2

Part of 2001 Cono Sur Olympiad

Problems(2)

Increasing sequence of positive integers w/ a_{3n+1}=2a_n+1

Source: XII Cono Sur Mathematical Olympiad (2001)

a_1,a_2,\ldots

of positive integers satisfies the following properties.[*]

a_1 = 1

a_{3n+1} = 2a_n + 1

a_{n+1}\ge a_n

a_{2001} = 200

Find the value of

a_{1000}

.
Note. In the original statement of the problem, there was an extra condition:[*]every positive integer appears at least once in the sequence.However, with this extra condition, there is no solution, i.e., no such sequence exists. (Try to prove it.) The problem as written above does have a solution.

algebra unsolvedalgebra

Solve 2001*(sum of the digits of m) = m

Source: XII Cono Sur Mathematical Olympiad (2001)

Find all positive integers

m

2001\cdot S (m) = m

S(m)

denotes the sum of the digits of

m

number theory unsolvednumber theory