Sample Problems

The following problems give a flavour of the style of problem considered in the enrichment classes.

Problem 1 (IrMO 2011)

Suppose abc ≠ 0. Express in terms of a, b, and c the solutions x, y, z, u, v, w of the equations

x + y = a;   z + u = b;   v + w = c;   ay = bz;   ub = cv;   wc = ax .

Problem 2 (IrMO 2010)

Find the least k for which the number 2010 can be expressed as the sum of the squares of k integers.

Problem 3 (IrMO 2009)

Hamilton Avenue has eight houses. On one side of the street are the houses numbered 1, 3, 5, 7 and directly opposite are houses 2, 4, 6, 8 respectively. An eccentric postman starts deliveries at house 1 and delivers letters to each of the houses, finally returning to house 1 for a cup of tea. Throughout the entire journey he must observe the following rules. The numbers of the houses delivered to must follow an odd-even-odd-even pattern throughout, each house except house 1 is visited exactly once (house 1 is visited twice) and the postman at no time is allowed to cross the road to the house directly opposite. How many different delivery sequences are possible?