geometry problems from Australian Mathematical Olympiads (AMO)
with aops links in the names
2012 Australian MO P3 (also)
Let $P$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$.Let $X,Y,Z$ be points on the interior of $AB,BC,CD$ respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that $XY$ is tangent to the circumcircle of $\triangle CYZ$ and that $Y Z$ is tangent to the circumcircle of $\triangle BXY$.Show that $\angle APD=\angle XYZ$.
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 $\angle MAN$.
Let $ABCD$ be a parallelogram, and let $P$ be a point on the side $CD$. Let the line through $P$ that is parallel to $AD$ intersect the diagonal $AC$ at $Q$. Prove that: $[BCP]^2 = [QBP] \cdot [ABP]$
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 (also)
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 Australian 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
sources:
www.amt.edu.au
Australian Mathematical Olympiads Book 1 1979-1995 (AMT)
Australian Mathematical Olympiads Book 2 1996-2011 (AMT)
with aops links in the names
Mathematics Contests - The Australian Scene with solutions and bookmarks:
1979 - 2020
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
$O$ 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 $AP+BQ+CR>AB+BC+CA$.
$ABC$ 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 $DL = DM$.
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.
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.
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.
$ABC$ 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 $SD+SE+SF=2(SA+SB+SC)$.
$ABC$ 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 $M(r) > 2$.
$GKA$ 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 $xb = y^2$.
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 $AB = x^3$.
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 $BC$.
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.
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.
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.
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 $\frac{r}{R}=\left(2sin\frac{\phi}{2}\right)^2$
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 $M\ge m\sqrt{2}$.
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.
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 $.
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 $BC$.
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 $AB$.
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 $DE$.
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 $ABG$.
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 $\angle CBE =2 \angle ABC$.
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 $\frac{AH}{AD}+\frac{BH}{BE}+\frac{CH}{CF}=2$
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$.
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 $DL= BM$.
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 $GF = HE$ .
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 $VM = MW$ .
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.
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 $PQ$.
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 $EA+EB+EC< E'A+E'B+E'C$.
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 P3
$ABC$ is a triangle with perimeter $p$. $D$ lies on $AB$ and $E$ lies on $AC$ with $DE$ parallel to $BC$ (and distinct from it) such that $DE$ touches the incircle. Show that $DE \le p/8$.
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 P1
Let $A$, $B$, $C$ and $D$ be four points lying on a circle $\Gamma$ (in that order), such that $AB$ is parallel to $DC$ and $|BC|=|CD|$. Let line $\ell$ be the tangent to $\Gamma$ at $B$, and let $P$ be a point on $\ell$ with $\angle CBP$ acute. Prove that $\angle ABP$ is trisected by the lines $BC$ and $BD$; in other words, prove that $\angle ABD=\angle DBC=\angle CBP$.
1999 Australian MO P4
Show that the sum of the altitudes of a triangle is at least nine times the inradius.
1999 Australian MO P6
$ABC$ is a triangle. $A'$ lies on the opposite side of $BC$ to $A$ and is chosen such that $A'BC$ is equilateral. $B'$ and $C'$ are chosen similarly. The lines $BC'$ and $B'C$ intersect at $A''$. The points $B''$ and $C''$ are de ned similarly. Show that the lines $A'A'', B'B''$ and $C'C''$ are parallel.
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.
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.
2001 Australian MO P6 (Archimedes Broken Chord)
$ABC$ is a triangle with $AC > BC$. $M$ is the midpoint of the arc $AB$ of the circumcircle which contains C. $X$ is the point on $AC$ such that $MX$ is perpendicular to $AC$. Show that $AX = XC + CB$.
2002 Australian MO P3 (also)
$ABC$ is a triangle. Show that all lines through a vertex which divide $ABC$ into two triangles with equal perimeter are concurrent.
2002 Australian MO P6
$ABCD$ is a rectangle, $E$ is a point on $BC$ and $F$ is a point on $CD$, such that $AEF$ is equilateral. Show that $[CEF] = [ABE] + [AFD]$
1998 Australian MO P3
$ABC$ is a triangle with perimeter $p$. $D$ lies on $AB$ and $E$ lies on $AC$ with $DE$ parallel to $BC$ (and distinct from it) such that $DE$ touches the incircle. Show that $DE \le p/8$.
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 P1
Let $A$, $B$, $C$ and $D$ be four points lying on a circle $\Gamma$ (in that order), such that $AB$ is parallel to $DC$ and $|BC|=|CD|$. Let line $\ell$ be the tangent to $\Gamma$ at $B$, and let $P$ be a point on $\ell$ with $\angle CBP$ acute. Prove that $\angle ABP$ is trisected by the lines $BC$ and $BD$; in other words, prove that $\angle ABD=\angle DBC=\angle CBP$.
1999 Australian MO P4
Show that the sum of the altitudes of a triangle is at least nine times the inradius.
1999 Australian MO P6
$ABC$ is a triangle. $A'$ lies on the opposite side of $BC$ to $A$ and is chosen such that $A'BC$ is equilateral. $B'$ and $C'$ are chosen similarly. The lines $BC'$ and $B'C$ intersect at $A''$. The points $B''$ and $C''$ are de ned similarly. Show that the lines $A'A'', B'B''$ and $C'C''$ are parallel.
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.
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.
$ABC$ is a triangle with $AC > BC$. $M$ is the midpoint of the arc $AB$ of the circumcircle which contains C. $X$ is the point on $AC$ such that $MX$ is perpendicular to $AC$. Show that $AX = XC + CB$.
2002 Australian MO P3 (also)
$ABC$ is a triangle. Show that all lines through a vertex which divide $ABC$ into two triangles with equal perimeter are concurrent.
2002 Australian MO P6
$ABCD$ is a rectangle, $E$ is a point on $BC$ and $F$ is a point on $CD$, such that $AEF$ is equilateral. Show that $[CEF] = [ABE] + [AFD]$
2003 Australian MO P3
Let $ABC$ be a triangle such that $\angle ACB = 2 \angle ABC$. Let $D$ be a point in the interior of $ABC$ satisfying $AD=AC$ and $DB=DC$. Prove that $\angle BAC=3\angle BAD$.
2003 Australian MO P6
Let $ABC$ be a triangle such that $\angle ACB = 2 \angle ABC$. Let $D$ be a point in the interior of $ABC$ satisfying $AD=AC$ and $DB=DC$. Prove that $\angle BAC=3\angle BAD$.
$D$ is the midpoint of $BC$. $E$ lies on the line $AD$ such that $\angle CEA = 90^o$ and $\angle ACE = \angle B$. Show that $AB = AC$ or $\angle A = 90^o$.
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 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.
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 P1
Let $ABC$ be a right-angled triangle with the right angle at $C$. Let $BCDE$ and $ACFG$ be squares external to the triangle. Furthermore, let $AE$ intersect $BC$ at $H$, and $BG$ intersect $AC$ at $K$. Find the size of the angle $DKH$.
Let $ABC$ be a right-angled triangle with the right angle at $C$. Let $BCDE$ and $ACFG$ be squares external to the triangle. Furthermore, let $AE$ intersect $BC$ at $H$, and $BG$ intersect $AC$ at $K$. Find the size of the angle $DKH$.
2005 Australian MO P2
Consider a polyhedron whose faces are convex polygons. Show that it has at least two faces with the same number of edges.
Consider a polyhedron whose faces are convex polygons. Show that it has at least two faces with the same number of edges.
Let $ABC$ be a triangle. Let $D, E, F$ be points on the line segments $BC, CA$ and $AB$, respectively, such that line segments $AD, BE$ and $CF$ meet in a single point. Suppose that $ACDF$ and $BCEF$ are cyclic quadrilaterals. Prove that $AD$ is perpendicular to $BC, BE$ is perpendicular to $AC, CF$ is perpendicular to $AB$.
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 $KF$ = $FG$ = $GL$
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.
In the triangle $ABC$ let $AB$ be the shortest side. Let the midpoints of $BC$ and $AC$ be $X$ and $Y$ respectively. Suppose $P$ is on $AC$ such that $PX$ is perpendicular to $BC$. The circle passing through $A, B$ and $P$ meets the side BC again at $Q$. Prove that $QY$ is perpendicular to $AC$.
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 $AC$.
Let $K$ be a circle with $PQ$ as its diameter. Let $C$ be a circle with centre on $K$ and with $PQ$ tangent to $C$. Prove that the other tangents to $C$ from $P$ and $Q$ are parallel.
Let $ABCD$ be a convex quadrilateral. Suppose there is a point $P$ on the segment $AB$ with $\angle APD = \angle BPC = 45^o$. If $Q$ is the intersection of the line $AB$ with the perpendicular bisector of $CD$, prove that $\angle CQD = 90^o$.
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?
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 $AB$.
Let $I$ be the incentre of a triangle $ABC$ in which $AC \ne BC$. Let $\Gamma$ be the circle passing through $A, I$ and $B$. Suppose $\Gamma$ intersects the line $AC$ at $A$ and at $X$ and intersects the line $BC$ at $B$ and at $Y$ . Show that $AX = BY$ .
Let $ABC$ be a triangle, and let $X, Y$ and $Z$ be points on the sides $BC, CA$ and $AB$ respectively. Let $T$ be the area of triangle $XYZ$, and $T_1, T_2$ and $T_3$ be the areas of triangles $AY Z, BZX$ and $CXY$ respectively. Prove that $\frac{3}{T} \le \frac{1}{T_1}+ \frac{1}{T_2}+\frac{1}{T_3}$
2010 Australian MO P3
Consider triangle $ABC$ with $AB \ne AC$. Points $P, Q, R$ and $S$ are on the line through $B$ and $C$. $P$ is the midpoint of $BC, AQ$ bisects $\angle BAC, AR$ and $BC$ are perpendicular, $AS$ and $AQ$ are perpendicular. Prove that $PR \cdot QS = AB \cdot AC$.
Consider triangle $ABC$ with $AB \ne AC$. Points $P, Q, R$ and $S$ are on the line through $B$ and $C$. $P$ is the midpoint of $BC, AQ$ bisects $\angle BAC, AR$ and $BC$ are perpendicular, $AS$ and $AQ$ are perpendicular. Prove that $PR \cdot QS = AB \cdot AC$.
Let $K$ and $L$ be concentric circles with radius $r$ and $s$, respectively, and $r < s$. Let $BA$ be a chord of $K$. Let $BC$ be a chord of $L$ passing through $P$ and perpendicular to $PA$. Find a formula, in terms of $r$ and $s$, for $PA^2 + PB^2 + PC^2$.
Let $O$ be the circuncentre and $K$ the orthocentre of a triangle $ABC$ with $\angle A < \angle B < \angle C < 90^o$Prove that the incentre of a triangle $ABC$ lies inside triangle $BHO$.
2011 Australian MO P2
The vertices of the regular polygon $P_1P_2...P_{2n}$ lie on a circle of radius $1$. For $1 \le i \le n - 1$, let $a_i$ be the length of the line segment $P_1P_{i+1}$. Prove that $\left( \frac{4}{a_1^2}-1\right)\left( \frac{4}{a_2^2}-1\right) ... \left( \frac{4}{a_{n-1}^2}-1\right)=1$
The vertices of the regular polygon $P_1P_2...P_{2n}$ lie on a circle of radius $1$. For $1 \le i \le n - 1$, let $a_i$ be the length of the line segment $P_1P_{i+1}$. Prove that $\left( \frac{4}{a_1^2}-1\right)\left( \frac{4}{a_2^2}-1\right) ... \left( \frac{4}{a_{n-1}^2}-1\right)=1$
2011 Australian MO P3
Let $A,B,C$ be three distinct points on a circle of radius $r$. Prove that the triangle $ABC$ has an obtuse angle if and only if there exists a point $X$ in the plane such that the distances $AX, BX$ and $CX$ are all less than $r$.
Let $A,B,C$ be three distinct points on a circle of radius $r$. Prove that the triangle $ABC$ has an obtuse angle if and only if there exists a point $X$ in the plane such that the distances $AX, BX$ and $CX$ are all less than $r$.
Let $PQRS$ be a rectangle with centre $O$. Let $E$ and $F$ be the midpoints of $PQ$ and $QR$, respectively. Let $A, B, C, D$ be points on sides $PQ, QR, RS, SP$, respectively, such that $ABCD$ is a rhombus and $A$ lies between $E$ and $Q$. Let $K$ be the intersection of $AB$ and $EF$. Prove that $OK$ is perpendicular to $AB$.
Let $P$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$.Let $X,Y,Z$ be points on the interior of $AB,BC,CD$ respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that $XY$ is tangent to the circumcircle of $\triangle CYZ$ and that $Y Z$ is tangent to the circumcircle of $\triangle BXY$.Show that $\angle APD=\angle XYZ$.
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 P1Let $ABCD$ be a parallelogram, and let $P$ be a point on the side $CD$. Let the line through $P$ that is parallel to $AD$ intersect the diagonal $AC$ at $Q$. Prove that: $[BCP]^2 = [QBP] \cdot [ABP]$
2013 Australian MO P7
Let $\vartriangle ABC$ be acute-angled with $\angle ABC = 35^o$. Suppose thet $I$ is the incentre of $\vartriangle ABC$ and that $AI + AC = BC$. Let $P$ be a point such that $PB$ and $PC$ are tangent to the circumcircle of $\vartriangle ABC$. Let $Q$ be the point on the line $AB$ such that $PQ$ is parallel to $AC$. Find the value of $\angle AQC$.
2014 Australian MO P2Let $\vartriangle ABC$ be acute-angled with $\angle ABC = 35^o$. Suppose thet $I$ is the incentre of $\vartriangle ABC$ and that $AI + AC = BC$. Let $P$ be a point such that $PB$ and $PC$ are tangent to the circumcircle of $\vartriangle ABC$. Let $Q$ be the point on the line $AB$ such that $PQ$ is parallel to $AC$. Find the value of $\angle AQC$.
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 $ABC$ 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 (also)
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 Australian 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.
Let $A,B,C,D,E$ be five points in order on a circle $K$. Suppose that $AB = CD$ and $BC = DE$. Let the chords $AD$ and $BE$ intersect at the point $P$. Prove that the circumcentre of triangle $AEP$ lies on $K$.
Let $ABC$ be a triangle with $\angle ACB = 90^o$ . Suppose that the tangent line at $C$ to the circle passing through $A, B, C$ intersects the line AB at $D$. Let $E$ be the midpoint of $CD$ and let $F$ be the point on the line $EB$ such that $AF$ is parallel to $CD$. Prove that the lines $AB$ and $CF$ are perpendicular.
Let $ABCD$ be a square. For a point $P$ inside $ABCD$, a $\textit{windmill}$ centered at $P$ consists of two perpendicular lines $\ell_1$ and $\ell_2$ passing through P, such that
1. $\ell_1$ intersects the sides $AB$ and $CD$ at $W$ and $Y$ respectively,
2. $\ell_2$ intersects the sides $BC$ and $DA$ at $X$ and $Z$ respectively.
A windmill is called $\textit{round}$ if the quadrilateral $WXYZ$ is cyclic. \\
Determine all points $P$ inside $ABCD$ such that every windmill centered at $P$ is round.
Let $K$ be the circle passing through all four corners of a square $ABCD$. Let $P$ be a point on the minor arc $CD$, different from $C$ and $D$. The line $AP$ meets the line $BD$ at $X$ and the line $CP$ meets the line $BD$ at $Y$. Let $M$ be the midpoint of $XY$. Prove that $MP$ is tangent to $K$.
1. $\ell_1$ intersects the sides $AB$ and $CD$ at $W$ and $Y$ respectively,
2. $\ell_2$ intersects the sides $BC$ and $DA$ at $X$ and $Z$ respectively.
A windmill is called $\textit{round}$ if the quadrilateral $WXYZ$ is cyclic. \\
Determine all points $P$ inside $ABCD$ such that every windmill centered at $P$ is round.
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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