Let $D'$ be the point of $\Omega$ diametrically opposite to $D$. Show that $I, D'$ and the midpoint $M$ of $EF$ lie on a line.

2000 National Olympiad First Round, 13

Tags: geometry , trigonometry , pythagorean theorem

Let $d$ be one of the common tangent lines of externally tangent circles $k_1$ and $k_2$. $d$ touches $k_1$ at $A$. Let $[AB]$ be a diameter of $k_1$. The tangent from $B$ to $k_2$ touches $k_2$ at $C$. If $|AB|=8$ and the diameter of $k_2$ is $7$, then what is $|BC|$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6\sqrt 2 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 5\sqrt 3 $

2003 Oral Moscow Geometry Olympiad, 2

Tags: ratio , geometry , right angle , equal segments , equal angles

In a convex quadrilateral $ABCD$, $\angle ABC = 90^o$ , $\angle BAC = \angle CAD$, $AC = AD, DH$ is the alltitude of the triangle $ACD$. In what ratio does the line $BH$ divide the segment $CD$?

2013 Harvard-MIT Mathematics Tournament, 1

Tags: geometry

Jarris the triangle is playing in the $(x, y)$ plane. Let his maximum $y$ coordinate be $k$. Given that he has side lengths $6$, $8$, and $10$ and that no part of him is below the $x$-axis, find the minimum possible value of $k$.

2012 Sharygin Geometry Olympiad, 3

Tags: geometry , centroid , incenter , perpendicular

Let $M$ and $I$ be the centroid and the incenter of a scalene triangle $ABC$, and let $r$ be its inradius. Prove that $MI = r/3$ if and only if $MI$ is perpendicular to one of the sides of the triangle. (A.Karlyuchenko)

2010 IMO, 4

Tags: geometry , circumcircle , triangle geometry

Let $P$ be a point interior to triangle $ABC$ (with $CA \neq CB$). The lines $AP$, $BP$ and $CP$ meet again its circumcircle $\Gamma$ at $K$, $L$, respectively $M$. The tangent line at $C$ to $\Gamma$ meets the line $AB$ at $S$. Show that from $SC = SP$ follows $MK = ML$. [i]Proposed by Marcin E. Kuczma, Poland[/i]

2016 USA Team Selection Test, 2

Tags: mixtilinear incircle , isogonal lines , reflection , excircle , geometry

Let $ABC$ be a scalene triangle with circumcircle $\Omega$, and suppose the incircle of $ABC$ touches $BC$ at $D$. The angle bisector of $\angle A$ meets $BC$ and $\Omega$ at $E$ and $F$. The circumcircle of $\triangle DEF$ intersects the $A$-excircle at $S_1$, $S_2$, and $\Omega$ at $T \neq F$. Prove that line $AT$ passes through either $S_1$ or $S_2$. [i]Proposed by Evan Chen[/i]

2021 Philippine MO, 8

Tags: geometry

In right triangle $ABC$, $\angle ACB = 90^{\circ}$ and $\tan A > \sqrt{2}$. $M$ is the midpoint of $AB$, $P$ is the foot of the altitude from $C$, and $N$ is the midpoint of $CP$. Line $AB$ meets the circumcircle of $CNB$ again at $Q$. $R$ lies on line $BC$ such that $QR$ and $CP$ are parallel, $S$ lies on ray $CA$ past $A$ such that $BR = RS$, and $V$ lies on segment $SP$ such that $AV = VP$. Line $SP$ meets the circumcircle of $CPB$ again at $T$. $W$ lies on ray $VA$ past $A$ such that $2AW = ST$, and $O$ is the circumcenter of $SPM$. Prove that lines $OM$ and $BW$ are perpendicular.

2018 India IMO Training Camp, 2

Tags: homothety , power of a point , excircle , geometry , tangent circle

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

2005 MOP Homework, 3

Tags: power of a point , radical axis , geometry

Circles $S_1$ and $S_2$ meet at points $A$ and $B$. A line through $A$ is parallel to the line through the centers of $S_1$ and $S_2$ and meets $S_1$ and $S_2$ again $C$ and $D$ respectively. Circle $S_3$ having $CD$ as its diameter meets $S_1$ and $S_2$ again at $P$ and $Q$ respectively. Prove that lines $CP$, $DQ$, and $AB$ are concurent.

1985 Vietnam National Olympiad, 3

Tags: 3d geometry , pyramid , geometry

A triangular pyramid $ O.ABC$ with base $ ABC$ has the property that the lengths of the altitudes from $ A$, $ B$ and $ C$ are not less than $ \frac{OB \plus{}OC}{2}$, $ \frac{OC \plus{} OA}{2}$ and $ \frac{OA \plus{} OB}{2}$, respectively. Given that the area of $ ABC$ is $ S$, calculate the volume of the pyramid.

2013 Online Math Open Problems, 36

Tags: geometry , trapezoid , incenter , inequalities , pythagorean theorem , triangle inequality

Let $ABCD$ be a nondegenerate isosceles trapezoid with integer side lengths such that $BC \parallel AD$ and $AB=BC=CD$. Given that the distance between the incenters of triangles $ABD$ and $ACD$ is $8!$, determine the number of possible lengths of segment $AD$. [i]Ray Li[/i]

2023 Bangladesh Mathematical Olympiad, P7

Tags: geometry , reflection , power of a point , perpendicular bisector

Let $\Delta ABC$ be an acute triangle and $\omega$ be its circumcircle. Perpendicular from $A$ to $BC$ intersects $BC$ at $D$ and $\omega$ at $K$. Circle through $A$, $D$ and tangent to $BC$ at $D$ intersect $\omega$ at $E$. $AE$ intersects $BC$ at $T$. $TK$ intersects $\omega$ at $S$. Assume, $SD$ intersects $\omega$ at $X$. Prove that $X$ is the reflection of $A$ with respect to the perpendicular bisector of $BC$.

2013 Baltic Way, 15

Tags: rectangle , arithmetic sequence , geometry

Four circles in a plane have a common center. Their radii form a strictly increasing arithmetic progression. Prove that there is no square with each vertex lying on a different circle.

2023 Romanian Master of Mathematics, 4

Tags: geometry , inequalities , geometry inequality

An acute triangle $ABC$ is given and $H$ and $O$ be its orthocenter and circumcenter respectively. Let $K$ be the midpoint of $AH$ and $\ell$ be a line through $O. $ Let $P$ and $Q$ be the projections of $B$ and $C$ on $\ell. $ Prove that$$KP+KQ\ge BC$$

2003 Romania National Olympiad, 3

Tags: geometry , complex geometry , complex number

Let be a circumcircle of radius $ 1 $ of a triangle whose centered representation in the complex plane is given by the affixes of $ a,b,c, $ and for which the equation $ a+b\cos x +c\sin x=0 $ has a real root in $ \left( 0,\frac{\pi }{2} \right) . $ prove that the area of the triangle is a real number from the interval $ \left( 1,\frac{1+\sqrt 2}{2} \right] . $ [i]Gheorghe Iurea[/i]

2003 Bulgaria Team Selection Test, 5

Tags: asymptote , trigonometry , trig identities , law of sines , geometry

Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its in center on $AC$. Prove that $\angle {APB}=\angle {APD}$

2024 Saint Petersburg Mathematical Olympiad, 3

Tags: geometry

The triangle $ABC$ is inscribed in a circle. Two ants crawl out of points $B$ and $C$ at the same time. They crawl along the arc $BC$ towards each other so that the product of the distances from them to point $A$ remains unchanged. Prove that during their movement (until the moment of meeting), the straight line passing through the ants touches some fixed circle.

1992 IMO Longlists, 42

Tags: trigonometry , geometry , law of sines , triangle

In a triangle $ ABC,$ let $ D$ and $ E$ be the intersections of the bisectors of $ \angle ABC$ and $ \angle ACB$ with the sides $ AC,AB,$ respectively. Determine the angles $ \angle A,\angle B, \angle C$ if $ \angle BDE \equal{} 24 ^{\circ},$ $ \angle CED \equal{} 18 ^{\circ}.$

1999 Taiwan National Olympiad, 5

Tags: circumcircle , geometry

Let $AD,BE,CF$ be the altitudes of an acute triangle $ABC$ with $AB>AC$. Line $EF$ meets $BC$ at $P$, and line through $D$ parallel to $EF$ meets $AC$ and $AB$ at $Q$ and $R$, respectively. Let $N$ be any poin on side $BC$ such that $\widehat{NQP}+\widehat{NRP}<180^{0}$. Prove that $BN>CN$.

2021 Olimphíada, 4

Tags: incircle , concurrency , orthocenter , incenter , geometry

Let $H$ be the orthocenter of the triangle $ABC$ and let $D$, $E$, $F$ be the feet of heights by $A$, $B$, $C$. Let $\omega_D$, $\omega_E$, $\omega_F$ be the incircles of $FEH$, $DHF$, $HED$ and let $I_D$, $I_E$, $I_F$ be their centers. Show that $I_DD$, $I_EE$ and $I_FF$ compete.