Netherlands (Dutch) 1990 - 2022 (DMO) 52p

geometry problems from Dutch Mathematical Olympiad (2nd round)
with aops links in the names

Dutch MO and TST 2008-20 with solutions

1990 - 2022

1990 Dutch MO P4

If $ABCDEFG$ is a regular $7$-gon with side $1$, show that: $\frac{1}{AC}+\frac{1}{AD}=1$.

1991 Dutch MO P2

An angle with vertex $A$ and measure $\alpha$ and a point $P_0$ on one of its rays are given so that $AP_0=2$. Point $P_1$ is chose on the other ray. The sequence of points $P_1,P_2,P_3,...$ is defined so that $P_n$ lies on the segment $AP_{n-2}$ and the triangle $P_n P_{n-1} P_{n-2}$ is isosceles with $P_n P_{n-1}=P_n P_{n-2}$ for all $n \ge 2$.

$(a)$ Prove that for each value of $\alpha$ there is a unique point $P_1$ for which the sequence $P_1,P_2,...,P_n,...$ does not terminate.

$(b)$ Suppose that the sequence $P_1,P_2,...$ does not terminate and that the length of the polygonal line $P_0 P_1 P_2 ... P_k$ tends to $5$ when $k \rightarrow \infty$. Compute the length of $P_0 P_1$.

1991 Dutch MO P5

Let $H$ be the orthocenter, $O$ the circumcenter, and $R$ the circumradius of an acute-angled triangle $ABC$. Consider the circles $k_a,k_b,k_c,k_h,k$, all with radius $R$, centered at $A,B,C,H,M,$ respectively. Circles $k_a$ and $k_b$ meet at $M$ and $F$; $k_a$ and $k_c$ meet at $M$ and $E$; and $k_b$ and $k_c$ meet at $M$ and $D$.

$(a)$ Prove that the points $D,E,F$ lie on the circle $k_h$.

$(b)$ Prove that the set of the points inside $k_h$ that are inside exactly one of the circles $k_a,k_b,k_c$ has the area twice the area of $\triangle ABC$.

1992 Dutch MO P3

Consider the configuration of six squares as shown on the picture. Prove that the sum of the area of the three outer squares ($I,II$ and $III$) equals three times the sum of the areas of the three inner squares ($IV,V$ and $VI$).

1992 Dutch MO P5

We consider regular $n$-gons with a fixed circumference $4$. Let $r_n$ and $a_n$ respectively be the distances from the center of such an $n$-gon to a vertex and to an edge.

(a) Determine $a_4,r_4,a_8,r_8$.

(b) Give an appropriate interpretation for $a_2$ and $r_2$

(c) Prove that $a_{2n}=\frac{1}{2} (a_n+r_n)$ and $r_{2n}=\sqrt{a_2n r_n}.$

(d) Define $u_0=0, u_1=1$ and $u_n=\frac{1}{2}(u_{n-2}+u_{n-1})$ for $n$ even or $u_n=\sqrt{u_{n-2} u_{n-1}}$ for $n$ odd. Determine $\displaystyle\lim_{n\to\infty}u_n$.

1993 Dutch MO P2

In a triangle $ABC$ with $\angle A=90^{\circ}$, $D$ is the midpoint of $BC$, $F$ that of $AB$, $E$ that of $AF$ and $G$ that of $FB$. Segment $AD$ intersects $CE,CF$ and $CG$ in $P,Q$ and $R$, respectively. Determine the ratio: $\frac{PQ}{QR}$.

1993 Dutch MO P4

Let $C$ be a circle with center $M$ in a plane $V$, and $P$ be a point not on the circle $C$.

(a) If $P$ is fixed, prove that $AP^2+BP^2$ is a constant for every diameter $AB$ of the circle $C$.

(b) Let $AB$ be a fixed diameter of $C$ and $P$ a point on a fixed sphere $S$ not intersecting $V$. Determine the points $P$ on $S$ that minimize $AP^2+BP^2$.

1994 Dutch MO P1

A unit square is divided into two rectangles in such a way that the smaller rectangle can be put on the greater rectangle with every vertex of the smaller on exactly one of the edges of the greater. Calculate the dimensions of the smaller rectangle.

1994 Dutch MO P4

Let $P$ be a point on the diagonal $BD$ of a rectangle $ABCD$, $F$ be the projection of $P$ on $BC$, and $H \not= B$ be the point on $BC$ such that $BF=FH$. If lines $PC$ and $AH$ intersect at $Q$, prove that the areas of triangles $APQ$ and $CHQ$ are equal.

1995 Dutch MO P2

For any point $P$ on a segment $AB$, isosceles and right-angled triangles $AQP$ and $PRB$ are constructed on the same side of $AB$, with $AP$ and $PB$ as the bases. Determine the locus of the midpoint $M$ of $QR$ when $P$ describes the segment $AB$.

1995 Dutch MO P4

A number of spheres with radius $1$ are being placed in the form of a square pyramid. First, there is a layer in the form of a square with $n^2$ spheres. On top of that layer comes the next layer with $(n-1)^2$ spheres, and so on. The top layer consists of only one sphere. Compute the height of the pyramid.

1996 Dutch MO P4

A line $l$ intersects the segment $AB$ perpendicular to $C$. Three circles are drawn successively with $AB, AC$ and $BC$ as the diameter. The largest circle intersects $l$ in $D$. The segments $DA$ and $DB$ still intersect the two smaller circles in $E$ and $F$.

a. Prove that quadrilateral $CFDE$ is a rectangle.

b. Prove that the line through $E$ and $F$ touches the circles with diameters $AC$ and $BC$ in $E$ and $F$.

1997 Dutch MO P2

The lines $AD , BE$ and $CF$ intersect in $S$ within a triangle $ABC$ .

It is given that $AS: DS = 3: 2$ and $BS: ES = 4: 3$ . Determine the ratio $CS: FS$ .

1997 Dutch MO P5

Given is a triangle $ABC$ and a point $K$ within the triangle. The point $K$ is mirrored in the sides of the triangle: $P , Q$ and $R$ are the mirrorings of $K$ in $AB , BC$ and $CA$, respectively . $M$ is the center of the circle passing through the vertices of triangle $PQR$. $M$ is mirrored again in the sides of triangle $ABC$: $P', Q'$ and $R'$ are the mirror of $M$ in $AB$ respectively, $BC$ and $CA$.

a. Prove that $K$ is the center of the circle passing through the vertices of triangle $P'Q'R'$ .

b. Where should you choose $K$ within triangle $ABC$ so that $M$ and $K$ coincide? Prove your answer.

1998 Dutch MO P2

Let $TABCD$ be a pyramid with top vertex $T$, such that its base $ABCD$ is a square of side length 4. It is given that, among the triangles $TAB$, $TBC$, $TCD$ and $TDA$, one can find an isosceles triangle and a right-angled triangle. Find all possible values for the volume of the pyramid.

1998 Dutch MO P4

Let $ABCD$ be a convex quadrilateral such that $AC \perp BD$.

(a) Prove that $AB^2 + CD^2 = BC^2 + DA^2$.

(b) Let $PQRS$ be a convex quadrilateral such that $PQ = AB$, $QR = BC$, $RS = CD$ and $SP = DA$. Prove that $PR \perp QS$.

1999 Dutch MO P3

Let $ABCD$ be a square and let $\ell$ be a line. Let $M$ be the centre of the square. The diagonals of the square have length 2 and the distance from $M$ to $\ell$ exceeds 1. Let $A',B',C',D'$ be the orthogonal projections of $A,B,C,D$ onto $\ell$. Suppose that one rotates the square, such that $M$ is invariant. The positions of $A,B,C,D,A',B',C',D'$ change. Prove that the value of $AA'^2 + BB'^2 + CC'^2 + DD'^2$ does not change.

2000 Dutch MO P3

Isosceles, similar triangles $QPA$ and $SPB$ are constructed (outwards) on the sides of parallelogram $PQRS$ (where $PQ = AQ$ and $PS = BS$). Prove that triangles $RAB$, $QPA$ and $SPB$ are similar.

2001 Dutch MO P3

A wooden beam $EFGH$ $ABCD$ is with three cuts in $8$ smaller ones sawn beams. Each cut is parallel to one of the three pair of opposit sides. Each pair of saw cuts is shown perpendicular to each other. The smaller bars at the corners $A, C, F$ and $H$ have a capacity of $9, 12, 8, 24$ respectively. Calculate content of the entire bar.

(The proportions in the picture are not correct!!)

2002 Dutch MO P3

$A, B$ and $C$ are points in the plane with integer coordinates. The lengths of the sides of triangle $ABC$ are integer numbers. Prove that the perimeter of the triangle is an even number.

2002 Dutch MO P5

In triangle $ABC$, angle $A$ is twice as large as angle $B$. $AB = 3$ and $AC = 2$. Calculate $BC$.

2003 Dutch MO P1

A Pythagorean triangle is a right triangle whose three sides are integers. The best known example is the triangle with rectangular sides $3$ and $4$ and hypotenuse $5$.Determine all Pythagorean triangles whose area is twice the perimeter.

2003 Dutch MO P2

Two squares with side $12$ lie exactly on top of each other. One square is rotated around a corner point through an angle of $30$ degrees relative to the other square. Determine the area of the common piece of the two squares.

2003 Dutch MO P4

In a circle with center $M$, two chords $AC$ and $BD$ intersect perpendicularly.

The circle of diameter $AM$ intersects the circle of diameter $BM$ besides $M$ also in point $P$. The circle of diameter $BM$ intersects the circle with diameter $CM$ besides $M$ also in point $Q$. The circle of diameter $CM$ intersects the circle of diameter $DM$ besides $M$ also in point $R$. The circle of diameter $DM$ intersects the circle of diameter $AM$ besides $M$ also in point $S$. Prove that quadrilateral $PQRS$ is a rectangle.

2004 Dutch MO P2

Two circles $A$ and $B$, both with radius $1$, touch each other externally.

Four circles $P, Q, R$ and $S$, all four with the same radius $r$, lie such that

$P$ externally touches on $A, B, Q$ and $S$,

$Q$ externally touches on $P, B$ and $R$,

$R$ externally touches on $A, B, Q$ and $S$,

$S$ externally touches on $P, A$ and $R$.

Calculate the length of $r.$

2004 Dutch MO P4

Two circles $C_1$ and $C_2$ touch each other externally in a point $P$. At point $C_1$ there is a point $Q$ such that the tangent line in $Q$ at $C_1$ intersects the circle $C_2$ at points $A$ and $B$. The line $QP$ still intersects $C_2$ at point $C$.

Prove that triangle $ABC$ is isosceles.

2005 Dutch MO P2

Let $P_1P_2P_3\dots P_{12}$ be a regular dodecagon. Show that $\left|P_1P_2\right|^2 + \left|P_1P_4\right|^2 + \left|P_1P_6\right|^2 + \left|P_1P_8\right|^2 + \left|P_1P_{10}\right|^2 + \left|P_1P_{12}\right|^2$ is equal to $\left|P_1P_3\right|^2 + \left|P_1P_5\right|^2 + \left|P_1P_7\right|^2 + \left|P_1P_9\right|^2 + \left|P_1P_{11}\right|^2.$

2005 Dutch MO P4

Let $ABCD$ be a quadrilateral with $AB \parallel CD$, $AB > CD$. Prove that the line passing through $AC \cap BD$ and $AD \cap BC$ passes through the midpoints of $AB$ and $CD$.

2006 Dutch MO P2

Given is a acute angled triangle $ABC$. The lengths of the altitudes from $A, B$ and $C$ are successively $h_A, h_B$ and $h_C$. Inside the triangle is a point $P$. The distance from $P$ to $BC$ is $1/3 h_A$ and the distance from $P$ to $AC$ is $1/4 h_B$. Express the distance from $P$ to $AB$ in terms of $h_C$

2006 Dutch MO P4

Given is triangle $ABC$ with an inscribed circle with center $M$ and radius $r$.

The tangent to this circle parallel to $BC$ intersects $AC$ in $D$ and $AB$ in $E$.

The tangent to this circle parallel to $AC$ intersects $AB$ in $F$ and $BC$ in $G$.

The tangent to this circle parallel to $AB$ intersects $BC$ in $H$ and $AC$ in $K$.

Name the centers of the inscribed circles of triangle $AED$, triangle $FBG$ and triangle $KHC$ successively $M_A, M_B, M_C$ and the rays successively $r_A, r_B$ and $r_C$.

Prove that $r_A + r_B + r_C = r$.

2007 Dutch MO P1

Consider the equilateral triangle ABC with $|BC| = |CA| = |AB| = 1$.  

On the extension of side $BC$, we define points $A_1$ (on the same side as B) and $A_2$ (on the same side as C) such that $|A_1B| = |BC| = |CA_2| = 1$. Similarly, we define $B_1$ and $B_2$ on the extension of side $CA$ such that $|B_1C| = |CA| =|AB_2| = 1$, and $C_1$ and $C_2$ on the extension of side $AB$ such that $|C_1A| = |AB| = |BC_2| = 1$. Now the circumcentre of 4ABC is also the centre of the circle that passes through the points $A_1,B_2,C_1,A_2,B_1$ and $C_2$.

Calculate the radius of the circle through $A_1,B_2,C_1,A_2,B_1$ and $C_2$.

2007 Dutch MO P5

A triangle $ABC$ and a point $P$ inside this triangle are given. Define $D, E$ and $F$ as the midpoints of $AP, BP$ and $CP$, respectively. Furthermore, let $R$ be the intersection of $AE$ and $BD, S$ the intersection of $BF$ and $CE$, and $T$ the intersection of $CD$ and $AF$. Prove that the area of hexagon $DRESFT$ is independent of the position of $P$ inside the triangle.

2008 Dutch MO P1

Suppose we have a square $ABCD$ and a point $S$ in the interior of this square. Under homothety with centre $S$ and ratio of magnification $k > 1$, this square becomes another square $A'B'C'D'$.

Prove that the sum of the areas of the two quadrilaterals $A'ABB'$ and $C'CDD'$ are equal to the sum of the areas of the two quadrilaterals $B'BCC'$ and $D'DAA'$.

2008 Dutch MO P4

Three circles $C_1,C_2,C_3$, with radii $1, 2, 3$ respectively, are externally tangent.

In the area enclosed by these circles, there is a circle $C_4$ which is externally tangent to all three circles. Find the radius of $C_4$.

2009 Dutch MO P4

Let $ABC$ be an arbitrary triangle. On the perpendicular bisector of $AB$, there is a point $P$ inside of triangle $ABC$. On the sides $BC$ and $CA$, triangles $BQC$ and $CRA$ are placed externally. These triangles satisfy $\vartriangle BPA \sim  \vartriangle BQC  \sim  \vartriangle CRA$. (So $Q$ and $A$ lie on opposite sides of $BC$, and $R$ and $B$ lie on opposite sides of $AC$.) Show that the points $P, Q, C$ and $R$ form a parallelogram.

2010 Dutch MO P1

Consider a triangle $ABC$ such that $\angle A = 90, \angle C =60^o$ and $|AC|= 6$. Three circles with centers $A, B$ and $C$ are pairwise tangent in points on the three sides of the triangle.

Determine the area of the region enclosed by the three circles (the grey area in the figure).

2010 Dutch MO P3
Consider a triangle $XYZ$ and a point $O$ in its interior. Three lines through $O$ are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from $O$ to the sides of the triangle. The lengths of these segments are integer numbers $a, b, c, d, e$ and $f$ (see figure). Prove that the product $a \cdot b \cdot c\cdot d \cdot e \cdot f$ is a perfect square.

2011 Dutch MO P2

Let $ABC$ be a triangle. Points $P$ and $Q$ lie on side $BC$ and satisfy $|BP| =|PQ| = |QC| = \frac13 |BC|$. Points $R$ and $S$ lie on side $CA$ and satisfy $|CR| =|RS| = |SA| = 1 3 |CA|$. Finally, points $T$ and $U$ lie on side $AB$ and satisfy $|AT| = |TU| = |UB| =\frac13 |AB|$. Points $P, Q,R, S, T$ and $U$ turn out to lie on a common circle. Prove that $ABC$ is an equilateral triangle

2012 Dutch MO P4

We are given an acute triangle $ABC$ and points $D$ on $BC$ and $E$ on $AC$ such that $AD$ is perpendicular to $BC$ and $BE$ is perpendicular to $AC$. The intersection of $AD$ and BE is called $H$. A line through $H$ intersects line segment $BC$ in $P$, and intersects line segment $AC$ in $Q$. Furthermore, $K$ is a point on $BE$ such that $PK$ is perpendicular to $BE$, and $L$ is a point on $AD$ such that $QL$ is perpendicular to $AD$. Prove that $DK$ and $EL$ are parallel.

2013 Dutch MO P3

The sides $BC$ and $AD$ of a quadrilateral $ABCD$ are parallel and the diagonals intersect in $O$. For this quadrilateral $|CD| =|AO|$ and $|BC| = |OD|$ hold. Furthermore $CA$ is the angular bisector of angle $BCD$. Determine the size of angle $ABC$.

2014 Dutch MO P2 Junior

Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be a point on the line $AB$, distinct from $B$, such that $|CG| = |CB|$. Let $H$ be a point on the line $BC$, distinct from $B$, such that $|AB| =|AH|$. Prove that triangle $DGH$ is isosceles.

2014 Dutch MO P2 Senior

On the sides of triangle $ABC$, isosceles right-angled triangles $AUB, CVB$, and $AWC$ are placed. These three triangles have their right angles at vertices $U, V$ , and $W$, respectively. Triangle $AUB$ lies completely inside triangle $ABC$ and triangles $CVB$ and $AWC$ lie completely outside $ABC$. See the figure. Prove that quadrilateral $UVCW$ is a parallelogram.

2015 Dutch MO P3 juniors

In quadrilateral $ABCD$ sides $BC$ and $AD$ are parallel. In each of the four vertices we draw an angular bisector. The angular bisectors of angles $A$ and $B$ intersect in point $P$, those of angles $B$ and $C$ intersect in point $Q$, those of angles $C$ and $D$ intersect in point $R$, and those of angles $D$ and $A$ intersect in point S. Suppose that $PS$ is parallel to $QR$. Prove that $|AB| =|CD|$. 

2015 Dutch MO P3 seniors

Points $A, B$, and $C$ are on a line in this order. Points $D$ and $E$ lie on the same side of this line, in such a way that triangles $ABD$ and $BCE$ are equilateral. The segments $AE$ and $CD$ intersect in point $S$. Prove that $\angle ASD = 60^o$.

2016 Dutch MO P4 juniors

In a quadrilateral $ABCD$ the intersection of the diagonals is called $P$. Point $X$ is the orthocentre of triangle $PAB$. (The orthocentre of a triangle is the point where the three altitudes of the triangle intersect.) Point $Y$ is the orthocentre of triangle $PCD$. Suppose that $X$ lies inside triangle $PAB$ and $Y$ lies inside triangle $PCD$. Moreover, suppose that $P$ is the midpoint of line segment $XY$ . Prove that $ABCD$ is a parallelogram.

2016 Dutch MO P4 seniors

In the acute triangle $ABC$, the midpoint of side $BC$ is called $M$. Point $X$ lies on the angle bisector of $\angle AMB$ such that $\angle BXM = 90^o$. Point $Y$ lies on the angle bisector of $\angle AMC$ such that $\angle CYM = 90^o$. Line segments $AM$ and $XY$ intersect in point $Z$. Prove that $Z$ is the midpoint of $XY$ .

2017 Dutch MO P2

A parallelogram $ABCD$ with $|AD| =|BD|$ has been given. A point $E$ lies on line segment $|BD|$ in such a way that $|AE| = |DE|$. The (extended) line $AE$ intersects line segment $BC$ in $F$. Line $DF$ is the angle bisector of angle $CDE$. Determine the size of angle $ABD$.

2018 Dutch MO P4

In triangle $ABC, \angle A$ is smaller than $\angle C$. Point $D$ lies on the (extended) line $BC$ (with $B$ between $C$ and $D$) such that $|BD| = |AB|$. Point $E$ lies on the bisector of $\angle ABC$ such that $\angle BAE = \angle ACB$. Line segment $BE$ intersects line segment $AC$ in point $F$. Point $G$ lies on line segment $AD$ such that $EG$ and $BC$ are parallel.
Prove that $|AG| =|BF|$.

2019 Dutch MO P3

Points $A, B$, and $C$ lie on a circle with centre $M$. The reflection of point $M$ in the line $AB$ lies inside triangle $ABC$ and is the intersection of the angle bisectors of angles $A$ and $B$. Line $AM$ intersects the circle again in point $D$. Show that $|CA| \cdot |CD| = |AB| \cdot |AM|$.

2020 Dutch MO P3

Given is a parallelogram $ABCD$ with $\angle A < 90^o$ and $|AB| < |BC|$. The angular bisector of angle $A$ intersects side $BC$ in $M$ and intersects the extension of $DC$ in $N$. Point $O$ is the centre of the circle through $M, C$, and $N$. Prove that $\angle OBC = \angle ODC$.

2021 Dutch MO P4

In triangle ABC we have $\angle ACB = 90^o$. The point $M$ is the midpoint of $AB$. The line through $M$ parallel to $BC$ intersects $AC$ in $D$. The midpoint of line segment $CD$ is $E$. The lines $BD$ and $CM$ are perpendicular.

(a) Prove that triangles $CME$ and $ABD$ are similar.

(b) Prove that $EM$ and $AB$ are perpendicular.

2022 Dutch MO P4

In triangle $ABC$, the point $D$ lies on segment $AB$ such that $CD$ is the angle bisector of angle $\angle C$. The perpendicular bisector of segment $CD$ intersects the line $AB$ in $E$. Suppose that $|BE| = 4$ and $|AB| = 5$.

(a) Prove that $\angle BAC = \angle BCE$.

(b) Prove that $2|AD| = |ED|$.