2022 Japan MO Finals

Part of Japan MO Finals

Subcontests

(5)

1

999 numbers around a circle

Find the smallest positive integer

m

satisfying the following proposition:
There are

999

grids on a circle. Fill a real number in each grid, such that for any grid

A

and any positive integer

k\leq m

, at least one of the following two propositions will be true:

\bullet

The difference between the numbers in the grid

A

k

A

in clockwise direction is

k

\bullet

The difference between the numbers in the grid

A

k

A

in anticlockwise direction is

k

.
Then, there must exist a grid

S

with the real number

x

in it on the circle, such that at least one of the following two propositions will be true:

\bullet

For any positive integer

k<999

, the number in the

k

S

in clockwise direction is

x+k

\bullet

For any positive integer

k<999

, the number in the

k

S

in anticlockwise direction is

x+k

1

A Diophantine equation of the difference of two powers

Find all positive integer pairs

(x,y)

satisfying the following equation:

3^x-8^y=2xy+1

1

An isosceles triangle and an arbitrary interior point

In an isosceles triangle

ABC

AB=AC

O

inside the triangle (not on the boundary). Draw a circle

\omega

O

and passing through

C

\omega\cap BC=\{C,D\}

\omega\cap AC=\{C,E\}

. Denote the circumcircle of

\triangle AEO

\Gamma

\Gamma\cap\omega=\{E,F\}

. Prove that the circumcenter of

\triangle BDF

\Gamma

1

A functional equation with iteration

Find all functions

f:\mathbb{Z}_+\to\mathbb{Z}_+

, such that for any two positive integers

m,n

f^{f(n)}(m)+mn=f(m)f(n)

f^k(n)=\underbrace{f(f(\ldots f(}_{k}n)\ldots))

1

One-dimensional Reversi

2022

grids in a row. Two people A and B play a game with these grids. At first, they mark each odd-numbered grid from the left with A's name, and each even-numbered grid from the left with B's name. Then, starting with the player A, they take turns performing the following action:

\bullet

One should select two grids marked with his/her own name, such that these two grids are not adjacent, and all grids between them are marked with the opponent's name. Then, change the name in all the grids between those two grids to one's own name.
Two people take turns until one can not perform anymore. Find the largest positive integer

m

satisfying the following condition:

\bullet

No matter how B acts, A can always make sure that there are at least

m

grids marked with A's name.