2023 Japan MO Finals

Part of Japan MO Finals

Subcontests

(5)

1

Hard problem

S=\{1,2,\dots,3000\}

. Determine the maximum possible integer

X

that satisfies the condition: For all bijective function

f:S\rightarrow S

, there exists bijective function

g:S\rightarrow S

\displaystyle\sum_{k=1}^{3000}\left(\max\{f(f(k)),f(g(k)),g(f(k)),g(g(k))\}-\min\{f(f(k)),f(g(k)),g(f(k)),g(g(k))\}\right)\geq X

1

Nice NT problem

Determine all positive integers

n

n

\phi(n)^{d(n)}+1

d(n)^5

does not divide

n^{\phi(n)}-1

1

Weird sequence problem

c

be a nonnegative integer. Find all infinite sequences of positive integers

\{a_{n}\}_{n\geq 1}

satisfying the following condition: for all positive integer

n

, there are exactly

a_n

positive integers

i

a_i\leq a_{n+1}+c

1

Standard geo

In an acute triangle

ABC

D,E,F

be the midpoints of

BC,CA,AB

respectively. Let

X,Y

be the feet of altitude from

D

AB,AC

respectively. The line through

F

XY

DY

P

AD\perp EP

1

Overlapping game

5\times 5

squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.