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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
9 months, 1 week ago

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