Olympiad problems

2010 Germany Team Selection Test, 2

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

2001 All-Russian Olympiad, 2

Tags: circumcircle , geometric transformation , ratio , trigonometry , homothety , power of a point , geometry

Let the circle $ {\omega}_{1}$ be internally tangent to another circle $ {\omega}_{2}$ at $ N$.Take a point $ K$ on $ {\omega}_{1}$ and draw a tangent $ AB$ which intersects $ {\omega}_{2}$ at $ A$ and $ B$. Let $M$ be the midpoint of the arc $ AB$ which is on the opposite side of $ N$. Prove that, the circumradius of the $ \triangle KBM$ doesnt depend on the choice of $ K$.

2019 IFYM, Sozopol, 4

Tags: geometry , equal angles , power of a point

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect in point $M$. The angle bisector of $\angle ACD$ intersects the ray $\overrightarrow{BA}$ in point $K$. If $MA.MC+MA.CD=MB.MD$, prove that $\angle BKC=\angle CDB$.

2014 Canada National Olympiad, 4

Tags: geometry , power of a point , circumcircle , radical axis

The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle P AB = \angle P BC = \angle P CD = \angle P DA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $ABCD$.

2003 Moldova Team Selection Test, 3

Tags: radical axis , circumcircle , power of a point , geometry

Let $ ABCD$ be a quadrilateral inscribed in a circle of center $ O$. Let M and N be the midpoints of diagonals $ AC$ and $ BD$, respectively and let $ P$ be the intersection point of the diagonals $ AC$ and $ BD$ of the given quadrilateral .It is known that the points $ O,M,Np$ are distinct. Prove that the points $ O,N,A,C$ are concyclic if and only if the points $ O,M,B,D$ are concyclic. [i]Proposer[/i]: [b]Dorian Croitoru[/b]

2007 Bulgaria Team Selection Test, 1

Tags: trigonometry , geometry , homothety , power of a point , radical axis , geometric transformation

In isosceles triangle $ABC(AC=BC)$ the point $M$ is in the segment $AB$ such that $AM=2MB,$ $F$ is the midpoint of $BC$ and $H$ is the orthogonal projection of $M$ in $AF.$ Prove that $\angle BHF=\angle ABC.$

2008 AMC 10, 10

Tags: trigonometry , power of a point , geometry , pythagorean theorem

Points $ A$ and $ B$ are on a circle of radius $ 5$ and $ AB\equal{}6$. Point $ C$ is the midpoint of the minor arc $ AB$. What is the length of the line segment $ AC$? $ \textbf{(A)}\ \sqrt{10} \qquad \textbf{(B)}\ \frac{7}{2} \qquad \textbf{(C)}\ \sqrt{14} \qquad \textbf{(D)}\ \sqrt{15} \qquad \textbf{(E)}\ 4$

2006 Hungary-Israel Binational, 1

Tags: geometry , quadratic , power of a point

A point $ P$ inside a circle is such that there are three chords of the same length passing through $ P$. Prove that $ P$ is the center of the circle.

2010 Peru IMO TST, 8

Tags: projective geometry , cyclic quadrilateral , tangent , power of a point , geometry

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

2000 Macedonia National Olympiad, 3

Tags: circumcircle , power of a point , geometry , inequalities

In a triangle with sides $a,b,c,t_a,t_b,t_c$ are the corresponding medians and $D$ the diameter of the circumcircle. Prove that \[\frac{a^2+b^2}{t_c}+\frac{b^2+c^2}{t_a}+\frac{c^2+a^2}{t_b}\le 6D\]

2011 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: geometric transformation , angle bisector , geometry , radical axis , circumcircle , power of a point

Let $ABC$ an acute triangle and $H$ its orthocenter. Let $E$ and $F$ be the intersection of lines $BH$ and $CH$ with $AC$ and $AB$ respectively, and let $D$ be the intersection of lines $EF$ and $BC$. Let $\Gamma_1$ be the circumcircle of $AEF$, and $\Gamma_2$ the circumcircle of $BHC$. The line $AD$ intersects $\Gamma_1$ at point $I \neq A$. Let $J$ be the feet of the internal bisector of $\angle{BHC}$ and $M$ the midpoint of the arc $\stackrel{\frown}{BC}$ from $\Gamma_2$ that contains the point $H$. The line $MJ$ intersects $\Gamma_2$ at point $N \neq M$. Show that the triangles $EIF$ and $CNB$ are similar.

JBMO Geometry Collection, 2008

Tags: trigonometry , angle bisector , power of a point , geometry , perpendicular bisector

The vertices $ A$ and $ B$ of an equilateral triangle $ ABC$ lie on a circle $k$ of radius $1$, and the vertex $ C$ is in the interior of the circle $ k$. A point $ D$, different from $ B$, lies on $ k$ so that $ AD\equal{}AB$. The line $ DC$ intersects $ k$ for the second time at point $ E$. Find the length of the line segment $ CE$.

2016 Indonesia TST, 3

Tags: homothety , geometry , circumcircle , incenter , power of a point , geometric transformation

Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$

2006 All-Russian Olympiad, 6

Tags: tetrahedron , power of a point , 3d geometry , radical axis , geometry , sphere

Consider a tetrahedron $SABC$. The incircle of the triangle $ABC$ has the center $I$ and touches its sides $BC$, $CA$, $AB$ at the points $E$, $F$, $D$, respectively. Let $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ be the points on the segments $SA$, $SB$, $SC$ such that $AA^{\prime}=AD$, $BB^{\prime}=BE$, $CC^{\prime}=CF$, and let $S^{\prime}$ be the point diametrically opposite to the point $S$ on the circumsphere of the tetrahedron $SABC$. Assume that the line $SI$ is an altitude of the tetrahedron $SABC$. Show that $S^{\prime}A^{\prime}=S^{\prime}B^{\prime}=S^{\prime}C^{\prime}$.

2007 Bulgaria Team Selection Test, 1

Tags: trigonometry , homothety , power of a point , radical axis , geometry , geometric transformation

In isosceles triangle $ABC(AC=BC)$ the point $M$ is in the segment $AB$ such that $AM=2MB,$ $F$ is the midpoint of $BC$ and $H$ is the orthogonal projection of $M$ in $AF.$ Prove that $\angle BHF=\angle ABC.$

2014 Harvard-MIT Mathematics Tournament, 2

Tags: analytic geometry , power of a point , geometry

Point $P$ and line $\ell$ are such that the distance from $P$ to $\ell$ is $12$. Given that $T$ is a point on $\ell$ such that $PT = 13$, find the radius of the circle passing through $P$ and tangent to $\ell$ at $T$.

2012 ELMO Shortlist, 2

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

In triangle $ABC$, $P$ is a point on altitude $AD$. $Q,R$ are the feet of the perpendiculars from $P$ to $AB,AC$, and $QP,RP$ meet $BC$ at $S$ and $T$ respectively. the circumcircles of $BQS$ and $CRT$ meet $QR$ at $X,Y$. a) Prove $SX,TY, AD$ are concurrent at a point $Z$. b) Prove $Z$ is on $QR$ iff $Z=H$, where $H$ is the orthocenter of $ABC$. [i]Ray Li.[/i]

2001 Tuymaada Olympiad, 3

Tags: power of a point , circumcircle , radical axis , geometry

$ABCD$ is a convex quadrilateral; half-lines $DA$ and $CB$ meet at point $Q$; half-lines $BA$ and $CD$ meet at point $P$. It is known that $\angle AQB=\angle APD$. The bisector of angle $\angle AQB$ meets the sides $AB$ and $CD$ of the quadrilateral at points $X$ and $Y$, respectively; the bisector of angle $\angle APD$ meets the sides $AD$ and $BC$ at points $Z$ and $T$, respectively. The circumcircles of triangles $ZQT$ and $XPY$ meet at point $K$ inside the quadrilateral. Prove that $K$ lies on the diagonal $AC$. [i]Proposed by S. Berlov[/i]

1996 Canadian Open Math Challenge, 7

Tags: geometry , trigonometry , pythagorean theorem , power of a point

Triangle $ABC$ is right angled at $A$. The circle with center $A$ and radius $AB$ cuts $BC$ and $AC$ internally at $D$ and $E$ respectively. If $BD = 20$ and $DC = 16$, determine $AC^2$.

2001 All-Russian Olympiad, 3

Tags: geometry , trigonometry , circumcircle , ratio , homothety , power of a point , geometric transformation

Let the circle $ {\omega}_{1}$ be internally tangent to another circle $ {\omega}_{2}$ at $ N$.Take a point $ K$ on $ {\omega}_{1}$ and draw a tangent $ AB$ which intersects $ {\omega}_{2}$ at $ A$ and $ B$. Let $M$ be the midpoint of the arc $ AB$ which is on the opposite side of $ N$. Prove that, the circumradius of the $ \triangle KBM$ doesnt depend on the choice of $ K$.

2023 Bangladesh Mathematical Olympiad, P7

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

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 $BC$.

2011 USA Team Selection Test, 7

Tags: geometric transformation , geometry , incenter , radical axis , power of a point

Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$. Circle $\omega$, centered at $O$, passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$, respectively. Point $P$ lies on major arc $BAC$ of $\Omega$. Prove that lines $BD, CE, OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.

2012 France Team Selection Test, 3

Tags: geometry , parallelogram , circumcircle , perpendicular bisector , power of a point

Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

2010 Indonesia TST, 3

Tags: geometry , parallelogram , power of a point , radical axis

Let $ABCD$ be a convex quadrilateral with $AB$ is not parallel to $CD$. Circle $\omega_1$ with center $O_1$ passes through $A$ and $B$, and touches segment $CD$ at $P$. Circle $\omega_2$ with center $O_2$ passes through $C$ and $D$, and touches segment $AB$ at $Q$. Let $E$ and $F$ be the intersection of circles $\omega_1$ and $\omega_2$. Prove that $EF$ bisects segment $PQ$ if and only if $BC$ is parallel to $AD$.

2006 Romania Team Selection Test, 3

Tags: circumcircle , incenter , geometry , power of a point , radical axis

Let $\gamma$ be the incircle in the triangle $A_0A_1A_2$. For all $i\in\{0,1,2\}$ we make the following constructions (all indices are considered modulo 3): $\gamma_i$ is the circle tangent to $\gamma$ which passes through the points $A_{i+1}$ and $A_{i+2}$; $T_i$ is the point of tangency between $\gamma_i$ and $\gamma$; finally, the common tangent in $T_i$ of $\gamma_i$ and $\gamma$ intersects the line $A_{i+1}A_{i+2}$ in the point $P_i$. Prove that a) the points $P_0$, $P_1$ and $P_2$ are collinear; b) the lines $A_0T_0$, $A_1T_1$ and $A_2T_2$ are concurrent.