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# 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 $$BH$$.

Note by Sharky Kesa
1 year, 5 months ago

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