geometry problems from Australian Mathematical Olympiads (AMO)

with aops links in the names

Let $ABC$ be a triangle with $\angle BAC < 90^o$. Let $k$ be the circle through $A$ that is tangent to $BC$ at $C$. Let $M$ be the midpoint of $BC$, and let $AM$ intersect $k$ a second time at $D$. Finally, let $BD$ (extended) intersect $k$ a second time at $E$. Prove that $\angle BAC =\angle CAE$.

Let $\Gamma$ be a fixed circle with centre $O$ and radius $r$. Let $B$ and $C$ be distinct fixed points on $\Gamma$ . Let $A$ be a variable point on $\Gamma$ , distinct from $B$ and $C$. Let $P$ be the point such that the midpoint of $OP$ is $A$. The line through $O$ parallel to $AB$ intersects the line through $P$ parallel to $AC$ at the point $D$.

(a) Prove that, as $A$ varies over the points of the circle $\Gamma$ (other than $B$ or $C$), $D$ lies on a fixed circle whose radius is greater than or equal to $r$.

(b) Prove that equality occurs in part (a) if and only if $BC$ is a diameter of $\Gamma$ .

2015 Australian MO P5

Let $ABC$ be a triangle with $ACB = 90^o$. The points $D$ and $Z$ lie on the side $AB$ such that $CD$ is perpendicular to $AB$ and $AC = AZ$. The line that bisects $BAC$ meets $CB$ and $CZ$ at $X$ and $Y$ , respectively. Prove that the quadrilateral $BXYD$ is cyclic.

2017 Australian MO P4

2017 Australian MO P6

The circles $K_1$ and $K_2$ intersect at two distinct points $A$ and $M$. Let the tangent to $K_1$ at $A$ meet $K_2$ again at $B$, and let the tangent to $K_2$ at $A$ meet $K_1$ again at $D$. Let $C$ be the point such that $M$ is the midpoint of $AC$. Prove that the quadrilateral$ ABCD$ is cyclic.

2018 Ausralian MO P3

Let $ABCDEFGHIJKLMN$ be a regular tetradecagon.

Prove that the three lines $AE, BG$ and $CK$ intersect at a point.

(A regular tetradecagon is a convex polygon with $14$ sides, such that all sides have the same length and all angles are equal.)

2018 Australian MO P7

with aops links in the names

Mathematics Contests - The Australian Scene with solutions and bookmarks:

2014 - 2018

2014 Australian MO P2Let $ABC$ be a triangle with $\angle BAC < 90^o$. Let $k$ be the circle through $A$ that is tangent to $BC$ at $C$. Let $M$ be the midpoint of $BC$, and let $AM$ intersect $k$ a second time at $D$. Finally, let $BD$ (extended) intersect $k$ a second time at $E$. Prove that $\angle BAC =\angle CAE$.

Let $ABC$ be a triangle. Let $P$ and $Q$ be points on the sides $AB$ and $AC$, respectively, such that $BC$ and $PQ$ are parallel. Let $D$ be a point inside triangle $APQ$. Let $E$ and $F$ be the intersections of $PQ$ with $BD$ and $CD$, respectively. Finally, let $O_E$ and $O_F$ be the circumcentres of triangle $DEQ$ and triangle $DFP$, respectively. Prove that $O_EO_F$ is perpendicular to $AD$.

2015 Australian MO P4Let $\Gamma$ be a fixed circle with centre $O$ and radius $r$. Let $B$ and $C$ be distinct fixed points on $\Gamma$ . Let $A$ be a variable point on $\Gamma$ , distinct from $B$ and $C$. Let $P$ be the point such that the midpoint of $OP$ is $A$. The line through $O$ parallel to $AB$ intersects the line through $P$ parallel to $AC$ at the point $D$.

(a) Prove that, as $A$ varies over the points of the circle $\Gamma$ (other than $B$ or $C$), $D$ lies on a fixed circle whose radius is greater than or equal to $r$.

(b) Prove that equality occurs in part (a) if and only if $BC$ is a diameter of $\Gamma$ .

2015 Australian MO P5

Let $ABC$ be a triangle with $ACB = 90^o$. The points $D$ and $Z$ lie on the side $AB$ such that $CD$ is perpendicular to $AB$ and $AC = AZ$. The line that bisects $BAC$ meets $CB$ and $CZ$ at $X$ and $Y$ , respectively. Prove that the quadrilateral $BXYD$ is cyclic.

2016 Australian MO P1

Let $ABC$ be a triangle. A circle intersects side $BC$ at points $U$ and $V$ , side $CA$ at points $W$ and $X$, and side $AB$ at points $Y$ and $Z$. The points $U, V,W,X,Y,Z$ lie on the circle in that order. Suppose that $AY = BZ$ and $BU = CV$ . Prove that $CW = AX$.

Let $ABC$ be a triangle. A circle intersects side $BC$ at points $U$ and $V$ , side $CA$ at points $W$ and $X$, and side $AB$ at points $Y$ and $Z$. The points $U, V,W,X,Y,Z$ lie on the circle in that order. Suppose that $AY = BZ$ and $BU = CV$ . Prove that $CW = AX$.

2016 Australian MO P8

Three given lines in the plane pass through a point $P$.

(a) Prove that there exists a circle that contains $P$ in its interior and intersects the three lines at six points $A,B,C,D,E, F$ in that order around the circle such that $AB = CD = EF$.

(b) Suppose that a circle contains $P$ in its interior and intersects the three lines at six points $A,B, C,D,E, F$ in that order around the circle such that $AB = CD = EF$. Prove that:

$\frac{1}{2}$ area (hexagon $ABCDEF$) $\ge $ area( $\triangle APB$) + area( $\triangle CPD$) $+$ area($ \triangle EPF$).

Three given lines in the plane pass through a point $P$.

(a) Prove that there exists a circle that contains $P$ in its interior and intersects the three lines at six points $A,B,C,D,E, F$ in that order around the circle such that $AB = CD = EF$.

(b) Suppose that a circle contains $P$ in its interior and intersects the three lines at six points $A,B, C,D,E, F$ in that order around the circle such that $AB = CD = EF$. Prove that:

$\frac{1}{2}$ area (hexagon $ABCDEF$) $\ge $ area( $\triangle APB$) + area( $\triangle CPD$) $+$ area($ \triangle EPF$).

2017 Australian MO P4

Suppose that $S$ is a set of $2017$ points in the plane that are not all collinear. Prove that $S$ contains three points that form a triangle whose circumcentre is not a point in $S$.

2017 Australian MO P6

The circles $K_1$ and $K_2$ intersect at two distinct points $A$ and $M$. Let the tangent to $K_1$ at $A$ meet $K_2$ again at $B$, and let the tangent to $K_2$ at $A$ meet $K_1$ again at $D$. Let $C$ be the point such that $M$ is the midpoint of $AC$. Prove that the quadrilateral$ ABCD$ is cyclic.

2018 Ausralian MO P3

Let $ABCDEFGHIJKLMN$ be a regular tetradecagon.

Prove that the three lines $AE, BG$ and $CK$ intersect at a point.

(A regular tetradecagon is a convex polygon with $14$ sides, such that all sides have the same length and all angles are equal.)

2018 Australian MO P7

Let $P, Q$ and R be three points on a circle C, such that $PQ = PR$ and $PQ > QR$. Let $D$ be the circle with centre $P$ that passes through $Q$ and $R$. Suppose that the circle with centre $Q$ and passing through $R$ intersects $C$ again at $X$ and $D$ again at $Y$ . Prove that $P, X$ and $Y$ lie on a line.

Two circles in a plane intersect. Let $A$ and $B$ be the two points of intersection. Starting simultaneously from $A$ two points $P$ and $Q$ move with constant speeds around different circles, each point travelling along its own circle in the same sense as the other point. The two points return to $A$ simultaneously after one revolution. Prove

(i) $P$, $B$ and $Q$ are always collinear (on the same straight line);

(ii) that there is a fixed point $S$ in the plane such that, at any time, the distances from $S$ to the moving points are equal.

it did not take place in 1980

1981 Australian MO P3

$O$ is the midpoint of the base $BC$ of an isosceles triangle $ABC$. A circle is drawn with centre $O$ and tangent to the equal sides $AB, AC. P$ is a point on $AB, Q$ is a point on $AC$.

If $PQ$ is also a tangent to this circle, prove that $PB \cdot CQ =\left(\frac 12 BC \right)^2$

Discuss the converse of this result.

1982 Australian MO P3

Let $\triangle ABC$ be a triangle and let the internal bisector of the angle $\angle BAC$ intersects the circumcircle again at $P$. Similarly define $Q$ and $R$. Prove that $AP+BQ+CR>AB+BC+CA$.

1983 Australian MO P2

$ABC$ is a triangle and $P$ is a point inside it such that $\angle PAC=\angle PBC$. The perpendiculars from $P$ to $BC$ and $CA$ meet these sides at $L$ and $N$ respectively, and $D$ is the midpoint of $AB$. Prove that $DL = DM$.

1983 Australian MO P6

The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$ and $\angle CAB = \angle C_1AB_1$. $M$ is the point of intersection of the lines $BC_1$ and $CB_1$. Prove that if the lines $AM$ and $CC_1$ exist, then they are perpendicular.

1984 Australian MO P2

Given an equilateral triangle $ABC$, draw a semicircle on $BC$ as diameter on the side of $BC$ remote from $A$. Let the points $P, Q$ trisect the interval $BC$ and $AP, AQ$ produced cut the arc in $K, L$ respectively. Prove that $K, L$ trisect the semicircular arc.

1984 Australian MO P5

On the edges of a triangle $ABC$ are drawn three similar isosceles triangles $APB$ (with $AP = PB$), $AQC$ (with $AQ = QC$) and $BRC$ (with $BR = RC$). The triangles $APB$ and $AQC$ lie outside the triangle $ABC$ and the triangle $BRC$ is lying on the same side of the line $BC$ as the triangle $ABC$. Prove that the quadrilateral $PAQR$ is a parallelogram.

1985 Australian MO P4

$ABC$ is a triangle whose angles are smaller than $120^0$. Equilateral triangles $AFB$, $BDC$ and $CEA$ are constructed on the sides of and exterior to $ABC$.

(a) Prove that the lines $AD, BE$ and $CF$ pass through the one point $S$

(b) Prove that $SD+SE+SF=2(SA+SB+SC)$.

1986 Australian MO P3

$ABC$ is a triangle. The internal bisector of the angle $A$ meets the circumcircle again at $P. Q$ and $R$ are similarly defined with B and C respectively.

Prove that $AP + BQ + CR > AB + BC + CA$.

1986 Australian MO P4 (ILL 1979-48)

In the plane a circle $C$ of unit radius is given. For any line $l$, a number $s(l)$ is defined in the following way: If $l$ and $C$ intersect in two points, $s(l)$ is their distance; otherwise, $s(l) = 0$. Let $P$ be a point at distance $r$ from the center of $C$. One defines $M(r)$ to be the maximum value of the sum $s(m) + s(n)$, where $m$ and $n$ are variable mutually orthogonal lines through $P$. Determine the values of $r$ for which $M(r) > 2$.

1987 Australian MO P1

$GKA$ is an isosceles triangle with base $GK$ of length $2b$. $GA$ and $AK$ each have length $a$. Let $C$ be the midpoint of AK and $z$ be the circumcircle of the triangle $GCK$. Let $Y$ be the point on the extension of $AK$ such that if $E$ is the intersection of $YG$ with $z$ then $EY$ is of length $a/2$. Prove that if $x$ is the length of $EC$ and $y$ is the length of $KY$ then $ay = x^2$ and $xb = y^2$.

1987 Australian MO P4

In the interior of the triangle $ABC$, points $O$ and $P$ are chosen such that angles $ABO$ and $CBP$ are equal, and angles $BCO$ and $ACP$ are also equal. Prove that angles $CAO$ and $BAP$ are equal.

1988 Australian MO P2

The triangles $ABC$ and $AEF$ are in the same plane. Between them, the following conditions hold:

(1) The midpoint of $AB$ is $E$,

(2) The points $A, G$ and $F$ are on the same line,

(3) There is a point $C$ at which $BG$ and $EF$ intersect,

(4) $CE = 1$ and $AC = AE = FG$.

Show that if $AG = x$ then $AB = x^3$.

1989 Australian MO P2

Suppose $BP$ and $CQ$ are the bisectors of the angles $B, C$ of triangle $ABC$ and suppose $AH, AK$ are the perpendiculars from $A$ to $BP, CQ$. Prove that $KH$ is parallel to $BC$.

1989 Australian MO P6

Four rods $AB, BC, CD, DA$ are freely jointed at $A, B, C$ and $D$ and move in a plane so that the shape of the quadrilateral can be varied. $P, Q$ and $R$ are the mid-points of $AB, BC$ and $CD$ respectively. In one position of the rods, the angle $PQR$ is acute. Show that this angle remains acute no matter how the shape of $ABCD$ is changed.

1989 Australian MO P8

Points $X, Y$ and $Z$ on sides $BC, CA$ and $AB$ respectively of triangle $ABC$ are such that triangles $ABC$ and $XYZ$ are similar, the angles at $X, Y$ and $Z$ being equal to those at $A, B$ and $C$ respectively. Find $X, Y$ and $Z$ so that triangle $XYZ$ has minimum area.

1990 Australian MO P3

Let $ABC$ be a triangle and $k_1$ be a circle through the points $A$ and $C$ such that $k_1$ intersects $AB$ and $BC$ a second time in the points $K$ and $N$ respectively, $K$ and $N$ being different. Let $O$ be the centre of $k_1$. Let $k_2$ be the circumcircle of the triangle $KBN$, and let the circumcircle of the triangle $ABC$ intersect $k_2$ also in $M$, a point different from$ B$.

Prove that $OM$ and$ MB$ are perpendicular.

1990 Australian MO P5

In a given plane, let $K$ and $k$ be circles with radii $R$ and $r$, respectively, and suppose that $K$ and $k$ intersect in precisely two points $S$ and $T$. Let the tangent to $k$ through $S$ intersect $K$ also in $B$, and suppose that $B$ lies on the common tangent to $k$ and $K$.

Prove that if $\phi$ is the (interior) angle between the tangents of $K$ and $k$ at $S$, then $\frac{r}{R}=\left(2sin\frac{\phi}{2}\right)^2$

1991 Australian MO P1

Let $ABCD$ be a convex quadrilateral. Denote the least and the greatest of the distances $AB,\ AC,\ AD,\ BC,\ BD$ and $CD$ by $m$ and $M$ respectively. Prove that $M\ge m\sqrt{2}$.

1991 Australian MO P3

Let $A, B, C$ be three points in the$ x-y$-plane and $X, Y, Z$ the midpoints of the line segments $AB, BC, AC$, respectively. Furthermore, let $P$ be a point on the line $BC$ so that $\angle CPZ = \angle YXZ$. Prove that $AP$ and $BC$ intersect in a right angle.

1991 Australian MO P5

Let $P_1, P_2, ..., P_n$ be $n$ different points in a given plane such that each triangle $P_iP_jP_k$ ($i \ne j \ne k \ne i$) has an area not greater than $1$. Prove that there exists a triangle $\Delta $ in this plane such that

(a) $\Delta $ has an area not greater than $4$; and

(b) each of the points $P_1, P_2, ..., P_n$ lies in the interior or on the boundary of $\Delta $.

1991 Australian MO P7

In triangle $ABC$, let $M$ be the midpoint of $BC$, and let $P$ and $R$ be points on $AB$ and $AC$ respectively. Let $Q$ be the intersection of $AM$ and PR. Prove that if $Q$ is the midpoint of $PR$, then $PR$ is parallel to $BC$.

1992 Australian MO P1

Let $N$ be a regular nonagon, i.e. a regular polygon with nine edges, having $O$ as the centre of its circumcircle, and let $PQ$ and $QR$ be adjacent edges of $N$. The midpoint of $PQ$ is $A$ and the midpoint of the radius perpendicular to $QR$ is $B$. Determine the angle between $AO$ and $AB$.

1992 Australian MO P3

Let $ABCDE$ be a convex pentagon such that $AB =BC$ and $\angle BCD = \angle EAB = 90^o$. Let $X$ be a point inside the pentagon such that $AX$ is perpendicular to $BE$ and $CX$ is perpendicular to $BD$. Show that $BX$ is perpendicular to $DE$.

1992 Australian MO P5

Extend a given line segment $AB$ in a straight line to $D$, where the length $BD$ may be chosen arbitrarily (see diagram). Draw a semicircle with diameter $AD$, and let $H$ be its centre. Let $G$ be a point on the semicircle such that $\angle ABG$ is acute. Draw $EZ$ parallel to $BG$, where $E$ is chosen such that $EH \cdot ED = EZ^2$. Then draw $ZH$ as well as the point $T$ on the semi-circle such that $BT$ and $ZH$ are parallel. Prove: angle $TBG$ is one third of angle $ABG$.

1992 Australian MO P1

In triangle $ABC$, the angle $ACB$ is greater than $90^o$. Point $D$ is the foot of the perpendicular from $C$ to $AB, M$ is the midpoint of $AB, E$ is the point on $AC$ extended such that $EM = BM, F$ is the point of intersection of BC and $DE$, moreover $BE = BF$. Prove that $\angle CBE =2 \angle ABC$.

1993 Australian MO P6

In the acute-angled triangle $ABC$, let $D, E, F$ be the feet of altitudes through $A, B, C$ respectively, and $H$ the orthocentre. Prove that $\frac{AH}{AD}+\frac{BH}{BE}+\frac{CH}{CF}=2$

1993 Australian MO P8 (also)

The vertices of triangle $ ABC$ in the $ x-y$ plane have integer coordinates, and its sides do not contain any other points having integer coordinates. The interior of triangle $ ABC$ contains only one point , $ G$, that has integer coordinates. Prove that $ G$ is the centroid of triangle $ ABC$.

1994 Australian MO P1

Let $ABC$ be a triangle and $M$ and $N$ points on $BC$ such that $BM = MN = NC$. A line parallel to $AC$ meets lines $AB, AM$ and $AN$ in points $D, E$ and $F$ respectively. Show that $EF = 3DE$.

1994 Australian MO P3

Let $ABC$ be a triangle with side lengths being integers and $AB$ and $AC$ being relatively prime. Let the tangent at A to the circumcircle of $ABC$ meet $BC$ produced at $D$. Prove that both $AD$ and $CD$ are rational, but that neither is an integer.

1994 Australian MO P8

Let $ABCD$ be a parallelogram, $E$ a point on $AB$ and $F$ a point on $CD$. Let $AF$ intersect $ED$ in $G$ and $EC$ intersect $FB$ in $H$. Further let $GH$ produced intersect $AD$ in $L$ and $BC$ in $M$. Prove that $DL= BM$.

1995 Australian MO P3

A straight line cuts two concentric circles in the points $A, B, C$ and $D$ in that order, $AE$ and $BF$ are parallel chords, one in each circle, $GC$ is perpendicular to $BF$ at $G$ , and $DH$ is perpendicular to $AE$ at $H$ . Prove that $GF = HE$ .

1995 Australian MO P7

The lines joining the three vertices of triangle $ABC$ to a point in its plane cut the sides opposite vertices $A, B, C$ in the points $K, L, M$ respectively. A line through $M$ parallel to $KL$ cuts $BC$ at $V$ and $AK$ at $W$ . Prove that $VM = MW$ .

1996 - 2013

1996 Australian MO P1 (also) (ISL 1988)

Let $ABCDE$ be a convex pentagon such that $BC = CD = DE$ and each diagonal of the pentagon is parallel to one of its sides. Prove that all the angles in the pentagon are equal, and that all sides are equal.

1996 Australian MO P6

Let $ABCD$ be a cyclic quadrilateral and let $P$ and $Q$ be points on the sides $AB$ and $AD$ respectively such that $AP = CD$ and $AQ = BC$. Let $M$ be the point of intersection of $AC$ and $PQ$. Show that $M$ is the midpoint of $PQ$.

1997 Australian MO P1

Let $ABC$ be a triangle with $AB=AC$ and $\angle BAC<120^o$. Let $D$ be the midpoint of $BC$. $E$ is the point on $AD$ such that $\angle AEB=120^o$. Let $E'$ be any point $AD$ distinct from $E$. Prove that $EA+EB+EC< E'A+E'B+E'C$.

1997 Australian MO P8

Let $ABC$ be a triangle with $\angle ABC= 60^o$ and $\angle BAC= 40^o$. Let $P$ be the point on AB such that $\angle BCP=70^o$ and let $Q$ be the point on $AC$ such that $\angle CBQ=40^o$. Let $BQ$ intersect $CP$ at $R$. Prove that $AR$ (extended) is perpendicular to $BC$.

1998 Australian MO P

1998 Australian MO P7

Let $ABC$ be a triangle whose area is $1998$ cm$^2$. Let $G$ be the centroid of the triangle $ABC$. Each line through $G$ cuts the triangle $ABC$ into two regions having areas $A_1$ and $A_2$, say. Determine (with proof) the largest possible value of $A_1-A_2$.

1999 Australian MO P

1999 Australian MO P

2000 Australian MO P

2000 Australian MO P4 (also)

Let $A$,$B$ ,$C$, $A'$,$B'$ and $C'$ be points on a circle so that $AA'$ is perpendicular to $BC$,$BB'\perp CA$ and $CC'\perp AB$.Further let $D$ be a point on that circle and let $DA'$ intersect $BC$ in $A''$,$DB'$ intersects $CA$ at $B''$ and $DC'$ intersects $AB$ at $C''$. (Segments being extended where required.) Prove that $A''$,$B''$ and $C''$ as well as the orthocenter of $\triangle ABC$ are collinear.

2000 Australian MO P

2001 Australian MO P2

Let $ABC$ be an isosceles triangle, with $AC=BC$. Let $P,Q,R$ be points on $AB,BC$ and $AC$, respectively, such that $PQ$ is parallel to $AC$ and $PR$ is parallel to $BC$. Further, let $O$ be the circumcentre of $ABC$. Prove that the quadrilateral $CPOQ$ is cyclic.

2002 Australian MO P

2002 Australian MO P

2003 Australian MO P

2003 Australian MO P

2004 Australian MO P4

Let $ABC$ be an equilateral triangle and let $D$ be a point on $AB$ between $A$ and $B$. Next, let $E$ be a point on $AC$ with $DE$ parallel to $BC$. Further, let $F$ be the midpoint of $CD$ and $G$ the circumcentre of triangle $ADE$. Determine the angles of triangle $BFG$.

2004 Australian MO P8

Let $ABCD$ be a parallelogram. Suppose there exists a point $P$ in the interior of $ABCD$ such that $\angle ABP = 2\angle ADP$ and $\angle DCP = 2\angle DAP$. Prove that $AB = BP = CP$.

2005 Australian MO P

2005 Australian MO P

2006 Australian MO P3

Let $PRUS$ be a trapezium such that $\angle PSR = 2\angle RSU$ and $\angle SPU = 2 \angle UPR$. Let $Q$ and $T$ be on $PR$ and $SU$ respectively such that $SQ$ and $PT$ bisect $\angle PSR$ and $\angle SPU$ respectively. Let $PT$ meet $SQ$ at $E$. The line through $E$ parallel to $SR$ meets $PU$ in $F$ and the line through $E$ parallel to $PU$ meets $SR$ in $G$. Let $FG$ meet $PR$ and $SU$ in $K$ and $L$ respectively. Prove that $KF$ = $FG$ = $GL$

2006 Australian MO P5

In a square $ABCD$, $E$ is a point on diagonal $BD$. $P$ and $Q$ are the circumcentres of $\triangle ABE$ and $\triangle ADE$ respectively. Prove that $APEQ$ is a square.

2007 Australian MO P

2007 Australian MO P7

Let $ABC$ be an acute angled triangle. For each point $M$ on the segment $AC$, let $S_1$ be the circle through $A,B$ and $M$, and let $S_2$ be the circle through $M,B$ and $C$. Let $D_1$ be the disk bounded by $S_1$, and let $D_2$ be the disk bounded by $S_2$. Show that the area of the intersection of $D_1$ and $D_2$ is smallest when $BM$ is perpendicular to $AC$.

2008 Australian MO P

Let $ABC$ and $DEF$ be triangles in the plane. If $p=AB+BC+CA+DE+EF+FA$ and $q=AD+AE+AF+BD+BE+BF+CD+CE+CF$, prove that $4q\geq{3p}$ (for this just apply triangle inequality to all triangles that have one side as a side of either $\triangle{ABC}$ or $\triangle{DEF}$ and the opposite vertice to that side as a vertice of the other triangle), when does equality occur?

2008 Australian MO P1

2009 Australian MO P1

Let $ABC$ be an acute-angled triangle , and let $P$ and $Q$ be points on sides $AC$ and $BC$, respectively, such that $APQB$ is a cyclic quadrilateral. Ler $R$ be the point such that $PR$ is perpendicular to $AC$ and $QR$ is perpendicular to $BC$. Prove that the line through $C$ and $R$ is perpendicular to $AB$.

2009 Australian MO P

2010 Australian MO P

2010 Australian MO P

2011 Australian MO P

2011 Australian MO P

2012 Australian MO P

2012 Australian MO P8

Two circles $C_1$ and $C_2$ intersect at distinct points $A$ and $B$. Let $P$ be the point on $C_1$ and let $Q$ be the point on $C_2$ such that $PQ$ is the common tangent closer to $B$ than to $A$. Let $BQ$ intersect $C_1$ again at $R$, and let $BP$ intersect $C_2$ again at $S$. Let $M$ be the midpoint of $PR$, and let $N$ be the midpoint of $QS$. Prove that $AB$ bisects $\angle MAN$.

sources:

www.amt.edu.au

Australian Mathematical Olympiads Book 1 1979-1995 (AMT)

Australian Mathematical Olympiads Book 2 1996-2011 (AMT)

1979 - 1995

1979 Australian MO P3Two circles in a plane intersect. Let $A$ and $B$ be the two points of intersection. Starting simultaneously from $A$ two points $P$ and $Q$ move with constant speeds around different circles, each point travelling along its own circle in the same sense as the other point. The two points return to $A$ simultaneously after one revolution. Prove

(i) $P$, $B$ and $Q$ are always collinear (on the same straight line);

(ii) that there is a fixed point $S$ in the plane such that, at any time, the distances from $S$ to the moving points are equal.

it did not take place in 1980

1981 Australian MO P3

$O$ is the midpoint of the base $BC$ of an isosceles triangle $ABC$. A circle is drawn with centre $O$ and tangent to the equal sides $AB, AC. P$ is a point on $AB, Q$ is a point on $AC$.

If $PQ$ is also a tangent to this circle, prove that $PB \cdot CQ =\left(\frac 12 BC \right)^2$

Discuss the converse of this result.

Let $\triangle ABC$ be a triangle and let the internal bisector of the angle $\angle BAC$ intersects the circumcircle again at $P$. Similarly define $Q$ and $R$. Prove that $AP+BQ+CR>AB+BC+CA$.

1983 Australian MO P2

$ABC$ is a triangle and $P$ is a point inside it such that $\angle PAC=\angle PBC$. The perpendiculars from $P$ to $BC$ and $CA$ meet these sides at $L$ and $N$ respectively, and $D$ is the midpoint of $AB$. Prove that $DL = DM$.

1983 Australian MO P6

The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$ and $\angle CAB = \angle C_1AB_1$. $M$ is the point of intersection of the lines $BC_1$ and $CB_1$. Prove that if the lines $AM$ and $CC_1$ exist, then they are perpendicular.

1984 Australian MO P2

Given an equilateral triangle $ABC$, draw a semicircle on $BC$ as diameter on the side of $BC$ remote from $A$. Let the points $P, Q$ trisect the interval $BC$ and $AP, AQ$ produced cut the arc in $K, L$ respectively. Prove that $K, L$ trisect the semicircular arc.

1984 Australian MO P5

On the edges of a triangle $ABC$ are drawn three similar isosceles triangles $APB$ (with $AP = PB$), $AQC$ (with $AQ = QC$) and $BRC$ (with $BR = RC$). The triangles $APB$ and $AQC$ lie outside the triangle $ABC$ and the triangle $BRC$ is lying on the same side of the line $BC$ as the triangle $ABC$. Prove that the quadrilateral $PAQR$ is a parallelogram.

1985 Australian MO P4

$ABC$ is a triangle whose angles are smaller than $120^0$. Equilateral triangles $AFB$, $BDC$ and $CEA$ are constructed on the sides of and exterior to $ABC$.

(a) Prove that the lines $AD, BE$ and $CF$ pass through the one point $S$

(b) Prove that $SD+SE+SF=2(SA+SB+SC)$.

1986 Australian MO P3

$ABC$ is a triangle. The internal bisector of the angle $A$ meets the circumcircle again at $P. Q$ and $R$ are similarly defined with B and C respectively.

Prove that $AP + BQ + CR > AB + BC + CA$.

1986 Australian MO P4 (ILL 1979-48)

In the plane a circle $C$ of unit radius is given. For any line $l$, a number $s(l)$ is defined in the following way: If $l$ and $C$ intersect in two points, $s(l)$ is their distance; otherwise, $s(l) = 0$. Let $P$ be a point at distance $r$ from the center of $C$. One defines $M(r)$ to be the maximum value of the sum $s(m) + s(n)$, where $m$ and $n$ are variable mutually orthogonal lines through $P$. Determine the values of $r$ for which $M(r) > 2$.

1987 Australian MO P1

$GKA$ is an isosceles triangle with base $GK$ of length $2b$. $GA$ and $AK$ each have length $a$. Let $C$ be the midpoint of AK and $z$ be the circumcircle of the triangle $GCK$. Let $Y$ be the point on the extension of $AK$ such that if $E$ is the intersection of $YG$ with $z$ then $EY$ is of length $a/2$. Prove that if $x$ is the length of $EC$ and $y$ is the length of $KY$ then $ay = x^2$ and $xb = y^2$.

1987 Australian MO P4

In the interior of the triangle $ABC$, points $O$ and $P$ are chosen such that angles $ABO$ and $CBP$ are equal, and angles $BCO$ and $ACP$ are also equal. Prove that angles $CAO$ and $BAP$ are equal.

The triangles $ABC$ and $AEF$ are in the same plane. Between them, the following conditions hold:

(1) The midpoint of $AB$ is $E$,

(2) The points $A, G$ and $F$ are on the same line,

(3) There is a point $C$ at which $BG$ and $EF$ intersect,

(4) $CE = 1$ and $AC = AE = FG$.

Show that if $AG = x$ then $AB = x^3$.

1989 Australian MO P2

Suppose $BP$ and $CQ$ are the bisectors of the angles $B, C$ of triangle $ABC$ and suppose $AH, AK$ are the perpendiculars from $A$ to $BP, CQ$. Prove that $KH$ is parallel to $BC$.

1989 Australian MO P6

Four rods $AB, BC, CD, DA$ are freely jointed at $A, B, C$ and $D$ and move in a plane so that the shape of the quadrilateral can be varied. $P, Q$ and $R$ are the mid-points of $AB, BC$ and $CD$ respectively. In one position of the rods, the angle $PQR$ is acute. Show that this angle remains acute no matter how the shape of $ABCD$ is changed.

1989 Australian MO P8

Points $X, Y$ and $Z$ on sides $BC, CA$ and $AB$ respectively of triangle $ABC$ are such that triangles $ABC$ and $XYZ$ are similar, the angles at $X, Y$ and $Z$ being equal to those at $A, B$ and $C$ respectively. Find $X, Y$ and $Z$ so that triangle $XYZ$ has minimum area.

1990 Australian MO P3

Let $ABC$ be a triangle and $k_1$ be a circle through the points $A$ and $C$ such that $k_1$ intersects $AB$ and $BC$ a second time in the points $K$ and $N$ respectively, $K$ and $N$ being different. Let $O$ be the centre of $k_1$. Let $k_2$ be the circumcircle of the triangle $KBN$, and let the circumcircle of the triangle $ABC$ intersect $k_2$ also in $M$, a point different from$ B$.

Prove that $OM$ and$ MB$ are perpendicular.

1990 Australian MO P5

In a given plane, let $K$ and $k$ be circles with radii $R$ and $r$, respectively, and suppose that $K$ and $k$ intersect in precisely two points $S$ and $T$. Let the tangent to $k$ through $S$ intersect $K$ also in $B$, and suppose that $B$ lies on the common tangent to $k$ and $K$.

Prove that if $\phi$ is the (interior) angle between the tangents of $K$ and $k$ at $S$, then $\frac{r}{R}=\left(2sin\frac{\phi}{2}\right)^2$

1991 Australian MO P1

Let $ABCD$ be a convex quadrilateral. Denote the least and the greatest of the distances $AB,\ AC,\ AD,\ BC,\ BD$ and $CD$ by $m$ and $M$ respectively. Prove that $M\ge m\sqrt{2}$.

1991 Australian MO P3

Let $A, B, C$ be three points in the$ x-y$-plane and $X, Y, Z$ the midpoints of the line segments $AB, BC, AC$, respectively. Furthermore, let $P$ be a point on the line $BC$ so that $\angle CPZ = \angle YXZ$. Prove that $AP$ and $BC$ intersect in a right angle.

1991 Australian MO P5

Let $P_1, P_2, ..., P_n$ be $n$ different points in a given plane such that each triangle $P_iP_jP_k$ ($i \ne j \ne k \ne i$) has an area not greater than $1$. Prove that there exists a triangle $\Delta $ in this plane such that

(a) $\Delta $ has an area not greater than $4$; and

(b) each of the points $P_1, P_2, ..., P_n$ lies in the interior or on the boundary of $\Delta $.

1991 Australian MO P7

In triangle $ABC$, let $M$ be the midpoint of $BC$, and let $P$ and $R$ be points on $AB$ and $AC$ respectively. Let $Q$ be the intersection of $AM$ and PR. Prove that if $Q$ is the midpoint of $PR$, then $PR$ is parallel to $BC$.

1992 Australian MO P1

Let $N$ be a regular nonagon, i.e. a regular polygon with nine edges, having $O$ as the centre of its circumcircle, and let $PQ$ and $QR$ be adjacent edges of $N$. The midpoint of $PQ$ is $A$ and the midpoint of the radius perpendicular to $QR$ is $B$. Determine the angle between $AO$ and $AB$.

1992 Australian MO P3

Let $ABCDE$ be a convex pentagon such that $AB =BC$ and $\angle BCD = \angle EAB = 90^o$. Let $X$ be a point inside the pentagon such that $AX$ is perpendicular to $BE$ and $CX$ is perpendicular to $BD$. Show that $BX$ is perpendicular to $DE$.

1992 Australian MO P5

Extend a given line segment $AB$ in a straight line to $D$, where the length $BD$ may be chosen arbitrarily (see diagram). Draw a semicircle with diameter $AD$, and let $H$ be its centre. Let $G$ be a point on the semicircle such that $\angle ABG$ is acute. Draw $EZ$ parallel to $BG$, where $E$ is chosen such that $EH \cdot ED = EZ^2$. Then draw $ZH$ as well as the point $T$ on the semi-circle such that $BT$ and $ZH$ are parallel. Prove: angle $TBG$ is one third of angle $ABG$.

1992 Australian MO P1

In triangle $ABC$, the angle $ACB$ is greater than $90^o$. Point $D$ is the foot of the perpendicular from $C$ to $AB, M$ is the midpoint of $AB, E$ is the point on $AC$ extended such that $EM = BM, F$ is the point of intersection of BC and $DE$, moreover $BE = BF$. Prove that $\angle CBE =2 \angle ABC$.

1993 Australian MO P6

In the acute-angled triangle $ABC$, let $D, E, F$ be the feet of altitudes through $A, B, C$ respectively, and $H$ the orthocentre. Prove that $\frac{AH}{AD}+\frac{BH}{BE}+\frac{CH}{CF}=2$

1993 Australian MO P8 (also)

The vertices of triangle $ ABC$ in the $ x-y$ plane have integer coordinates, and its sides do not contain any other points having integer coordinates. The interior of triangle $ ABC$ contains only one point , $ G$, that has integer coordinates. Prove that $ G$ is the centroid of triangle $ ABC$.

1994 Australian MO P1

Let $ABC$ be a triangle and $M$ and $N$ points on $BC$ such that $BM = MN = NC$. A line parallel to $AC$ meets lines $AB, AM$ and $AN$ in points $D, E$ and $F$ respectively. Show that $EF = 3DE$.

Let $ABC$ be a triangle with side lengths being integers and $AB$ and $AC$ being relatively prime. Let the tangent at A to the circumcircle of $ABC$ meet $BC$ produced at $D$. Prove that both $AD$ and $CD$ are rational, but that neither is an integer.

Let $ABCD$ be a parallelogram, $E$ a point on $AB$ and $F$ a point on $CD$. Let $AF$ intersect $ED$ in $G$ and $EC$ intersect $FB$ in $H$. Further let $GH$ produced intersect $AD$ in $L$ and $BC$ in $M$. Prove that $DL= BM$.

1995 Australian MO P3

A straight line cuts two concentric circles in the points $A, B, C$ and $D$ in that order, $AE$ and $BF$ are parallel chords, one in each circle, $GC$ is perpendicular to $BF$ at $G$ , and $DH$ is perpendicular to $AE$ at $H$ . Prove that $GF = HE$ .

1995 Australian MO P7

The lines joining the three vertices of triangle $ABC$ to a point in its plane cut the sides opposite vertices $A, B, C$ in the points $K, L, M$ respectively. A line through $M$ parallel to $KL$ cuts $BC$ at $V$ and $AK$ at $W$ . Prove that $VM = MW$ .

1996 - 2013

under construction

1996 Australian MO P1 (also) (ISL 1988)

Let $ABCDE$ be a convex pentagon such that $BC = CD = DE$ and each diagonal of the pentagon is parallel to one of its sides. Prove that all the angles in the pentagon are equal, and that all sides are equal.

1996 Australian MO P6

Let $ABCD$ be a cyclic quadrilateral and let $P$ and $Q$ be points on the sides $AB$ and $AD$ respectively such that $AP = CD$ and $AQ = BC$. Let $M$ be the point of intersection of $AC$ and $PQ$. Show that $M$ is the midpoint of $PQ$.

1997 Australian MO P1

Let $ABC$ be a triangle with $AB=AC$ and $\angle BAC<120^o$. Let $D$ be the midpoint of $BC$. $E$ is the point on $AD$ such that $\angle AEB=120^o$. Let $E'$ be any point $AD$ distinct from $E$. Prove that $EA+EB+EC< E'A+E'B+E'C$.

1997 Australian MO P8

Let $ABC$ be a triangle with $\angle ABC= 60^o$ and $\angle BAC= 40^o$. Let $P$ be the point on AB such that $\angle BCP=70^o$ and let $Q$ be the point on $AC$ such that $\angle CBQ=40^o$. Let $BQ$ intersect $CP$ at $R$. Prove that $AR$ (extended) is perpendicular to $BC$.

1998 Australian MO P

1998 Australian MO P7

Let $ABC$ be a triangle whose area is $1998$ cm$^2$. Let $G$ be the centroid of the triangle $ABC$. Each line through $G$ cuts the triangle $ABC$ into two regions having areas $A_1$ and $A_2$, say. Determine (with proof) the largest possible value of $A_1-A_2$.

1999 Australian MO P

1999 Australian MO P

2000 Australian MO P

2000 Australian MO P4 (also)

Let $A$,$B$ ,$C$, $A'$,$B'$ and $C'$ be points on a circle so that $AA'$ is perpendicular to $BC$,$BB'\perp CA$ and $CC'\perp AB$.Further let $D$ be a point on that circle and let $DA'$ intersect $BC$ in $A''$,$DB'$ intersects $CA$ at $B''$ and $DC'$ intersects $AB$ at $C''$. (Segments being extended where required.) Prove that $A''$,$B''$ and $C''$ as well as the orthocenter of $\triangle ABC$ are collinear.

2000 Australian MO P

2001 Australian MO P2

Let $ABC$ be an isosceles triangle, with $AC=BC$. Let $P,Q,R$ be points on $AB,BC$ and $AC$, respectively, such that $PQ$ is parallel to $AC$ and $PR$ is parallel to $BC$. Further, let $O$ be the circumcentre of $ABC$. Prove that the quadrilateral $CPOQ$ is cyclic.

2002 Australian MO P

2003 Australian MO P

2003 Australian MO P

2004 Australian MO P4

Let $ABC$ be an equilateral triangle and let $D$ be a point on $AB$ between $A$ and $B$. Next, let $E$ be a point on $AC$ with $DE$ parallel to $BC$. Further, let $F$ be the midpoint of $CD$ and $G$ the circumcentre of triangle $ADE$. Determine the angles of triangle $BFG$.

2004 Australian MO P8

Let $ABCD$ be a parallelogram. Suppose there exists a point $P$ in the interior of $ABCD$ such that $\angle ABP = 2\angle ADP$ and $\angle DCP = 2\angle DAP$. Prove that $AB = BP = CP$.

2005 Australian MO P

2005 Australian MO P

2006 Australian MO P3

Let $PRUS$ be a trapezium such that $\angle PSR = 2\angle RSU$ and $\angle SPU = 2 \angle UPR$. Let $Q$ and $T$ be on $PR$ and $SU$ respectively such that $SQ$ and $PT$ bisect $\angle PSR$ and $\angle SPU$ respectively. Let $PT$ meet $SQ$ at $E$. The line through $E$ parallel to $SR$ meets $PU$ in $F$ and the line through $E$ parallel to $PU$ meets $SR$ in $G$. Let $FG$ meet $PR$ and $SU$ in $K$ and $L$ respectively. Prove that $KF$ = $FG$ = $GL$

2006 Australian MO P5

In a square $ABCD$, $E$ is a point on diagonal $BD$. $P$ and $Q$ are the circumcentres of $\triangle ABE$ and $\triangle ADE$ respectively. Prove that $APEQ$ is a square.

2007 Australian MO P

2007 Australian MO P7

Let $ABC$ be an acute angled triangle. For each point $M$ on the segment $AC$, let $S_1$ be the circle through $A,B$ and $M$, and let $S_2$ be the circle through $M,B$ and $C$. Let $D_1$ be the disk bounded by $S_1$, and let $D_2$ be the disk bounded by $S_2$. Show that the area of the intersection of $D_1$ and $D_2$ is smallest when $BM$ is perpendicular to $AC$.

2008 Australian MO P

Let $ABC$ and $DEF$ be triangles in the plane. If $p=AB+BC+CA+DE+EF+FA$ and $q=AD+AE+AF+BD+BE+BF+CD+CE+CF$, prove that $4q\geq{3p}$ (for this just apply triangle inequality to all triangles that have one side as a side of either $\triangle{ABC}$ or $\triangle{DEF}$ and the opposite vertice to that side as a vertice of the other triangle), when does equality occur?

2008 Australian MO P1

2009 Australian MO P1

Let $ABC$ be an acute-angled triangle , and let $P$ and $Q$ be points on sides $AC$ and $BC$, respectively, such that $APQB$ is a cyclic quadrilateral. Ler $R$ be the point such that $PR$ is perpendicular to $AC$ and $QR$ is perpendicular to $BC$. Prove that the line through $C$ and $R$ is perpendicular to $AB$.

2009 Australian MO P

2010 Australian MO P

2010 Australian MO P

2011 Australian MO P

2011 Australian MO P

2012 Australian MO P

2012 Australian MO P8

Two circles $C_1$ and $C_2$ intersect at distinct points $A$ and $B$. Let $P$ be the point on $C_1$ and let $Q$ be the point on $C_2$ such that $PQ$ is the common tangent closer to $B$ than to $A$. Let $BQ$ intersect $C_1$ again at $R$, and let $BP$ intersect $C_2$ again at $S$. Let $M$ be the midpoint of $PR$, and let $N$ be the midpoint of $QS$. Prove that $AB$ bisects $\angle MAN$.

by Ivan Guo (Sydney)

2013 Australian MO P
2013 Australian MO P

sources:

www.amt.edu.au

Australian Mathematical Olympiads Book 1 1979-1995 (AMT)

Australian Mathematical Olympiads Book 2 1996-2011 (AMT)

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