2022 Nordic

Part of Nordic

Subcontests

(4)

1

Nice geometry:three circles concur

ABC

be an acute-angled triangle with circumscribed circle

k

and centre of the circumscribed circle

O

. A line through

O

intersects the sides

AB

AC

D

E

B'

C'

the reflections of

B

C

O

, respectively. Prove that the circumscribed circles of

ODC'

OEB'

k

1

A combinatorial game strategy including number theory

Anton and Britta play a game with the set

M=\left \{ 1,2,\dots,n-1 \right \}

n \geq 5

is an odd integer. In each step Anton removes a number from

M

and puts it in his set

A

, and Britta removes a number from

M

and puts it in her set

B

A

B

are empty to begin with). When

M

is empty, Anton picks two distinct numbers

x_1, x_2

A

and shows them to Britta. Britta then picks two distinct numbers

y_1, y_2

B

. Britta wins if

(x_1x_2(x_1-y_1)(x_2-y_2))^{\frac{n-1}{2}}\equiv 1\mod n

otherwise Anton wins. Find all

n

for which Britta has a winning strategy.

1

a combinatorics problem about graph theory

In Wonderland, the towns are connected by roads, and whenever there is a direct road between two towns there is also a route between these two towns that does not use that road. (There is at most one direct road between any two towns.) The Queen of Hearts ordered the Spades to provide a list of all ”even” subsystems of the system of roads, that is, systems formed by subsets of the set of roads, where each town is connected to an even number of roads (possibly none). For each such subsystem they should list its roads. If there are totally

n

roads in Wonderland and

x

subsystems on the Spades’ list, what is the number of roads on their list when each road is counted as many times as it is listed?

1

A functional equation

Find all functions

f:\mathbb{R}\to\mathbb{R}

f(f(x)f(1-x))=f(x)

f(f(x))=1-f(x)

x