3

Part of 2008 Cuba MO

Problems(2)

three numbers in a blackboard, x-> a+b-1

Source: : 2008 Cuba MO 1.3

A boy write three times the natural number

n

in a blackboard. He then performed an operation of the following type several times:
He erased one of the numbers and wrote in its place the sum of the two others minus

1

. After several moves, one of the three numbers in the blackboard is

900

.
Find all the posible values of

n

combinatoricsnumber theory

(m+1)/n +(n+1)/ m is integer

Source: 2008 Cuba 2.3

Prove that there are infinitely many ordered pairs of positive integers

(m, n)

\frac{m+1}{n}+\frac{n+1}{m}

is a positive integer.

number theoryInteger