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Sweden 1962 - 2021 79p

  geometry problems from Swedish Mathematical Competitions [2nd / final round] with aops links

here x year stands for x  / x+1 competition, as x is the year it took place
e.g. 2006 stands for 2006-2007 year, but took place in 2006


1962 - 2021 

$ABCD$ is a square side $1$. $P$ and $Q$ lie on the side $AB$ and $R$ lies on the side $CD$. What are the possible values for the circumradius of $PQR$?

Find the largest cube which can be placed inside a regular tetrahedron with side $1$ so that one of its faces lies on the base of the tetrahedron.

The squares of a chessboard have side $4$. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?

A road has constant width. It is made up of finitely many straight segments joined by corners, where the inner corner is a point and the outer side is a circular arc. The direction of the straight sections is always between $NE$ ($45^o$) and $SSE$ ($157 1/2^o$). A person wishes to walk along the side of the road from point $A$ to point $B$ on the same side. He may only cross the street perpendicularly. What is the shortest route?

[figure missing]

Find the side lengths of the triangle $ABC$ with area $S$ and $\angle BAC = x$ such that the side $BC$ is as short as possible.

Points $H_1, H_2, ... , H_n$ are arranged in the plane so that each distance $H_iH_j \le 1$. The point $P$ is chosen to minimise $\max (PH_i)$. Find the largest possible value of $\max (PH_i)$ for $n = 3$. Find the best upper bound you can for $n = 4$.

The feet of the altitudes in the triangle $ABC$ are $A', B', C'$. Find the angles of $A'B'C'$ in terms of the angles $A, B, C$. Show that the largest angle in $A'B'C'$ is at least as big as the largest angle in $ABC$. When is it equal?
You are given a ruler with two parallel straight edges a distance $d$ apart. It may be used
(1) to draw the line through two points,
(2) given two points a distance $\ge d$ apart, to draw two parallel lines, one through each point,
(3) to draw a line parallel to a given line, a distance d away.
One can also (4) choose an arbitrary point in the plane, and (5) choose an arbitrary point on a line.
Show how to construct 
(A) the bisector of a given angle, and
(B) the perpendicular to the midpoint of a given line segment.

The vertices of a triangle are lattice points. There are no lattice points on the sides (apart from the vertices) and $n$ lattice points inside the triangle. Show that its area is $n + \frac12$. Find the formula for the general case where there are also $m$ lattice points on the sides (apart from the vertices).

Show that the sum of the squares of the sides of a quadrilateral is at least the sum of the squares of the diagonals. When does equality hold?

$6$ open disks in the plane are such that the center of no disk lies inside another. Show that no point lies inside all $6$ disks.

A $3\times 1$ paper rectangle is folded twice to give a square side $1$. The square is folded along a diagonal to give a right-angled triangle. A needle is driven through an interior point of the triangle, making $6$ holes in the paper. The paper is then unfolded. Where should the point be in order to maximise the smallest distance between any two holes?

$ABC$ is a triangle with $\angle A = 90^\circ$, $\angle B = 60^\circ$. The points $A_1$, $B_1$, $C_1$ on $BC$, $CA$, $AB$ respectively are such that $A_1B_1C_1$ is equilateral and the perpendiculars (to $BC$ at $A_1$, to $CA$ at $B_1$ and to $AB$ at $C_1$) meet at a point $P$ inside the triangle. Find the ratios $PA_1:PB_1:PC_1$.

$P_1$, $P_2$, $P_3$, $Q_1$, $Q_2$, $Q_3$ are distinct points in the plane. The distances $P_1Q_1$, $P_2Q_2$, $P_3Q_3$ are equal. $P_1P_2$ and $Q_2Q_1$ are parallel (not antiparallel), similarly $P_1P_3$ and $Q_3Q_1$, and $P_2P_3$ and $Q_3Q_2$. Show that $P_1Q_1$, $P_2Q_2$ and $P_3Q_3$ intersect in a point.

There is a point inside an equilateral triangle side $d$ whose distance from the vertices is $3, 4, 5$. Find $d$.

Two satellites are orbiting the earth in the equatorial plane at an altitude $h$ above the surface. The distance between the satellites is always $d$, the diameter of the earth. For which $h$ is there always a point on the equator at which the two satellites subtend an angle of $90^\circ$?

Find the sharpest inequalities of the form $a\cdot AB < AG < b\cdot AB$ and $c\cdot AB < BG < d\cdot AB$ for all triangles $ABC$ with centroid $G$ such that $GA > GB > GC$.

Find the smallest constant $c$ such that for every $4$ points in a unit square there are two a distance $\leq c$ apart.

$ABC$ is a triangle. $X$, $Y$, $Z$ lie on $BC$, $CA$, $AB$ respectively. Show that area $XYZ$ cannot be smaller than each of area $AYZ$, area $BZX$, area $CXY$.

Show that there is a point $P$ inside the quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal area. Show that $P$ must lie on one of the diagonals.

$ABC$ is a triangle with $AB = 33$, $AC = 21$ and $BC = m$, an integer. There are points $D$, $E$ on the sides $AB$, $AC$ respectively such that $AD = DE = EC = n$, an integer. Find $m$.

$C$, $C'$ are concentric circles with radii $R$, $R'$. A rectangle has two adjacent vertices on $C$ and the other two vertices on $C'$. Find its sides if its area is as large as possible.

Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$.What is the smallest possible radius?

Let $A$ and $B$ be two points inside a circle $C$. Show that there exists a circle that contains $A$ and $B$ and lies completely inside $C$.

Points $A,B,C$ with $AB = BC$ are given on a circle with radius $r$, and $D$ is a point inside the circle such that the triangle $BCD$ is equilateral. The line $AD$ meets the circle again at $E$. Show that $DE = r$.

In a rectangular coordinate system, $O$ is the origin and $A(a,0)$, $B(0,b)$ and $C(c,d)$ the vertices of a triangle. Prove that $AB+BC+CA \ge  2CO$.

The diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect at $O$. If $S_1$ and $S_2$ are the areas of triangles $AOB$ and $COD$ and S that of $ABCD$, show that $\sqrt{S_1} + \sqrt{S_2} \le \sqrt{S}$. Prove that equality holds if and only if $AB$ and $CD$ are parallel.

A circle of radius $R$ is divided into two parts of equal area by an arc of another circle. Prove that the length of this arc is greater than $2R$.

Let $a > b > c$ be sides of a triangle and $h_a,h_b,h_c$ be the corresponding altitudes.
Prove that $a+h_a > b+h_b > c+h_c$.

Six ducklings swim on the surface of a pond, which is in the shape of a circle with radius $5$ m. Show that at every moment, two of the ducklings swim on the distance of at most $5$ m from each other.

Let $ABCD$ be a regular tetrahedron. Find the positions of point $P$ on the edge $BD$ such that the edge $CD$ is tangent to the sphere with diameter $AP$.

The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$. $P$ is chosen to minimise $\sum PA_i$. Find the minimum.

$ABCD$ is a quadrilateral. The bisectors of $\angle A$ and $\angle B$ meet at $E$. The line through $E$ parallel to $CD$ meets $AD$ at $L$ and $BC$ at $M$. Show that $LM = AL + BM$.

Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original triangle.

A triangle has sides $a, b, c$ with longest side $c$, and circumradius $R$. Show that if $a^2 + b^2 = 2cR$, then the triangle is right-angled.

A triangle with sides $a,b,c$ and perimeter $2p$ is given. Is possible, a new triangle with sides $p-a$, $p-b$, $p-c$ is formed. The process is then repeated with the new triangle. For which original triangles can this process be repeated indefinitely?

In the triangle $ABC$, the medians from $B$ and $C$ are perpendicular. Show that $\cot B + \cot C \ge  \frac23$.

The vertex $B$ of the triangle $ABC$ lies in the plane $P$. The plane of the triangle meets the plane in a line $L$. The angle between $L$ and $AB$ is a, and the angle between $L$ and $BC$ is $b$. The angle between the two planes is $c$. Angle $ABC$ is $90^o$. Show that $\sin^2c = \sin^2a + \sin^2b$.

On a circle with center $O$ and radius $r$ are given points $A,B,C,D$ in this order such that $AB, BC$ and $CD$ have the same length $s$ and the length of $AD$ is $s+ r$.Assume that $s < r$. Determine the angles of quadrilateral $ABCD$.

Through an arbitrary point inside a triangle, lines parallel to the sides of the triangle are drawn, dividing the triangle into three triangles with areas $T_1,T_2,T_3$ and three parallelograms. If $T$ is the area of the original triangle, prove that
$T=(\sqrt{T_1}+\sqrt{T_2}+\sqrt{T_3})^2$
The angles at $A,B,C,D,E$ of a pentagon $ABCDE$ inscribed in a circle form an increasing sequence. Show that the angle at $C$ is greater than $\pi/2$, and that this lower bound cannot be improved.

Let $AC$ be a diameter of a circle and $AB$ be tangent to the circle. The segment $BC$ intersects the circle again at $D$. Show that if $AC = 1$, $AB = a$, and $CD = b$, then$$\frac{1}{a^2+ \frac12 }< \frac{b}{a}< \frac{1}{a^2}$$

Let $D$ be the point on side $AC$ of a triangle $ABC$ such that $BD$ bisects $\angle B$, and $E$ be the point on side $AB$ such that $3\angle  ACE = 2\angle  BCE$. Suppose that $BD$ and $CE$ intersect at a point $P$ with $ED = DC = CP$. Determine the angles of the triangle

$ABC$ is a triangle. Show that $c \ge (a+b) \sin \frac{C}{2}$

$ABCD$ is a quadrilateral with $\angle A = 90o$, $AD = a$, $BC = b$, $AB = h$, and area $\frac{(a+b)h}{2}$. What can we say about $\angle B$?

Circle $C$ center $O$ touches externally circle $C'$ center $O'$. A line touches $C$ at $A$ and $C'$ at $B$. $P$ is the midpoint of $AB$. Show that $\angle OPO' = 90^o$.

An equilateral triangle of side $x$ has its vertices on the sides of a square side $1$. What are the possible values of $x$?

The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.

Show that if $b = \frac{a+c}{2}$ in the triangle $ABC$, then $\cos (A-C) + 4 \cos B = 3$.

$ABC$ is a triangle. A circle through $A$ touches the side $BC$ at $D$ and intersects the sides $AB$ and $AC$ again at $E, F$ respectively. $EF$ bisects $\angle AFD$ and $\angle ADC = 80^o$. Find $\angle ABC$.

A tetrahedron has five edges of length $3$ and circumradius $2$. What is the length of the sixth edge?

Given two positive numbers $a, b$, how many non-congruent plane quadrilaterals are there such that $AB = a$, $BC = CD = DA = b$ and $\angle B = 90^o$ ?

Two circles in the plane, both of radius $R$, intersect at a right angle. Compute the area of the intersection of the interiors of the two circles.

Prove that every convex $n$-gon of area $1$ contains a quadrilateral of area at least $\frac12 $.

In a triangle $ABC$ the bisectors of angles $A$ and $C$ meet the opposite sides at $D$ and $E$ respectively. Show that if the angle at $B$ is greater than $60^\circ$, then $AE +CD <AC$.

A regular tetrahedron of edge length $1$ is orthogonally projected onto a plane. Find the largest possible area of its image.

In a triangle $ABC$, point $P$ is the incenter and $A'$, $B'$, $C'$ its orthogonal projections on $BC$, $CA$, $AB$, respectively. Show that $\angle B'A'C'$ is acute.

In the plane, a triangle is given. Determine all points $P$ in the plane such that each line through $P$ that divides the triangle into two parts with the same area must pass through one of the corners of the triangle.

2008 Sweden p1
A rhombus is inscribed in a convex quadrilateral. The sides of the rhombus are parallel with the diagonals of the quadrilateral, which have the lengths $d_1$ and $d_2$. Calculate the length of side of the rhombus , expressed in terms of $d_1$ and $d_2$.

2009 Sweden p1
Five square carpets have been bought for a square hall with a side of $6$ m , two with the side $2$ m, one with the side $2.1$ m and two with the side $2.5$ m. Is it possible to place the five carpets so that they do not overlap in any way each other? The edges of the carpets do not have to be parallel to the cradles in the hall.

2009 Sweden p5
A semicircular arc and a diameter $AB$ with a length of $2$ are given. Let $O$ be the midpoint of the diameter. On the radius perpendicular to the diameter, we select a point $P$ at the distance $d$ from the midpoint of the diameter $O$, $0 <d <1$. A line through $A$ and $P$ intersects the semicircle at point $C$. Through point $P$ we draw another line at right angle against $AC$ that intersects the semicircle at point $D$.  Through point $C$ we draw a line $l_1$, parallel to $PD$ and then a line $l_2$, through $D$ parallel to $PC$. The lines $l_1$ and $l_2$ intersect at point $E$. Show that the distance between $O$ and $E$ is equal to $\sqrt{2- d^2}$

2010 Sweden p1
Exists a triangle whose three altitudes have lengths $1, 2$ and $3$ respectively?

2010 Sweden p5
Consider the number of triangles where the side lengths $a,b,c$ satisfy $(a + b + c) (a + b -c) = 2b^2$. Determine the angles in the triangle for which the angle opposite the side with the length $a$  is as big as possible.

Given a triangle $ABC$, let $P$ be a point inside the triangle such that $| BP | > | AP |, | BP | > | CP |$. Show that $\angle ABC <90^o$

The catheti $AC$ and $BC$ in a right-angled triangle $ABC$ have lengths $b$ and $a$, respectively. A circle centered at $C$ is tangent to hypotenuse $AB$ at point $D$. The tangents to the circle through points $A$ and $B$ intersect the circle at points $E$ and $F$, respectively (where $E$ and $F$ are both different from $D$). Express the length of the section $EF$ in terms of $a$ and $b$.

A circle is inscribed in an trapezoid. Show that the diagonals of the trapezoid intersect at a point on the diameter of the circle perpendicular to the two parallel sides.

The paper folding art origami is usually performed with square sheets of paper. Someone folds the sheet once along a line through the center of the sheet in orde to get a nonagon. Let $p$ be the perimeter of the nonagon minus the length of the fold, i.e. the total length of the eight sides that are not folds, and denote by s the original side length of the square. Express the area of the nonagon in terms of $p$ and $s$.

2013 Sweden p4 A robotic lawnmower is located in the middle of a large lawn. Due a manufacturing defect, the robot can only move straight ahead and turn in directions that are multiples of $60^o$.  A fence must be set up so that it delimits the entire part of the lawn that the robot can get to, by traveling along a curve with length no more than $10$ meters from its starting position, given that it is facing north when it starts. How long must the fence be?

2014 Sweden p2
Three circles that touch each other externally have all their centers on one fourth circle with radius $R$. Show that the total area of the three circle disks is smaller than $4\pi R^2$.

2015 Sweden p1
Given the acute triangle $ABC$. A diameter of the circumscribed circle of the triangle intersects the sides $AC$ and $BC$, dividing the side $BC$ in half. Show that the same diameter divides the side $AC$ in a ratio of $1: 3$, calculated from $A$, if and only if  $\tan B = 2 \tan C$.

In a garden there is an $L$-shaped fence, see figure. You also have at your disposal two finished straight fence sections that are $13$ m and $14$ m long respectively. From point $A$ you want to delimit a part of the garden with an area of at least $200$ m$^2$ . Is it possible to do this?
2016 Sweden p3
The quadrilateral $ABCD$ is an isosceles trapezoid, where $AB\parallel CD$. The trapezoid is inscribed in a circle with radius $R$ and center on side $AB$. Point $E$ lies on the circumscribed circle and is such that  $\angle DAE = 90^o$. Given that  $\frac{AE}{AB}=\frac34$, calculate the length of the sides of the isosceles trapezoid.

Given the segments $AB$ and $CD$ not necessarily on the same plane. Point $X$ is the midpoint of the segment $AB$, and the point $Y$ is the midpoint of $CD$. Given that point $X$ is not on line $CD$, and that point $Y$ is not on line $AB$, prove that  $2 | XY | \le  | AD | + | BC |$. When is equality achieved?

Let $D$ be the foot of the altitude towards $BC$ in the triangle $ABC$. Let $E$ be the intersection of $AB$ with the bisector of angle $\angle C$. Assume that the angle $\angle AEC = 45^o$ . Determine the angle $\angle EDB$.

Let the $ABCD$ be a quadrilateral without parallel sides, inscribed in a circle. Let $P$ and $Q$ be the points of intersection between the lines containing the quadrilateral opposite sides. Show that the bisectors of the angles at $P$ and $Q$ are parallel to the bisectors of the angles at the point of intersection of the diagonals of the quadrilateral.

In a triangle $ABC$, two lines are drawn that together trisect the angle at $A$. These intersect the side $BC$ at points $P$ and $Q$ so that $P$ is closer to $B$ and $Q$ is closer to $C$. Determine the smallest constant $k$ such that $| P Q | \le k (| BP | + | QC |)$, for all such triangles. Determine if there are triangles for which equality applies.

Segment $AB$ is the diameter of a circle. Points $C$ and $D$ lie on the circle. The rays $AC$ and $AD$ intersect the tangent to the circle at point $B$ at points $P$ and $Q$, respectively. Show that points $C, D, P$ and $Q$ lie on a circle.

The medians of the sides $AC$ and $BC$ in the triangle $ABC$ are perpendicular to each other. Prove that  $\frac12 <\frac{|AC|}{|BC|}<2$.

In a triangle, both the sides and the angles form arithmetic sequences. Determine the angles of the triangle.

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