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 \(OH\).