European Mathematical Cup 2016 problem 1 junior division
Source:
December 31, 2016
Problem Statement
A grasshopper is jumping along the number line. Initially it is situated at zero. In -th step, the
length of his jump is .
a) If the jump length is even, then it jumps to the left, otherwise it jumps to the right (for example, firstly
it jumps one step to the right, then two steps to the left, then three steps to the right, then four steps to
the left...). Will it visit on every integer at least once?
b) If the jump length is divisible by three, then it jumps to the left, otherwise it jumps to the right (for
example, firstly it jumps one step to the right, then two steps to the right, then three steps to the left,
then four steps to the right...). Will it visit every integer at least once?Proposed by Matko Ljulj