MathDB

4

Part of 1980 Bundeswettbewerb Mathematik

Problems(2)

Bundeswettbewerb Mathematik 1980 Problem 1.4

Source: Bundeswettbewerb Mathematik 1980 Round 1

9/23/2022
Consider the sequence a1,a2,a3,…a_1, a_2, a_3, \ldots with an=1n(n+1). a_n = \frac{1}{n(n+1)}. In how many ways can the number 11980\frac{1}{1980} be represented as the sum of finitely many consecutive terms of this sequence?
SequenceSumnumber theory
Bundeswettbewerb Mathematik 1980 Problem 2.4

Source: Bundeswettbewerb Mathematik 1980 Round 2

9/23/2022
A sequence of integers a1,a2,…a_1,a_2,\ldots is defined by a1=1,a2=2a_1=1,a_2=2 and for n≥1n\geq 1, an+2={5an+1−3an,if an⋅an+1 is even,an+1−an,if an⋅an+1 is odd,a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right. (a) Prove that the sequence contains infinitely many positive terms and infinitely many negative terms.
(b) Prove that no term of the sequence is zero.
(c) Show that if n=2k−1n = 2^k - 1 for k≥2k\geq 2, then ana_n is divisible by 77.
number theorySequenceIntegers