My problems are from the Australian School of Excellence I went to last year from the worksheets.

**1.** Acute triangle \(ABC\) has circumcircle \(\Gamma\). The tangent at \(A\) to \(\Gamma\) intersects \(BC\) at \(P\). Let \(M\) be the midpoint of the segment \(AP\). Let \(R\) be the second intersection point of \(BM\) with \(\Gamma\). Let \(S\) be the second intersection point of \(PR\) with \(\Gamma\). Prove \(CS\) is parallel to \(PA\).

**2.** Let \(O\) be the circumcentre of acute \(\Delta ABC\), \(H\) be the orthocentre. Let \(AD\) be the altitude of \(\Delta ABC\) from \(A\), and let the perpendicular bisector of \(AO\) intersect \(BC\) at \(E\). Prove that the circumcircle of \(\Delta ADE\) passes through the midpoint of \(OH\).

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TopNewestQuestion 2. Let's call the midpoint of \(AO\) and \(OH\), \(P\) and \(N\) respectively. Since \(\angle APE = \angle ADE \), this implies that quadrilateral \(APDE\) is cyclic. Therefore it suffices to prove that quadrilateral \(APND\) is cyclic as well.

Since \(\frac{OP}{AO} = \frac{ON}{OH} \), we know that \(PN || AD\).

We also know that \(N\) is the center of the nine-point circle, and \(D\) is on the circle, therefore \(ND = \frac{R}{2} \). We also know that \(PA = \frac{AO}{2} = \frac{R}{2} \). This implies that \(APND\) is a isosceles trapezoid which implies that this is cyclic and we are done. – Alan Yan · 1 year, 1 month ago

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– Sharky Kesa · 1 year, 1 month ago

Nicely done.Log in to reply

@Nihar Mahajan ,@Mehul Arora , @Surya Prakash , @Agnishom Chattopadhyay , @Alan Yan , @Shivam Jadhav , @Swapnil Das , @Mardokay Mosazghi , @Kushagra Sahni , @Xuming Liang Post your solutions please. Sorry for the late submission. – Sharky Kesa · 1 year, 1 month ago

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– Nihar Mahajan · 1 year, 1 month ago

So , finally you posted.I was waiting for you shraky :PLog in to reply

– Sharky Kesa · 1 year, 1 month ago

I had no internet for the last 2 days. I was transitioning to a more permanent wi-fi connection.Log in to reply

Hints: 1. power of a point/radical axis configuration/similarity 2. reflect a certain point. – Xuming Liang · 1 year, 1 month ago

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