Australian School of Excellence 2015 Geometry Exam

• Each question is worth 7 points

• Time allowed is 4 hours

• No books, notes or calculators permitted

• Write full proofs with your answers

1) Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Let $\omega$ be a circle passing through $A$ and tangent to $BC$ at $X$. Suppose that $\omega$ intersects $\Gamma$ for a second time at $Y$ where $Y$ lies on the minor arc $AC$ of $\Gamma$. The line $AX$ intersects $\Gamma$ for a second time at $W$. The line $XY$ intersects $\Gamma$ for a second time at $Z$.

Prove that the minor arcs $CW$ and $ZB$ of $\Gamma$ are equal in length.

2) Let $M$ be a point on side $AB$ of equilateral triangle $ABC$. The point $N$ is such that triangle $AMN$ is equilateral but $N$ does not lie on $AC$. Let $D$ be the intersection of lines $AC$ and $BN$. Let $K$ be the intersection of lines $CM$ and $AN$.

Prove $KA = KD$.

3) Let $X$, $Y$ and $Z$ be points on the sides $AD$, $AB$ and $AC$ respectively of rectangle $ABCD$.

Given that $AX = CZ$, prove that $XY + YZ \geq AC$.

4) Triangle $ABC$ satisfies $\angle ABC = 90^{\circ}$. Point $P$ lies on side $BC$, point $Q$ lies on side $AB$ and point $R$ lies inside triangle $ABC$ such that

$\angle PAB = \angle RAP = \angle CAR \quad \text{ and } \quad \angle BCQ = \angle QCR = \angle RCA.$

Prove that triangle $PQR$ is equilateral.

5) Let $H$ be the orthocentre of triangle $ABC$. The circle with diameter $AC$ intersects circle $ABH$ for a second time at point $K$.

Prove that line $CK$ bisects segment $BH$.