geometry problems from Dutch Team Selection Tests for IMO, Benelux MO and European's Girl MO with aops links in the names

Let $\Gamma_1$ and $\Gamma_2$ be two intersecting circles with midpoints respectively $O_1$ and $O_2$, such that $\Gamma_2$ intersects the line segment $O_1O_2$ in a point $A$. The intersection points of $\Gamma_1$ and $\Gamma_2$ are $C$ and $D$. The line $AD$ intersects $\Gamma_1$ a second time in $S$. The line $CS$ intersects $O_1O_2$ in $F$. Let $\Gamma_3$ be the circumcircle of triangle $AD$. Let $E$ be the second intersection point of $\Gamma_1$ and $\Gamma_3$. Prove that $O_1E$ is tangent to $\Gamma_3$.

Netherlands IMO TST 2006 - 2019

Circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. Let A be a point on $\Gamma_1$ different from $P$ and $Q$. The lines $AP$ and $AQ$ intersect $\Gamma_2$ again at $B$ and $C$. Prove that the line of the altitude from $A$ in triangle $ABC$, passes through a fixed point that is independent of the choice of $A$.

Four points $P, Q, R$ and $S$ lie on a circle in this order, so that $\angle PSR = 90^o$. Let $H$ and $K$ be the feet of the perpendicular lines from $Q$ on $PR$ and $PS$, respectively. $T$ is the intersection of $HK$ and $QS$. Prove that $| ST | = | T Q |$.

Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$ and $|AB| > |BC|$, and let $\Gamma$ be the semicircle with diameter $AB$ that lies on the same side as $C$. Let $P$ be a point on $\Gamma$ such that $|BP| = |BC|$ and let $Q$ be on $AB$ such that $|AP| = |AQ|$. Prove that the midpoint of $CQ$ lies on $\Gamma$.

Let $ABC$ be a triangle, $P$ the midpoint of $BC$, and $Q$ a point on segment $CA$ such that $|CQ| = 2|QA|$. Let $S$ be the intersection of $BQ$ and $AP$. Prove that $|AS| = |SP|$.

Suppose that we are given an $n$-gon of which all sides have the same length, and of which all the vertices have rational coordinates. Prove that $n$ is even.

Suppose that we are given an $n$-gon of which all sides have the same length, and of which all the vertices have rational coordinates. Prove that $n$ is even.

Let $ABC$ be an acute triangle such that $\angle BAC = 45^o$. Let $D$ a point on $AB$ such that $CD \perp AB$. Let $P$ be an internal point of the segment $CD$. Prove that $AP\perp BC$ if and only if $|AP| = |BC|$.

Let $ABCD$ be a cyclic quadrilateral satisfying $\angle ABD = \angle DBC$. Let $E$ be the intersection of the diagonals $AC$ and $BD$. Let $M$ be the midpoint of $AE$, and $N$ be the midpoint of $DC$. Show that $MBCN$ is a cyclic quadrilateral.

Let $ABCD$ be a square with circumcircle $\Gamma_1$. Let $P$ be a point on the arc $AC$ that also contains $B$. A circle $\Gamma_2$ touches $\Gamma_1$ in $P$ and also touches the diagonal $AC$ in $Q$. Let $R$ be a point on $\Gamma_2$ such that the line $DR$ touches $\Gamma_2$. Proof that $|DR| = |DA|$.

The circles $\Gamma_1$ and $\Gamma_2$ intersect at $D$ and $P$. The common tangent line of the two circles closest to point $D$ touches $\Gamma_1$ in A and $\Gamma_2$ in $B$. The line $AD$ intersects $\Gamma_2$ for the second time in $C$. Let $M$ be the middle of line segment $BC$. Prove that $\angle DPM = \angle BDC$.

Let $ABC$ be a triangle with $|AB|> |BC|$. Let $D$ be the midpoint of $AC$. Let $E$ be the intersection of the angular bisector of $\angle ABC$ and the line $AC$. Let $F$ be the point on $BE$ such that $CF$ is perpendicular to $BE$. Finally, let $G$ be the intersection of $CF$ and $BD$. Prove that $DF$ divides the line segment $EG$ into two equal parts.

Let $I$ be the incentre of triangle $ABC$. A line through $I$ intersects the interior of line segment $AB$ in $M$ and the interior of line segment $BC$ in $N$. We assume that $BMN$ is an acute triangle. Let $K$ and $L$ be points on line segment $AC$ such that $\angle BMI = \angle ILA$ and $\angle BNI = \angle IKC$. Prove that $|AM| + |KL| + |CN| = |AC|$.

Let $\Gamma$ be the circumcircle of the acute triangle $ABC$. The angle bisector of angle $ABC$ intersects $AC$ in the point $B_1$ and the short arc $AC$ of $\Gamma$ in the point $P$. The line through $B_1$ perpendicular to $BC$ intersects the short arc $BC$ of $\Gamma$ in $K$. The line through $B$ perpendicular to $AK$ intersects $AC$ in $L$. Prove that $K, L$ and $P$ lie on a line.

Let $\vartriangle ABC$ be a triangle. The angle bisector of $\angle CAB$ intersects$ BC$ at $L$. On the interior of line segments $AC$ and $AB$, two points, $M$ and $N$, respectively, are chosen in such a way that the lines $AL, BM$ and $CN$ are concurrent, and such that $\angle AMN = \angle ALB$. Prove that $\angle NML = 90^o$.

Fix a triangle $ABC$. Let $\Gamma_1$ the circle through $B$, tangent to edge in $A$. Let $\Gamma_2$ the circle through C tangent to edge $AB$ in $A$. The second intersection of $\Gamma_1$ and $\Gamma_2$ is denoted by $D$. The line $AD$ has second intersection $E$ with the circumcircle of $\vartriangle ABC$. Show that $D$ is the midpoint of the segment $AE$.

Let $P$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$.Let $X,Y,Z$ be points on the interior of $AB,BC,CD$ respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that $XY$ is tangent to the circumcircle of $\triangle CYZ$ and that $Y Z$ is tangent to the circumcircle of $\triangle BXY$.Show that $\angle APD=\angle XYZ$

Let $ABCDEF$ be a cyclic hexagon satisfying $AB\perp BD$ and $BC=EF$.Let $P$ be the intersection of lines $BC$ and $AD$ and let $Q$ be the intersection of lines $EF$ and $AD$.Assume that $P$ and $Q$ are on the same side of $D$ and $A$ is on the opposite side.Let $S$ be the midpoint of $AD$.Let $K$ and $L$ be the incentres of $\triangle BPS$ and $\triangle EQS$ respectively.Prove that $\angle KDL=90^0$.

Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.

Let $\triangle ABC$ be a triangle with $AC=2AB$ and let $O$ be its circumcenter. Let $D$ be the intersection of the angle bisector of $\angle A$ and $BC$. Let $E$ be the orthogonal projection of $O$ on $AD$ and $F \neq D$ be a point on $AD$ satisfying $CD=CF$. Prove that $\angle EBF=\angle ECF$.

Let $H$ be the orthocentre of an acute triangle $ABC$. The line through $A$ perpendicular to $AC$ and the line through $B$ perpendicular to $BC$ intersect in $D$. The circle with centre $C$ through $H$ intersects the circumcircle of triangle $ABC$ in the points $E$ and $F$. Prove that $|DE| = |DF| = |AB|$.

In a quadrilateral $ABCD$ we have $\angle A = \angle C = 90^o$. Let $E$ be a point in the interior of $ABCD$. Let $M$ be the midpoint of $BE$. Prove that $\angle ADB = \angle EDC$ if and only if $|MA| = |MC|$.

Let $\Gamma_1$ and $\Gamma_2$ be circles - with respective centres $O_1$ and $O_2$ - that intersect each other in $A$ and $B$. The line $O_1A$ intersects $\Gamma_2$ in $A$ and $C$ and the line $O_2A$ intersects $\Gamma_1$ in $A$ and $D$. The line through $B$ parallel to $AD$ intersects $\Gamma_1$ in $B$ and $E$. Suppose that $O_1A$ is parallel to $DE$. Show that $CD$ is perpendicular to $O_2C$.

An equilateral triangle $ABC$ is given. On the line through $B$ parallel to $AC$ there is a point $D$, such that $D$ and $C$ are on the same side of the line $AB$. The perpendicular bisector of $CD$ intersects the line $AB$ in $E$. Prove that triangle $CDE$ is equilateral.

Let $\triangle ABC$ be a acute triangle. Let $H$ the foot of the C-altitude in $AB$ such that $AH=3BH$, let $M$ and $N$ the midpoints of $AB$ and $AC$ and let $P$ be a point such that $NP=NC$ and $CP=CB$ and $B$, $P$ are located on different sides of the line $AC$. Prove that $\measuredangle APM=\measuredangle PBA$.

Let $\vartriangle ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $|BF| = |BE|$ and such that $ED$ is the internal angle bisector of $\angle BEC$. Prove that $|BD|= |EF|$ if and only if $|AF| = |EC|$.

Let $\Gamma_1$ be a circle with centre $A$ and $\Gamma_2$ be a circle with centre $B$, with $A$ lying on $\Gamma_2$. On $\Gamma_2$ there is a (variable) point $P$ not lying on $AB$. A line through $P$ is a tangent of $\Gamma_1$ at $S$, and it intersects $\Gamma_2$ again in $Q$, with $P$ and $Q$ lying on the same side of $AB$. A different line through $Q$ is tangent to $\Gamma_1$ at $T$. Moreover, let $M$ be the foot of the perpendicular to $AB$ through $P$. Let $N$ be the intersection of $AQ$ and $MT$. Show that $N$ lies on a line independent of the position of $P$ on $\Gamma_2$.

Let $ABC$ be a triangle, let $M$ be the midpoint of $AB$, and let $N$ be the midpoint of $CM$. Let $X$ be a point satisfying both $\angle XMC = \angle MBC$ and $\angle XCM = \angle MCB$ such that $X$ and $B$ lie on opposite sides of $CM$. Let $\omega$ be the circumcircle of triangle $AMX$.

(a) Show that $CM$ is tangent to $\omega$.

(b) Show that the lines $NX$ and $AC$ intersect on $\omega$

(a) Show that $CM$ is tangent to $\omega$.

(b) Show that the lines $NX$ and $AC$ intersect on $\omega$

The incircle of a triangle scalene $ABC$ with center $I$, touch the sides $BC$ and $CA$ in $D$ and $E$ respectively. Denote with $H$ the orthocenter of $ABI$. $K=AI\cap BH$ and $L=BI\cap AH$. Prove that the circumcircles of the triangles $DKH$ and $ELH$ they are cut at a point of the incircle of $ABC$.

A circle $\omega$ with diameter $AK$ is given. The point $M$ lies in the interior of the circle, but not on $AK$. The line $AM$ intersects $\omega$ in $A$ and $Q$. The tangent to $\omega$ at $Q$ intersects the line through $M$ perpendicular to $AK$, at $P$. The point $L$ lies on $\omega$, and is such that $PL$ is tangent to $\omega$ and $L\neq Q$. Show that $K, L$, and $M$ are collinear.

Suppose a triangle $\vartriangle ABC$ with $\angle C = 90^o$ is given. Let $D$ be the midpoint of $AC$, and let $E$ be the foot of the altitude through $C$ on $BD$. Show that the tangent in $C$ of the circumcircle of $\vartriangle AEC$ is perpendicular to $AB$.

Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude through $A$. On $AD$, there are distinct points $E$ and $F$ such that $|AE| = |BE|$ and $|AF| =|CF|$. A point$ T \ne D$ satis es $\angle BTE = \angle CTF = 90^o$. Show that $|TA|^2 =|TB| \cdot |TC|$.

In a non-isosceles triangle $ABC$ the centre of the incircle is denoted by $I$. The other intersection point of the angle bisector of $\angle BAC$ and the circumcircle of $\vartriangle ABC$ is $D$. The line through $I$ perpendicular to $AD$ intersects $BC$ in $F$. The midpoint of the circle arc $BC$ on which $A$ lies, is denoted by $M$. The other intersection point of the line $MI$ and the circle through $B, I$ and $C$, is denoted by $N$. Prove that $FN$ is tangent to the circle through $B, I$ and $C$.

Let $\Delta ABC$ be a scalene triangle. Points $D,E$ lie on side $\overline{AC}$ in the order, $A,E,D,C$. Let the parallel through $E$ to $BC$ intersect $\odot (ABD)$ at $F$, such that, $E$ and $F$ lie on the same side of $AB$. Let the parallel through $E$ to $AB$ intersect $\odot (BDC)$ at $G$, such that, $E$ and $G$ lie on the same side of $BC$. Prove, Points $D,F,E,G$ are concyclic.

Let $ABC$ be an acute angles triangle with $O$ the center of the circumscribed circle. Point $Q$ lies on the circumscribed circle of $\vartriangle BOC$ so that $OQ$ is a diameter. Point $M$ lies on $CQ$ and point $N$ lies internally on line segment $BC$ so that $ANCM$ is a parallelogram. Prove that the circumscribed circle of $\vartriangle BOC$ and the lines $AQ$ and $NM$ pass through the same point.

Let $ABCD$ be a cyclic quadrilateral (In the same order) inscribed into the circle $\odot (O)$. Let $\overline{AC}$ $\cap$ $\overline{BD}$ $=$ $E$. A randome line $\ell$ through $E$ intersects $\overline{AB}$ at $P$ and $BC$ at $Q$. A circle $\omega$ touches $\ell$ at $E$ and passes through $D$. Given, $\omega$ $\cap$ $\odot (O)$ $=$ $R$. Prove, Points $B,Q,R,P$ are concyclic.

Netherlands BxMO - EGMO TST 2010-19

only the years 2012-2015 the BxMO TST counted also for EGMO,

the following years no extra TST took place for EGMO, while the BxMO keeps on happening

2010 Dutch BxMO TST P1

Let $ABCD$ be a trapezoid with $AB // CD$, $2|AB| = |CD|$ and $BD \perp BC$. Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$. Let $O$ be the intersection of $AM$ and $BD$. Let $N$ be the intersection of $OE$ and $AB$.

(a) Prove that $ABMD$ is a rhombus.

(b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$.

2012 Dutch BxMO / EGMO TST P2

Let $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.

2012 Dutch BxMO / EGMO TST P4

Let $ABCD$ a convex quadrilateral (this means that all interior angles are smaller than $180^o$), such that there exist a point $M$ on line segment $AB$ and a point $N$ on line segment $BC$ having the property that $AN$ cuts the quadrilateral in two parts of equal area, and such that the same property holds for $CM$. Prove that $MN$ cuts the diagonal $BD$ in two segments of equal length.

2014 Dutch BxMO / EGMO TST P3

only the years 2012-2015 the BxMO TST counted also for EGMO,

the following years no extra TST took place for EGMO, while the BxMO keeps on happening

2010 Dutch BxMO TST P1

Let $ABCD$ be a trapezoid with $AB // CD$, $2|AB| = |CD|$ and $BD \perp BC$. Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$. Let $O$ be the intersection of $AM$ and $BD$. Let $N$ be the intersection of $OE$ and $AB$.

(a) Prove that $ABMD$ is a rhombus.

(b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$.

2010 Dutch BxMO TST P4

The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $P$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.

In an acute triangle $ABC$ the angle $\angle C$ is greater than $\angle A$. Let $E$ be such that $AE$ is a diameter of the circumscribed circle $\Gamma$ of \vartriangle ABC. Let $K$ be the intersection of $AC$ and the tangent line at $B$ to $\Gamma$. Let $L$ be the orthogonal projection of $K$ on $AE$ and let $D$ be the intersection of $KL$ and $AB$. Prove that $CE$ is the bisector of $\angle BCD$.The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $P$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.

A trapezoid $ABCD$ is given with $BC // AD$. Assume that the bisectors of the angles $BAD$ and $CDA$ intersect on the perpendicular bisector of the line segment $BC$. Prove that $|AB|= |CD|$ or $|AB| +|CD| =|AD|$.

Let $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.

2012 Dutch BxMO / EGMO TST P4

Let $ABCD$ a convex quadrilateral (this means that all interior angles are smaller than $180^o$), such that there exist a point $M$ on line segment $AB$ and a point $N$ on line segment $BC$ having the property that $AN$ cuts the quadrilateral in two parts of equal area, and such that the same property holds for $CM$. Prove that $MN$ cuts the diagonal $BD$ in two segments of equal length.

In quadrilateral $ABCD$ the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of diagonal $AC$. Suppose that triangles $ABM$ and $ACD$ have equal area. Prove that $DM // BC$.

Let $ABCD$ be a cyclic quadrilateral for which $|AD| =|BD|$. Let $M$ be the intersection of $AC$ and $BD$. Let $I$ be the incentre (centre of the inscribed circle) of $\triangle BCM$. Let $N$ be the second point of intersection of $AC$ and the circumscribed circle of $\triangle BMI$. Prove that $|AN| \cdot |NC| = |CD | \cdot |BN|$.

2014 Dutch BxMO / EGMO TST P3

In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$. Prove that $|AR|\cdot |BQ|=|P I|^2$

In a triangle $ABC$ the point $D$ is the intersection of the interior angle bisector of $\angle BAC$ and side $BC$. Let $P$ be the second intersection point of the exterior angle bisector of $\angle BAC$ with the circumcircle of $\angle ABC$. A circle through $A$ and $P$ intersects line segment $BP$ internally in $E$ and line segment $CP$ internally in $F$. Prove that $\angle DEP = \angle DFP$.

2016 Dutch BxMO TST P3

Let $\vartriangle ABC$ be a right-angled triangle with $\angle A = 90^o$ and circumcircle $\Gamma$. The inscribed circle is tangent to $BC$ in point $D$. Let $E$ be the midpoint of the arc $AB$ of $\Gamma$ not containing $C$ and let $F$ be the midpoint of the arc $AC$ of $\Gamma$ not containing $B$.

(a) Prove that $\vartriangle ABC \sim \vartriangle DEF$.

(b) Prove that $EF$ goes through the points of tangency of the incircle to $AB$ and $AC$.

2019 Dutch BxMO TST P2

Let $\Delta ABC$ be a triangle with an inscribed circle centered at $I$. The line perpendicular to $AI$ at $I$ intersects $\odot (ABC)$ at $P,Q$ such that, $P$ lies closer to $B$ than $C$. Let $\odot (BIP) \cap \odot (CIQ) =S$. Prove that, $SI$ is the angle bisector of $\angle PSQ$

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2016 Dutch BxMO TST P3

Let $\vartriangle ABC$ be a right-angled triangle with $\angle A = 90^o$ and circumcircle $\Gamma$. The inscribed circle is tangent to $BC$ in point $D$. Let $E$ be the midpoint of the arc $AB$ of $\Gamma$ not containing $C$ and let $F$ be the midpoint of the arc $AC$ of $\Gamma$ not containing $B$.

(a) Prove that $\vartriangle ABC \sim \vartriangle DEF$.

(b) Prove that $EF$ goes through the points of tangency of the incircle to $AB$ and $AC$.

2017 Dutch BxMO TST P3

Let $ABC$ be a triangle with $\angle A = 90$ and let $D$ be the orthogonal projection of $A$ onto $BC$. The midpoints of $AD$ and $AC$ are called $E$ and $F$, respectively. Let $M$ be the circumcentre of $BEF$. Prove that $AC$ and $ BM$ are parallel.

2018 Dutch BxMO TST P2

Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.

Let $ABC$ be a triangle with $\angle A = 90$ and let $D$ be the orthogonal projection of $A$ onto $BC$. The midpoints of $AD$ and $AC$ are called $E$ and $F$, respectively. Let $M$ be the circumcentre of $BEF$. Prove that $AC$ and $ BM$ are parallel.

2018 Dutch BxMO TST P2

Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.

In a non-isosceles triangle $\vartriangle ABC$ we have $\angle BAC = 60^o$. Let $D$ be the intersection of the angular bisector of $\angle BAC$ with side $BC, O$ the centre of the circumcircle of $\vartriangle ABC$ and $E$ the intersection of $AO$ and $BC$. Prove that $\angle AED + \angle ADO = 90^o$.

Let $\Delta ABC$ be a triangle with an inscribed circle centered at $I$. The line perpendicular to $AI$ at $I$ intersects $\odot (ABC)$ at $P,Q$ such that, $P$ lies closer to $B$ than $C$. Let $\odot (BIP) \cap \odot (CIQ) =S$. Prove that, $SI$ is the angle bisector of $\angle PSQ$

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