geometry problems from Danish Math Olympiads (Mohr Contest) [2nd / final round] with aops links
collected inside aops here
1991- 2023
A right-angled triangle has perimeter $60$ and the altitude of the hypotenuse has a length $12$. Determine the lengths of the sides.
Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.
In a right-angled triangle, $a$ and $b$ denote the lengths of the two catheti. A circle with radius $r$ has the center on the hypotenuse and touches both catheti. Show that $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$..
Let $a, b$ and $c$ denote the side lengths and $m_a, m_b$ and $m_c$ of the median's lengths in an arbitrary triangle. Show that $$\frac34 < \frac{m_a + m_b + m_c}{a + b + c}<1$$ Also show that there is no narrower range that for each triangle that contains the fraction
$$\frac{m_a + m_b + m_c}{a + b + c}$$
A
rectangular piece of paper has the side lengths $12$ and $15$. A corner is bent
about as shown in the figure. Determine the area of the gray triangle.
In triangle $ABC$, points $D, E$, and $F$ intersect one-third of the respective sides. Show that the sum of the areas of the three gray triangles is equal to the area of middle triangle.
A wine glass with a cross section as shown has the property of an orange in shape as a sphere with a radius of $3$ cm just can be placed in the glass without protruding above glass. Determine the height $h$ of the glass.
In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$. Determine the length of the hypotenuse
In a right-angled and isosceles triangle, the two catheti are both length $1$. Find the length of the shortest line segment dividing the triangle into two figures with the same area, and specify the location of this line segment
A trapezoid has side lengths as indicated in the figure (the sides with length $11$ and $36$ are parallel). Calculate the area of the trapezoid.
From the vertex $C$ in triangle $ABC$, draw a straight line that bisects the median from $A$. In what ratio does this line divide the segment $AB$?
In the plan, six circles are given so that none of the circles contain one the center of the other. Show that there is no point that lies in all the circles.
IIn triangle $ABC$, angle $C$ is right and the two catheti are both length $1$. For one given the choice of the point $P$ on the cathetus $BC$, the point $Q$ on the hypotenuse and the point $R$ are plotted on the second cathetus so that $PQ$ is parallel to $AC$ and $QR$ is parallel to $BC$. Thereby the triangle is divided into three parts. Determine the locations of point $P$ for which the rectangular part has a larger area than each of the other two parts.
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This year's gift idea from BabyMath consists of a series of nine colored plastic containers of decreasing size, alternating in shape like a cube and a sphere. All containers can open and close with a convenient hinge, and each container can hold just about anything next in line. The largest and smallest container are both cubes. Determine the relationship between the edge lengths of these cubes.
Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area.
About pentagon $ABCDE$ is known that angle $A$ and angle $C$ are right and that the sides $| AB | = 4$, $| BC | = 5$, $| CD | = 10$, $| DE | = 6$. Furthermore, the point $C'$ that appears by mirroring $C$ in the line $BD$, lies on the line segment $AE$. Find angle $E$.
In the figure shown, the small circles have radius $1$. Calculate the area of the gray part of the figure.
The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square.
1999 Denmark Mohr p1
In a coordinate system, a circle with radius $7$ and center is on the y-axis placed inside the parabola with equation $y = x^2$ , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.
2000 Denmark Mohr p1
2001 Denmark Mohr p3
2002 Denmark Mohr p1
2002 Denmark Mohr p4
The quadrilateral $ABCD$ is a square of sidelength $1$, and the points $E, F, G, H$ are the midpoints of the sides. Determine the area of quadrilateral $PQR$.
Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom?
A rectangular floor is covered by a certain number of equally large quadratic tiles. The tiles along the edge are red, and the rest are white. There are equally many red and white tiles. How many tiles can there be?
In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$.
Is it possible to place within a square an equilateral triangle whose area is larger than $9/ 20$ of the area of the square?
An interior point in a rectangle is connected by line segments to the midpoints of its four sides. Thus four domains (polygons) with the areas $a, b, c$ and $d$ appear (see the figure).
Prove that $a + c = b + d$.
In triangle $ABC$ we have $\angle C = 90^o$ and $AC = BC$. Furthermore $M$ is an interior pont in the triangle so that $MC = 1 , MA = 2$ and $MB =\sqrt2$. Determine $AB$
2003 Denmark Mohr p1
2004 Denmark Mohr p1
2005 Denmark Mohr p1
2005 Denmark Mohr p3
2006 Denmark Mohr p1
In a right-angled triangle, the sum $a + b$ of the sides enclosing the right angle equals $24$ while the length of the altitude $h_c$ on the hypotenuse $c$ is $7$. Determine the length of the hypotenuse.
Georg and his mother love pizza. They buy a pizza shaped as an equilateral triangle. Georg demands to be allowed to divide the pizza by a straight cut and then make the first choice. The mother accepts this reluctantly, but she wants to choose a point of the pizza through which the cut must pass. Determine the largest fraction of the pizza which the mother is certain to get by this procedure.
The width of rectangle $ABCD$ is twice its height, and the height of rectangle $EFCG$ is twice its width. The point $E$ lies on the diagonal $BD$. Which fraction of the area of the big rectangle is that of the small one?
This figure is cut out from a sheet of paper. Folding the sides upwards along the dashed lines, one gets a (non-equilateral) pyramid with a square base. Calculate the area of the base.
The point $P$ lies inside $\vartriangle ABC$ so that $\vartriangle BPC$ is isosceles, and angle $P$ is a right angle. Furthermore both $\vartriangle BAN$ and $\vartriangle CAM$ are isosceles with a right angle at $A$, and both are outside $\vartriangle ABC$. Show that $\vartriangle MNP$ is isosceles and right-angled.
The star shown is symmetric with respect to each of the six diagonals shown. All segments connecting the points $A_1, A_2, . . . , A_6$ with the centre of the star have the length $1$, and all the angles at $B_1, B_2, . . . , B_6$ indicated in the figure are right angles. Calculate the area of the star.
We consider an acute triangle $ABC$. The altitude from $A$ is $AD$, the altitude from $D$ in triangle $ABD$ is $DE,$ and the altitude from $D$ in triangle $ACD$ is $DF$.
a) Prove that the triangles $ABC$ and $AF E$ are similar.
b) Prove that the segment $EF$ and the corresponding segments constructed from the vertices $B$ and $C$ all have the same length.
2007 Denmark Mohr p1
2008 Denmark Mohr p4
Triangle $ABC$ lies in a regular decagon as shown in the figure.
What is the ratio of the area of the triangle to the area of the entire decagon?
Write the answer as a fraction of integers.
The figure shows a $60^o$ angle in which are placed $2007$ numbered circles (only the first three are shown in the figure). The circles are numbered according to size. The circles are tangent to the sides of the angle and to each other as shown. Circle number one has radius $1$. Determine the radius of circle number $2007$.
In triangle $ABC$ we have $AB = 2, AC = 6$ and $\angle A = 120^o$ . The bisector of angle $A$ intersects the side BC at the point $D$. Determine the length of $AD$. The answer must be given as a fraction with integer numerator and denominator.
In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$. If angle $D$ is $60^o$ and angle $E$ is $40^o$, how large is then angle $u$?
Let $E$ be an arbitrary point different from $A$ and $B$ on the side $AB$ of a square $ABCD$, and let $F$ and $G$ be points on the segment $CE$ so that $BF$ and $DG$ are perpendicular to $CE$. Prove that $DF = AG$.
Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown.
How large a fraction does the area of the small circle make up of that of the big one?
An equilateral triangle $ABC$ is given. With $BC$ as diameter, a semicircle is drawn outside the triangle. On the semicircle, points $D$ and $E$ are chosen such that the arc lengths $BD, DE$ and $EC$ are equal. Prove that the line segments $AD$ and $AE$ divide the side $BC$ into three equal parts.
In the octagon below all sides have the length $1$ and all angles are equal.
Determine the distance between the corners $A$ and $B$.
2012 Denmark Mohr p1
Inside a circle with radius $6$ lie four smaller circles with centres $A, B, C$ and $D$. The circles
touch each other as shown. The point where the circles with centres $A$ and $C$ touch each other is the centre of the big circle. Calculate the area of quadrilateral $ABCD$.
In the hexagon $ABCDEF$, all angles are equally large. The side lengths satisfy $AB = CD = EF = 3$ and $BC = DE = F A = 2$. The diagonals $AD$ and $CF$ intersect each other in the point $G$. The point $H$ lies on the side $CD$ so that $DH = 1$. Prove that triangle $EGH$ is equilateral.
2013 Denmark Mohr p3
2022 Denmark Mohr p3
2013 Denmark Mohr p3
The figure shows a rectangle, its circumscribed circle and four semicircles, which have the rectangle’s sides as diameters. Prove that the combined area of the four dark gray crescentshaped regions is equal to the area of the light gray rectangle.
The angle bisector of $A$ in triangle $ABC$ intersects $BC$ in the point $D$. The point $E$ lies on the side $AC$, and the lines $AD$ and $BE$ intersect in the point $F$. Furthermore, $\frac{|AF|}{|F D|}= 3$ and $\frac{|BF|}{|F E|}=\frac{5}{3}$. Prove that $|AB| = |AC|$.
2014 Denmark Mohr p3
The points $C$ and $D$ lie on a halfline from the midpoint $M$ of a segment $AB$, so that $|AC| = |BD|$. Prove that the angles $u = \angle ACM$ and $v = \angle BDM$ are equal.
2018 Denmark Mohr p2
The figure shows a large circle with radius $2$ m and four small circles with radii $1$ m. It is to be painted using the three shown colours. What is the cost of painting the figure?
2018 Denmark Mohr p5
In triangle $ABC$ the angular bisector from $A$ intersects the side $BC$ at the point $D$, and the angular bisector from $B$ intersects the side $AC$ at the point $E$. Furthermore $|AE| + |BD| = |AB|$. Prove that $\angle C = 60^o$
The points $C$ and $D$ lie on a halfline from the midpoint $M$ of a segment $AB$, so that $|AC| = |BD|$. Prove that the angles $u = \angle ACM$ and $v = \angle BDM$ are equal.
2015 Denmark Mohr p3
Triangle $ABC$ is equilateral. The point $D$ lies on the extension of $AB$ beyond $B$, the point $E$ lies on the extension of $CB$ beyond $B$, and $|CD| = |DE|$. Prove that $|AD| = |BE|$.
2016 Denmark Mohr p3
Prove that all quadrilaterals $ABCD$ where $\angle B = \angle D = 90^o$, $|AB| = |BC|$ and $|AD| + |DC| = 1$, have the same area.
2017 Denmark Mohr p3
The figure shows an arc $\ell$ on the unit circle and two regions $A$ and $B$.Prove that the area of $A$ plus the area of $B$ equals the length of $\ell$.
2018 Denmark Mohr p2
The figure shows a large circle with radius $2$ m and four small circles with radii $1$ m. It is to be painted using the three shown colours. What is the cost of painting the figure?
In triangle $ABC$ the angular bisector from $A$ intersects the side $BC$ at the point $D$, and the angular bisector from $B$ intersects the side $AC$ at the point $E$. Furthermore $|AE| + |BD| = |AB|$. Prove that $\angle C = 60^o$
2019 Denmark Mohr p5
In the figure below the triangles $BCD, CAE$ and $ABF$ are equilateral, and the triangle $ABC$ is right-angled with $\angle A = 90^o$. Prove that $|AD| = |EF|$.
2020 Denmark Mohr p2
A quadrilateral is cut from a piece of gift wrapping paper, which has equally wide white and gray stripes. The grey stripes in the quadrilateral have a combined area of $10$. Determine the area of the quadrilateral.
2021 Denmark Mohr p4
Given triangle $ABC$ with $|AC| > |BC|$. The point $M$ lies on the angle bisector of angle $C$, and $BM$ is perpendicular to the angle bisector. Prove that the area of triangle AMC is half of the area of triangle $ABC$.
2022 Denmark Mohr p1
The figure shows a glass prism which is partially filled with liquid. The surface of the prism consists of two isosceles right triangles, two squares with side length $10$ cm and a rectangle. The prism can lie in three different ways. If the prism lies as shown in figure $1$, the height of the liquid is $5$ cm.
The figure shows a glass prism which is partially filled with liquid. The surface of the prism consists of two isosceles right triangles, two squares with side length $10$ cm and a rectangle. The prism can lie in three different ways. If the prism lies as shown in figure $1$, the height of the liquid is $5$ cm.
a) What is the height of the liquid when it lies as shown in figure $2$?
b) What is the height of the liquid when it lies as shown in figure$ 3$?
The square $ABCD$ has side length $1$. The point $E$ lies on the side $CD$. The line through $A$ and $E$ intersects the line through $B$ and $C$ at the point $F$. Prove that $\frac{1}{|AE|^2}+\frac{1}{|AF|^2}= 1.$
In the $9$-gon $ABCDEFGHI$, all sides have equal lengths and all angles are equal. Prove that $|AB| + |AC| = |AE|$.
source: http://georgmohr.dk/
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