Olympiad problems

1994 Tournament Of Towns, (418) 6

Tags: geometry , collinear

Consider a convex quadrilateral $ABCD$. Pairs of its opposite sides are continued until they intersect: $BA$ and $CD$ at the point $P$, $BC$ and $AD$ at the point $Q$. Let $K$ be the intersection point of the exterior bisectors of the angles $A$ and $C$ of the quadrilateral, $L$ be the intersection point of the exterior bisectors of the angles $B$ and $D$ of the quadrilateral, and $M$ be the intersection point of the exterior bisectors of the angles $P$ and $Q$ (the exterior bisector of an angle $X$ is the line passing through X and perpendicular to its ordinary bisector). Prove that the points $K$, $L$ and $M$ lie on a straight line. (S Markelov)

2005 Georgia Team Selection Test, 11

Tags: rhombus , geometry

On the sides $ AB, BC, CD$ and $ DA$ of the rhombus $ ABCD$, respectively, are chosen points $ E, F, G$ and $ H$ so, that $ EF$ and $ GH$ touch the incircle of the rhombus. Prove that the lines $ EH$ and $ FG$ are parallel.

2011 Belarus Team Selection Test, 1

Tags: chord , parabola , circles , parallel , geometry , conic

$AB$ and $CD$ are two parallel chords of a parabola. Circle $S_1$ passing through points $A,B$ intersects circle $S_2$ passing through $C,D$ at points $E,F$. Prove that if $E$ belongs to the parabola, then $F$ also belongs to the parabola. I.Voronovich

KoMaL A Problems 2021/2022, A. 817

Tags: geometry , mixtilinear incircle

Let $ABC$ be a triangle. Let $T$ be the point of tangency of the circumcircle of triangle $ABC$ and the $A$-mixtilinear incircle. The incircle of triangle $ABC$ has center $I$ and touches sides $BC,CA$ and $AB$ at points $D,E$ and $F,$ respectively. Let $N$ be the midpoint of line segment $DF.$ Prove that the circumcircle of triangle $BTN,$ line $TI$ and the perpendicular from $D$ to $EF$ are concurrent. [i]Proposed by Diaconescu Tashi, Romania[/i]

2010 Saudi Arabia BMO TST, 2

Tags: equal segments , cyclic quadrilateral , geometry

Quadrilateral $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle. Altitude $DE$ in triangle $ABD$ intersects diagonal $AC$ in $F$. Prove that $FB = BC$

2021 Korea Winter Program Practice Test, 2

Tags: geometry

Let $ABC$ be a triangle with $\angle A=60^{\circ}$. Point $D, E$ in lines $\overrightarrow{AB}, \overrightarrow{AC}$ respectively satisfies $DB=BC=CE$. ($A,B,D$ lies on this order, and $A,C,E$ likewise) Circle with diameter $BC$ and circle with diameter $DE$ meets at two points $X, Y$. Prove that $\angle XAY\ge 60^{\circ}$

V Soros Olympiad 1998 - 99 (Russia), 9.10

Tags: geometry , tangent circle

On the bisector of angle $A$ of triangle $ABC$, points $D$ and $F$ are taken inside the triangle so that $\angle DBC = \angle FBA$. Prove that: a) $\angle DCB = \angle FCA$ b) a circle passing through $B$ and $F$ and tangent to the segment $BC$ is tangle to the circumscribed circle of the triangle $ABC$.

2017 JBMO Shortlist, G4

Tags: geometry

Let $ABC $ be an acute triangle such that $AB\neq AC$ ,with circumcircle $ \Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $BC$ and $D$ be a point on $ \Gamma$ such that $AD \perp BC$. let $T$ be a point such that $BDCT$ is a parallelogram and $Q$ a point on the same side of $BC$ as $A$ such that $\angle{BQM}=\angle{BCA}$ and $\angle{CQM}=\angle{CBA}$. Let the line $AO$ intersect $ \Gamma$ at $E$ $(E\neq A)$ and let the circumcircle of $\triangle ETQ$ intersect $ \Gamma$ at point $X\neq E$. Prove that the point $A,M$ and $X$ are collinear.

2024 BMT, 4

Tags: geometry

Two circles, $\omega_1$ and $\omega_2$, are internally tangent at $A.$ Let $B$ be the point on $\omega_2$ opposite of $A.$ The radius of $\omega_1$ is $4$ times the radius of $\omega_2.$ Point $P$ is chosen on the circumference of $\omega_1$ such that the ratio $\tfrac{AP}{BP}=\tfrac{2\sqrt{3}}{\sqrt{7}}.$ Let $O$ denote the center of $\omega_2.$ Determine $\tfrac{OP}{AO}.$

2008 Mathcenter Contest, 7

Tags: area of a triangle , geometric inequality , area , geometry

$ABC$ is a triangle with an area of $1$ square meter. Given the point $D$ on $BC$, point $E$ on $CA$, point $F$ on $AB$, such that quadrilateral $AFDE$ is cyclic. Prove that the area of $DEF \le \frac{EF^2}{4 AD^2}$. [i](holmes)[/i]

2015 Princeton University Math Competition, A6/B8

Tags: geometry

Triangle $ABC$ is inscribed in a unit circle $\omega$. Let $H$ be its orthocenter and $D$ be the foot of the perpendicular from $A$ to $BC$. Let $\triangle XY Z$ be the triangle formed by drawing the tangents to $\omega$ at $A, B, C$. If $\overline{AH} = \overline{HD}$ and the side lengths of $\triangle XY Z$ form an arithmetic sequence, the area of $\triangle ABC$ can be expressed in the form $\tfrac{p}{q}$ for relatively prime positive integers $p, q$. What is $p + q$?

2012 Finnish National High School Mathematics Competition, 1

Tags: ratio , quadratic , trigonometry , algebra , trig identities , law of cosines , geometry

A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is $5:9$. Compute the ratio of the secant line versus circle radius.

Russian TST 2019, P1

Tags: geometry

A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold: [list=1] [*] each triangle from $T$ is inscribed in $\omega$; [*] no two triangles from $T$ have a common interior point. [/list] Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.

2005 Sharygin Geometry Olympiad, 18

Tags: symmetry , line , circumcenter , concurrency , geometry

On the plane are three straight lines $\ell_1, \ell_2,\ell_3$, forming a triangle, and the point $O$ is marked, the center of the circumscribed circle of this triangle. For an arbitrary point X of the plane, we denote by $X_i$ the point symmetric to the point X with respect to the line $\ell_i, i = 1,2,3$. a) Prove that for an arbitrary point $M$ the straight lines connecting the midpoints of the segments $O_1O_2$ and $M_1M_2, O_2O_3$ and $M_2M_3, O_3O_1$ and $M_3M_1$ intersect at one point, b) where can this intersection point lie?

2010 Sharygin Geometry Olympiad, 3

Tags: circumcircle , geometry

Points $A', B', C'$ lie on sides $BC, CA, AB$ of triangle $ABC.$ for a point $X$ one has $\angle AXB =\angle A'C'B' + \angle ACB$ and $\angle BXC = \angle B'A'C' +\angle BAC.$ Prove that the quadrilateral $XA'BC'$ is cyclic.

1951 AMC 12/AHSME, 9

Tags: geometry , perimeter

An equilateral triangle is drawn with a side of length $ a$. A new equilateral triangle is formed by joining the midpoints of the sides of the first one. Then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. The limit of the sum of the perimeters of all the triangles thus drawn is: $ \textbf{(A)}\ \text{Infinite} \qquad\textbf{(B)}\ 5\frac {1}{4}a \qquad\textbf{(C)}\ 2a \qquad\textbf{(D)}\ 6a \qquad\textbf{(E)}\ 4\frac {1}{2}a$

Kyiv City MO Juniors 2003+ geometry, 2012.7.4

Tags: geometry , perimeter , isosceles , geometric inequality

Given an isosceles triangle $ABC$ with a vertex at the point $B$. Based on $AC$, an arbitrary point $D $ is selected, different from the vertices $A$ and $C $. On the line $AC $ select the point $E $ outside the segment $AC$, for which $AE = CD$. Prove that the perimeter $\Delta BDE$ is larger than the perimeter $\Delta ABC$.

2023 Olimphíada, 1

Tags: circumcircle , geometry , tangent

Let $ABC$ be a triangle and $H$ and $D$ be the feet of the height and bisector relative to $A$ in $BC$, respectively. Let $E$ be the intersection of the tangent to the circumcircle of $ABC$ by $A$ with $BC$ and $M$ be the midpoint of $AD$. Finally, let $r$ be the line perpendicular to $BC$ that passes through $M$. Show that $r$ is tangent to the circumcircle of $AHE$.

2020 Junior Balkan Team Selection Tests - Moldova, 6

Tags: geometry , inscribed circle , cyclic quadrilateral

The inscribed circle inside triangle $ABC$ intersects side $AB$ in $D$. The inscribed circle inside triangle $ADC$ intersects sides $AD$ in $P$ and $AC$ in $Q$.The inscribed circle inside triangle $BDC$ intersects sides $BC$ in $M$ and $BD$ in $N$. Prove that $P , Q, M, N$ are cyclic.

2024 Germany Team Selection Test, 2

Tags: geometry

Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$. Prove that line $AO$ passes through the midpoint of segment $BE$.

2022 Indonesia TST, G

Tags: geometry , orthocenter , collinear , perpendicular , extension , acute , triangle

Given an acute triangle $ABC$. with $H$ as its orthocenter, lines $\ell_1$ and $\ell_2$ go through $H$ and are perpendicular to each other. Line $\ell_1$ cuts $BC$ and the extension of $AB$ on $D$ and $Z$ respectively. Whereas line $\ell_2$ cuts $BC$ and the extension of $AC$ on $E$ and $X$ respectively. If the line through $D$ and parallel to $AC$ and the line through $E$ parallel to $AB$ intersects at $Y$, prove that $X,Y,Z$ are collinear.

2023 Romania Team Selection Test, P1

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.

2011 China Second Round Olympiad, 6

Tags: analytic geometry , trigonometry , 3d geometry , tetrahedron , geometry

In a tetrahedral $ABCD$, given that $\angle ADB=\angle BDC =\angle CDA=\frac{\pi}{3}$, $AD=BD=3$, and $CD=2$. Find the radius of the circumsphere of $ABCD$.

2004 Harvard-MIT Mathematics Tournament, 2

Tags: geometry

A parallelogram has $3$ of its vertices at $(1, 2)$, $(3,8)$, and $(4, 1)$. Compute the sum of the possible $x$-coordinates for the $4$th vertex.

2014 India IMO Training Camp, 3

Tags: circumcircle , geometric transformation , trigonometry , function , angle bisector , geometry

In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.