Switzerland 2004-22 (SMO) 47p

geometry problems from Swiss Mathematical Olympiads, Final Round (SMO)
with aops links in the names 

collected inside aops here

                                          2004 - 2022

2004 Swiss MO p1
Let $\Gamma$ be a circle and $P$ a point outside of $\Gamma$ . A tangent from $P$ to the circle intersects it in $A$. Another line through $P$ intersects $\Gamma$ at the points $B$ and $C$. The bisector of $\angle APB$ intersects $AB$ at $D$ and $AC$ at $E$. Prove that the triangle $ADE$ is isosceles.

2004 Swiss MO p9
Let $ABCD$ be a cyclic quadrilateral, so that $|AB| + |CD|  = |BC|$. Show that the intersection of the bisector of $\angle DAB$ and $\angle CDA$ lies on the side $BC$.

2005 Swiss MO p1
Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.

2005 Swiss MO p8
Let $ABC$ be an acute-angled triangle. $M ,N$ are any two points on the sides $AB , AC$ respectively. The circles with the diameters $BN$ and $CM$ intersect at points $P$ and $Q$. Show that the points $P, Q$ and the orthocenter of the triangle $ABC$ lie on a straight line.

2006 Swiss MO p2
Let $ABC$ be an equilateral triangle and let $D$ be an inner point of the side $BC$. A circle is tangent to $BC$ at $D$ and intersects the sides $AB$ and $AC$ in the inner points $M, N$ and $P, Q$ respectively. Prove that $|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|$.

2006 Swiss MO p5
A circle $k_1$ lies within a second circle $k_2$ and touches it at point $A$. A line through $A$ intersects $k_1$ again in $B$ and $k_2$ in $C$. The tangent to $k_1$ through $B$ intersects $k_2$ at points $D$ and $E$. The tangents at $k_1$ passing through $C$ intersects $k_1$ in points $F$ and $G$. Prove that $D, E, F$ and $G$ lie on a circle.

2006 Swiss MO p7
Let $ABCD$ be a cyclic quadrilateral with $\angle ABC = 60^o$ and $| BC | = | CD |$. Prove that $|CD| + |DA| = |AB|$

2007 Swiss MO p4
Let $ABC$ be an acute-angled triangle with $AB> AC$ and orthocenter $H$. Let $D$ the projection of $A$ on $BC$. Let $E$ be the reflection of $C$ wrt $D$. The lines $AE$ and $BH$ intersect at point $S$. Let $N$ be the center of $AE$ and let $M$ be the midpoint of $BH$. Prove that $MN$ is perpendicular to $DS$.

2007 Swiss MO p6
Three equal circles $k_1, k_2, k_3$ intersect non-tangentially at a point $P$. Let $A$ and $B$ be the centers of circles $k_1$ and $k_2$. Let $D$ and $C$ be the intersection of $k_3$ with $k_1$ and $k_2$ respectively, which is different from $P$. Show that $ABCD$ is a parallelogram.

2008 Swiss MO p1

Let $ABC$ be a triangle with $\angle BAC \ne 45^o$ and $\angle ABC \ne 135^o$. Let $P$ be the point on the line $AB$ with $\angle CPB = 45^o$. Let $O_1$ and $O_2$ be the centers of the circumcircles of the triangles $ACP$ and $BCP$ respectively. Show that the area of the square $CO_1P O_2$ is equal to the area of the triangle $ABC$.

2008 Swiss MO p5

Let $ABCD$ be a square with side length $1$. Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.

2008 Swiss MO p8

Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

2009 Swiss MO p5

Let $ABC$ be a triangle with $AB \ne AC$ and incenter $I$. The incircle touches $BC$ at $D$. Let $M$ be the midpoint of $BC$ . Show that the line $IM$ bisects segment $AD$ .

2009 Swiss MO p7

Points $A, M_1, M_2$ and $C$ are on a line in this order. Let $k_1$ the circle with center $M_1$ passing through $A$ and $k_2$ the circle with center $M_2$ passing through $C$. The two circles intersect at points $E$ and $F$. A common tangent of $k_1$ and $k_2$, touches $k_1$ at $B$ and $k_2$ at $D$. Show that the lines $AB, CD$ and $EF$ intersect at one point.

2010 Swiss MO p2

Let $\triangle{ABC}$ be a triangle with $AB\not=AC$. The incircle with centre $I$ touches $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Furthermore let $M$ the midpoint of $EF$ and $AD$ intersect the incircle at $P\not=D$. Show that $PMID$ ist cyclic.

2010 Swiss MO p9

Let $k$ and $k'$ two concentric circles centered at $O$, with $k'$ being larger than $k$. A line through $O$ intersects $k$ at $A$ and $k'$ at $B$ such that $O$ seperates $A$ and $B$. Another line through $O$ intersects $k$ at $E$ and $k'$ at $F$ such that $E$ separates $O$ and $F$. Show that the circumcircle of $\triangle{OAE}$ and the circles with diametres $AB$ and $EF$ have a common point.

2011 Swiss MO p2

Let $\triangle{ABC}$ be an acute-angled triangle and let $D$, $E$, $F$ be points on $BC$, $CA$, $AB$, respectively, such that $$\angle{AFE}=\angle{BFD}\mbox{,}\quad\angle{BDF}=\angle{CDE}\quad\mbox{and}\quad\angle{CED}=\angle{AEF}\mbox{.}$$ Prove that $D$, $E$ and $F$ are the feet of the perpendiculars through $A$, $B$ and $C$ on $BC$, $CA$ and $AB$, respectively.

2011 Swiss MO p5

Let $\triangle{ABC}$ be a triangle with circumcircle $\tau$. The tangentlines to $\tau$ through $A$ and $B$ intersect at $T$. The circle through $A$, $B$ and $T$ intersects $BC$ and $AC$ again at $D$ and $E$, respectively; $CT$ and $BE$ intersect at $F$. Suppose $D$ is the midpoint of $BC$. Calculate the ratio $BF:BE$.

2011 Swiss MO p8

Let $ABCD$ be a parallelogram and $H$ the Orthocentre of $\triangle{ABC}$. The line parallel to $AB$ through $H$ intersects $BC$ at $P$ and $AD$ at $Q$ while the line parallel to $BC$ through $H$ intersects $AB$ at $R$ and $CD$ at $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.

2012 Swiss MO p1

The circles $k_1$ and $k_2$ intersect at points $D$ and $P$. The common tangent of the two circles on the side of $D$ touches $k_1$ at $A$ and $k_2$ at $B$. The straight line $AD$ intersects $k_2$ for a second time at $C$. Let $M$ be the center of the segment $BC$. Show that $\angle DPM = \angle BDC$ .

2012 Swiss MO p6

Let $ABCD$ be a parallelogram with at least an angle not equal to $90^o$ and $k$ the circumcircle of the triangle $ABC$. Let E be the diametrically opposite point of $B$. Show that the circumcircle of the triangle $ADE$ and $k$ have the same radius.

2012 Swiss MO p10

Let $O$ be an inner point of an acute-angled triangle $ABC$. Let $A_1, B_1$ and $C_1$ be the projections of $O$ on the sides $BC, AC$ and $AB$ respectively . Let $P$ be the intersection of the perpendiculars on $B_1C_1$ and $A_1C_1$ from points$A$ and $B$ respectilvey. Let $H$ be the projection of $P$ on $AB$. Show that points $A_1, B_1, C_1$ and $H$ lie on a circle.

2013 Swiss MO p3

Let $ABCD$ be a cyclic quadrilateral with $\angle ADC = \angle DBA$. Furthermore, let $E$ be the projection of $A$ on $BD$. Show that $BC = DE - BE$ .

2013 Swiss MO p7
Let $O$ be the center of the circle of the triangle $ABC$ with $AB \ne AC$. Furthermore, let $S$ and $T$ be points on the rays $AB$ and $AC$, such that $\angle ASO = \angle ACO$ and $\angle ATO = \angle ABO$. Show that $ST$ bisects the segment $BC$.

2013 Swiss MO p10
Let $ABCD$ be a tangential quadrilateral with $BC> BA$. The point $P$ is on the segment $BC$, such that $BP = BA$ . Show that the bisector of $\angle BCD$, the perpendicular on line $BC$ through $P$ and the perpendicular on $BD$ through $A$, intersect at one point.

2014 Swiss MO p1
The points $A, B, C$ and $D$ lie in this order on the circle $k$. Let $t$ be the tangent at $k$ through $C$ and $s$ the reflection of $AB$ at $AC$. Let $G$ be the intersection of the straight line $AC$ and $BD$ and $H$ the intersection of the straight lines $s$ and $CD$. Show that $GH$ is parallel to $t$.

2014 Swiss MO p8
In the acute-angled triangle $ABC$, let $M$ be the midpoint of the atlitude $h_b$ through $B$ and $N$ be the midpoint of the height $h_c$ through $C$. Further let $P$ be the intersection of $AM$ and $h_c$ and $Q$ be the intersection of $AN$ and $h_b$. Show that $M, N, P$ and $Q$ lie on a circle.

2014 Swiss MO p10

Let $k$ be a circle with diameter $AB$. Let $C$ be a point on the straight line $AB$, so that $B$ between $A$ and $C$ lies. Let $T$ be a point on $k$ such that $CT$ is a tangent to $k$. Let $l$ be the parallel to $CT$ through $A$ and $D$ the intersection of $l$ and the perpendicular to $AB$ through $T$. Show that the line $DB$ bisects segment $CT$.

2015 Swiss MO p1
Let ABC be an acute-angled triangle with $AB \ne  BC$ and radius $k$. Let $P$ and $Q$ be the points of intersection of $k$ with the internal bisector and the external bisector of $\angle CBA$ respectively. Let $D$ be the intersection of $AC$ and $PQ$. Find the ratio $AD: DC$.

2015 Swiss MO p4
Given a circle $k$ and two points $A$ and $B$ outside the circle. Specify how to can construct a circle with a compass and ruler, so that $A$ and $B$ lie on that circle and that circle is tangent to $k$.

2015 Swiss MO p8
Let $ABCD$ be a trapezoid, where $AB$ and $CD$ are parallel. Let $P$ be a point on the side $BC$. Show that the parallels to $AP$ and $PD$ intersect through $C$ and $B$ to $DA$, respectively.

2016 Swiss MO p1
Let $ABC$ be a triangle with $\angle BAC = 60^o$. Let $E$ be the point on the side $BC$ , such that $2 \angle BAE = \angle ACB$ . Let $D$ be the second intersection of $AB$ and the circumcircle of the triangle $AEC$ and $P$ be the second intersection of $CD$ and the circumcircle of the triangle $DBE$. Calculate the angle $\angle BAP$.

2016 Swiss MO p5
Let $ABC$ be a right triangle with $\angle ACB = 90^o$ and M the center of $AB$. Let $G$ br any point on the line $MC$ and $P$ a point on the line $AG$, such that $\angle CPA = \angle BAC$ . Further let $Q$ be a point on the straight line $BG$, such that $\angle BQC = \angle CBA$ . Show that the circles of the triangles $AQG$ and $BPG$ intersect on the segment $AB$.

2016 Swiss MO p8  (2015 ISL G1)
Let $ABC$ be an acute-angled triangle with height intersection $H$. Let $G$ be the intersection of parallel of $AB$ through $H$ with the parallel of $AH$ through $B$. Let $I$ be the point on the line $GH$, so that $AC$ bisects segment $HI$. Let $J$ be the second intersection of $AC$ and the circumcircle of the triangle $CGI$. Show that $IJ = AH$

2017 Swiss MO p1

Let $A$ and $B$ be points on the circle $k$ with center $O$, so that $AB> AO$. Let $C$ be the intersection of the bisectors of $\angle OAB$ and $k$,  different from $A$. Let $D$ be the intersection of the straight line $AB$ with the circumcircle of the triangle $OBC$, different from $B$. Show that $AD = AO$ .

2017 Swiss MO p5

Let $ABC$ be a triangle with $AC> AB$. Let $P$ be the intersection of $BC$ and the tangent through $A$ around the triangle $ABC$. Let $Q$ be the point on the straight line $AC$, so that $AQ = AB$ and $A$ is between $C$ and $Q$. Let $X$ and $Y$ be the center of $BQ$ and $AP$. Let $R$ be the point on $AP$ so that $AR = BP$ and $R$ is between $A$ and $P$. Show that $BR = 2XY$.

2017 Swiss MO p8

Let $ABC$ be an isosceles triangle with vertex $A$ and $AB> BC$. Let $k$ be the circle with center $A$  passsing through $B$ and $C$. Let $H$ be the second intersection of $k$ with the altitude of the triangle $ABC$ through $B$. Further let $G$ be the second intersection of $k$ with the median through $B$ in triangle $ABC$. Let $X$ be the intersection of the lines $AC$ and $GH$. Show that $C$ is the midpoint of $AX$.

2018 Swiss MO p4

Let $D$ be a point inside an acute triangle $ABC$, such that $\angle BAD = \angle DBC$ and $\angle DAC = \angle BCD$. Let $P$ be a point on the circumcircle of the triangle $ADB$. Suppose $P$ are itself outside the triangle $ABC$. A line through $P$ intersects the ray $BA$ in $X$ and ray $CA$ in $Y$, so that $\angle XPB = \angle PDB$. Show that $BY$ and $CX$ intersect on $AD$.

2018 Swiss MO p6

Let $k$ be the incircle of the triangle $ABC$ with the center of the incircle $I$. The circle $k$ touches the sides $BC, CA$ and $AB$ in points $D, E$ and $F$. Let $G$ be the intersection of the straight line $AI$ and the circle $k$, which lies between $A$ and $I$. Assume $BE$ and $FG$ are parallel. Show that $BD = EF$.

2019 Swiss MO p1

Let $A$ be a point and let k be a circle through $A$. Let $B$ and $C$ be two more points on $k$. Let $X$ be the intersection of the bisector of  $\angle ABC$ with $k$. Let $Y$ be the reflection of $A$ wrt point $X$,  and $D$ the intersection of the straight line $YC$ with $k$. Prove that point $D$ is independent of the choice of $B$ and $C$ on the circle $k$. 

2019 Swiss MO p7

Let $ABC$ be a triangle with $\angle  CAB = 2 \angle ABC$. Assume that a point $D$ is inside the triangle $ABC$ exists such that $AD = BD$ and $CD = AC$. Show that $\angle  ACB = 3 \angle DCB$.

2020 Swiss MO p2

Let $ABC$ be an acute triangle. Let $M_A, M_B$ and $M_C$ be the midpoints of sides $BC,CA$, respectively $AB$. Let $M'_A , M'_B$ and $M'_C$ be the the midpoints of the arcs  $BC, CA$ and $AB$ respectively of the circumscriberd circle of triangle  $ABC$. Let $P_A$ be the intersection of the straight line $M_BM_C$ and the perpendicular to $M'_BM'_C$ through $A$.  Define $P_B$ and $P_C$ similarly.  Show that the straight line $M_AP_A, M_BP_B$ and $M_CP_C$ intersect at one point.

2020 Swiss MO p7

Let $ABCD$ be an isosceles trapezoid with bases $AD> BC$. Let $X$ be the intersection of the bisectors of $\angle BAC$ and $BC$. Let $E$ be the intersection of$DB$ with the parallel to the bisector of $\angle CBD$ through $X$ and let $F$ be the intersection of $DC$ with the parallel to the bisector of $\angle DCB$ through $X$. Show that quadrilateral $AEFD$ is cyclic.

2021 Swiss MO p2

Let $\triangle ABC$ be an acute triangle with $AB =AC$ and let $D$ be a point on the side $BC$. The circle with centre $D$ passing through $C$ intersects $\odot(ABD)$ at points $P$ and $Q$, where $Q$ is the point closer to $B$. The line $BQ$ intersects $AD$ and $AC$ at points $X$ and $Y$ respectively. Prove that quadrilateral $PDXY$ is cyclic.

2021 Swiss MO p8

Let $\triangle ABC$ be a triangle with $AB =AC$ and $\angle BAC = 20^{\circ}$. Let $D$ be point on the side $AB$ such that $\angle BCD = 70^{\circ}$. Let $E$ be point on the side $AC$ such that $\angle CBE = 60^{\circ}$. Determine the value of angle $\angle CDE$.

2022 Swiss MO p1

Let $k$ be a circle with centre $M$ and let $AB$ be a diameter of $k$. Furthermore, let $C$ be a point on $k$ such that $AC = AM$. Let $D$ be the point on the line $AC$ such that $CD = AB$ and $C$ lies between $A$ and $D$. Let $E$ be the second intersection of the circumcircle of $BCD$ with line $AB$ and $F$ be the intersection of the lines $ED$ and $BC$. The line $AF$ cuts the segment $BD$ in $X$. Determine the ratio $BX/XD$.

2022 Swiss MO p8

Let $ABC$ be a triangle and let $P$ be a point in the interior of the side $BC$. Let $I_1$ and $I_2$ be the incenters of the triangles $AP B$ and $AP C$, respectively. Let $X$ be the closest point to $A$ on the line $AP$ such that $XI_1$ is perpendicular to $XI_2$. Prove that the distance $AX$ is independent of the choice of $P$.

source: https://imosuisse.ch/