### Greetings

From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How many such students are there?

### Writ Large

Suppose you had to begin the never ending task of writing out the natural numbers: 1, 2, 3, 4, 5.... and so on. What would be the 1000th digit you would write down.

### Mathland Election

A political commentator summed up an election result. Given that there were just four candidates and that the figures quoted were exact find the number of votes polled for each candidate.

# Euromaths

##### Age 11 to 14Challenge Level
You might like to look at Counting on Letters before you try this problem.

How many ways can you write the word EUROMATHS by starting at the top left hand corner and taking the next letter by stepping one step down or one step to the right in this array?

 E U R O M U R O M A R O M A T O M A T H M A T H S

Thank you to Henriett Keri from Miskolc in Hungary for bringing this task for discussion at our EuroMaths conference, with Josephine Burgess from Bedfordshire in England, and Inga Bendtsen and Karina Leth Hansen from Aarhus in Denmark. The task was contributed by Eva Pinter Kovacsne from Tiszaujvaros, Hungary.