MathDB

1

Part of 2016 Bundeswettbewerb Mathematik

Problems(2)

101010...0101 is not prime

Source: Germany - BWM Round 1 2016, #1

11/12/2016
A number with 20162016 zeros that is written as 101010…0101101010 \dots 0101 is given, in which the zeros and ones alternate. Prove that this number is not prime.
prime numbersnumber theoryGermanydecimal representation
Distinct sums and consecutive numbers

Source: 2016 Bundeswettbewerb Mathematik Round 2 - #1

9/7/2016
There are n(n+1)2\tfrac{n(n+1)}{2} distinct sums of two distinct numbers, if there are nn numbers. For which n (n≥3)n \ (n \geq 3) do there exist nn distinct integers, such that those sums are n(n−1)2\tfrac{n(n-1)}{2} consecutive numbers?
number theorynumber theory proposedconsecutiveSumSummation