1

Part of 2009 Portugal MO

Problems(2)

Circumference invariance - Portuguese MO, Problem 1, 2009

Source:

A circumference was divided in

n

equal parts. On each of these parts one number from

1

n

was placed such that the distance between consecutive numbers is always the same. Numbers

11

4

17

were in consecutive positions. In how many parts was the circumference divided?

modular arithmeticnumber theory proposednumber theory

Giant sum of products - Portuguese MO, Problem 4, 2009

Source:

João calculated the product of the non zero digits of each integer from

1

10^{2009}

and then he summed these

10^{2009}

products. Which number did he obtain?

number theory proposednumber theory