1

Part of 2010 Irish Math Olympiad

Problems(2)

Squares Sum to 2010

Source: 2010 IrMO Paper 1 Problem 1

k

for which the number

2010

can be expressed as the sum of the squares of

k

Source:

14

boys in a class. Each boy is asked how many other boys in the class have his first name, and how many have his last name. It turns out that each number from

0

6

occurs among the answers.
Prove that there are two boys in the class with the same first name and the same last name.