2019 Switzerland - Final Round

Part of Switzerland - Final Round

Subcontests

(5)

1

f(m + f(n)) = f(m) - n

Show that there exists no function

f : Z \to Z

such that for all

m, n \in Z

f(m + f(n)) = f(m) - n.

1

2019 Switzerland MO P5

A group of children is sitting around a round table . At first, each child has an even number of candies. Each turn, each child gives half of his candies to the child sitting at his right. If, after a turn, a child has an odd number of candies, the teacher gives him\her an extra candy. Show that after a finite number of rounds all children will have the same number of candies.

1

2019 Switzerland MO P3

Find all periodic sequences

x_1,x_2,\dots

of strictly positive real numbers such that

\forall n \geq 1

x_{n+2}=\frac{1}{2} \left( \frac{1}{x_{n+1}}+x_n \right)

1

2019 Switzerland MO P2

\mathbb{P}

be the set of all primes and let

M

\mathbb{P}

with at least three elements. Suppose that for all

k \geq 1

and for all subsets

A=\{p_1,p_2,\dots ,p_k \}

M

A\neq M

, all prime factors of

p_1p_2\dots p_k-1

M

M=\mathbb{P}

1

Resistant integers

n\ge2

is called resistant, if it is coprime to the sum of all its divisors (including

1

n

). Determine the maximum number of consecutive resistant numbers.
For instance: *

n=5

has sum of divisors

S=6

and hence is resistant. *

n=6

has sum of divisors

S=12

and hence is not resistant. *

n=8

has sum of divisors

S=15

and hence is resistant. *

n=18

has sum of divisors

S=39

and hence is not resistant.