South Africa 1995 - 2021 (SAMO) 47p

geometry problems from South African Mathematical Olympiads  (SAMO)
with aops links in the names

SAMO Juniors 2015 - 1998 EN Questions , SAMO Juniors 2015 - 1998 EN Solutions
SAMO Seniors 2015 - 1997 EN Questions, SAMO Seniors 2015 - 1997 EN Solutions

1995 - 2021

1995 South African P2
$ABC$ is a triangle with $\hat{A}<\hat{C}$, and $D$ is the point on $BC$ such that $B\hat{A}D=A\hat{C}B$. The perpendicular bisectors of $AD$ and $AC$ intersect in the point $E$. Prove that $B\hat{A}E=90^\circ$.

1995 South African P4
Three circles, with radii $p$, $q$ and $r$ and centres $A$, $B$ and $C$ respectively, touch one another externally at points $D$, $E$ and $F$. Prove that the ratio of the areas of $\triangle DEF$ and $\triangle ABC$ equals $\frac{2pqr}{(p+q)(q+r)(r+p)}.$

1995 South African P5
The convex quadrilateral $ABCD$ has area $1$, and $AB$ is produced to $E$, $BC$ to $F$, $CD$ to $G$ and $DA$ to $H$, such that $AB=BE$, $BC=CF$, $CD=DG$ and $DA=AH$. Find the area of the quadrilateral $EFGH$.

1995 South African P7

The circumcircle of $\triangle ABC$ has radius $1$ and centre $O$ and $P$ is a point inside the triangle such that $OP=x$. Prove that $AP\cdot BP\cdot CP\le(1+x)^2(1-x),$ with equality if and only if $P=O$.

1996 South African P3
The sides of triangle $ABC$ has integer lengths. Given that $AC=6$ and $\angle BAC=120^\circ$, determine the lengths of the other two sides.

1996 South African P5
$ABC$ is a triangle with sides $1$, $2$ and $\sqrt3$. Determine the smallest possible area of an equilateral triangle with a vertex on each side of triangle $ABC$.

1997 South African P1
From an initial triangle $\Delta A_0B_0C_0$, a sequence of triangles $\Delta A_1B_1C_1$, $A_2B_2C_2$, ... is formed such that, at each stage, $A_{k + 1}$, $B_{k + 1}$ and $C_{k + 1}$ are the points where the incircle of $\Delta A_kB_kC_k$ touches the sides $B_kC_k$, $C_kA_k$ and $A_kB_k$ respectively.
(a) Express $\angle A_{k + 1}B_{k + 1}C_{k + 1}$ in terms of $\angle A_kB_kC_k$.
(b) Deduce that, as $k$ increases, $\angle A_kB_kC_k$ tends to $60^{\circ}$.

1997 South African P5
A circle and a point $P$ higher than the circle lie in the same vertical plane. A particle moves along a straight line under gravity from $P$ to a point $Q$ on the circle. Given that the distance travelled from $P$ in time $t$ is equal to $\dfrac{1}{2}gt^2 \sin{\alpha}$, where $\alpha$ is the angle of inclination of the line $PQ$ to the horizontal, give a geometrical characterization of the point $Q$ for which the time taken from $P$ to $Q$ is a minimum.

1998 South African P3
$A,\ B,\ C,\ D,\ E$ and $F$ lie (in that order) on the circumference of a circle. The chords $AD,\ BE$ and $CF$ are concurrent. $P,\ Q$ and $R$ are the midpoints of $AD,\ BE$ and $CF$ respectively. Two further chords $AG \parallel BE$ and $AH \parallel CF$ are drawn. Show that $PQR$ is similar to $DGH$.

1999 South African P2
$A,\ B,\ C$ and $D$ are points on a given straight line, in that order. Show how to construct a square $PQRS$, with all of $P,\ Q,\ R$ and $S$ on the same side of $AD$, such that $A,\ B,\ C$ and $D$ lie on $PQ,\ SR,\ QR$ and $PS$ produced respectively.

1999 South African P3
The bisector of $\angle{BAD}$ in the parallellogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$ respectively. Prove that the centre of the circle passing through the points $C,\ K$ and $L$ lies on the circle passing through the points $B,\ C$ and $D$.

2000 South African P4
$ABCD$ is a square of side 1. $P$ and $Q$ are points on $AB$ and $BC$ such that $\widehat{PDQ} = 45^{\circ}$. Find the perimeter of $\Delta PBQ$.

2001 South African P1
$ABCD$ is a convex quadrilateral with perimeter $p$. Prove that $\dfrac{1}{2}p < AC + BD < p.$
(A polygon is convex if all of its interior angles are less than $180^\circ$.)

2002 South African P1
Given a quadrilateral $ABCD$ such that $AB^2 + CD^2 = AD^2 + BC^2$, prove that $AC \perp BD$.

2002 South African P3
A small square $PQRS$ is contained in a big square. Produce $PQ$ to $A$, $QR$ to $B$, $RS$ to $C$ and $SP$ to $D$ so that $A$, $B$, $C$ and $D$ lie on the four sides of the large square in order, produced if necessary. Prove that $AC = BD$ and $AC \perp BD$.

2002 South African P5

In acute-angled triangle $ABC$, a semicircle with radius $r_a$ is constructed with its base on $BC$ and tangent to the other two sides. $r_b$ and $r_c$ are defined similarly. $r$ is the radius of the incircle of $ABC$. Show that $\frac{2}{r} = \frac{1}{r_a} + \frac{1}{r_b} + \frac{1}{r_c}.$

2003 South African P2
Given a parallelogram $ABCD$, join $A$ to the midpoints $E$ and $F$ of the opposite sides $BC$ and $CD$. $AE$ and $AF$ intersect the diagonal $BD$ in $M$ and $N$. Prove that $M$ and $N$ divide $BD$ into three equal parts.

2003 South African P4
In a given pentagon $ABCDE$, triangles $ABC$, $BCD$, $CDE$, $DEA$ and $EAB$ all have the same area. The lines $AC$ and $AD$ intersect $BE$ at points $M$ and $N$. Prove that $BM = EN$.

2004 South African P4
Let $A_1$ and $B_1$ be two points on the base $AB$ of isosceles triangle $ABC$ (with $\widehat{C}>60^\circ$) such that $\widehat{A_1CB_1}=\widehat{BAC}$. A circle externally tangent to the circumcircle of triangle $\triangle A_1B_1C$ is tangent also to rays $CA$ and $CB$ at points $A_2$ and $B_2$ respectively. Prove that $A_2B_2=2AB$.

2005 South African P4
The inscribed circle of triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$ respectively. Let $Q$ denote the other point of intersection of $AD$ and the inscribed circle. Prove that $EQ$ extended passes through the midpoint of $AF$ if and only if $AC = BC$.

2006 South African P2
Triangle $ABC$ has $BC=1$ and $AC=2$. What is the maximum possible value of $\hat{A}$.

2006 South African P4
In triangle $ABC$, $AB=AC$ and $B\hat{A}C=100^\circ$. Let $D$ be on $AC$ such that $A\hat{B}D=C\hat{B}D$. Prove that $AD+DB=BC$.

2007 South African P3
In acute-angled triangle $ABC$, the points $D,E,F$ are on sides $BC,CA,AB$, respectively such that $\angle AFE = \angle BFD, \angle FDB = \angle EDC, \angle DEC = \angle FEA$. Prove that $AD$ is perpendicular to $BC$.

2007 South African P4
Let $ABC$ be a triangle and $PQRS$ a square with $P$ on $AB$, $Q$ on $AC$, and $R$ and $S$ on $BC$. Let $H$ on $BC$ such that $AH$ is the altitude of the triangle from  $A$ to base $BC$. Prove that:
(a) $\frac{1}{AH} +\frac{1}{BC}=\frac{1}{PQ}$
(b) the area of $ABC$ is twice the area of $PQRS$ iff $AH=BC$

2008 South African P2
Let $ABCD$ be a convex quadrilateral with the property that $AB$ extended and $CD$ extended intersect at a right angle. Prove that $AC\cdot BD>AD\cdot BC$.

2008 South African P5
Triangle $ABC$ has orthocentre $H$. The feet of the perpendiculars from $H$ to the internal and external bisectors of $\hat{A}$ are $P$ and $Q$ respectively. Prove that $P$ is on the line that passes through $Q$ and the midpoint of $BC$. (Note: The ortohcentre of a triangle is the point where the three altitudes intersect.)

2009 South African P2
Let $ABCD$ be a rectangle and $E$ the reflection of $A$ with respect to the diagonal $BD$. If $EB = EC$, what is the ratio $\frac{AD}{AB}$ ?

2010 South African P2
Consider a triangle $ABC$ with $BC = 3$. Choose a point $D$ on $BC$ such that $BD = 2$. Find the value of $AB^2 + 2AC^2 - 3AD^2.$

2011 South African P6
In triangle $ABC$, the incircle touches $BC$ in $D$, $CA$ in $E$ and $AB$ in $F$. The bisector of $\angle BAC$ intersects $BC$ in $G$. The lines $BE$ and $CF$ intersect in $J$. The line through $J$ perpendicular to $EF$ intersects $BC$ in $K$. Prove that $\frac{GK}{DK}=\frac{AE}{CE}+\frac{AF}{BF}$

2012 South African P2
Let $ABCD$ be a square and $X$ a point such that $A$ and $X$ are on opposite sides of $CD$. The lines $AX$ and $BX$ intersect $CD$ in $Y$ and $Z$ respectively. If the area of $ABCD$ is $1$ and the area of $XYZ$ is $\frac{2}{3}$, determine the length of $YZ$

2012 South African P5
Let $ABC$ be a triangle such that $AB\neq AC$. We denote its orthocentre by $H$, its circumcentre by $O$ and the midpoint of $BC$ by $D$. The extensions of $HD$ and $AO$ meet in $P$. Prove that triangles $AHP$ and $ABC$ have the same centroid.

2013 South African P3
Let ABC be an acute-angled triangle and AD one of its altitudes (D on BC). The line through D parallel to AB is denoted by $l$, and t is the tangent to the circumcircle of ABC at A. Finally, let E be the intersection of $l$ and t. Show that CE and t are perpendicular to each other.

2013 South African P6
Let $ABC$ be an acute-angled triangle with $AC \neq BC$, and let $O$ be the circumcentre and $F$ the foot of the altitude through $C$. Furthermore, let $X$ and $Y$ be the feet of the perpendiculars dropped from $A$ and $B$ respectively to (the extension of) $CO$. The line $FO$ intersects the circumcircle of $FXY$ a second time at  $P$. Prove that $OP<OF$.

2014 South African P3
In obtuse triangle $ABC$, with the obtuse angle at $A$, let $D$, $E$, $F$ be the feet of the altitudes through $A$,  $B$, $C$ respectively. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle BAC$. Find the angles of the triangle.

2015 South African P1

Points $E$ and $F$ lie inside a square $ABCD$ such that the two triangles $ABF$ and $BCE$ are equilateral. Show that $DEF$ is an equilateral triangle.

2015 South African P4
Let $ABC$ be an acute-angled triangle with $AB < AC$, and let points $D$ and $E$ be chosen on the side $AC$ and $BC$ respectively in such a way that $AD = AE = AB$. The circumcircle of $ABE$ intersects the line $AC$ at $A$ and $F$ and the line $DE$ at $E$ and $P$. Prove that $P$ is the circumcentre of $BDF$.

2016 South African P3
The inscribed circle of triangle $ABC$, with centre $I$, touches sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$, respectively. Let $P$ be a point, on the same side of $FE$ as $A$, for which $\angle PFE = \angle BCA$ and $\angle PEF = \angle ABC$. Prove that $P$, $I$ and $D$ lie on a straight line.

2017 South African P2
Let $ABCD$ be a rectangle with side lengths $AB = CD = 5$ and $BC = AD = 10$. $W, X, Y, Z$ are points on $AB, BC, CD$ and $DA$ respectively chosen in such a way that $WXYZ$ is a kite, where $\angle ZWX$ is a right angle. Given that $WX = WZ = \sqrt{13}$ and $XY = ZY$, determine the length of $XY$.

2017 South African P5
Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $D$ be a point on segment $BC$ such that $\angle BAD = \angle DAC$, and let $M$ and $N$ be points on segments $BD$ and $CD$, respectively, such that $\angle MAD = \angle DAN$. Let $S, P$ and $Q$ (all different from $A$) be the intersections of the rays $AD$, $AM$ and $AN$ with $\Gamma$, respectively. Show that the intersection of $SM$ and $QD$ lies on $\Gamma$.

2018 South African P2
In a triangle $ABC$, $AB = AC$, and $D$ is on $BC$. A point $E$ is chosen on $AC$, and a point $F$ is chosen on $AB$, such that $DE = DC$ and $DF = DB$. It is given that $\frac{DC}{BD} = 2$ and $\frac{AF}{AE} = 5$. Determine that value of $\frac{AB}{BC}$.

2018 South African P4
Let $ABC$ be a triangle with circumradius $R$, and let $\ell_A, \ell_B, \ell_C$ be the altitudes through $A, B, C$ respectively. The altitudes meet at $H$. Let $P$ be an arbitrary point in the same plane as $ABC$. The feet of the perpendicular lines through $P$ onto $\ell_A, \ell_B, \ell_C$ are $D, E, F$ respectively. Prove that the areas of $DEF$ and $ABC$ satisfy the following equation: area $(DEF) = \frac{{PH}^2}{4R^2} \cdot$ area $(ABC)$.

2019 South African P3
Let $A$, $B$, $C$ be points on a circle whose centre is $O$ and whose radius is $1$, such that $\angle BAC = 45^\circ$. Lines $AC$ and $BO$ (possibly extended) intersect at $D$, and lines $AB$ and $CO$ (possibly extended) intersect at $E$. Prove that $BD \cdot CE = 2$.

2020 South African P2

Let $S$ be a square with sides of length $2$ and $R$ be a rhombus with sides of length $2$ and angles measuring $60^\circ$ and $120^\circ$. These quadrilaterals are arranged to have the same centre and the diagonals of the rhombus are parallel to the sides of the square. Calculate the area of the region on which the figures overlap.

2020 South African P5

Let $ABC$ be a triangle, and let $T$ be a point on the extension of $AB$ beyond $B$, and $U$ a point on the extension of $AC$ beyond $C$, such that $BT = CU$. Moreover, let $R$ and $S$ be points on the extensions of $AB$ and $AC$ beyond $A$ such that $AS = AT$ and $AR = AU$. Prove that $R$, $S$, $T$, $U$ lie on a circle whose centre lies on the circumcircle of $ABC$.

2021 South African P2

Let $PAB$ and $PBC$ be two similar right-angled triangles (in the same plane) with $\angle PAB = \angle PBC = 90^\circ$ such that $A$ and $C$ lie on opposite sides of the line $PB$. If $PC = AC$, calculate the ratio $\frac{PA}{AB}$.

2021 South African P4

Let $ABC$ be a triangle with $\angle ABC \neq 90^\circ$ and $AB$ its shortest side. Denote by $H$ the intersection of the altitudes of triangle $ABC$. Let $K$ be the circle through $A$ with centre $B$. Let $D$ be the other intersection of $K$ and $AC$. Let $K$ intersect the circumcircle of $BCD$ again at $E$. If $F$ is the intersection of $DE$ and $BH$, show that $BD$ is tangent to the circle through $D$, $F$, and $H$.

source for pdfs:erudits.com.ng