2021 Japan MO Finals

Part of Japan MO Finals

Subcontests

(5)

1

Japan MO Finals 2021 P5

n

be a positive integer. Find all integers

k

1,2,\dots,2n^2

which satisfy the following condition: ・There is a

2n\times 2n

grid of square unit cells. When

k

different cells are painted black while the other cells are painted white, the minimum possible number of

2\times 2

squares that contain both black and white cells is

2n-1

1

Japan MO Finals 2021 P4

a_1,a_2,\dots,a_{2021}

2021

integers which satisfy

a_{n+5}+a_n>a_{n+2}+a_{n+3}

for all integers

n=1,2,\dots,2016

. Find the minimum possible value of the difference between the maximum value and the minimum value among

a_1,a_2,\dots,a_{2021}

1

Japan MO Finals 2021 P3

D,E

AB,AC

of an acute-angled triangle

ABC

respectively satisfy

BD=CE

. Furthermore, points

P

DE

Q

BC

ABC

A

BP:PC=EQ:QD

A,B,C,D,E,P,Q

are pairwise distinct. Prove that

\angle BPC=\angle BAC+\angle EQD

1

Japan MO Finals 2021 P2

n\geq 2

be an integer. Players

A

B

play a game using

n\times 2021

grid of square unit cells. Firstly,

A

paints each cell either black of white.

B

places a piece in one of the cells in the uppermost row, and designates one of the cells in the lowermost row as the goal. Then,

A

repeats the following operation

n-1

times: ・When the cell with the piece is painted white,

A

moves the piece to the cell one below. ・Otherwise,

A

moves the piece to the next cell on the left or right, and then to the cell one below. Find the minimum possible value of

n

A

can always move a piece to the goal, regardless of

B

1

Japan MO Finals 2021 P1

Find all functions from positive integers to themselves, such that for any positive integers

m,n

the two conditions below are equivalent:
[*]

n

m

f(n)

f(m)-n