Australia (AMO) 2014-18 10p

geometry problems from Australian Mathematical Olympiads (AMO)
with aops links in the names

Mathematics Contests -  The Australian Scene with solutions and bookmarks:

2014 - 2018 
2012-2013  under construction 

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

2014 Australian MO P2
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 P4
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.

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 $CW = AX$.

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$).

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.
source:
www.amt.edu.au

Δεν υπάρχουν σχόλια:

Δημοσίευση σχολίου