2022 Japan TST

Part of Japan TST

Subcontests

(4)

1

The intrigue of the chairman

m\geq 3

be an odd integer and

n\geq 3

be an integer. For each integer

i

1\leq i\leq m

a_{i,1},a_{i,2},\dots,a_{i,n}

be a permutation of

1,2,\dots,n

. The chairman and

m

team leaders (team leader

1,

2,\dots,

m

) have gathered to hold the jury meeting to choose the problems for this year's IMO. There are

n

shortlisted problems (problem

1,

2,\dots,

n

) submitted to the meeting. Each team leader has a positive integer score of impression between

1

n

for each problem, and initially team leader

i

a_{i,j}

j

. The chairman can repeatedly perform the following operation:
Choose an integer

i

1\leq i\leq m

j,k

1\leq j,k\leq n

such that the difference between the scores of problem

j

k

for team leader

i

1

, and swap the scores of problem

j

k

for team leader

i

.
Regardless of the

mn

a_{i,j}

1\leq i\leq m

1\leq j\leq n

), find the minimum non-negative integer

L

such that the chairman can make the following condition true by performing the operation at most

L

times:
For any two distinct integers

x

y

1\leq x,y\leq n

, there exists a sequence of integers

p_1,p_2,\dots,p_s

s\geq 2

p_1=x

p_s=y

p_i

1\leq i\leq s

is an integer between

1

n

, and for all integers

t

1\leq t\leq s-1

, the number of team leaders who prefer problem

p_t

p_{t+1}

\frac{m+1}{2}

1

R FE with given f(-1)=-1

Find all functions

f:\mathbb{R} \rightarrow \mathbb{R}

f(x)f(y)-f(x+y)=yf(x)-f(x+f(y))

x, y \in \mathbb{R}

f(-1)=-1

1

Three concurrent lines in a configuration with two circles

Given is a circle

\Gamma

MN

A

\Gamma

. The circle with center

N

, passing through

A

\Gamma

B

C

P, Q \in BC

\angle BAP= \angle QAC

NP, NQ

\Gamma

X, Y

, respectively. Prove that

AM, PY, QX

are concurrent.

1

Long R FE

Find all functions

f:\mathbb{R} \rightarrow \mathbb{R}

f(xf(y)+f(f(y)))+yf(f(x))=f((f(f(x))+1)f(y))+xy

x, y \in \mathbb{R}