Norway 1993 - 2022 (Abel) 44p

geometry problems from Norwegian Math Olympiads (Abel Contest)
with aops links

1993 - 2021

1993 Norway Abel p1a
Let $ABCD$ be a convex quadrilateral and $A',B'C',D'$ be the midpoints of $AB,BC,CD,DA$, respectively. Let $a,b,c,d$ denote the areas of quadrilaterals into which lines $A'C'$ and $B'D'$ divide the quadrilateral $ABCD$ (where a corresponds to vertex $A$ etc.). Prove that $a+c = b+d$.
1993 Norway Abel p1b
Given a triangle with sides of lengths $a,b,c$, prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}< 2$

1994 Norway Abel p1a
In a half-ball of radius $3$ is inscribed a cylinder with base lying on the base plane of the half-ball, and another such cylinder with equal volume. If the base-radius of the first cylinder is $\sqrt3$, what is the base-radius of the other one?

1994 Norway Abel p1b
Let $C$ be a point on the prolongation of the diameter $AB$ of a circle. A line through $C$ is tangent to the circle at point $N$. The bisector of $\angle ACN$ meets the lines $AN$ and $BN$ at $P$ and $Q$ respectively. Prove that $PN = QN$.

1995 Norway Abel p2a
Two circles $k_1,k_2$ touch each other at $P$, and touch a line $\ell$ at $A$ and $B$ respectively. Line $AP$ meets $k_2$ at $C$. Prove that $BC$ is perpendicular to $\ell$.

1995 Norway Abel p2b
Two circles of the same radii intersect in two distinct points $P$ and $Q$. A line passing through $P$, not touching any of the circles, intersects the circles again at $A$ and $B$. Prove that $Q$ lies on the perpendicular bisector of $AB$.

1996 Norway Abel p1
Let $S$ be a circle with center $C$ and radius $r$, and let $P \ne C$ be an arbitrary point.
A line $\ell$ through $P$ intersects the circle in $X$ and $Y$. Let $Z$ be the midpoint of $XY$.
Prove that the points $Z$, as $\ell$ varies, describe a circle. Find the center and radius of this circle.

1997 Norway Abel p2a
Let $P$ be an interior point of an equilateral triangle $ABC$, and let $Q,R,S$ be the feet of perpendiculars from $P$ to $AB,BC,CA$, respectively. Show that the sum $PQ+PR+PS$ is independent of the choice of $P$.

1997 Norway Abel p2b
Let $A,B,C$ be different points on a circle such that $AB = AC$.
Point $E$ lies on the segment $BC$, and $D \ne A$ is the point of intersection of the circle and line $AE$. Show that the product $AE \cdot AD$ is independent of the choice of $E$.

1998 Norway Abel p4
Let $A,B,P$ be points on a line $\ell$, with $P$ outside the segment $AB$. Lines $a$ and $b$ pass through $A$ and $B$ and are perpendicular to $\ell$. A line $m$ through $P$, which is neither parallel nor perpendicular to $\ell$, intersects $a$ and $b$ at $Q$ and $R$, respectively. The perpendicular from $B$ to $AR$ meets $a$ and $AR$ at $S$ and $U$, and the perpendicular from $A$ to $BQ$ meets $b$ and $BQ$ at $T$ and $V$, respectively.
(a) Prove that $P,S,T$ are collinear.
(b) Prove that $P,U,V$ are collinear.

1999 Norway Abel p3
An isosceles triangle $ABC$ with $AB = AC$ and $\angle A = 30^o$ is inscribed in a circle with center $O$. Point $D$ lies on the shorter arc $AC$ so that $\angle DOC = 30^o$, and point $G$ lies on the shorter arc $AB$ so that $DG = AC$ and $AG < BG$. The line $BG$ intersects $AC$ and $AB$ at $E$ and $F$, respectively.
(a) Prove that triangle $AFG$ is equilateral.
(b) Find the ratio between the areas of triangles $AFE$ and $ABC$.

2000 Norway Abel p4
For some values of c, the equation $x^c + y^c = z^c$ can be illustrated geometrically.
For example, the case $c = 2$ can be illustrated by a right-angled triangle. By this we mean that, x, y, z is a solution of the equation $x^2 + y^2 = z^2$ if and only if there exists a right-angled triangle with catheters $x$ and $y$ and hypotenuse $z$.
In this problem we will look at the cases $c = -\frac{1}{2}$ and $c = - 1$.
a) Let $x, y$ and $z$ be the radii of three circles intersecting each other and a line, as shown, in the figure. Show that,
$x^{-\frac{1}{2}}+ y^{-\frac{1}{2}} = z^{-\frac{1}{2}}$

b) Draw a geometric figure that illustrates the case in a similar way, $c = - 1$. The figure must be able to be constructed with a compass and a ruler. Describe such a construction and prove that, in the figure, lines $x, y$ and $z$ satisfy $x^{-1}+ y^{-1} = z^{-1}$. (All positive solutions of this equation should be possible values for $x, y$, and $z$ on such a figure, but you don't have to prove that.)

2001 Norway Abel p3a
What is the largest possible area of a quadrilateral with sidelengths $1, 4, 7$ and $8$ ?

2001 Norway Abel p3b
The diagonals $AC$ and $BD$ in the convex quadrilateral $ABCD$ intersect in $S$. Let $F_1$ and $F_2$ be the areas of $\vartriangle  ABS$ and $\vartriangle CSD$. and let $F$ be the area of the quadrilateral $ABCD$. Show that $\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}$

2002 Norway Abel p3a
A circle with center in $O$ is given. Two parallel tangents tangent to the circle at points $M$ and $N$. Another tangent intersects the first two tangents at points $K$ and $L$. Show that the circle having the line segment $KL$ as diameter passes through $O$.

2002 Norway Abel p3b
Six line segments of lengths $17, 18, 19, 20, 21$ and $23$ form the side edges of a triangular pyramid (also called a tetrahedron). Can there exist a sphere tangent to all six lines?

2003 Norway Abel p3
Let $ABC$ be a triangle with $AC> BC$, and let $S$ be the circumscribed circle of the triangle. $AB$ divides $S$ into two arcs. Let $D$ be the midpoint of the arc containing $C$.
(a) Show that $\angle ACB +2 \cdot \angle ACD = 180^o$.
(b) Let $E$ be the foot of the altitude from $D$ on $AC$. Show that $BC +CE = AE$.

2004 Norway Abel p3
In a quadrilateral $ABCD$ with $\angle A = 60^o, \angle B = 90^o, \angle C = 120^o$, the point $M$ of intersection of the diagonals satisfies $BM = 1$ and $MD = 2$.
(a) Prove that the vertices of $ABCD$ lie on a circle and find the radius of that circle.
(b) Find the area of quadrilateral $ABCD$.

2005 Norway Abel p3a
In the isosceles triangle $\vartriangle ABC$ is $AB = AC$. Let $D$ be the midpoint of the segment $BC$. The points $P$ and $Q$ are respectively on the lines $AD$ and $AB$ (with $Q \ne B$) so that $PQ = PC$. Show that $\angle PQC =\frac12  \angle A$

2005 Norway Abel p3b
In the parallelogram $ABCD$, all sides are equal, and $\angle A = 60^o$. Let $F$ be a point on line $AD, H$ a point on line $DC$, and $G$ a point on diagonal $AC$ such that $DFGH$ is a parallelogram. Show that then $\vartriangle BHF$ is equilateral.

2006 Norway Abel p4
Let $\gamma$ be the circumscribed circle about a right-angled triangle $ABC$ with right angle $C$. Let $\delta$ be the circle tangent to the sides $AC$ and $BC$ and tangent to the circle $\gamma$ internally.
(a) Find the radius $i$ of $\delta$ in terms of $a$ when $AC$ and $BC$ both have length $a$.
(b) Show that the radius $i$ is twice the radius of the inscribed circle of $ABC$.

2007 Norway Abel p2
The vertices of a convex pentagon $ABCDE$ lie on a circle $\gamma_1$.
The diagonals $AC , CE, EB, BD$, and $DA$ are tangents to another circle $\gamma_2$ with the same centre as $\gamma_1$.
(a) Show that all angles of the pentagon $ABCDE$ have the same size and that all edges of the pentagon have the same length.
(b) What is the ratio of the radii of the circles $\gamma_1$ and $\gamma_2$? (The answer should be given in terms of integers, the four basic arithmetic operations and extraction of roots only.)

2008 Norway Abel p4a
Three distinct points $A, B$, and $C$ lie on a circle with centre at $O$. The triangles $AOB, BOC$ , and $COA$ have equal area. What are the possible magnitudes of the angles of the triangle $ABC$ ?

2008 Norway Abel p4b
A point $D$ lies on the side $BC$ , and a point $E$ on the side $AC$ , of the triangle $ABC$ , and $BD$ and $AE$ have the same length. The line through the centres of the circumscribed circles of the triangles $ADC$ and $BEC$ crosses $AC$ in $K$ and $BC$ in $L$. Show that $KC$ and $LC$ have the same length.

2009 Norway Abel p3a
In the triangle $ABC$ the edge $BC$ has length $a$, the edge $AC$ length $b$, and the edge $AB$ length $c$. Extend all the edges at both ends – by the length $a$ from the vertex $A, b$ from $B$, and $c$ from $C$. Show that the six endpoints of the extended edges all lie on a common circle.

2009 Norway Abel p3b
Show for any positive integer $n$ that there exists a circle in the plane such that there are exactly $n$ grid points within the circle. (A grid point is a point having integer coordinates.)

2010 Norway Abel p1a
The point $P$ lies on the edge $AB$ of a quadrilateral $ABCD$. The angles $BAD, ABC$ and $CPD$ are right, and $AB = BC + AD$. Show that $BC = BP$ or $AD = BP$.

2010 Norway Abel p1b

The edges of the square in the figure have length $1$.
Find the area of the marked region in terms of $a$, where $0 \le  a  \le 1$.

2011 Norway Abel p2a
In the quadrilateral $ABCD$ the side $AB$ has length $7, BC$ length $14, CD$ length $26$, and $DA$ length $23$. Show that the diagonals are perpendicular.

You may assume that the quadrilateral is convex (all internal angles are less than $180^o$).

2012 Norway Abel p2b
The diagonals $AD, BE$, and $CF$ of a convex hexagon $ABCDEF$ intersect in a common point.
Show that $a(ABE) a(CDA) a(EFC) = a(BCE) a(DEA) a(FAC)$,
where $a(KLM)$ is the area of the triangle $KLM$/

2013 Norway Abel p1
In a triangle $T$, all the angles are less than $90^o$, and the longest side has length $s$. Show that for every point $p$ in $T$ we can pick a corner $h$ in $T$ such that the distance from $p$ to $h$ is less than or equal to $s/\sqrt3$.

2014 Norway Abel p2
The points $P$ and $Q$ lie on the sides $BC$ and $CD$ of the parallelogram ABCD so that $BP = QD$. Show that the intersection point between the lines $BQ$ and $DP$ lies on the line bisecting $\angle BAD$.

2015 Norway Abel p3
The five sides of a regular pentagon are extended to lines $\ell_1, \ell_2, \ell_3, \ell_4$, and $\ell_5$.
Denote by $d_i$ the distance from a point $P$ to $\ell_i$. For which point(s) in the interior of the pentagon is the product $d_1d_2d_3d_4d_5$ maximal?

2016 Norway Abel p3a
Three circles $S_A, S_B$, and $S_C$ in the plane with centers in $A, B$, and $C$, respectively, are mutually tangential on the outside. The point of tangency between $S_A$ and $S_B$ we call $C'$, the one $S_A$ between $S_C$ we call $B'$, and the one between $S_B$ and $S_C$ we call $A'$. The common tangent between $S_A$ and $S_C$ (passing through B') we call $\ell_B$, and the common tangent between $S_B$ and $S_C$ (passing through $A'$) we call  $\ell_A$. The point of intersection of  $\ell_A$ and  $\ell_B$ is called $X$. The point $Y$ is located so that $\angle XBY$ and $\angle YAX$ are both right angles. Show that the points $X, Y$, and $C'$ lie on a line if and only if $AC = BC$.

2016 Norway Abel p3b
Let $ABC$ be an acute triangle with $AB < AC$. The points $A_1$ and $A_2$ are located on the line $BC$ so that $AA_1$ and $AA_2$ are the inner and outer angle bisectors at $A$ for the triangle $ABC$. Let $A_3$ be the mirror image $A_2$ with respect to $C$, and let $Q$ be a point on AA1 such that $\angle A_1QA_3 = 90^o$. Show that $QC // AB$.

2017 Norway Abel p4

Let $a > 0$ and $0 < \alpha <\pi$ be given. Let $ABC$ be a triangle with $BC = a$ and $\angle BAC = \alpha$ , and call the cicumcentre $O$, and the orthocentre $H$. The point $P$ lies on the ray from $A$ through $O$. Let $S$ be the mirror image of $P$ through $AC$, and $T$ the mirror image of $P$ through $AB$. Assume that $SATH$ is cyclic. Show that the length $AP$ depends only on $a$ and $\alpha$.

2018 Norway Abel p2
The circumcentre of a triangle $ABC$ is called $O$. The points $A',B'$ and $C'$ are the reflections of $O$ in $BC, CA$, and $AB$, respectively. Show that the three lines $AA' , BB'$, and $CC'$ meet in a common point.

2019 Norway Abel p3a
Three circles are pairwise tangent, with none of them lying inside another.
The centres of the circles are the corners of a triangle with circumference $1$.
What is the smallest possible value for the sum of the areas of the circles?

2019 Norway Abel p4

The diagonals of a convex quadrilateral $ABCD$ intersect at $E$. The triangles $ABE, BCE, CDE$ and $DAE$ have centroids $K,L,M$ and $N$, and orthocentres  $Q,R,S$ and $T$. Show that the quadrilaterals $QRST$ and $LMNK$ are similar.

2020 Norway Abel p4a

The midpoint of the side $AB$ in the triangle $ABC$ is called $C'$. A point on the side $BC$ is called $D$, and $E$ is the point of intersection of $AD$ and $CC'$. Assume that $AE/ED = 2$. Show that $D$ is the midpoint of $BC$.

2020 Norway Abel p4b

The triangle $ABC$ has a right angle at $A$. The centre of the circumcircle is called $O$, and the base point of the normal from $O$ to $AC$ is called $D$. The point $E$ lies on $AO$ with $AE = AD$. The angle bisector of $\angle CAO$ meets $CE$ in $Q$. The lines $BE$ and $OQ$ intersect in $F$. Show that the lines $CF$ and $OE$ are parallel.

2021 Norway Abel p4a

A tetrahedron $ABCD$ satisfies $\angle BAC=\angle CAD=\angle DAB=90^o$. Show that the areas of its faces satisfy the equation $area(BAC)^2 + area(CAD)^2 + area(DAB)^2 = area(BCD)^2$.

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2021 Norway Abel p4b

The tangent at $C$ to the circumcircle of triangle $ABC$ intersects the line through $A$ and $B$ in a point $D$. Two distinct points $E$ and $F$ on the line through $B$ and $C$ satisfy $|BE| = |BF | =\frac{||CD|^2 - |BD|^2|}{|BC|}$. Show that either $|ED| = |CD|$ or $|FD| = |CD|$.

2022 Norway Abel p2

(a) A triangle $ABC$ with circumcircle $\omega$ satisfies $|AB| > |AC|$. Points $X$ and $Y$ on $\omega$ are different from $A$, such that the line $AX$ passes through the midpoint of $BC, AY$ is perpendicular to $BC$, and $XY$ is parallel to $BC$. Find $\angle BAC.$

(b) Triangles $ABC$ and $DEF$ have pairwise parallel sides: $EF || BC, FD || CA,$

and $DE || AB.$ The line $m_{A}$ is the reflection of $EF$ through $BC$, similarly $m_{B}$ is the reflection of $FD$ through $CA,$ and $m_{C}$ the reflection of DE through $AB.$ Assume that the lines $m_{A}, m_{B}$ and $m_{C}$ meet in a common point. What is the ratio between the areas of triangles $ABC$ and $DEF?$

source:
https://abelkonkurransen.no/