1

Part of 1981 Bundeswettbewerb Mathematik

Problems(2)

Bundeswettbewerb Mathematik 1981 Problem 1.1

Source: Bundeswettbewerb Mathematik 1981 Round 1

a

n

be positive integers and

s = a + a^2 + \cdots + a^n

. Prove that the last digit of

s

1

if and only if the last digits of

a

n

are both equal to

1

Digitsnumber theorypowers

Bundeswettbewerb Mathematik 1981 Problem 2.1

Source: Bundeswettbewerb Mathematik 1981 Round 2

a_1, a_2, a_3, \ldots

is defined as follows:

a_1

is a positive integer and

a_{n+1} = \left\lfloor \frac{3}{2} a_n \right\rfloor +1

n \in \mathbb{N}

a_1

be chosen in such a way that the first

100000

terms of the sequence are even, but the

100001

-th term is odd?

floor functionSequenceoddalgebra