3

Part of 1988 Iran MO (2nd round)

Problems(2)

Function f(f(m)+f(n))=m+n - [Iran Second Round 1988]

Source:

f : \mathbb N \to \mathbb N

be a function satisfying f(f(m)+f(n))=m+n, \forall m,n \in \mathbb N. Prove that

f(x)=x

x \in \mathbb N

functionnumber theory proposednumber theory

All of the numbers are equal - [Iran Second Round 1988]

Source:

n

be a positive integer.

1369^n

positive rational numbers are given with this property: if we remove one of the numbers, then we can divide remain numbers into

1368

sets with equal number of elements such that the product of the numbers of the sets be equal. Prove that all of the numbers are equal.

logarithmsnumber theory proposednumber theory