geometry problems from Flanders Math Olympiads (VWO - Vlaamse Wiskunde Olympiade)
with aops links
with aops links
collected inside aops here
started in 1986
1986 - 2022
A circle with radius $R$ is divided into twelve equal parts. The twelve dividing points are connected with the centre of the circle, producing twelve rays. Starting from one of the dividing points a segment is drawn perpendicular to the next ray in the clockwise sense; from the foot of this perpendicular another perpendicular segment is drawn to the next ray, and the process is continued ad infinitum. What is the limit of the sum of these segments (in terms of $R$)?
To put a marble with radius $1$ cm in a cube it is obvious that the cube must have an edge with at least a length of $2$ cm. What is the minimum length of the edge of a cube which can contain two marbles of radius $1$ cm? Prove your answer.
Two parallel lines $a$ and $b$ meet two other lines $c$ and $d$. Let $A$ and $A'$ be the points of intersection of $a$ with $c$ and $d$, respectively. Let $B$ and $B'$ be the points of intersection of $b$ with $c$ and $d$, respectively. If $X$ is the midpoint of the line segment $A A'$ and $Y$ is the midpoint of the segment $BB'$, prove that
$$|XY| \le \frac{|AB|+|A'B'|}{2}.$$
A 3-dimensional cross is made up of 7 cubes, one central cube and 6 cubes that share a face with it. The cross is inscribed in a circle with radius 1. What's its volume?
1988 Flanders p4
Be $R$ a positive real number. If $R, 1, R+\frac12$ are triangle sides, call $\theta$ the angle between $R$ and $R+\frac12$ (in rad). Prove $2R\theta$ is between $1$ and $\pi$.
1989 Flanders p2
When drawing all diagonals in a regular pentagon, one gets an smaller pentagon in the middle. What's the ratio of the areas of those pentagons?
1990 Flanders p1
On the standard unit circle, draw 4 unit circles with centers [0,1],[1,0],[0,-1],[-1,0].
You get a figure as below, find the area of the colored part.
If $Area\left(\Delta ABC\right) = 2 \cdot Area\left(\Delta XYZ\right)$, find the ratio of $\frac{AX}{XB},\frac{BY}{YC},\frac{CZ}{ZA}$.
1992 Flanders p3
A sphere is inscribed in a cone having apothema equal to $A$. The tangent circle of the sphere and the cone, determines the upper end of a cylinder which is inscribed in the sphere. Assume that the total area of the cone (mantle as well as base) equals nine times the area of a great circle of the sphere. Assume also that the apothema of the cone is larger than half the perimeter of the base of the cone. Determine the height of the cylinder as a function of $A$.
1994 Flanders p2
A jeweler covers the diagonal of a unit square with small golden squares in the following way:
- the sides of all squares are parallel to the sides of the unit square
- for each neighbour is their sidelength either half or double of that square (squares are neighbour if they share a vertex)
- each midpoint of a square has distance to the vertex of the unit square equal to $\dfrac12, \dfrac14, \dfrac18, ...$ of the diagonal. (so real length: $\times \sqrt2$)
- all midpoints are on the diagonal
(a) What is the side length of the middle square?
(b) What is the total gold-plated area?
Let $b$ a line perpendicular to line $a_0$ at point $O$, and for $n \ge 0$, let $a_{n+1}$ be the bisector of the acute angle between the lines $a_n$ and $b$. Point $A_0$ with $OA_0 = 1$ is taken on $a_0$, and for all $n \ge0, A_{n+1}$ is the orthogonal projection of $A_n$ onto $a_{n+1}$.
Determine $\lim_{n\to + \infty} OA_n$.
1994 Flanders p3
Two regular tetrahedrons $A$ and $B$ are made with the 8 vertices of a unit cube. (this way is unique). What's the volume of $A\cup B$?
1995 Flanders p3
Points $A,B,C,D$ are on a circle with radius $R$. $|AC|=|AB|=500$, while the ratio between $|DC|, |DA|, |DB|$ is $1,5,7$. Find $R$.
1996 Flanders p1
In triangle $\Delta ADC$ we got $AD=DC$ and $D=100^\circ$.
In triangle $\Delta CAB$ we got $CA=AB$ and $A=20^\circ$.
Prove that $AB=BC+CD$.
1997 Flanders p3
$\Delta oa_1b_1$ is isosceles with $\angle a_1ob_1 = 36^\circ$. Construct $a_2,b_2,a_3,b_3,...$ as below, with $|oa_{i+1}| = |a_ib_i|$ and $\angle a_iob_i = 36^\circ$, Call the summed area of the first $k$ triangles $A_k$.
Let $S$ be the area of the isocseles triangle, drawn in - - -, with top angle $108^\circ$ and $|oc|=|od|=|oa_1|$, going through the points $b_2$ and $a_2$ as shown on the picture.
(yes, $cd$ is parallel to $a_1b_1$ there)
Show $A_k < S$ for every positive integer $k$.
Given a cube with edges of length 1, $e$ the midpoint of $[bc]$, and $m$ midpoint of the face $cdc_1d_1$, as on the figure. Find the area of intersection of the cube with the plane through the points $a,m,e$.
A billiard table. (see picture)
A white ball is on $p_1$ and a red ball is on $p_2$. The white ball is shot towards the red ball as shown on the pic, hitting 3 sides first. Find the minimal distance the ball must travel.
2000 Flanders p2
Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.
2001 Flanders p2
Consider a triangle and 2 lines that each go through a corner and intersects the opposing segment, such that the areas are as on the figure. Find the "?"
2002 Flanders p4
A lamp is situated at point $A$ and shines inside the cube. A (massive) square is hung on the midpoints of the 4 vertical faces. What's the area of its shadow?
Two circles $C_1$ and $C_2$ intersect at $S$.
The tangent in $S$ to $C_1$ intersects $C_2$ in $A$ different from $S$.
The tangent in $S$ to $C_2$ intersects $C_1$ in $B$ different from $S$.
Another circle $C_3$ goes through $A, B, S$.
The tangent in $S$ to $C_3$ intersects $C_1$ in $P$ different from $S$ and $C_2$ in $Q$ different from $S$. Prove that the distance $PS$ is equal to the distance $QS$.
Consider a triangle with side lengths 501m, 668m, 835m. How many lines can be drawn with the property that such a line halves both area and perimeter?
2004 Flanders p4
Each cell of a beehive is constructed from a right regular 6-angled prism, open at the bottom and closed on the top by a regular 3-sided pyramidical mantle. The edges of this pyramid are connected to three of the rising edges of the prism and its apex $T$ is on the perpendicular line through the center $O$ of the base of the prism (see figure). Let $s$ denote the side of the base, $h$ the height of the cell and $\theta$ the angle between the line $TO$ and $TV$.
(a) Prove that the surface of the cell consists of 6 congruent trapezoids and 3 congruent rhombi.
(b) the total surface area of the cell is given by the formula $6sh - \dfrac{9s^2}{2\tan\theta} + \dfrac{s^2 3\sqrt{3}}{2\sin\theta}$
Let $\triangle ABC$ be an equilateral triangle and let $P$ be a point on $\left[AB\right]$.
$Q$ is the point on $BC$ such that $PQ$ is perpendicular to $AB$. $R$ is the point on $AC$ such that $QR$ is perpendicular to $BC$. And $S$ is the point on $AB$ such that $RS$ is perpendicular to $AC$. $Q'$ is the point on $BC$ such that $PQ'$ is perpendicular to $BC$. $R'$ is the point on $AC$ such that $Q'R'$ is perpendicular to $AC$. And $S'$ is the point on $AB$ such that $R'S'$ is perpendicular to $AB$. Determine $\frac{|PB|}{|AB|}$ if $S=S'$.
2007 Flanders p2
Given is a half circle with midpoint $O$ and diameter $AB$. Let $Z$ be a random point inside the half circle, and let $X$ be the intersection of $OZ$ and the half circle, and $Y$ the intersection of $AZ$ and the half circle. If $P$ is the intersection of $BY$ with the tangent line in $X$ to the half circle, show that $PZ \perp BX$.
Let $ABCD$ be a square with side $10$. Let $M$ and $N$ be the midpoints of $[AB]$ and $[BC]$ respectively. Three circles are drawn: one with midpoint $D$ and radius $|AD|$, one with midpoint $M$ and radius $|AM|$, and one with midpoint $N$ and radius $|BN|$. The three circles intersect in the points $R, S$ and $T$ inside the square. Determine the area of $\triangle RST$.
A square with sides $1$ and four circles of radius $1$ considered each having a vertex of have the square as the center. Find area of the shaded part (see figure).
Consider a line segment $[AB]$ with midpoint $M$ and perpendicular bisector $m$. For each point$ X \ne M$ on m consider we are the intersection point $Y$ of the line $BX$ with the bisector from the angle $\angle BAX$. As $X$ approaches $M$, then approaches $Y$ to a point of $[AB]$. Which point?
A parallelogram with an angle of $60^o$ has $a$ as the longest side and a shortest side $b$. Let's take the perpendiculars down from the vertices of the obtuse angles to the longest diagonal, then it is divided into three equal parts. Determine the ratio $\frac{a}{b}$.
In a triangle $ABC$, $\angle B= 2\angle A \ne 90^o$ . The inner bisector of $B$ intersects the perpendicular bisector of $[AC]$ at a point $D$. Prove that $AB \parallel CD$.
The area of the ground plane of a truncated cone $K$ is four times as large as the surface of the top surface. A sphere $B$ is circumscribed in $K$, that is to say that $B$ touches both the top surface and the base and the sides. Calculate ratio volume $B :$ Volume $K$.
Given is a triangle $ABC$ and points $D$ and $E$, respectively on $] BC [$ and $] AB [$. $F$ it is intersection of lines $AD$ and $CE$. We denote as $| CD | = a, | BD | = b, | DF | = c$ and $| AF | = d$. Determine the ratio $\frac{| BE |}{|AE |}$ in terms of $a, b, c$ and $d$
In $\vartriangle ABC, \angle A = 66^o$ and $| AB | <| AC |$. The outer bisector in $A$ intersects $BC$ in $D$ and $| BD | = | AB | + | AC |$. Determine the angles of $\vartriangle ABC$.
Consider (in the plane) three concentric circles with radii $1, 2$ and $3$ and equilateral triangle $\Delta$ such that on each of the three circles is one vertex of $\Delta$ . calculate the length of the side of $\Delta$ .
(a) Prove the parallelogram law that says that in a parallelogram the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of the two diagonals.
(b) The edges of a tetrahedron have lengths $a, b, c, d, e$ and $f$. The three line segments connecting the centers of intersecting edges have lengths $x, y$ and $z$. Prove that
$4 (x^2 + y^2 + z^2) = a^2 + b^2 + c^2 + d^2 + e^2 + f^2$
Let $PQRS$ be a quadrilateral with $| P Q | = | QR | = | RS |$, $\angle Q= 110^o$ and $\angle R = 130^o$ . Determine $\angle P$ and $\angle S$ .
Consider two points $Y$ and $X$ in a plane and a variable point $P$ which is not on $XY$. Let the parallel line to $YP$ through $X$ intersect the internal angle bisector of $\angle XYP$ in $A$, and let the parallel line to $XP$ through $Y$ intersect the internal angle bisector of $\angle YXP$ in $B$. Let $AB$ intersect $XP$ and $YP$ in $S$ and $T$ respectively. Show that the product $|XS| \cdot |YT|$ does not depend on the position of $P$.
In the quadrilateral $ABCD$ is $AD \parallel BC$ and the angles $\angle A$ and $\angle D$ are acute. The diagonals intersect in $P$. The circumscribed circles of $\vartriangle ABP$ and $\vartriangle CDP$ intersect the line $AD$ again at $S$ and $T$ respectively. Call $M$ the midpoint of $[ST]$. Prove that $\vartriangle BCM$ is isosceles.
Three line segments divide a triangle into five triangles.
The area of these triangles is called $u, v, x,$ yand $z$, as in the figure.
(a) Prove that $uv = yz$.
(b) Prove that the area of the great triangle is at most $ \frac{xz}{y}$
On the
parabola $y = x^2$ lie three different points $P, Q$ and $R$. Their projections
$P', Q'$ and $R'$ on the $x$-axis are equidistant and equal to $s$ , i.e. $|
P'Q'| = | Q'R'| = s$. Determine the area of $\vartriangle PQR$ in terms of $s$
In triangle $\vartriangle ABC$, $\angle A = 50^o, \angle B = 60^o$ and $\angle C = 70^o$. The point $P$ is on the side $[AB]$ (with $P \ne A$ and $P \ne B$). The inscribed circle of $\vartriangle ABC$ intersects the inscribed circle of $\vartriangle ACP$ at points $U$ and $V$ and intersects the inscribed circle of $\vartriangle BCP$ at points $X$ and $Y$. The rights $UV$ and $XY$ intersect in $K$. Calculate the $\angle UKX$.
In the triangle $\vartriangle ABC$ we have $| AB |^3 = | AC |^3 + | BC |^3$. Prove that $\angle C> 60^o$ .
Two touching balls with radii $a$ and $b$ are enclosed in a cylindrical tin of diameter $d$ . Both balls hit the top surface and the shell of the cylinder. The largest ball also hits the bottom surface. Show that $\sqrt{d} =\sqrt{a} +\sqrt{b}$
In triangle $\vartriangle ABC$ holds $\angle A= 40^o$ and $\angle B = 20^o$ . The point $P$ lies on the line $AC$ such that $C$ is between $A$ and $P$ and $| CP | = | AB | - | BC |$. Calculate the $\angle CBP$.
The point $M$ is the center of a regular pentagon $ABCDE$. The point $P$ is an inner point of the line segment $[DM]$. The circumscribed circle of triangle $\vartriangle ABP$ intersects the side $[AE]$ at point $Q$ (different from $A$). The perpendicular from $P$ on $CD$ intersects the side $[AE] $ at point $S$. Prove that $PS$ is the bisector of $\angle APQ$.
Catherine lowers five matching wooden discs over bars placed on the vertices of a regular pentagon. Then she leaves five smaller congruent checkers these rods drop. Then she stretches a ribbon around the large discs and a second ribbon around the small discs. The first ribbon has a length of $56$ centimeters and the second one of $50$ centimeters. Catherine looks at her construction from above and sees an area demarcated by the two ribbons. What is the area of that area?
The points $A, B, C, D$ lie in that order on a circle. The segments $AC$ and $BD$ intersect at the point $P$. The point $B'$ lies on the line $AB$ such that $A$ is between $B$ and $B'$ and $|AB'| = |DP |$.The point $C'$ lies on the line $CD$ such that $D$ is between $C$ and $C'$ lies and $|DC' | = |AP|$. Prove that $\angle B'PC' = \angle ABD'$.
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