MathDB

Let

m\geq 3

be an odd integer and

n\geq 3

be an integer. For each integer

i

with

1\leq i\leq m

, let

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,

team leader

2,\dots,

team leader

m

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

n

shortlisted problems (problem

1,

problem

2,\dots,

problem

n

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

1

and

n

for each problem, and initially team leader

i

has a score of

a_{i,j}

for problem

j

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

i

with

1\leq i\leq m

and integers

j,k

with

1\leq j,k\leq n

such that the difference between the scores of problem

j

and problem

k

for team leader

i

1

, and swap the scores of problem

j

and problem

k

for team leader

i

.
Regardless of the

mn

integers

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

and

y

with

1\leq x,y\leq n

, there exists a sequence of integers

p_1,p_2,\dots,p_s

with

s\geq 2

such that

p_1=x

p_s=y

, and each

p_i

with

1\leq i\leq s

is an integer between

1

and

n

, and for all integers

t

with

1\leq t\leq s-1

, the number of team leaders who prefer problem

p_t

to problem

p_{t+1}

is at least

\frac{m+1}{2}

Problem Statement