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 be the circumcircle of acute triangle . Let be a circle passing through and tangent to at . Suppose that intersects for a second time at where lies on the minor arc of . The line intersects for a second time at . The line intersects for a second time at .
Prove that the minor arcs and of are equal in length.
2) Let be a point on side of equilateral triangle . The point is such that triangle is equilateral but does not lie on . Let be the intersection of lines and . Let be the intersection of lines and .
3) Let , and be points on the sides , and respectively of rectangle .
Given that , prove that .
4) Triangle satisfies . Point lies on side , point lies on side and point lies inside triangle such that
Prove that triangle is equilateral.
5) Let be the orthocentre of triangle . The circle with diameter intersects circle for a second time at point .
Prove that line bisects segment .