Switzerald TST 1997 - 2022 52p

geometry problems from Swiss IMO Team Selection Tests (TST)
with aops links in the names 

(only those not in IMO Shortlist)

collected inside aops here

                                        1997 - 2022

1997 Switzerald TST p2

Let $ABCD$ be a convex quadrilateral. Find the necessary and sufficient condition for the existence of point $P$ inside the quadrilateral such that the triangles $ABP,BCP,CDP,DAP$ have the same area.

1998 Switzerald TST p5

Points $A$ and $B$ are chosen on a circle $k$. Let AP and $BQ$ be segments of the same length tangent to $k$, drawn on different sides of line $AB$. Prove that the line $AB$ bisects the segment $PQ$.

1998 Switzerald TST p8
Let $\vartriangle ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP,BP,CP$ meet the sides $BC,CA,AB$ in the points $X,Y,Z$ respectively. Prove that $XY \cdot YZ\cdot ZX \ge XB\cdot YC\cdot ZA$.

1999 Switzerald TST p1
Two circles intersect at points $M$ and $N$. Let $A$ be a point on the first circle, distinct from $M,N$. The lines $AM$ and $AN$ meet the second circle again at $B$ and $C$, respectively. Prove that the tangent to the first circle at $A$ is parallel to $BC$.

1999 Switzerald TST p5

In a rectangle $ABCD, M$ and $N$ are the midpoints of $AD$ and $BC$ respectively and $P$ is a point on line $CD$. The line $PM$ meets $AC$ at $Q$. Prove that MN bisects the angle $\angle QNP$.

2000 Switzerald TST p1
A convex quadrilateral $ABCD$ is inscribed in a circle. Show that the line connecting the midpoints of the arcs $AB$ and $CD$ and the line connecting the midpoints of the arcs $BC$ and $DA$ are perpendicular.

2000 Switzerald TST p9
Two given circles $k_1$ and $k_2$ intersect at points $P$ and $Q$. Construct a segment $AB$ through $P$ with the endpoints at $k_1$ and $k_2$ for which $AP \cdot PB$ is maximal.

2000 Switzerald TST p13
The incircle of a triangle $ABC$ touches the sides $AB,BC,CA$ at points $D,E,F$ respectively. Let $P$ be an internal point of triangle $ABC$ such that the incircle of triangle $ABP$ touches $AB$ at $D$ and the sides $AP$ and $BP$ at $Q$ and $R$. Prove that the points $E,F,R,Q$ lie on a circle.

2001 Switzerald TST p3
In a convex pentagon every diagonal is parallel to one side. Show that the ratios between the lengths of diagonals and the sides parallel to them are equal and find their value.

2001 Switzerald TST p7
Let $ABC$ be an acute-angled triangle with circumcenter $O$. The circle $S$ through $A,B$, and $O$ intersects $AC$ and $BC$ again at points $P$ and $Q$ respectively. Prove that $CO \perp PQ$.

2002 Switzerald TST p2
A point$O$ inside a parallelogram $ABCD$ satisfies $\angle AOB + \angle COD = \pi$. Prove that $\angle CBO = \angle CDO$.

2003 Switzerald TST p2

In an acute-angled triangle $ABC, E$ and $F$ are the feet of the altitudes from $B$ and $C$, and $G$ and $H$ are the projections of $B$ and $C$ on $EF$, respectively. Prove that $HE = FG$.

2003 Switzerald TST p8
Let $A_1A_2A_3$ be a triangle and $\omega_1$ be a circle passing through $A_1$ and $A_2$. Suppose that there are circles $\omega_2,...,\omega_7$ such that:
(a) $\omega_k$ passes through $A_k$ and $A_{k+1}$ for $k = 2,3,...,7$, where $A_i = A_{i+3}$,
(b) $\omega_k$ and $\omega_{k+1}$ are externally tangent for $k = 1,2,...,6$.
Prove that $\omega_1 = \omega_7$.

2004 Switzerald TST p10
In an acute-angled triangle $ABC$ the altitudes $AU,BV,CW$ intersect at $H$.
Points $X,Y,Z$, different from $H$, are taken on segments $AU,BV$, and $CW$, respectively.
(a) Prove that if $X,Y,Z$ and $H$ lie on a circle, then the sum of the areas of triangles $ABZ, AYC, XBC$ equals the area of $ABC$.
(b) Prove the converse of (a).

2005 Switzerald TST p4 (Slovenia TST 1997)
Circles $K_1$ and $K_2$ are externally tangent to each other at $A$ and are internally tangent to a circle $K$ at $A_1$ and $A_2$ respectively. The common tangent to $K_1$ and $K_2$ at $A$ meets $K$ at point $P$. Line $PA_1$ meets $K_1$ again at $B_1$ and $PA_2$ meets $K_2$ again at $B_2$. Show that $B_1B_2$ is a common tangent of $K_1$ and $K_2$.

2005 Switzerald TST p10 (German TST 2004)

Let $ABC$ be a triangle and $H$ its orthocenter. Take $M,N$ on $CA$ such $MN= AC$. $D,E$ is projection of $M,N$ on $BC,AB$. Prove that B,D,H,E is cyclic

2006 Switzerald TST p1
In the triangle $A,B,C$, let $D$ be the middle of $BC$ and $E$ the projection of $C$ on $AD$. Suppose $\angle ACE = \angle ABC$. Show that the triangle $ABC$ is isosceles or rectangle.

2006 Switzerald TST p4
Let $D$ be inside $\triangle ABC$ and $E$ on $AD$ different to $D$. Let $\omega_1$ and $\omega_2$ be the circumscribed circles of $\triangle BDE$ and $\triangle CDE$ respectively. $\omega_1$ and $\omega_2$ intersect $BC$ in the interior points $F$ and $G$ respectively. Let $X$ be the intersection between $DG$ and $AB$ and $Y$ the intersection between $DF$ and $AC$. Show that $XY$ is $\|$ to $BC$.

2006 Switzerald TST p9
Let $\triangle ABC$ be an acute-angled triangle with $AB \not= AC$. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ be the midpoint of the side $BC$. Let $D$ be a point on the side $AB$ and $E$ a point on the side $AC$ such that $AE=AD$ and the points $D$, $H$, $E$ are on the same line. Prove that the line $HM$ is perpendicular to the common chord of the circumscribed circles of triangle $\triangle ABC$ and triangle $\triangle ADE$.

2007 were all shortlist problems

2008 Switzerald TST p3

Let $ABC$ be a triangle with $\angle  ABC \ne \angle  BCA$. The inscribed circle $k$ of the triangle $ABC$ is tangent to the sides $BC, CA,AB$ at points $D, E , F$ respectively. The segment $AD$ cut $k$ for second time at $P$. Let $Q$ be the point of intersection of $EF$ with the line perpendicular to $AD$ passing through $P$. Let $X,Y$ be the points of intersection of $AQ$ with $DE,DF$ respectively. Prove that $A$ is the midpoint  of the segment $XY$.

2008 Switzerald TST p4

Two circles $k_1$ and $k_2$ intersect at $A$ and $B$. Let $r$ be a straight line passing through $B$, intersecting $k_1$ at $C$ and $k_2$ at $D$, such that $B$ is between $C$ and $D$. Let $s$ be the line parallel to $AD$ which is tangent to $k_1$ at $E$ and which is at a minimum distance from $AD$. The line $AE$ cuts $k_2$ at $F$. Let $t$ be the straight line tangent to $k_2$ passing through $F$. Prove that: 

(a) The line $t$ is parallel to $AC$.

(b) The lines $r, s$ and $t$ intersect at a point.

2008 Switzerald TST p8

Let $ABC$ be a triangle and $D$ a point on the segment $BC$. Let $X$ be a point inside the segment $BD$ and let $Y$ be the point of intersection of $AX$ with the circumscribed circle of $ABC$. Let $P$ be the second point of intersection of the circumscribed circles of $ABC$ and $DXY$. Show that $P$ is independent of the choice of $X$.

2009 Switzerald TST p1

Let $GERMANYISHOT$ be a regular dodecagon and let $P$ be the point of intersection of $GN$ and $MI$. Prove that:

(a) the circumscribed circle of the triangle $GIP$ has the same size as the circumscribed circle of $GERMANYISHOT$.

(b) the $PA$ segment has the same length as the side of $GERMANYISHOT$ .

2010 Switzerald TST p2

Let $AB$ be a diameter of the circle $k$. Let $t$ be the tangent to $k$ at point $B$ and $C, D$ be  two points on $t$, so that $B$ lies between $C$ and $D$. The lines $AC$ and $AD$ cut $k$ again at points $E$ and $F$ respectively. The lines $DE$ and $CF$ cut again $k$ at points $G$ and $H$ respectively . Prove that the segments $AG$ and $AH$ have the same length.

2010 Switzerald TST p4

The points $X, Y, Z$ lie in this order on a line with $|XY|  \ne |Y Z|$. Let $k_1 ,k_2$ be the circles with diameter $XY ,Y Z$ respectively. Points $A_1$ and $B_1 ,  A_2$ and $B_2$ are on $k_1$ and $k_2$, respectively such that $\angle  A_1Y A_2 = \angle  B_1Y B_2 = 90^o$ . Show that the two lines $A_1A_2$ and $B_1B_2$ intersect on $XY$

2010 Switzerald TST p9

Let $ABC$ be an acute-angled triangle with orthocenter  $H$. A line passing through $H$ intersects $AB , AC$ at points $D,E$ respectively  so that $|AD|=|AE|$. The bisector of $\angle BAC$ intersects the circumcircle of the triangle $ADE$ at point $K \ne A$. Show that $HK$ bisects segment $BC$.

2011 Switzerald TST p2
Line $g$ intersects circle $k$ at points $A$ and $B$. The perpendicular bisector of segment $AB$ intersects $k$ at points $C$ and $D$. Let $P$ be a point of $g$ which lies outside the circle $k$. The parallels to $CA$ and $CP$ passing through $P$ intersect $CB$ and $CA$ at points $X$ and $Y$, respectively. Prove that $XY$ and $DP$ are perpendicular.

2011 Switzerald TST p9
In a triangle $ABC$ with $AB \ne AC$, let $D$ be the projection of $A$ on $BC$. Also let $E$ and $F$ be the midpoint of the segments $AD$ and $BC$ respectively and let $G$ be the projection of $B$ on $AF$. Show that the line $EF$ is the tangent at point $F$ to the circumscribed circle of the triangle $GFC$.

2011 Switzerald TST p10
Let $ABCD$ be a square and $M$ a point inside the segment$BC$. The angle bisector of $\angle BAM$ intersects the segment $BC$ at point $E$. In addition, the angle bisector of $\angle MAD$ intersects the line $CD$ at point $F$. Show that $AM$ and $EF$ are perpendicular.

2012 Switzerald TST p3
Let $ABCD$ be an cyclic quadrilateral and $k$ its circumscribed circle. Let $S$ be the point of intersection from $AB$ and $CD$ and $T$ the point of intersection of the tangents to $k$ at points $A$ and $C$. Show that $ADTS$ is a cyclic quadrilateral if and only if $BD$ bisects  the segment $AC$.

2012 Switzerald TST p4
Let $ABC$ be a triangle with $\angle BAC = 60^o$. Let $E$ be a point on the line $AB$ such that $B$ is between $A$ and $E$ and such that $BE = BC$. Similarly, let $F$ be a point on $AC$ such that $C$ is between $A$ and $F$ and such that $CF = BC$. Let $K$ be the point of intersection of the circumscribed circle of $ACE$ with the line $EF$. Show that $K$ lies on the bisector of the angle $\angle  BAC$.

2012 Switzerald TST p11
Let $ABC$ be a triangle. Let $I$ be the center of the inscribed circle and $AD$ the diameter of the circle circumscribed around $ABC$. Let $E$ and $F$ be points on the rays $BA$ and $CA$ such that $BE = CF =\frac{AB + BC + CA}{2}$. Show that the lines $EF$ and $DI$ are perpendicular.

2013 Switzerald TST p1
The incircle of the triangle $ABC$ touches the sides $AB, BC$ and $CA$ at the points $X, Y$ and $Z$ respectively. Let $I_1, I_2$ and $I_3$ be the incenters of the triangles $AXZ, BYX$ and $CZY$. Find all triples $(\alpha , \beta, \gamma) =(\angle BAC, \angle CBA, \angle ACB)$ they can take, so that the triangle $ABC$ is similar a triangle with vertices  $I_1, I_2$ and $I_3$ .

2013 Switzerald TST p11

Let $ABC$ be a triangle with $\angle ACB \ge 90^o$ and $k$ the circle with diameter $AB$ and center $O$. The incircle of $ABC$ touches $AC ,BC$ at $M,N$ respectively. $MN$ intersects $k$ at points $X$ and $Y$. Show that $\angle XOY = \angle ACB$.

2014 Switzerald TST p3
Let $4$ points in the plane be arranged so that the $4$ triangles they form have all have the same radius of the inscribed circle. Show that the $4$ triangles are equal.

2014 Switzerald TST p5
Let $ABC$ be a triangle in which $\alpha = \angle  BAC$ is the smallest angle (strictly).
Let $P$ be a point on the side $BC$ and $D$ a point on the line $AB$ such that $B$ lies between $A$ and $D$ and $\angle BPD =\alpha$ . Likewise, let $E$ be a point on the line $AC$ such that $C$ lies between $A$ and $E$ and $\angle  EPC =\alpha$ . Show that the lines $AP, BE$ and $CD$ are concurrent if and only if $AP$ is perpendicular to $BC$.

2014 Switzerald TST p7
Two circles  $\omega_1$ and $\omega_2$  are tangent at a point $A$ and lie at inside a circle $\Omega$. In addition, $\omega_ 1$ is tangent to $\Omega$  at point $B$ and $\omega_2$ is tangent to $\Omega$ at  point $C$. Line $AC$ cuts $\omega_1$ for second time at  point $D$. Show that the triangle $DBC$ is right if $A, B$ and $C$ are not collinear.


2015 Switzerald TST p3
Let $ABC$ be a triangle with $AB> AC$. Let $D$ be a point on $AB$ such that $DB = DC$ and $M$ the middle of $AC$. The parallel to $BC$ passing through $D$ intersects the line $BM$ in $K$. Show that $\angle KCD = \angle DAC$.

2015 Switzerald TST p10
Let $ABCD$ be a parallelogram. Suppose that there exists a point $P$ in the interior of the parallelogram which is on the perpendicular bisector of $AB$ and such that $\angle PBA = \angle ADP$. Show that $\angle CPD = 2 \angle BAP$ 

2016 Switzerald TST p8
Let $ABC$ be a triangle with $AB  \neq AC$ and let $M$ be the middle of $BC$. The bisector of $\angle BAC$ intersects the line $BC$ in $Q$. Let $H$ be the foot of $A$ on $BC$. The perpendicular to $AQ$ passing through $A$ intersects the line $BC$ in $S$. Show that $MH \times QS=AB \times AC$.

2016 Switzerald TST p10
Let $ABC$ be a non-rectangle triangle with $M$ the middle of $BC$. Let $D$ be a point on the line $AB$ such that $CA=CD$ and let $E$ be a point on the line $BC$ such that $EB=ED$. The parallel to $ED$ passing through $A$ intersects the line $MD$ at the point $I$ and the line $AM$ intersects the line $ED$ at the point $J$. Show that the points $C, I$ and $J$ are aligned.

2017 Switzerald TST p4
Let $k$ be a circle and $AB$ a chord of $k$ such that the center of $k$ is not found on $AB$. Let $C$ be point on $k$ different from $A$ and $B$. For each choice of $C$, let $P_C$ and $Q_C$ be the projections of $A$ on $BC$ and $B$ on $AC$, respectively. Let $O_C$ be the center of the circle circumscribed around the triangle $P_CQ_CC$. Prove that there is a circle $\omega$ as $O_C$ lies on $\omega$ for each choice of $C$.

2018 Switzerald TST p6
Let $A, B, C$ and $D$ be four points on a circle, placed in this order. Suppose there is a point $K$ on the segment $AB$ such that $BD$ bisects segment $KC$ and $AC$ bisects segment $KD$. Determine the minimum value that  $\frac{AB}{CD}$ can reach.

2018 Switzerald TST p10
Let $ABC$ be a triangle, $M$ the center of the segment $BC$ and $D$ a point on the straight line $AB$ so that $B$ is between $A$ and $D$. Let $E$ be a point on the other side of the straight $CD$ as $B$, so that $\angle EDC = \angle ACB$ and $\angle  DCE = \angle  BAC$. Let $F$ be the intersection of $CE$ with the parallel to $DE$ passing through $A$ and let $Z$ be the intersection of $AE$ with $DF$. Show that the lines $AC, BF$ and $MZ$ are concurrent.

2019 Switzerald TST p1 (2017 Japan MO)

Let $ABC$ be a triangle and $D, E, F$ be the foots of altitudes drawn from $A,B,C$ respectively. Let $H$ be the orthocenter of $ABC$. Lines $EF$ and $AD$ intersect at $G$. Let $K$ the point on circumcircle of $ABC$ such that $AK$ is a diameter of this circle. $AK$ cuts $BC$ in $M$. Prove that $GM$ and $HK$ are parallel.

2019 Switzerald TST p9 (2018 Polish MO)
Let $ABC$ be an acute triangle with $AB<AC$. $E,F$ are foots of the altitudes drawn from $B,C$ respectively. Let $M$ be the midpoint of segment $BC$. The tangent at $A$ to the circumcircle of $ABC$ cuts $BC$ in $P$ and $EF$ cuts the parallel to $BC$ from $A$ at $Q$. Prove that $PQ$ is perpendicular to $AM$.

2020 Switzerald TST p3

Let $k$ be a circle with center $O$. Let $AB$ be a chord of this circle with center $M \ne O$. The

tangents from $k$ to $A$ and $B$ intersect at $T$. The line $\ell$ goes through T and intersects $k$ at

$C$ and $D$, with $CT <DT$ and $BC = BM$. Prove that the circumcenter of the triangle $ADM$ is

the reflection of $O$ wrt the line $AD$.

2020 Switzerald TST p8 (2017 Belarus TST 3.1)

Let $I$ be the center of the incircle of a non-isosceles triangle $ABC$. Let $F$ be the intersection the perpendicular to $AI$ through $I$ with the line $BC$. Let M be the point on the circumcircle of $ABC$, so that $MB = MC$ and $M$ lies on the same side of the line $BC$ as $A$. Let $N$ be the second interection of the line $MI$ with the circumcircle of the triangle $BIC$. Show that $FN$ is a tangent to the circumcircle of the triangle $BIC$.

2020 Switzerald TST p10

Let $(ABC)$ be a triangle with circumcircle $w$.Let $A_1$ ,$B_1$ abd $C_1$ be points on the interior of the sides $BC$,$CA$ and $AB$ respectively.Let $X$ be  point on $w$ and denote by $Y$ the second intersection of the circumcircles of $(BC_1X)$ and $(CB_1X)$.Define the points $P$ and $Q$ to be the intersections of $BY$ with $B_1A_1$ and $CY$ with $C_1A_1$,respectively.Prove that $A$ lies on the line $PQ$.

2021 Switzerald TST p1

Let $ABC$ be a triangle where $\angle BAC = 90^o$, with circumcenter $O$ and incenter $I$. The angle bisector of $\angle BAC$ intersects the circumcircle of $ABC$ in $A$ and $P$. Let $Q$ be the projection of $P$ onto $AB$, and $R$ be the projection of $I$ onto $P Q$. Prove that $RO$ bisects $CI$.

2021 Switzerald TST p12

Let $ABC$ be an acute triangle, and $I$ its incenter. Let $A_1$ be the intersection of AI and $BC$, and $C_1$ the intersection of $CI$ and $AB$. Furthermore, let $M$ and $N$ be the midpoints of $AI$ and $CI$, respectively. Inside the triangles $AC_1I$ and $A_1CI$ we choose points $K$ and $L$ such that $\angle AKI = \angle CLI = \angle AIC$, $\angle AKM = \angle ICA$ and $\angle CLN = \angle IAC$. Prove that the radii of the circumcircles of the triangles $KIL$ and $ABC$ are equal.

2022 Switzerald TST p2

Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ is tangent to the line $CD$, and the circle with diameter $CD$ is tangent to the line $AB$. Prove that the two intersection points of these circles and the point $AC \cap BD$ are collinear.

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