2021 Switzerland - Final Round

Part of Switzerland - Final Round

Subcontests

(5)

1

combinatorics

m \ge n

be positive integers. Frieder is given

mn

posters of Linus with different integer dimensions of

k \times l

1 \ge k \ge m

1 \ge l \ge n

. He must put them all up one by one on his bedroom wall without rotating them. Every time he puts up a poster, he can either put it on an empty spot on the wall or on a spot where it entirely covers a single visible poster and does not overlap any other visible poster. Determine the minimal area of the wall that will be covered by posters.

1

game with dandelions

(m,n)

be pair of positive integers. Julia has carefully planted

m

n

dandelions in an

m \times n

array in her back garden. Now, Jana un Viviane decides to play a game with a lawnmower they just found. Taking alternating turns and starting with Jana, they can now mow down all the dandelions in a straight horizontal or vertical line (and they must mow down at least one dandelion ). The winner is the player who mows down the final dandelion. Determine all pairs of

(m,n)

for which Jana has a winning strategy.

1

Irritated function

\mathbb{N}

be the set of positive integers. Let

f: \mathbb{N} \rightarrow \mathbb{N}

be a function such that for every positive integer

n \in \mathbb{N}

f(n) -n<2021 \text{and} f^{f(n)}(n) =n Prove that

f(n)=n

for infinitely many

n \in \mathbb{N}

1

Integers in the row

For which integers

n \ge 2

can we arrange numbers

1,2, \ldots, n

in a row, such that for all integers

1 \le k \le n

the sum of the first

k

numbers in the row is divisible by

k

1

NT with finite set

Find all finite sets

S

of positive integers with at least

2

elements, such that if

m>n

are two elements of

S

\frac{n^2}{m-n}

is also an element of

S