# 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?