MathDB

1

Part of 2016 European Mathematical Cup

Problems(2)

European Mathematical Cup 2016 senior division problem 1

Source:

12/31/2016
Is there a sequence a1,...,a2016a_{1}, . . . , a_{2016} of positive integers, such that every sum ar+ar+1+...+as−1+asa_{r} + a_{r+1} + . . . + a_{s-1} + a_{s} (with 1≤r≤s≤20161 \le r \le s \le 2016) is a composite number, but: a) GCD(ai,ai+1)=1GCD(a_{i}, a_{i+1}) = 1 for all i=1,2,...,2015i = 1, 2, . . . , 2015; b) GCD(ai,ai+1)=1GCD(a_{i}, a_{i+1}) = 1 for all i=1,2,...,2015i = 1, 2, . . . , 2015 and GCD(ai,ai+2)=1GCD(a_{i}, a_{i+2}) = 1 for all i=1,2,...,2014i = 1, 2, . . . , 2014? GCD(x,y)GCD(x, y) denotes the greatest common divisor of xx, yy.
Proposed by Matija Bucić
number theory
European Mathematical Cup 2016 problem 1 junior division

Source:

12/31/2016
A grasshopper is jumping along the number line. Initially it is situated at zero. In kk-th step, the length of his jump is kk. 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