1 A booking office at a railway station sells tickets to 200 destinations. One day, tickets were issued to 3800 passengers. Show that (i) there are (at least) 6 destinations at which the passenger arrival numbers are the same; (ii) the statement in (i) becomes false if '6' is replaced by '7'.

2 A triangle ABC has BAC > BCA . A line AP is drawn so that PAC = BCA where P is inside the triangle. A point Q outside the triangle is constructed so that PQ is parallel to AB, and BQ is parallel to AC. R is the point on BC (separated from Q by the line AP) such thatPRQ = BCA. Prove that the circumcircle of ABC touches the circumcircle of PQR.

3 Suppose x , y and z are positive integers satisfying the equation 1/x - 1/y = 1/z , and let h be the highest common factor of x , y and z . Prove that hxyz is a perfect square. Prove also that h (y - x ) is a perfect square.

4 Find a solution of the simultaneous equations xy + yz + zx = 12 and xyz = 2 + x + y + z in which all of x , y and z are positive, and prove that it is the only such solution. Show that a solution exists in which x , y and z are real and distinct.