Olympiad problems

1984 AMC 12/AHSME, 27

In $\triangle ABC$, $D$ is on $AC$ and $F$ is on $BC$. Also, $AB \perp AC, AF \perp BC$, and $BD = DC = FC = 1$. Find $AC$. A. $\sqrt{2}$ B. $\sqrt{3}$ C. $\sqrt[3] {2}$ D. $\sqrt[3] {3}$ E. $\sqrt[4] {3}$

2000 India Regional Mathematical Olympiad, 5

Tags: geometry , angle bisector , perpendicular bisector , circumcircle

The internal bisector of angle $A$ in a triangle $ABC$ with $AC > AB$ meets the circumcircle $\Gamma$ of the triangle in $D$. Join$D$ to the center $O$ of the circle $\Gamma$ and suppose that $DO$ meets $AC$ in $E$, possibly when extended. Given that $BE$ is perpendicular to $AD$, show that $AO$ is parallel to $BD$.

2014 India National Olympiad, 5

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

In a acute-angled triangle $ABC$, a point $D$ lies on the segment $BC$. Let $O_1,O_2$ denote the circumcentres of triangles $ABD$ and $ACD$ respectively. Prove that the line joining the circumcentre of triangle $ABC$ and the orthocentre of triangle $O_1O_2D$ is parallel to $BC$.

2019 Austrian Junior Regional Competition, 2

Tags: geometry , perpendicular bisector

A square $ABCD$ is given. Over the side $BC$ draw an equilateral triangle $BCS$ on the outside. The midpoint of the segment $AS$ is $N$ and the midpoint of the side $CD$ is $H$. Prove that $\angle NHC = 60^o$. . (Karl Czakler)

2023 Stanford Mathematics Tournament, 3

Tags: geometry , perpendicular bisector

Triangle $\vartriangle ABC$ has side lengths $AB = 5$, $BC = 8$, and $CA = 7$. Let the perpendicular bisector of $BC$ intersect the circumcircle of $\vartriangle ABC$ at point $D$ on minor arc $BC$ and point $E$ on minor arc $AC$, and $AC$ at point $F$. The line parallel to $BC$ passing through $F$ intersects $AD$ at point $G$ and $CE$ at point $H$. Compute $\frac{[CHF]}{[DGF]}$ . (Given a triangle $\vartriangle ABC$, $[ABC]$ denotes its area.)

2003 Estonia Team Selection Test, 6

Tags: orthocenter , ratio , projection , circumcenter , perpendicular bisector , geometry

Let $ABC$ be an acute-angled triangle, $O$ its circumcenter and $H$ its orthocenter. The orthogonal projection of the vertex $A$ to the line $BC$ lies on the perpendicular bisector of the segment $AC$. Compute $\frac{CH}{BO}$ . (J. Willemson)

2001 Turkey MO (2nd round), 2

Tags: perpendicular bisector , trigonometry , circumcircle , geometric transformation , geometry

Two nonperpendicular lines throught the point $A$ and a point $F$ on one of these lines different from $A$ are given. Let $P_{G}$ be the intersection point of tangent lines at $G$ and $F$ to the circle through the point $A$, $F$ and $G$ where $G$ is a point on the given line different from the line $FA$. What is the locus of $P_{G}$ as $G$ varies.

2009 Ukraine National Mathematical Olympiad, 4

Tags: circumcircle , perpendicular bisector , trapezoid , geometry

In the trapezoid $ABCD$ we know that $CD \perp BC, $ and $CD \perp AD .$ Circle $w$ with diameter $AB$ intersects $AD$ in points $A$ and $P,$ tangent from $P$ to $w$ intersects $CD$ at $M.$ The second tangent from $M$ to $w$ touches $w$ at $Q.$ Prove that midpoint of $CD$ lies on $BQ.$

1998 IMO, 1

Tags: circumcircle , quadrilateral , perpendicular bisector , geometry

A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.

2009 China Team Selection Test, 2

Tags: perpendicular bisector , angle bisector , law of sines , geometry

In convex quadrilateral $ ABCD$, $ CB,DA$ are external angle bisectors of $ \angle DCA,\angle CDB$, respectively. Points $ E,F$ lie on the rays $ AC,BD$ respectively such that $ CEFD$ is cyclic quadrilateral. Point $ P$ lie in the plane of quadrilateral $ ABCD$ such that $ DA,CB$ are external angle bisectors of $ \angle PDE,\angle PCF$ respectively. $ AD$ intersects $ BC$ at $ Q.$ Prove that $ P$ lies on $ AB$ if and only if $ Q$ lies on segment $ EF$.

2014 China Team Selection Test, 1

Tags: cyclic quadrilateral , geometry , perpendicular bisector

$ABCD$ is a cyclic quadrilateral, with diagonals $AC,BD$ perpendicular to each other. Let point $F$ be on side $BC$, the parallel line $EF$ to $AC$ intersect $AB$ at point $E$, line $FG$ parallel to $BD$ intersect $CD$ at $G$. Let the projection of $E$ onto $CD$ be $P$, projection of $F$ onto $DA$ be $Q$, projection of $G$ onto $AB$ be $R$. Prove that $QF$ bisects $\angle PQR$.

2007 India IMO Training Camp, 1

Tags: perpendicular bisector , euler , trigonometry , geometry

Show that in a non-equilateral triangle, the following statements are equivalent: $(a)$ The angles of the triangle are in arithmetic progression. $(b)$ The common tangent to the Nine-point circle and the Incircle is parallel to the Euler Line.

2003 Baltic Way, 13

Tags: rectangle , perpendicular bisector , circumcircle , geometry

In a rectangle $ABCD$ be a rectangle and $BC = 2AB$, let $E$ be the midpoint of $BC$ and $P$ an arbitrary inner point of $AD$. Let $F$ and $G$ be the feet of perpendiculars drawn correspondingly from $A$ to $BP$ and from $D$ to $CP$. Prove that the points $E,F,P,G$ are concyclic.

2002 Pan African, 2

Tags: perpendicular bisector , circumcircle , geometry

$\triangle{AOB}$ is a right triangle with $\angle{AOB}=90^{o}$. $C$ and $D$ are moving on $AO$ and $BO$ respectively such that $AC=BD$. Show that there is a fixed point $P$ through which the perpendicular bisector of $CD$ always passes.

2009 Indonesia MO, 4

Tags: perpendicular bisector , circumcircle , angle bisector , trigonometry , geometry

Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$. (a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$. (b) Show that $ A_1KML$ is a cyclic quadrilateral.

2017 Puerto Rico Team Selection Test, 2

Tags: geometry , perpendicular bisector , altitude

For an acute triangle $ ABC $ let $ H $ be the point of intersection of the altitudes $ AA_1 $, $ BB_1 $, $ CC_1 $. Let $ M $ and $ N $ be the midpoints of the $ BC $ and $ AH $ segments, respectively. Show that $ MN $ is the perpendicular bisector of segment $ B_1C_1 $.

Geometry Mathley 2011-12, 15.3

Tags: perpendicular bisector , circumcircle , geometry , symmedian

Triangle $ABC$ has circumcircle $(O,R)$, and orthocenter $H$. The symmedians through $A,B,C$ meet the perpendicular bisectors of $BC,CA,AB$ at $D,E, F$ respectively. Let $M,N, P$ be the perpendicular projections of H on the line $OD,OE,OF.$ Prove that $$\frac{OH^2}{R^2} =\frac{\overline{OM}}{\overline{OD}}+\frac{\overline{ON}}{\overline{OE}} +\frac{\overline{OP}}{\overline{OF}}$$ Đỗ Thanh Sơn

2011 China Girls Math Olympiad, 2

Tags: perpendicular bisector , rectangle , circumcircle , geometry , trapezoid , projective geometry , geometric transformation

The diagonals $AC,BD$ of the quadrilateral $ABCD$ intersect at $E$. Let $M,N$ be the midpoints of $AB,CD$ respectively. Let the perpendicular bisectors of the segments $AB,CD$ meet at $F$. Suppose that $EF$ meets $BC,AD$ at $P,Q$ respectively. If $MF\cdot CD=NF\cdot AB$ and $DQ\cdot BP=AQ\cdot CP$, prove that $PQ\perp BC$.

2023 Germany Team Selection Test, 1

Tags: perpendicular bisector , circumcircle , geometry

Let $ABC$ be an acute triangle and let $\omega$ be its circumcircle. Let the tangents to $\omega$ through $B,C$ meet each other at point $P$. Prove that the perpendicular bisector of $AB$ and the parallel to $AB$ through $P$ meet at line $AC$.

2018 Thailand TST, 3

Tags: perpendicular bisector , geometry

Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.

2007 Korea Junior Math Olympiad, 4

Tags: geometry , perpendicular bisector , collinear , concyclic

Let $P$ be a point inside $\triangle ABC$. Let the perpendicular bisectors of $PA,PB,PC$ be $\ell_1,\ell_2,\ell_3$. Let $D =\ell_1 \cap \ell_2$ , $E=\ell_2 \cap \ell_3$, $F=\ell_3 \cap \ell_1$. If $A,B,C,D,E,F$ lie on a circle, prove that $C, P,D$ are collinear.

2023 Yasinsky Geometry Olympiad, 5

Tags: perpendicular bisector , geometry

Let $I$ be the center of the circle inscribed in triangle $ABC$. The inscribed circle is tangent to side $BC$ at point $K$. Let $X$ and $Y$ be points on segments $BI$ and $CI$ respectively, such that $KX \perp AB $ and $KY\perp AC$. The circumscribed circle around triangle $XYK$ intersects line $BC$ at point $D$. Prove that $AD \perp BC$. (Matthew Kurskyi)

2013 NIMO Summer Contest, 7

Tags: perpendicular bisector , geometry

Circle $\omega_1$ and $\omega_2$ have centers $(0,6)$ and $(20,0)$, respectively. Both circles have radius $30$, and intersect at two points $X$ and $Y$. The line through $X$ and $Y$ can be written in the form $y = mx+b$. Compute $100m+b$. [i]Proposed by Evan Chen[/i]

2011 USA TSTST, 2

Tags: perpendicular bisector

Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. Line $\ell$ is tangent to $\omega_1$ at $P$ and to $\omega_2$ at $Q$ so that $A$ is closer to $\ell$ than $B$. Let $X$ and $Y$ be points on major arcs $\overarc{PA}$ (on $\omega_1$) and $AQ$ (on $\omega_2$), respectively, such that $AX/PX = AY/QY = c$. Extend segments $PA$ and $QA$ through $A$ to $R$ and $S$, respectively, such that $AR = AS = c\cdot PQ$. Given that the circumcenter of triangle $ARS$ lies on line $XY$, prove that $\angle XPA = \angle AQY$.

2007 China Team Selection Test, 1

Tags: perpendicular bisector , parallelogram , circumcircle , geometry

Let convex quadrilateral $ ABCD$ be inscribed in a circle centers at $ O.$ The opposite sides $ BA,CD$ meet at $ H$, the diagonals $ AC,BD$ meet at $ G.$ Let $ O_{1},O_{2}$ be the circumcenters of triangles $ AGD,BGC.$ $ O_{1}O_{2}$ intersects $ OG$ at $ N.$ The line $ HG$ cuts the circumcircles of triangles $ AGD,BGC$ at $ P,Q$, respectively. Denote by $ M$ the midpoint of $ PQ.$ Prove that $ NO \equal{} NM.$