Olympiad problems

1991 Greece National Olympiad, 2

Tags: geometry , angle

Let $\widehat{xOy}$ be an acute angle , $A$ a point on ray $Oy$ and $B$ a point on ray $Ox$ such that $AB \perp OX$ .Prove that there are two points on $Ox$, each of the equidistant from $A$ and $Ox$.

2015 Iran Geometry Olympiad, 5

Tags: geometry

a) Do there exist 5 circles in the plane such that every circle passes through centers of exactly 3 circles? b) Do there exist 6 circles in the plane such that every circle passes through centers of exactly 3 circles?

2023 ISL, G7

Tags: geometry

Let $ABC$ be an acute, scalene triangle with orthocentre $H$. Let $\ell_a$ be the line through the reflection of $B$ with respect to $CH$ and the reflection of $C$ with respect to $BH$. Lines $\ell_b$ and $\ell_c$ are defined similarly. Suppose lines $\ell_a$, $\ell_b$, and $\ell_c$ determine a triangle $\mathcal T$. Prove that the orthocentre of $\mathcal T$, the circumcentre of $\mathcal T$, and $H$ are collinear. [i]Fedir Yudin, Ukraine[/i]

1986 Flanders Math Olympiad, 4

Tags: geometry , 3d geometry , sphere , algebra

Given a cube in which you can put two massive spheres of radius 1. What's the smallest possible value of the side - length of the cube? Prove that your answer is the best possible.

2009 Ukraine National Mathematical Olympiad, 3

Tags: circumcircle , parallelogram , cyclic quadrilateral , geometric transformation , geometry

In triangle $ABC$ points $M, N$ are midpoints of $BC, CA$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PCA = \angle MAC .$ Prove that $\angle PNA = \angle AMB .$

1996 Korea National Olympiad, 8

Tags: geometry , angle bisector , circumcircle

Let $\triangle ABC$ be the acute triangle such that $AB\ne AC.$ Let $V$ be the intersection of $BC$ and angle bisector of $\angle A.$ Let $D$ be the foot of altitude from $A$ to $BC.$ Let $E,F$ be the intersection of circumcircle of $\triangle AVD$ and $CA,AB$ respectively. Prove that the lines $AD, BE,CF$ is concurrent.

The Golden Digits 2024, P3

Tags: geometry

Let $ABC$ be an acute scalene triangle with orthocentre $H{}$ and circumcentre $O.{}$ Let $P{}$ be an arbitrary point on the segment $OH$ and $O_a$ be the circumcentre of $PBC.{}$ The line $PO_a$ intersects the line $HA$ at $X_a.{}$ Define $X_b$ and $X_c$ similarly. Let $Q{}$ be the isogonal conjugate of $P{}$ and $X{}$ be the circumcentre of $X_aX_bX_c.{}$ Prove that $PQ$ and $HX$ are parallel. [i]Proposed by David Anghel[/i]

2016 Korea National Olympiad, 2

Tags: geometry , incenter

A non-isosceles triangle $\triangle ABC$ has its incircle tangent to $BC, CA, AB$ at points $D, E, F$. Let the incenter be $I$. Say $AD$ hits the incircle again at $G$, at let the tangent to the incircle at $G$ hit $AC$ at $H$. Let $IH \cap AD = K$, and let the foot of the perpendicular from $I$ to $AD$ be $L$. Prove that $IE \cdot IK= IC \cdot IL$.

2014 India Regional Mathematical Olympiad, 1

Tags: geometry , circumcircle , asymptote , perpendicular bisector

Let $ABC$ be a triangle with $\angle ABC $ as the largest angle. Let $R$ be its circumcenter. Let the circumcircle of triangle $ARB$ cut $AC$ again at $X$. Prove that $RX$ is perpendicular to $BC$.

2019 Yasinsky Geometry Olympiad, p2

Tags: geometry , perimeter , 3d geometry , pyramid , rectangle

The base of the quadrilateral pyramid $SABCD$ lies the $ABCD$ rectangle with the sides $AB = 1$ and $AD = 10$. The edge $SA$ of the pyramid is perpendicular to the base, $SA = 4$. On the edge of $AD$, find a point $M$ such that the perimeter of the triangle of $SMC$ was minimal.

1965 IMO Shortlist, 3

Tags: geometry , 3d geometry , tetrahedron , prism

Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\epsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.

2017 IFYM, Sozopol, 2

Tags: geometry

Point $F$ lies on the circumscribed circle around $\Delta ABC$, $P$ and $Q$ are projections of point $F$ on $AB$ and $AC$ respectively. Prove that, if $M$ and $N$ are the middle points of $BC$ and $PQ$ respectively, then $MN$ is perpendicular to $FN$.

2010 Turkey MO (2nd round), 1

Tags: circumcircle , geometry

Let $A$ and $B$ be two points on the circle with diameter $[CD]$ and on the different sides of the line $CD.$ A circle $\Gamma$ passing through $C$ and $D$ intersects $[AC]$ different from the endpoints at $E$ and intersects $BC$ at $F.$ The line tangent to $\Gamma$ at $E$ intersects $BC$ at $P$ and $Q$ is a point on the circumcircle of the triangle $CEP$ different from $E$ and satisfying $|QP|=|EP|. \: AB \cap EF =\{R\}$ and $S$ is the midpoint of $[EQ].$ Prove that $DR$ is parallel to $PS.$

2019 Kosovo National Mathematical Olympiad, 5

Tags: geometry

Let $ABCDE$ be a regular pentagon. Let point $F$ be intersection of segments $AC$ and $BD$. Let point $G$ be in segment $AD$ such that $2AD=3AG$. Let point $H$ be the midpoint of side $DE$. Show that the points $F,G,H$ lie on a line.

2002 Iran MO (3rd Round), 15

Tags: ratio , conic , function , geometry

Let A be be a point outside the circle C, and AB and AC be the two tangents from A to this circle C. Let L be an arbitrary tangent to C that cuts AB and AC in P and Q. A line through P parallel to AC cuts BC in R. Prove that while L varies, QR passes through a fixed point. :)

1956 AMC 12/AHSME, 37

Tags: geometry , rhombus

On a map whose scale is $ 400$ miles to an inch and a half, a certain estate is represented by a rhombus having a $ 60^{\circ}$ angle. The diagonal opposite $ 60^{\circ}$ is $ \frac {3}{16}$ in. The area of the estate in square miles is: $ \textbf{(A)}\ \frac {2500}{\sqrt {3}} \qquad\textbf{(B)}\ \frac {1250}{\sqrt {3}} \qquad\textbf{(C)}\ 1250 \qquad\textbf{(D)}\ \frac {5625\sqrt {3}}{2} \qquad\textbf{(E)}\ 1250\sqrt {3}$

2014 Turkey MO (2nd round), 3

Tags: inequalities , geometric inequality , geometry

Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$. Let a line $\ell$ through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$, respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$, respectively such that the equation \[ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} \] holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.

2019 Polish MO Finals, 1

Tags: concyclic , orthocenter , tangent , phantom point , geometry

Let $ABC$ be an acute triangle. Points $X$ and $Y$ lie on the segments $AB$ and $AC$, respectively, such that $AX=AY$ and the segment $XY$ passes through the orthocenter of the triangle $ABC$. Lines tangent to the circumcircle of the triangle $AXY$ at points $X$ and $Y$ intersect at point $P$. Prove that points $A, B, C, P$ are concyclic.

1976 Bundeswettbewerb Mathematik, 4

Tags: geometry , 3d geometry , sphere , geometric transformation , vector , combinatorics

Each vertex of the 3-dimensional Euclidean space either is coloured red or blue. Prove that within those squares being possible in this space with edge length 1 there is at least one square either with three red vertices or four blue vertices !

2018 Polish Junior MO Finals, 5

Tags: geometry , equilateral triangle , plane geometry

Point $M$ is middle of side $AB$ of equilateral triangle $ABC$. Points $D$ and $E$ lie on segments $AC$ and $BC$, respectively and $\angle DME = 60 ^{\circ}$. Prove that, $AD + BE = DE + \frac{1}{2}AB$.

Durer Math Competition CD 1st Round - geometry, 2018.C+2

Tags: geometry , isosceles , right triangle , angle

In an isosceles right-angled triangle $ABC$, the right angle is at $A$. $D$ lies so on the side $BC$ that $2CD = DB$. Let $E$ be the projection of $B$ onto $AD$. What is the measure fof angle $\angle CED $?

2011 Albania Team Selection Test, 3

Tags: ratio , hyperbola , trigonometry , analytic geometry , circumcircle , geometry , conic

In the acute angle triangle $ABC$ the point $O$ is the center of the circumscribed circle and the lines $OA,OB,OC$ intersect sides $BC,CA,AB$ respectively in points $M,N,P$ such that $\angle NMP=90^o$. [b](a)[/b] Find the ratios $\frac{\angle AMN}{\angle NMC}$,$\frac{\angle AMP}{\angle PMB}$. [b](b)[/b] If any of the angles of the triangle $ABC$ is $60^o$, find the two other angles.

2016 Latvia National Olympiad, 2

Tags: geometry

Triangle $ABC$ has median $AF$, and $D$ is the midpoint of the median. Line $CD$ intersects $AB$ in $E$. Prove that $BD = BF$ implies $AE = DE$!

2019 South Africa National Olympiad, 3

Tags: geometry

Let $A$, $B$, $C$ be points on a circle whose centre is $O$ and whose radius is $1$, such that $\angle BAC = 45^\circ$. Lines $AC$ and $BO$ (possibly extended) intersect at $D$, and lines $AB$ and $CO$ (possibly extended) intersect at $E$. Prove that $BD \cdot CE = 2$.

2011 Regional Olympiad of Mexico Center Zone, 6

Tags: geometry , inradius , geometric inequality

Given a circle $C$ and a diameter $AB$ in it, mark a point $P$ on $AB$ different from the ends. In one of the two arcs determined by $AB$ choose the points $M$ and $N$ such that $\angle APM = 60 ^ \circ = \angle BPN$. The segments $MP$ and $NP$ are drawn to obtain three curvilinear triangles; $APM $, $MPN$ and $NPB$ (the sides of the curvilinear triangle $APM$ are the segments $AP$ and $PM$ and the arc $AM$). In each curvilinear triangle a circle is inscribed, that is, a circle is built tangent to the three sides. Show that the sum of the radii of the three inscribed circles is less than or equal to the radius of $C$.